\documentstyle[11pt,farsi]{freport} \english \input{amssym} \renewcommand{\baselinestretch}{1.1} \newtheorem{theorem}{{\bf \InF{}ì‰Ì‰ƒ‰‚ \EnF{}}} \newtheorem{example}[theorem]{\bf \InF{}õ‰·‰‘ñ \EnF{}} \newtheorem{note}[theorem]{\bf \InF{}—‰Áî‰Â \EnF{}} \newtheorem{definition}[theorem]{\bf \InF{}—‰ã‰Âş‰Ó \EnF{}} \newtheorem{convention}[theorem]{\bf \InF{}ì‰Â¤¢¢ \EnF{}} \newtheorem{corollary}[theorem]{\bf \InF{}÷‰µ‰ƒ‰¹‰‚ \EnF{}} \newtheorem{lemma}[theorem]{\bf \InF{}ó‰İ \EnF{}} \newtheorem{proposition}[theorem]{\bf \InF{}ğ‰Ã¤ù \EnF{}} \newtheorem{question}[theorem]{\bf \InF{}¨‰ñ \EnF{}} \newtheorem{problem}[theorem]{\bf\InF{} õ‰Æ‰‘ó‰‚\EnF{}} \newtheorem{remark}[theorem]{\bf \InF{}—‰±‰Ê‰Âù \EnF{}} \newtheorem{conjecture}[theorem]{\bf \InF{}Ÿ‰À§ \EnF{}} \setlength{\textwidth}{160mm} \setlength{\textheight}{232mm} \setlength{\topmargin}{-16mm} \setlength{\evensidemargin}{0mm} \setlength{\oddsidemargin}{0mm} %\pagestyle{empty} \input{amssym} \farsi \begin{document} \vspace{-11mm} \hspace{-8mm}{\small\siah ¢÷‰È‰Ú‰‘ù ¬‰€‰ã‰µ‰ü ¬‰Ô‰ú‰‘ö \hspace{45mm}\small “‰Æ‰Ş‰‚ —‰ã‰‘ó‰ü}\\ {\siah ¢÷‰‹‰È‰Ø‰Àùı ä‰Ü‰‹‰ô ¤ş‰‘®‰ü}\\ \InE{}%\EnE{}{\large\siah —‰½‰Ê‰ƒ‰…– —‰Ø‰Ş‰ƒ‰Ü‰ü ¢÷‰È‰Ø‰Àù\hspace{2mm} {\large\sayedar •‰ƒ‰È‰€‰ú‰‘¢ş‰‚ı õ‰®‰á —‰½‰Ö‰ƒ‰Õ •‰‘ş‰‘ö÷‰‘õ‰‚ı ¤ª‰€‰‘¨‰ü¤ª‰À}} \vspace{3mm} \hrule\vspace{6mm} \hspace{-8mm} \noindent \english\noindent % English Tiltle of Thesis \farsi \vspace{6mm} \hspace{-8mm} {\large\siah õ‰Ö‰Àõ‰‚ }\\ ş‰× ê‰Âş‰€‰À —‰Ê‰‘¢ê‰ü, ¢÷‰±‰‘ó‰‚ı ¥ õ‰µ‰ç‰ƒ‰Âû‰‘ı \InE{}$X_t$\EnE{} ¨‰´  ¤øı ş‰× ê‰Ì‰‘ı Ÿ‰µ‰Ş‰‘ñ —‰ã‰Âş‰Ó ª‰Àù ÷‰À ø “‰Â ‰Æ‰° ÷‰Àş‰Å \InE{}$t$\EnE{}  õ‰ã‰Ş‰\nasb „ ö ¤ ÷‰Àş‰Å ¥õ‰‘ö ğ‰ş‰€‰À, õ‰Â—‰° ª‰Àù ¨‰´. ÷‰Àş‰Å ¥õ‰‘ö õ‰Ö‰‘¢ş‰Â ¡‰¢ ¤ ¥ õ‰¹‰Ş‰ä‰‚ ı ÷‰Àş‰Å ğ‰Á¤ı õ‰‘÷‰€‰À \InE{}$T$\EnE{} ¡‰µ‰ƒ‰‘¤ õ‰ü ‰À. “‰€‰‘“‰Âş‰ß \InE{}$\{X_t ~;~ t ~\in~ T \}$\EnE{} ¤ ş‰× ê‰Âş‰€‰À —‰Ê‰‘¢ê‰ü õ‰ü ÷‰‘õ‰€‰À. ê‰Âş‰€‰Àû‰‘ı —‰Ê‰‘¢ê‰ü “‰Âı õ‰Àó‰Æ‰‘¥ı “‰Æ‰ƒ‰‘¤ı ¥ ê‰Âş‰€‰Àû‰‘ ¢¤ ä‰Ü‰ô õ‰¿‰µ‰Ü‰Ó ꉃ‰Ãş‰×, õ‰ú‰€‰À¨‰ü, “‰ƒ‰ó‰¦ı ø 艃‰Âù ¤“‰Â¢ ğ‰Æ‰µ‰Â¢ù ¢¤¢. \\ õ‰¹‰Ş‰ä‰‚ ÷‰Àş‰Å ğ‰Á¤ \InE{}$T$\EnE{} õ‰Ş‰Ø‰ß ¨‰´ ğ‰Æ‰Æ‰µ‰‚ ş‰‘ •‰ƒ‰¨‰µ‰‚ “‰‘ª‰À  ¢¤ ş‰ß ¬‰¤– ê‰Âş‰€‰À õ‰µ‰€‰‘҉ ö ¤ ¥õ‰‘ö ğ‰Æ‰Æ‰µ‰‚ ş‰‘ ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ õ‰ü ÷‰‘õ‰€‰À.\\ {\large\siah õ‰·‰‘ñ 1}\\ ê‰Â­ ‰ƒ‰À ¢¤ \InE{}$n$\EnE{} “‰‘¤ •‰Â—‰‘’ ş‰× ¨‰Ø‰‚ \InE{}$X_n$\EnE{} —‰ã‰À¢ ¢ê‰ã‰‘—‰ü  ¤ø õ‰È‰‘û‰Àù õ‰ü ª‰¢ ¤ ÷‰È‰‘ö õ‰ü ¢û‰À. ¢¤ ş‰ß ¬‰¤– \InE{}$\{X_n ~;~ n=1,2,\ldots \}$\EnE{} ş‰× ê‰Âş‰€‰À —‰Ê‰‘¢ê‰ü ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ ¡‰û‰À “‰¢. ¢¤ ş‰ß õ‰·‰‘ñ \InE{}$X_n$\EnE{}û‰‘ ¥ ş‰Ø‰Àş‰Ú‰Â õ‰Æ‰µ‰Ö‰Û ÷‰ƒ‰Æ‰µ‰€‰À ø “‰‘ ş‰Ø‰Àş‰Ú‰Â ¤—‰±‰‘¯ ¢¤÷‰À. ¢¤ 䉪‰Û ÷‰ƒ‰Ã ¢¤ ê‰Âş‰€‰Àû‰‘ı —‰Ê‰‘¢ê‰ü õ‰¤¢ “‰Â¤¨‰ü ¤—‰±‰‘ωü “‰ƒ‰ß õ‰µ‰ç‰ƒ‰Âû‰‘ ø›‰¢ ¢¤¢  “‰‚ “‰Â¤¨‰ü ê‰Âş‰€‰À ‰× õ‰ü ‰À. %\vspace{6mm} \newline \hspace{-8mm} {\large\siah õ‰·‰‘ñ 2}\\ ê‰Â­ ‰ƒ‰À \InE{}$X_t$\EnE{} —‰ã‰À¢ õ‰‘ª‰ƒ‰ß û‰‘ı •‰‘¤í ª‰Àù ¢¤ ꉑ¬‰Ü‰‚ ¥õ‰‘÷‰ü \InE{}$(0,t]$\EnE{} ¢¤ ş‰× •‰‘¤î‰ƒ‰€‰Ù ¤ ÷‰È‰‘ö ¢û‰À, ¢¤ ş‰ß ¬‰¤– \InE{}$\{X_t~,~t\geq~0\}$\EnE{} ş‰× ê‰Âş‰€‰À ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ ¤ ÷‰È‰‘ö õ‰ü ¢û‰À. õ‰µ‰ç‰ƒ‰Âû‰‘ı \InE{}$X_t$\EnE{} õ‰Ö‰‘¢ş‰Â ¡‰¢ ¤ ¥ õ‰¹‰Ş‰ä‰‚ı  ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ ê‰Âş‰€‰À ÷‰‘õ‰ƒ‰Àù õ‰üª‰¢, ¡‰µ‰ƒ‰‘¤ õ‰ü‰À. ¢¤ ¬‰¤—‰ü  “‰‚ ¥ı û‰Â \InE{}$t$\EnE{}, \InE{}$X_t$\EnE{} ş‰× õ‰µ‰ç‰ƒ‰Â —‰Ê‰‘¢ê‰ü ğ‰Æ‰Æ‰µ‰‚ “‰‘ª‰À  ¢¤ ş‰ß ¬‰¤– \InE{}$S$\EnE{} ş‰× õ‰¹‰Ş‰ä‰‚ı ª‰Ş‰‘¤©•‰Áş‰Â ¡‰û‰À “‰¢. ö ğ‰‘ù ê‰Âş‰€‰À \InE{}$\{X_t ~;~ t ~\in~ T \}$\EnE{} ¤ ş‰× ¥÷‰¹‰ƒ‰Â —‰Ê‰‘¢ê‰ü õ‰ü÷‰‘õ‰€‰À. ¢¤ ¬‰¤—‰ü  \InE{}$S$\EnE{} ş‰× õ‰¹‰Ş‰ä‰‚ ª‰Ş‰‘¤ õ‰‘÷‰€‰À \InE{}$S=\{0,1,2,\ldots\}$\EnE{} “‰‘ª‰À, ş‰× ê‰Âş‰€‰À “‰‘ ø®‰ã‰ƒ‰´ ğ‰Æ‰Æ‰µ‰‚ ¡‰û‰ƒ‰İ ¢ª‰´. ¢¤ ¬‰¤—‰ü  \InE{}$S=R^k$\EnE{}, ÷‰Ú‰‘ù \InE{}$X_t$\EnE{} ş‰× ê‰Âş‰€‰À “‰Â¢¤ı \InE{}$k$\EnE{} “‰ã‰Àı ¡‰û‰À “‰¢. %\newlin \hspace{-8mm} {\large\siah õ‰·‰‘ñ:}\\ ê‰Â­ ‰ƒ‰À ş‰× ¨‰Ø‰‚ ¤ “‰‘ ¢ê‰ã‰‘– •‰ü ¢¤ •‰ü •‰Â—‰‘’ ‰ƒ‰İ ê‰Âş‰€‰À \InE{}$\{X_1,X_2,~\ldots\}$\EnE{} ê‰Âş‰€‰Àı ¨‰´  õ‰µ‰ç‰ƒ‰Â \InE{}$X_n$\EnE{} ÷‰µ‰ƒ‰¹‰‚ •‰Â—‰‘’ ¤ ¢¤ “‰‘¤ \InE{}$n$\EnE{}ô ÷‰È‰‘ö õ‰ü¢û‰À. \InE{}$X_n$\EnE{} ş‰× õ‰µ‰ç‰ƒ‰Â “‰Â÷‰ó‰ü ¨‰´ ø õ‰µ‰ç‰ƒ‰Âû‰‘ı õ‰¿‰µ‰Ü‰Ó \InE{}$iid$\EnE{} û‰Æ‰µ‰€‰À. ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ ş‰‘ Ÿ‰‘ó‰´ \InE{}$S=\{0,1\}$\EnE{} ø \InE{}$T=\{1,2,~\ldots\}$\EnE{} õ‰¹‰Ş‰ä‰‚ ÷‰Àş‰Å ğ‰Á¤ ¡‰û‰À “‰¢.\\\\ %\vspace{3mm}\\ %\vspace{6mm} %\newline \hspace{-8mm} {\large\siah õ‰Æ‰ƒ‰Â ÷‰Ş‰÷‰‚ ı}\\ ¢¤ ê‰Âş‰€‰À \InE{}$\{X_t~;~t \in T\}$\EnE{}, ğ‰Â \InE{}$x_t$\EnE{} ş‰× õ‰Ö‰À¤ õ‰Ş‰Ø‰€‰‚ı \InE{}$X_t$\EnE{} “‰‘ª‰À, \InE{}$\{~x_t~,~t \in ~T\}$\EnE{} ¤ ş‰× õ‰Æ‰ƒ‰Â ÷‰Ş‰÷‰‚ı ğ‰ş‰€‰À, ş‰ã‰€‰ü ¤—‰±‰‘ωü ¨‰´  “‰‚ û‰Â \InE{}$t ~\in~ T$\EnE{}, õ‰Ö‰À¤ õ‰Ş‰Ø‰€‰‚ı \InE{}$x_t$\EnE{} ¤ õ‰Â“‰¯ õ‰ü‰À. õ‰·‰\nasb … ¢¤ õ‰·‰‘ñ 쉱‰Û 0100001010011101 ş‰× õ‰Æ‰ƒ‰Â ÷‰Ş‰÷‰‚ı ¨‰´. ¢¤ õ‰·‰‘ñ )1( õ‰µ‰ç‰ƒ‰Âû‰‘ ¥ ş‰Ø‰Àş‰Ú‰Â õ‰Æ‰µ‰Ö‰Û “‰¢÷‰À, øó‰ü û‰Ş‰ƒ‰È‰‚ ş‰€‰Ú‰÷‰‚ ÷‰ƒ‰Æ‰´. ¢¤ è‰Ü‰° ê‰Âş‰€‰Àû‰‘ õ‰µ‰ç‰ƒ‰Âû‰‘ “‰‚ ş‰Ø‰Àş‰Ú‰Â ø“‰Æ‰µ‰‚ û‰Æ‰µ‰€‰À. %\vspace{6mm} %\newline \hspace{-8mm} {\large\siah ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó}\\ ş‰× ê‰Âş‰€‰À õ‰‘¤î‰é, ê‰Âş‰€‰Àı ¨‰´  ™‰Â ğ‰Áª‰µ‰‚ ê‰Âş‰€‰À “‰Â ş‰€‰Àùı ö “‰Â“‰Â “‰‘ ¡‰Âş‰ß 󉽉щ‚ı ğ‰Áª‰µ‰‚ ¨‰´. “‰‚ ¥“‰‘ö ¤ş‰‘®‰ü ê‰Âş‰€‰À \InE{}$\{X_t ~;~ t ~\in~ T \}$\EnE{}  ¤øı ê‰Ì‰‘ı Ÿ‰µ‰Ş‰‘ñ \InE{}$(\Omega , F , P)$\EnE{} —‰ã‰Âş‰Ó ª‰Àù ¨‰´, ê‰Âş‰€‰À õ‰‘¤î‰Ó ¨‰´ ğ‰Â:\\ \InE{}$p~(X_t \in A ~| ~X_u ~, ~\forall~ u\leq s) = p~(X_t~\in A ~|~X_s)$ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \EnE{} ~~~~~~~~~~~~~~~~~ \InE{}$\forall ~~u \leq s < t~\in ~T$ \EnE{} \vspace{6mm} \hspace{-8mm}\\ {\large\siah —‰ã‰Âş‰Ó:} ê‰Â­ ‰ƒ‰À \InE{}$S$\EnE{} ş‰× õ‰¹‰Ş‰ä‰‚ı ª‰Ş‰‘¤©•‰Áş‰Â ø \InE{}$(\Omega , F , P)$\EnE{} ş‰× ê‰Ì‰‘ı Ÿ‰µ‰Ş‰‘ñ “‰‘ª‰À. ê‰Âş‰€‰À \InE{}$\{X_n~,~n=0,1,2,\ldots\}$\EnE{} ¤ ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ¤øı \InE{}$S$\EnE{} ğ‰ş‰€‰À, û‰Â ğ‰‘ù “‰‚ ¥ı û‰Â \InE{}$i_0,i_1,\ldots,i,j ~\epsilon S$\EnE{} \InE{}$$p~(X_{n+1}= j~|X_0=i_0,~X_1=i_1,~...,~X_{n-1}=i_{n-1},~X_n=i) =p~(X_{n+1}=j~|X_n=i) $$ \EnE{} •‰ƒ‰È‰‘õ‰À \InE{}$X_n=i$\EnE{} “‰‚ ş‰ß õ‰ã‰€‰‘¨‰´  ¥÷‰¹‰ƒ‰Â ¢¤ 󉽉щ‚ı \InE{}$n$\EnE{} ¢¤ ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} ì‰Â¤ ¢¤¢.\\ ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ¥õ‰‘ö ğ‰Æ‰Æ‰µ‰‚ ¤ õ‰ü—‰ö “‰‚ ¬‰¤– Ÿ‰Â £¤ùı õ‰¹‰Æ‰İ î‰Â¢  õ‰Ø‰‘ö £¤ù ¢¤ 󉽉щ‚ \InE{}$n+1$\EnE{} ô “‰‚ ª‰Â¯ ¢÷‰Æ‰µ‰ß õ‰Æ‰ƒ‰Â £¤ù —‰‘ ÷‰µ‰Ö‰‘ñ \InE{}$n$\EnE{} ô ê‰Ö‰Í “‰‚ ø®‰ã‰ƒ‰´ ö ¢¤ ÷‰µ‰Ö‰‘ñ \InE{}$n$\EnE{} ô “‰Æ‰µ‰Ú‰ü ¢¤¢.\\ \InE{}$p~(X_{n+1}=j|X_n=i)$\EnE{} Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ¥ \InE{} $i$\EnE{}“‰‚ \InE{}$j$\EnE{} ¢¤ ş‰× õ‰ÂŸ‰Ü‰‚ ¨‰´  “‰‘ \InE{}$p_{ij}^{n n+1}$\EnE{} ÷‰È‰‘ö õ‰ü¢û‰ƒ‰İ. ¢¤ ¬‰¤—‰ü  Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ş‰× õ‰ÂŸ‰Ü‰‚ı “‰‚ \InE{}$n$\EnE{} “‰Æ‰µ‰Ú‰ü ÷‰Àª‰µ‰‚ “‰‘ª‰À, ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ¤ ¥õ‰‘ö-û‰Ş‰Ú‰ß ğ‰ş‰€‰À ø “‰‚ ş‰ß ¬‰¤– ÷‰È‰‘ö õ‰ü¢û‰ƒ‰İ : \InE{}$ p_{ij}^{n n+1} = p_{ij} $\EnE{}.\\ ¥ ş‰ß ì‰Æ‰Ş‰´ “‰‚ “‰ã‰À ê‰Ö‰Í “‰‚ ¥÷‰¹‰ƒ‰Âû‰‘ı õ‰‘¤î‰Ó ¥õ‰‘ö-û‰Ş‰Ú‰ß ¡‰û‰ƒ‰İ •‰Â¢¡‰´. “‰€‰‘“‰Âş‰ß \InE{}$p_{ij}$\EnE{} “‰‚ õ‰ã‰€‰‘ı Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ¥ ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} “‰‚ ø®‰ã‰ƒ‰´ \InE{}$j$\EnE{} ωü ş‰× õ‰ÂŸ‰Ü‰‚ ¨‰´. õ‰‘—‰Âş‰Å \InE{}$P=[p_{ij}]_{i,j~ \in S}$\EnE{} ¤ õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘„– ÷‰µ‰Ö‰‘ñ ş‰× õ‰ÂŸ‰Ü‰‚ı ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘ ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ \InE{}$S$\EnE{} ğ‰ş‰€‰À.\\ $$ P= \left(% \begin{array}{cccc} p_{00} & p_{01} & p_{02} & \ldots \\ p_{10} & p_{11} & p_{12} & \ldots \\ \vdots & \vdots& \vdots & \ddots\\ \end{array}% \right) $$ “‰Âı õ‰·‰‘ñ, õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ş‰× õ‰ÂŸ‰Ü‰‚ı ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘ ¢ø ø®‰ã‰ƒ‰´\InE{}$0$ \EnE{} ø \InE{}$1$\EnE{}, “‰Â“‰Â ¨‰´ “‰‘ $$ \left(% \begin{array}{cc} P(010) & P(110) \\ P(011) & P(111) \\ \end{array}% \right) = \left(% \begin{array}{cc} p_{00} & p_{01} \\ p_{10} & p_{11} \\ \end{array}% \right) = \left(% \begin{array}{cc} p & 1-p \\ q & 1-q \\ \end{array}% \right) $$ \hspace{-8mm} {\large\siah õ‰·‰‘ñ: } ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ì‰Àô ¥¢ö —‰Ê‰‘¢ê‰ü ¨‰‘¢ù: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ø®‰ã‰ƒ‰´ ê‰Â¢ ¢¤ ì‰Àô \InE{}$n$\EnE{} ô ÷‰Æ‰±‰´ “‰‚ õ‰±‰À\hamze :\InE{}$X_n$\EnE{} \InE{}$\{X_n~: ~ n=0,1,2,\ldots\}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$S=\{0,\pm1,\pm2,\pm3,\ldots\} = Z$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$p~(X_1=3~|~X_0=1)=0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$p~(X_5=4~|~X_4=3)={1\over2}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$p~(X_n=i~|~X_{n-1}=i+1)={1\over2}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$$p_{i i+1}={1\over2} ~~~~~,~~~~~~~~p_{i-1 i}={1\over2}$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} $$ p_{ij}= \left\{% \begin{array}{ll} {1\over2} & \hbox{$j$ ~=~$i$+1 $or$ $j$~=~$i$-1} \\\\ 0 & \hbox{$o.w$.} \\ \end{array}% \right. $$ \hspace{-8mm} {\large\siah ÷‰Ø‰µ‰‚ 1:} ¢¤ ¬‰¤—‰ü  ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ ¥÷‰¹‰ƒ‰Â õ‰µ‰€‰‘û‰ü “‰‘ª‰À õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ş‰× õ‰ÂŸ‰Ü‰‚ı õ‰Â“‰ã‰ü “‰‘ “‰ã‰À õ‰µ‰€‰‘û‰ü ¡‰û‰À “‰¢, ¢¤ 艃‰Â ş‰ß ¬‰¤– —‰ã‰À¢ ¨‰Î‰Âû‰‘ ø ¨‰µ‰öû‰‘ õ‰µ‰€‰‘û‰ü ¡‰û‰À “‰¢.\\ \hspace{-8mm} {\large\siah ÷‰Ø‰µ‰‚ 2:} ¢¤ş‰‚û‰‘ı õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ¢¤ı ¢ø ¡‰‘¬‰ƒ‰´ ¥ş‰Â û‰Æ‰µ‰€‰À:\\ \InE{}$p_{ij}\geq0$~~~~~~~~~~~~~~~$\forall~i,j\in S$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}~)1 \\ \InE{}$\sum_{j\in S}p_{ij} = 1~~~~~~~~\forall ~i\in S$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{})2\\ ¡‰‘¬‰ƒ‰´ ¢øô “‰‚ ş‰ß ¢ó‰ƒ‰Û ¨‰´  ¥÷‰¹‰ƒ‰Â ¢¤ û‰Â ø®‰ã‰ƒ‰´  “‰‘ª‰À ¢¤ õ‰ÂŸ‰Ü‰‚ “‰ã‰À “‰‚ û‰Â Ÿ‰‘ñ “‰‚ ş‰× ø®‰ã‰ƒ‰´ ¡‰û‰À ¤ê‰´.\\ \InE{}$\sum_{j\in S}p_{ij} = \sum_{j\in S}p~(X_1=j~|~X_0 = i) = p~(X_1\in~S~|X_0 = i) = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\large\siah —‰ã‰Âş‰Ó:} õ‰‘—‰Âş‰Å \InE{}$A=[a_{ij}]$\EnE{} ¤ õ‰‘—‰Âş‰Å —‰Ê‰‘¢ê‰ü ğ‰ş‰€‰À ğ‰Â:\\ 1(~~~\InE{}$\forall ~i,j ~\in S ~~~~a_{ij}\geq 0$\EnE{}\\ 2(\InE{}$\sum_{i\in S}a_{j i} = 1$~~~\EnE{}\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ:} ê‰Âş‰€‰À õ‰‘¤î‰Ó \InE{}$ X_0,X_1,X_2,\ldots$\EnE{} “‰‘ õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ \InE{}$P$\EnE{} ø ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ \InE{}$\{0,1,2\}$\EnE{} ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À:\\ $$ P= \left(% \begin{array}{ccc} 0/6 & 0/3 & 0/1 \\ 0/3 & 0/3 & 0/4 \\ 0/4 & 0/1 & 0/5 \\ \end{array}% \right) $$ ¢¤ ¬‰¤—‰ü  “‰À÷‰ƒ‰İ ê‰Âş‰€‰À “‰‘ \InE{}$X_0 = 1$\EnE{} ª‰Âøá ª‰Àù ¨‰´, ş‰‘ õ‰ü—‰÷‰ƒ‰İ \InE{}$P~(X_0 = 1 , X_1 = 0 , X_2 = 2)$\EnE{} ¤ õ‰½‰‘¨‰±‰‚ ‰ƒ‰İ?\\ \InE{}$P~(X_0=1,X_1=0,X_2=2)= P~(X_2 = 2 | X_1 =0 , X_0 =1) P~(X_1=0,X_0=1)$~~~~~~~~~~~~~~~~\EnE{} \InE{}$ = P~(X_2 = 2|X_1 = 0)P~(X_1=0,X_0=1)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ = 0/1 \times P~(X_1=0|X_0=1)P~(X_0=1)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$= 0/1 \times 0/3 \times P~(X_0=1)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ = 0/03 \times P~(X_0=1)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ ğ‰Â “‰À÷‰ƒ‰İ \InE{}${1\over2} = P~(X_0 =0) = P~(X_0=1)$\EnE{} “‰‘ª‰À ‰Î‰¤?\\ “‰€‰‘“‰Âş‰ß:\\ \InE{}$P~(X_0 =1,X_1=0,X-2=2)=0/015$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\large\siah ÷‰Ø‰µ‰‚:} “‰‘ õ‰ã‰Ü‰ô “‰¢ö õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ø —‰¥ş‰â õ‰µ‰ç‰ƒ‰Â ¥õ‰‘ö ª‰Âøá ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó û‰Â —‰¥ş‰â —‰¥ş‰â —‰\hamze ô õ‰µ‰€‰‘û‰ü󉱉ã‰À ê‰Âş‰€‰À õ‰ã‰Ü‰ô ¡‰û‰À “‰¢ ø ¢¤ ÷‰µ‰ƒ‰¹‰‚ ê‰Âş‰€‰À ¥ ÷‰Ñ‰Â Ÿ‰µ‰Ş‰‘ó‰ü õ‰ã‰Ü‰ô ¨‰´.\\ \InE{}$P~(X_0 = i) = \pi_i ~~~~~~~~ \forall ~i\in~S$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$\Rightarrow P(X_0 = i, X_1=i_1, \ldots, X_n=i_n) $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$= P(X_n=i_n ~|~ X_{n-1} =i_{n-1})P(X_{n-1}=i_{n-1}| X_{n-2}=i_{n-2}) \times...\times P(X_0=i)$~~~~~~~~~~~~~ \EnE{} \InE{}$ = p_{i_{n-1} i_{n}} \times p_{i_{n-2} i_{n-1}} \times \ldots \times p_{i i_1} \pi_i$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\large\siah ì‰Ì‰ƒ‰‚:} ¢¤ ş‰× ê‰Âş‰€‰À ¥õ‰‘ö - û‰Ş‰Ú‰ß \InE{}$P~(X_{m+n}=j | X_m = i)$\EnE{}  ÷‰Â Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ¢¤ \InE{}$n$\EnE{} õ‰ÂŸ‰Ü‰‚ ) \InE{}$p_{ij}^{(n)} = P~(X_{m+n}=j | X_m = i)$\EnE{}( ğ‰ş‰€‰À “‰‚ \InE{}$m$\EnE{} “‰Æ‰µ‰Ú‰ü ÷‰À¤¢ ø \InE{}$P^{(n)} = [p_{ij}^{(n)}]$\EnE{} ¤ õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ \InE{}$n$\EnE{} õ‰ÂŸ‰Ü‰‚ı ğ‰ş‰€‰À.\\ Ÿ‰µ‰Ş‰‘„– ÷‰µ‰Ö‰‘ñ \InE{}$n$\EnE{} õ‰ÂŸ‰Ü‰‚ı ¢¤ ¤“‰Î‰‚ ¥ş‰Â  ÷‰Â ‰³‰Ş‰ß-î‰Ü‰Ş‰ğ‰Âøé ğ‰ş‰€‰À, ¬‰Àë õ‰ü‰À:\\ \InE{}$p_{ij}^{(n)} = \sum_{k \in S}p_{ik}~~p_{kj}^{(n-1)}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} ~~~~~~~~~~~~~~~~~~~~~~ø쉵‰ü  $$ p_{ij}^{(n)}= \left\{% \begin{array}{ll} 1 & \hbox{$i = j$} \\ 0 & \hbox{$i \neq j$} \\ \end{array}% \right. $$ ™‰±‰‘– ş‰ß ¤“‰Î‰‚ ¤ ¢¤ ‰‘’ ¤ó‰ƒ‰ß-—‰ƒ‰Ü‰¤ \footnote{\InE{}Karlin-Taylor\EnE{}} õ‰ü—‰ö ¢ş‰À. “‰‘ —‰›‰‚ “‰‚ ş‰ß ¤“‰Î‰‚ ¡‰û‰ƒ‰İ ¢ª‰´:\\ \InE{}$P^{(n)} = P \times P^{(n-1)} = \ldots = P^{(n)}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\  û‰Ş‰‘ö õ‰‘—‰Âş‰Å \InE{}$P$\EnE{} “‰‚ —‰ö \InE{}$n$\EnE{} ¨‰´.\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ:} ê‰Â­ ‰ƒ‰À:\\ \InE{}$Y_0,Y_1,Y_2,\ldots \stackrel{ i.i.d}{\sim} Y$\EnE{} ø \InE{}$P(Y=i)=a_i$\EnE{} ê‰Âş‰€‰À \InE{}$\{X_0,X_1,X_2,\ldots\}$\EnE{} ş‰× ê‰Âş‰€‰À õ‰‘¤î‰Ó ¨‰´. \\ \InE{}$P(Y_n=j~|Y_0=i_0,Y_1=i_1, ... , Y_{n-1}=i_n,Y_n=i)=P(Y_{n+1}=j) = a_j = P(Y_n = j~|Y_n=i)$~~~~~~\EnE{} “‰€‰‘“‰Âş‰ß õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ş‰× õ‰ÂŸ‰Ü‰‚ı ş‰ß ¥÷‰¹‰ƒ‰Â “‰Ê‰¤– ¥ş‰Â ¨‰Î‰Âû‰‘ı “‰Â“‰Â ¢¤¢.\\\\ \InE{}$P = \bordermatrix{&0&1&2&\ldots \cr 0 &a_0&a_1&a_2&\ldots \cr 1 &a_0&a_1&a_2&\ldots \cr 2 &a_0&a_1&a_2&\ldots \cr \vdots &\vdots&\vdots&\vdots&\ddots }$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ ä‰Ø‰Å ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ÷‰ƒ‰Ã ¬‰½‰ƒ‰¼ ¨‰´. õ‰Æ‰‘øı “‰¢ö ¨‰Î‰Âû‰‘ õ‰±‰ƒ‰ß ¨‰µ‰Ö‰…ñ \InE{}$X_n$\EnE{} ø \InE{}$X_{n+1}$\EnE{} “‰‚ ¥ı û‰Â \InE{}$n\geq 0$\EnE{} ¨‰´.\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ:} ê‰Âş‰€‰À: \InE{}$X_n = Y_0 + \ldots +Y_n~~~n=0,1,2,\ldots$\EnE{} ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À  \InE{}$Y_i$\EnE{} û‰‘ û‰Ş‰‘ö õ‰µ‰ç‰ƒ‰Âû‰‘ı õ‰·‰‘ñ 쉱‰Û û‰Æ‰µ‰€‰À, ¢¤ ş‰ß ¬‰¤– \InE{}$\{~X_n ~, n=0,1,\ldots\}$\EnE{} ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰é ¨‰´.\\ \InE{}$P(X_{n+1}=j | X_n = i_n , X_{n-1} = i_{n-1} ,..., X_0 = i_0)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$P(Y_0 + ... +Y_{n+1} = j |~Y_0+ ...+Y_n=i,Y_0+\ldots +Y_{n-1}=i_{n-1},...,Y_0=i_0) $~~~~~~~~~~~~~~~\EnE{} \InE{}$ = P(Y_{n+1}=j-i) = a_{j-i}~~~~~~~~~~~~~~~ j-i \geq 0 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ $$ \Rightarrow ~~~~p_{ij} = \left\{% \begin{array}{ll} a_{j-i} & \hbox{$j-i\geq 0$} \\ 0 & \hbox{$o.w$} \\ \end{array}% \right. $$ \InE{}$$P = \bordermatrix{&0&1&2&\ldots \cr 0 &a_0&a_1&a_2&\ldots \cr 1 &0&a_0&a_1&\ldots \cr 2 &0&0&a_0&\ldots \cr \vdots &\vdots&\vdots&\ddots&\ddots}$$\EnE{} \hspace{-8mm} {\large\siah õ‰·‰‘ñ:} ê‰Â­ ‰ƒ‰À \InE{}$X = \{X_n , n=0,1,2,\ldots\}$\EnE{} ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰é “‰‘ ê‰Ì‰‘ı Ÿ‰‘ó‰´ \InE{}$S = \{a,b,c\}$\EnE{} ø õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ: \InE{}$$ P = \bordermatrix{&a&b&c \cr a & {1\over2} & {1\over4} & {1\over4} \cr b & {2\over3} & 0 & {1\over3} \cr c & {3\over5} & {2\over5} & 0}$$\EnE{} ¢¤ ş‰€‰Ê‰¤– :\\ \InE{}$P(X_1=b,X_2=c,X_4=c,X_5=a,X_6=c|X_0=c) $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$ =P_{cb}.P_{bc}.P_{cc}^{(2)}.P_{ca}^{(2)}.P_{ac} = {2\over5} \times {1\over3} \times {17\over60} \times {3\over5} \times {1\over4}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} $$ P^2 = P \times P = \left(% \begin{array}{ccc} {17\over30} & {9\over40} & {5\over24} \\ {8\over15} & {3\over10} & {1\over6} \\ {17\over30} & {3\over20} & {17\over60} \\ \end{array}% \right) $$ ş‰× ÷‰á ê‰Âş‰€‰À ì‰Àô ¥¢ö —‰Ê‰‘¢ê‰ü: $$ P = \left(% \begin{array}{cccccccc} 1 & 0 & 0 & \cdots & 0 & 0 & 0 &0 \\ q_1 & r_1 & p_1 & 0 & \cdots & 0 & 0 &0 \\ 0 & q_2 & r_2 & p_2 & 0 & \cdots & 0 & 0 \\ \vdots & 0 & \ddots & \ddots & \ddots & 0 & \ldots & 0 \\ 0 & 0 & \ddots & 0 & 0 & q_{a-1} & r_{a-1} & p_{a-1} \\ 0 & \vdots & \ddots & 0 & \ldots & 0 & q_a & r_a \\ \end{array}% \right) $$ “‰‚ ω¤ı  :\\ \InE{}$p_i +r_i +q_i = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ~\InE{}$r_a +q_a = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ğ‰Â ¥÷‰¹‰ƒ‰Â ø®‰ã‰ƒ‰´ \InE{}$0$\EnE{} ¤ õ‰…쉑– ‰À ¢¤ ş‰ß ø®‰ã‰ƒ‰´ ¡‰û‰À õ‰‘÷‰À. “‰‚ ‰€‰ƒ‰ß ø®‰ã‰ƒ‰µ‰ü, ø®‰ã‰ƒ‰´ ›‰‘£’ ğ‰ş‰€‰À. \hspace{-8mm} {\large\siah —‰ã‰Âş‰Ó :} ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} ¤ ›‰‘£’ ğ‰ş‰€‰À û‰Â ğ‰‘ù: \InE{}$p_{ii} = 1$\EnE{}.\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ:} ¥÷‰¹‰ƒ‰Â ì‰Àô ¥¢ö —‰Ê‰‘¢ê‰ü ¢¤ Ÿ‰‘󉵉ü  ê‰Ö‰Í õ‰Ö‰‘¢ş‰Â ¬‰½‰ƒ‰¼ ÷‰‘õ‰€‰Ô‰ü ä‰Ì‰ ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ û‰Æ‰µ‰€‰À ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À. ¢¤ ş‰ß Ÿ‰‘ó‰´ ø쉵‰ü £¤ù “‰‚ ø®‰ã‰ƒ‰´ \InE{}$0$\EnE{} õ‰ü¤¨‰À, “‰Âõ‰üğ‰Â¢¢ øó‰ü “‰‚ û‰Â Ÿ‰‘ñ ¥÷‰¹‰ƒ‰Âı ¨‰´  ¢¤ û‰Â ÷‰µ‰Ö‰‘ñ ş‰‘ ş‰× øŸ‰À ê‰Ãş‰Ç õ‰üş‰‘“‰À ş‰‘ ş‰× øŸ‰À û‰Ç ø ş‰‘ “‰Àøö —‰ç‰ƒ‰ƒ‰Â õ‰üõ‰‘÷‰À. $$ P = \left(% \begin{array}{ccccc} r_0 & p_0 & 0 & 0 & \cdots \\ q_1 & r_1 & p_1 & 0 & \cdots \\ 0 & q_2 & r_2 & p_2 & \cdots \\ \vdots & \ddots & \ddots & \ddots &\ddots \\ \end{array}% \right) $$ ø쉵‰ü :\\ \InE{}$p_i + r_i + q_i =1~~~~~~~~~~~~~~~~~~~ i \geq 1 ~~ , p_i , q_i > 0 ~~,r_i \geq 0$~~~~~~~~~~~~~~~~~~~~ \EnE{}\\ \InE{}$p_0 + r_0 = 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~r_0,p_0 \geq 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ:} ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ¬‰Ó “‰€‰Àı ğ‰Æ‰Æ‰µ‰‚:\\ ¨‰Âøş‰Å ¢û‰€‰Àùı ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À  ¢¤ û‰Â øŸ‰À ¥õ‰‘÷‰ü “‰‚ ş‰× õ‰È‰µ‰Âı ¨‰Âøş‰Å õ‰ü¢û‰À ø ¤ õ‰È‰µ‰Âı ¢¤ ş‰× øŸ‰À ¥õ‰‘ö “‰‚ •‰‘ş‰‘ö õ‰ü¤¨‰À. ¢¤ û‰Â øŸ‰À ¥õ‰‘÷‰ü —‰ã‰À¢ı —‰Ê‰‘¢ê‰ü õ‰È‰µ‰Âı ø¤¢ ¬‰Ó õ‰üª‰÷‰À ø ¨‰Âøş‰Å¢û‰€‰Àù “‰‚ —‰Â—‰ƒ‰° ø¤ø¢ ÷‰ú‰‘ “‰‚ ÷‰ú‰‘ ¨‰Âøş‰Å ¡‰û‰À ¢¢. ê‰Â­ ‰ƒ‰À —‰ã‰À¢ õ‰È‰µ‰Âş‰‘÷‰ü  ¢¤ øŸ‰À ¥õ‰‘÷‰ü \InE{}$n$\EnE{} ô ø¤¢ ¬‰Ó õ‰üª‰÷‰À, õ‰Æ‰µ‰Ö‰Û ¥ ¨‰‘ş‰Â ¥õ‰‘öû‰‘ “‰‘ª‰€‰À ø —‰¥ş‰â Ÿ‰µ‰Ş‰‘ñ ö ÷‰ƒ‰Ã “‰‚ \InE{}$n$\EnE{} “‰Æ‰µ‰Ú‰ü ÷‰Àª‰µ‰‚ “‰‘ª‰À. ğ‰Â \InE{}$\xi _n $\EnE{} ş‰ß õ‰µ‰ç‰ƒ‰Â “‰‘ª‰À, ¢¤ş‰İ:\\ \InE{}$P(\xi _n = k) = P(\xi = k) = a_k ~~~~~~~k=0,1,2,...~~~~~~~~~~~~~~~~~ $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$\sum_{k=0}^\infty a_k =1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} ğ‰Â \InE{}$X_n$\EnE{} —‰ã‰À¢ õ‰È‰µ‰Âş‰‘ö ¢¡‰Û ¬‰Ó ¢¤ øŸ‰À ¥õ‰‘÷‰ü \InE{}$n$\EnE{} “‰‘ª‰À, ÷‰Ú‰‘ù:\\ $$ X_{n+1} = \left\{% \begin{array}{cc} X_n - 1 + \xi _n &~~~~ \hbox{$X_n \geq 0$} \\ \xi _n &~~~~ \hbox{$X_n = 0$} \\ \end{array}% \right. $$ $$ P = \left(% \begin{array}{ccccc} a_0 & a_1 & a_2 & a_3 & \cdots \\ a_0 & a_1 & a_2 & a_3 & \cdots \\ 0 & a_0 & a_1 & a_2 & \cdots \\ 0 & 0 & a_0 & a_1 & \cdots \\ 0 & 0 & 0 & a_0 & \cdots \\ \vdots & \vdots & \vdots & \ddots & \ddots \\ \end{array}% \right) $$ \InE{}$P(X_{n+1} = j| X_n=i) = P(X_n - +\xi _n = j~|X_n=i) = P(\xi _{n+1} = j-i+1) = a_{j-i+1}~$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$ ~~~ ~~~~i \geq1 , ~j-i+1\geq0,~j\geq i-1 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$P(X_{n+1} =j | X_n = 0) = P(\xi _n = j) = a_j ~~~i=0 , ~j\geq0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \hspace{-8mm} {\large\siah ¤¢ù“‰€‰Àı ø®‰ã‰ƒ‰´û‰‘ı ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó: }\\ \hspace{-8mm} {\large\siah —‰ã‰Âş‰Ó :} ø®‰ã‰ƒ‰´ \InE{}$j$\EnE{} ¤ ¢¤ ¢¨‰µ‰Â§ ğ‰ş‰€‰À, ğ‰Â ¥÷‰¹‰ƒ‰Â “‰‘ ª‰Âøá ¥ \InE{}$i$\EnE{} “‰‘ Ÿ‰µ‰Ş‰‘ñ õ‰·‰±‰´ •‰Å ¥ —‰ã‰À¢ õ‰µ‰€‰‘û‰ü ÷‰µ‰Ö‰‘ñ “‰‚ ø®‰ã‰ƒ‰´ \InE{}$j$\EnE{} “‰Â¨‰À: \InE{}$$\exists ~n \geq 0 ~; ~p_{ij}^n > 0$$\EnE{} õ‰·‰\nasb … ¢¤ ¥÷‰¹‰ƒ‰Â ¬‰Ó “‰€‰Àı ğ‰Æ‰Æ‰µ‰‚ ğ‰Â \InE{}$\forall ~i~\geq 0 ~,~a_i > 0$\EnE{} ÷‰Ú‰‘ù —‰Ş‰‘ô ø®‰ã‰ƒ‰µ‰ú‰‘ ¢¤ ¢¨‰µ‰Â§ ş‰Ø‰Àş‰Ú‰Â÷‰À. “‰‚ ‰× ¤¨‰İ ÷‰Ş‰¢¤ ÷‰È‰‘ö ¢û‰€‰Àùı ÷‰µ‰Ö‰‘„– ş‰× õ‰ÂŸ‰Ü‰‚ı ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ¤ õ‰ü—‰ö ¢ş‰À.\\ \hspace{-8mm} {\large\siah —‰ã‰Âş‰Ó :} ¢ø ø®‰ã‰ƒ‰´  ¢¤ ¢¨‰µ‰Â§ û‰Æ‰µ‰€‰À ¤ õ‰Â—‰±‰Í ğ‰ş‰€‰À: \InE{}$$i~ \leftrightarrow ~ j ~~~~\exists ~ n \geq ~0~,~ p_{ij}^n ~>0$$\EnE{} \InE{}$i~ \leftrightarrow ~ j ~~~~\exists ~ m \geq ~0~,~ p_{ji}^m ~>0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ õ‰Â—‰±‰Í “‰¢ö ş‰× ¤“‰Î‰‚ı û‰İ¤¥ı ¨‰´:\\ 1( \InE{}$i~ \leftrightarrow ~ i$\EnE{}: û‰Â ø®‰ã‰ƒ‰´ “‰‘ ¡‰¢© ¢¤ ¤—‰±‰‘¯ ¨‰´ : \InE{}$p_{ii}^0=1$\EnE{} \\ 2( \InE{}$i~ \leftrightarrow ~ j$\EnE{}: ÷‰Ú‰‘ù \InE{}$j~ \leftrightarrow ~ i $\EnE{}\\ 3( \InE{}$i~ \leftrightarrow ~ j$\EnE{} ø \InE{}$j~ \leftrightarrow ~ k$\EnE{} ÷‰Ú‰‘ù \InE{}$i~ \leftrightarrow ~ k$\EnE{} ¥ş‰Â:\\ \InE{}$\exists ~ n_1 ~;~ p_{ij}^{n_1} >0 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$\exists ~ n_2 ~;~ p_{ji}^{n_2} >0 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$\exists ~ m_1 ~;~ p_{jk}^{m_1} >0 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$\exists ~ m_2 ~;~ p_{kj}^{m_2} >0 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} “‰‘ ¨‰µ‰Ô‰‘¢ù ¥ —‰Æ‰‘øı ‰³‰Ş‰ß-î‰Ü‰Ş‰ğ‰Âøé õ‰ü—‰ö ÷‰È‰‘ö ¢¢:\\ \InE{}$p_{ik}^{n_1+m_1}~\geq~ p_{ij}^{n_1}~p_{jk}^{m_1}~>0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} ø “‰‚ û‰Ş‰ƒ‰ß —‰Â—‰ƒ‰° :\\ \InE{}$p_{k_1}^{n_2+m_2}~>~0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \hspace{-8mm} {\large\siah —‰Ş‰Âş‰ß :} ÷‰‘õ‰Æ‰‘øı ¥ş‰Â ¤ ™‰±‰‘– ‰ƒ‰À:\\ \InE{}$p_{ik}^{n_1+m_1} = \sum_{\nu \in S} p_{i \nu}^{n_1}~p_{\nu k}^{m_1}~\geq~p_{ij}^{n_1} p_{jk}^{m_1} > 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} “‰‘ —‰›‰‚ “‰‚ ş‰€‰Ø‰‚ õ‰Â—‰±‰Í “‰¢ö ş‰× ¤“‰Î‰‚ı û‰İ¤¥ı ¨‰´, “‰€‰‘“‰Âş‰ß õ‰ü—‰ö ø®‰ã‰ƒ‰µ‰ú‰‘ ¤ “‰‚ ¨‰ú‰‘ı û‰İ¤¥ı —‰Ö‰Æ‰ƒ‰İ î‰Â¢. ¢¤ û‰Â ¤¢ù ş‰‘ § û‰İ¤¥ı ø®‰ã‰ƒ‰µ‰ú‰‘ş‰ü ø›‰¢ ¢¤÷‰À  “‰‘ û‰İ õ‰Â—‰±‰Í û‰Æ‰µ‰€‰À.\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ :} ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘ Ÿ‰µ‰Ş‰‘„– ÷‰µ‰Ö‰‘ñ \InE{}$P$\EnE{}:\\ $$ P = \left(% \begin{array}{ccccc} {1\over2} & {1\over2} & 0 & 0 & 0 \\ {1\over3} & {2\over3} & 0 & 0 & 0 \\ 0 & 0 & {1\over4} & {1\over4} & {1\over2} \\ 0 & 0 & {1\over3} & {1\over3} & {1\over3} \\ 0 & 0 & 0 & 0 & 1 \\ \end{array}% \right) = \left(% \begin{array}{cc} P_1 & 0 \\ 0 & P_2 \\ \end{array}% \right) $$ \\ \InE{}$\{1,2\} ~ , ~ \{3,4,5\}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} : §û‰‘ı û‰İ¤¥ı\\\\ ¢¤ øì‰â ¢ø ê‰Âş‰€‰À 艃‰Â õ‰Â—‰±‰Í “‰‘ û‰İ —‰Â‰° ª‰À÷‰À. ğ‰Â ş‰× ¥÷‰¹‰ƒ‰Â ê‰Ö‰Í ş‰× ¤¢ùı û‰İ¤¥ı ¢ª‰µ‰‚ “‰‘ª‰À ö ¤ ¥÷‰¹‰ƒ‰Â —‰½‰ş‰Û ÷‰‘•‰Áş‰Â ğ‰ş‰€‰À.\\ ¢¤ ì‰Àô ¥¢ö —‰Ê‰‘¢ê‰ü “‰‘ õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ¥ş‰Â : \begin{equation} P = \left(% \begin{array}{ccccccccc} 1 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\ q & 0 & p & 0 & 0 & \cdots & 0 & 0 & 0 \\ 0 & q & 0 & p & 0 & \cdots & 0 & 0 & 0 \\ 0 & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & q & 0 & p \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array}% \right) \end{equation} §û‰‘ı û‰İ¤¥ı “‰Â“‰Â ¨‰´ “‰‘: \InE{}$$\{a\}~,~ \{0\}~,~\{1,2,...,a-1\}$$\EnE{} \hspace{-8mm} {\large\siah õ‰·‰‘ñ :} ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘ õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ¥ş‰Â —‰½‰ş‰Û ÷‰‘•‰Áş‰Â ¨‰´: \begin{equation} P = \left(% \begin{array}{ccccc} p_0 & q_0 & 0 & 0 & \cdots \\ p_1 & 0 & q_1 & 0 & \cdots \\ p_2 & 0 & 0 & q_2 & \cdots \\ \vdots & \vdots & \vdots & \ddots & \ddots \\ \end{array}% \right) \end{equation} \InE{}$q_i~>0~,~p_i~>0~~~~~~i=0,1,2,...~~~~~~~~~~~~~~~~~~$~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ \exists ~n_j~; ~~p_{ij}^n~>0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \hspace{-8mm} {\large\siah —‰€‰‘ø’ ¥÷‰¹‰ƒ‰Âùı õ‰‘¤î‰Ó}\\ ¢ø¤ùı —‰€‰‘ø’ ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} “‰Â“‰Â ¨‰´ “‰‘\InE{}$d(i) = g.c.d \{~n\geq 1~,~p_{ii}^n > 0\}$ \EnE{}.\\ ğ‰Â \InE{}$ \forall~n~\geq 1~:~p_{ii}^n = 0$\EnE{} ÷‰Ú‰‘ù \InE{}$d(i) = 0$\EnE{}.\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ)1(:} ¢¤ ¥÷‰¹‰ƒ‰Â ì‰Àô ¥¢ö —‰Ê‰‘¢ê‰ü “‰‘ õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ )1( ¢¤ş‰İ:\\ \InE{}$d(0) = 1~,~d(a) = 1 ~,~d(i) = \{2,4,\ldots\} = 2 ~~~~~~~~~~~~i =1,2,...,a-1$~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ)2( :} ğ‰Â õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰é “‰‚ ¬‰¤– : \InE{}$$P=\bordermatrix{&0&1&2&3&\ldots&n\cr0&0&1&0&0&\cdots&0\cr1&0&0&1&0&\cdots&0\cr2&0&0&0&1&\cdots&0\cr\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots\cr{n-1}&0&0&\cdots&\cdots&0&1\cr n &1 & 0 & 0 & \cdots & \cdots & 0 }$$\EnE{} “‰‘ª‰À, ÷‰Ú‰‘ù \InE{}$d(i) = n$\EnE{}.\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ)3( :} ¢¤ ¥÷‰¹‰ƒ‰Â ª‰Ş‰‘¤ùı )2( ¢¤ş‰İ  :\\ \InE{}$d(0) = g.c.d \{1,2,3,\ldots\} = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$d(1) = g.c.d \{2,3,4,\ldots\} = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$d(i) = 1 ~~~~\forall~ i ~\geq 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ¢¤ øì‰â ¢ø¤ùı —‰€‰‘ø’ —‰Ş‰‘ô ø®‰ã‰ƒ‰´û‰‘ı ş‰× § û‰İ¤¥ı “‰Â“‰Â÷‰À. “‰€‰‘“‰Âş‰ß ğ‰Â ş‰× ¥÷‰¹‰ƒ‰Â —‰½‰ş‰Û÷‰‘•‰Áş‰Â “‰‘ª‰À ÷‰Ú‰‘ù ¢ø¤ùı —‰€‰‘ø’ û‰Ş‰‚ı ø®‰ã‰ƒ‰´û‰‘ “‰Â“‰Â ¨‰´.\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ)4( :} ÷‰È‰‘ö ¢û‰ƒ‰À ğ‰Â —‰ã‰À¢ ø®‰ã‰ƒ‰´û‰‘ı ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó \InE{}$m$\EnE{} “‰‘ª‰À ø ø®‰ã‰ƒ‰´ \InE{}$j$\EnE{} ¢¤ ¢¨‰µ‰Â§ \InE{}$i$\EnE{} “‰‘ª‰À ÷‰Ú‰‘ù ş‰ß ø®‰ã‰ƒ‰´ ¢¤ ‰µ‰Â ş‰‘ õ‰Æ‰‘øı \InE{}$m-1$\EnE{} õ‰ÂŸ‰Ü‰‚ ¢¨‰´ ş‰‘ꉵ‰€‰ü ¨‰´. \\ \hspace{-8mm} {\large\siah Ÿ‰Û :}\\ ê‰Â­ ‰ƒ‰À \InE{}$S$\EnE{} ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ “‰‘ª‰À:\\ \InE{}$~~~~~~\Rightarrow ~~ \exists ~n~ ; ~~~p_{ij}^n > 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$j$\EnE{} ¢¤ ¢¨‰µ‰Â§ \InE{}$i$\EnE{} ¨‰´ \\ \InE{}$\Rightarrow ~~~\exists ~k_1,k_2,\ldots,k_{n-1} \in S ~;~~ p_{ij}^n > p_{ik_1}p_{k_1k_2}p_{k_2k_3}...p_{k_{n-2} k_{n-1}}p_{k_{n-1}j} > 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} “‰€‰‘“‰Âş‰ß õ‰ü—‰ö \InE{}$k_r$\EnE{} û‰‘ ¤ ω¤ı ğ‰Âꉴ  \InE{}$k_r \neq k_{r'}$\EnE{} “‰Âı \InE{}$ r \neq~r'$\EnE{}. ¥ş‰Â ğ‰Â \InE{}$k_r$\EnE{} ø \InE{}$k_{r'}$\EnE{} “‰Â“‰Â “‰‘ª‰€‰À “‰€‰‘“‰Âş‰ß :\\ \InE{}$p_{ik_1}p_{k_1 k_2}\ldots p_{k_{r-1}k_{r}}p_{k_{r'}k_{r'+1}}\ldots p_{k_{n-1}j} > 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰€‰‘“‰Âş‰ß ωñ õ‰Æ‰ƒ‰Â \InE{}$r' - r$\EnE{} ‰‘ù—‰Â õ‰üª‰¢, ¥ş‰Â \InE{}$k_r\neq k_{r'}$\EnE{} “‰Âı û‰Â \InE{}$r = r' = 1,2,\ldots ,n-1$\EnE{}. •‰Å \InE{}$n-1 \leq m-2$\EnE{}. ‰ö ş‰ß \InE{}$n-1$\EnE{} ø®‰ã‰ƒ‰´ ¥ ş‰× õ‰¹‰Ş‰ä‰‚ı \InE{}$m-2$\EnE{} ä‰Ì‰ı ¡‰µ‰ƒ‰‘¤ õ‰üª‰¢, •‰Å “‰ƒ‰Ç ¥ \InE{}$m-2$\EnE{} ä‰Ì‰ õ‰µ‰Ô‰‘ø– 쉑“‰Û ¡‰µ‰ƒ‰‘¤ ÷‰ƒ‰Æ‰´ ø “‰€‰‘“‰Âş‰ß \InE{}$n ~\leq m-1$\EnE{}.\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ)5( :} §û‰‘ı ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰é ¥ş‰Â ¤ õ‰È‰¿‰É ‰ƒ‰À ø —‰ã‰ƒ‰ƒ‰ß ‰ƒ‰À ş‰‘ “‰‘¥ğ‰È‰µ‰ü û‰Æ‰µ‰€‰À ş‰‘ ğ‰Á¤? $$ P = \left(% \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ {1\over4} & {1\over2} & {1\over4} & 0 & 0 & 0 \\ 0 & {1\over5} & {2\over5} & {1\over5} & 0 & {1\over5} \\ 0 & 0 & 0 & {1\over6} & {1\over3} & {1\over2}\\ 0 & 0 & 0 & {1\over2} & 0 & {1\over2} \\ 0 & 0 & 0 & {1\over4} & 0 & {3\over4}\\ \end{array}% \right) $$ \InE{} ~~~~~~~~~~~~~~~~~~~~\EnE{}ø®‰ã‰ƒ‰´ \InE{} $0$\EnE{}“‰‘¥ğ‰È‰µ‰ü ¨‰´ \InE{}$f_{00}^1 = 1~~~f_{00}^2 = 0~~\ldots ~~\sum_{n=1}^\infty f_{00}^n = $1$ ~~~~~~~\Rightarrow $~~~~~~~~~~~ \EnE{}\\ \InE{}$f_{11}^1 = {1\over2}~~,~~f_{11}^2 = {1\over4} . ({1\over5}) = {1\over20} ~~, ~ f_{11}^n = ({1\over4}) .({2\over5})^{n-2} . ({1\over5})~~~~~~~n \geq 3 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\sum_{n=1}^\infty f_{11}^n = {1\over2} + {1\over20} + \sum_{n=3}^\infty ({1\over4}).({2\over5})^{n-2}({1\over5}) $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$= {1\over2} + {1\over20} + {1\over20}.({2\over5} / (1 - {2\over5})) = {1\over2} + {1\over20} + {1\over30} ~< 1 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ø®‰ã‰ƒ‰´ 1 ğ‰Á¤ ¨‰´. “‰€‰‘“‰Âş‰ß ø®‰ã‰ƒ‰´ 2 ÷‰ƒ‰Ã  ¢¤ û‰Ş‰‘ö ¤¢ùı 1 ¨‰´, ğ‰Á¤ ¨‰´.\\ \InE{}$f_{33}^{1} = {1\over 6} ~~,~~f_ {33}^2 = ({1\over3} . {1\over2}) + ({1\over2} . {1\over 4}) = {1\over6} + {1\over8} $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$f_{33}^1 = {1\over3} . {1\over2} . {1\over4} + {1\over2} . {3\over4} . {1\over4}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$f_{33}^n = ({1\over3} . {1\over2} . {1\over4} . ({3\over4})^{n-3} ) + ({1\over2} . {1\over4} . ({3\over4})^{n-2} )= ({3\over4})^{n-3}({1\over3} . {1\over2} . {1\over4} + {1\over2} . {3\over4}{1\over4}) $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$= ({3\over4})^{n-3}({1\over24} + {3\over 32}) ~~~~~~~~~~~~n\geq3$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\sum_{n=1}^\infty f_{33}^n = {1\over6} + {1\over6} + {1\over8} + ({1\over24} + {3\over32})\sum_{n=3}^\infty ({3\over4})^{n-3}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ = {1\over6} + {1\over6} + {1\over8} + ({1\over24} + {3\over32}).{1\over({1\over4})} = {1\over6} + {1\over6} + {1\over8} + {1\over6} + {3\over8} = {3\over6} + {1\over2} = 1$~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰€‰‘“‰Âş‰ß ø®‰ã‰ƒ‰´ 3 “‰‘¥ğ‰È‰µ‰ü ¨‰´ ø ¥ ş‰ß ¤ø 4 ø 5 ÷‰ƒ‰Ã “‰‘¥ğ‰È‰µ‰ü û‰Æ‰µ‰€‰À.\\ \hspace{-8mm} {\large\siah õ‰·‰‘ñ)6( :} ÷‰È‰‘ö ¢û‰ƒ‰À ğ‰Â ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} “‰‘¥ğ‰È‰µ‰ü “‰¢ù ø “‰‘ ø®‰ã‰ƒ‰´ \InE{}$j$\EnE{} ¢¤ ¤—‰±‰‘¯ ÷‰±‰‘ª‰À ÷‰Ú‰‘ù \InE{}$p_{ij} = 0$\EnE{}. “‰€‰‘“‰Âş‰ß ¥õ‰‘÷‰ü  ê‰Âş‰€‰À ø¤¢ ş‰× § “‰‘¥ğ‰È‰µ‰ü ¥ ø®‰ã‰ƒ‰´û‰‘ ª‰À û‰Âğ‰Ã ö ¤ —‰Âí ÷‰Ş‰ü‰À. “‰Àş‰ß ä‰Ü‰´ ş‰× § “‰‘¥ğ‰È‰µ‰ü è‰Ü‰° ş‰× § “‰Æ‰µ‰‚ ÷‰‘õ‰ƒ‰Àù õ‰üª‰¢.\\ \hspace{-8mm} {\large\siah Ÿ‰Û :}\\ \InE{}$ i \nleftrightarrow j$ ~~~$ \Rightarrow ~~ p_{ij} = 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}~ ø \InE{}$i$\EnE{} “‰‘¥ğ‰È‰µ‰ü\\ \InE{}$\sum_{n=1}^\infty f_{ii}^n = (\sum_{n=2}^\infty \sum_{k\neq i} p_{ik} f_{ki}^{n-1}) + p_{ii}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ğ‰Â \InE{}$p_{ij} \neq 0$\EnE{} “‰€‰‘“‰Âş‰ß \InE{}$\forall ~n~~~f_{ji}^n = 0$\EnE{} ‰ö \InE{}$i \nleftrightarrow j$\EnE{}\\ \InE{}$\Rightarrow~~\sum_{n=1}^\infty f_{ii}^n = p_{ii} + \sum_{n=2}^\infty \sum_{k\neq i,j} p_{ik} f_{ki}^{n-1} = p_{ii} + \sum_{k\neq i,j} p_{ik} \sum_{n=2}^\infty f_{ki}^{n-1} $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ —‰€‰‘ì‰Ë \InE{}$ \leq ~p_{ii} + \sum_{k\neq i,j} p_{ik} = \sum_{k \neq j} p_{ik} = 1- p_{ij} < 1~~$\EnE{}.\\\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚:} û‰Â ğ‰‘ù \InE{}$i\leftrightarrow j$\EnE{}, ÷‰Ú‰‘ù \InE{}$d(i) = d(j)$\EnE{}.\\ \hspace{-8mm} {\siah ™‰±‰‘–:}\\\\\\ {\siah ì‰Ì‰ƒ‰‚:}\\ ğ‰Â \InE{}$i$\EnE{} ¢ø¤ùı —‰€‰‘ø’ \InE{}$d(i)$\EnE{} “‰‘ª‰À:\\ \InE{}$\exists ~N_i~~;~ \forall~n> N_i , p_{ii}^{nd(i)}> 0 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah ÷‰µ‰ƒ‰¹‰‚:} û‰Â ğ‰‘ù \InE{}$p_{ji}^m > 0 $\EnE{} ÷‰Ú‰‘ù “‰‚ ¥ı û‰Â \InE{}$n$\EnE{} “‰‚ ì‰À¤ ê‰ü “‰Ã¤ï \InE{}$p_{ji}^{m+nd(i)} > 0$\EnE{}.\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó )(:} ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ô‰ü  ¢ø¤ù —‰€‰‘ø’ û‰Ş‰‚ı ø®‰ã‰ƒ‰´û‰‘ı ö “‰Â“‰Â ş‰× “‰‘ª‰À ş‰ã‰€‰ü \InE{}$d(i) = 1 : \forall i \in S $\EnE{}, ÷‰‘õ‰µ‰€‰‘ø’ ÷‰‘õ‰ƒ‰Àù õ‰üª‰¢.\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó )(:} ê‰Â­ ‰ƒ‰À “‰‘ ª‰Â¯ ª‰Âøá ¥ \InE{}$i$\EnE{} Ÿ‰µ‰Ş‰‘ñ ş‰€‰Ø‰‚ øó‰ƒ‰ß “‰‘¥ğ‰È‰´ “‰‚ \InE{}$i$\EnE{} ¢¤ ÷‰µ‰Ö‰‘ñ \InE{}$n$\EnE{} ô ¤  ¢û‰À ¤ “‰‘ \InE{}$f_{ii}^n$\EnE{} ÷‰È‰‘ö ¢û‰ƒ‰İ. ¢¤ ş‰ß ¬‰¤– :\\ \InE{}$f_{ii}^n = P(X_n = i , X_\nu = \neq i ~~~\nu = 1,2,...,n-1 |~X_0 = i)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$f_{11}^1 = p_{ii}~~~~,~~~f_{ii}^0 = 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} õ‰ü—‰ö ÷‰È‰‘ö ¢¢ :\\ \InE{}$p_{11}^n = \sum_{k=0}^n f_{ii}^k p_{ii}^{n-k}~~~~n\geq 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah ™‰±‰‘– :}\\ \InE{}$E_k$\EnE{} : •‰ƒ‰È‰‘õ‰À ş‰€‰Ø‰‚ \InE{}$X_n = 1$\EnE{} ø øó‰ƒ‰ß “‰‘¥ğ‰È‰´ ¢¤ 󉽉щ‚ \InE{}$k$\EnE{} ¤  ¢û‰À õ‰È‰Âø¯ “‰‚ ş‰€‰Ø‰‚ \InE{}$X_0 = i$\EnE{}\\ \InE{}$p_{ii}^n = P(X_n = i ~|~X_0 = i) = P(\cup_{k=0}^n E_k) = \sum_{k=0}^n P(E_k)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$ = \sum_{k=0}^n P(E_k) = \sum_{k=0}^n P(X_n = i| X_0 =i ,) f_{ii}^k$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ = \sum_{k=0}^n f_{ii}^k p_{ii}^{n-k}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah —‰“‰â õ‰ó‰À Ÿ‰µ‰Ş‰‘ñ :}\\\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} —‰‘“‰â —‰¥ş‰â \InE{}$p_{ij}(.)$\EnE{} —‰‘“‰â õ‰ó‰À ¢÷‰±‰‘ó‰‚ \InE{}$\{p_{ij}^n\}$\EnE{} 䉱‰‘¤– ¨‰´ ¥ :\\ \InE{}$p_{ij}(s) = \sum_{n=0}^\infty s^n~p_{ij}^n~~~~~~~~~~~~~~~~~~~~|s|<1 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} —‰‘“‰â \InE{}$F_{ij}(.)$\EnE{} —‰‘“‰â õ‰ó‰À ¢÷‰±‰‘ó‰‚ \InE{}$\{f_{ij}^n\}$\EnE{} 䉱‰‘¤– ¨‰´ ¥:\\ \InE{}$F_{ij}(s) = \sum_{n=0}^\infty f_{ij}^n~s^n~~~~~~~~~~~~~~~~~~~~|s|<1 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ õ‰ü—‰ö ™‰‘“‰´ î‰Â¢  :~~~~~\InE{}$p_{ii}(s) = $ $1\over 1-F_{ii}(s) $~~~~~$|s|<1$\EnE{}\\\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} ¤ “‰‘¥ğ‰È‰µ‰ü ğ‰ş‰€‰À ğ‰Â: \InE{}$\sum_{n=1}^\infty f_{ii}^n = 1 $\EnE{}.\\ ş‰ã‰€‰ü ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} “‰‘¥ğ‰È‰µ‰ü ¨‰´, ğ‰Â “‰‘ ª‰Âøá ¥ \InE{}$i$\EnE{}, “‰‘ Ÿ‰µ‰Ş‰‘ñ ş‰× “‰ã‰À ¥ —‰ã‰À¢ õ‰µ‰€‰‘û‰ü “‰‘¤ ÷‰µ‰Ö‰‘ñ “‰‚ \InE{}$i$\EnE{} “‰‘¥ğ‰Â¢¢.\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} ø®‰ã‰ƒ‰µ‰ü  “‰‘¥ğ‰È‰µ‰ü ÷‰±‰‘ª‰À, ğ‰Á¤ ¨‰´.\\ õ‰ü—‰ö ÷‰È‰‘ö ¢¢  “‰‘¥ğ‰È‰µ‰ü “‰¢ö ¤ “‰‘ —‰›‰‚ “‰‚ \InE{}$p_{ii}^n$\EnE{} û‰‘ õ‰ü—‰ö ÷‰È‰‘ö ¢¢.\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} “‰‘¥ğ‰È‰µ‰ü ¨‰´ ğ‰Â ø —‰€‰ú‰‘ ğ‰Â: \InE{}$$\sum_{n=1}^\infty p_{ii}^n = \infty $$\EnE{} \hspace{-8mm} {\siah õ‰·‰‘ñ :} ¢¤ ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘ õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ: \InE{}$$\bordermatrix{&0 &1 &2 &3 \cr 0&{1\over2}&{1\over2}&0&0 \cr 1&{1\over3}&{2\over3}&0&0 \cr 2 &0&0&{3\over4}&{1\over4} \cr 3 &0&0&1&0}$$ \EnE{} ø®‰ã‰ƒ‰´ \InE{}$0$\EnE{} “‰‘¥ğ‰È‰µ‰ü ¨‰´ ş‰‘ ğ‰Á¤?\\ “‰Âı ø®‰ã‰ƒ‰´ \InE{}$0$\EnE{} “‰‘ş‰À \InE{}$\sum_{n=1}^\infty f_{00}^n$\EnE{} ¤ õ‰½‰‘¨‰±‰‚ ‰ƒ‰İ :\\ \InE{}$f_{00}^1 = {1\over2}~~~~~~~~~~~~~~~~f_{00}^2 = {1\over2} . {1\over3} = {1\over6}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$f_{00}^n = {1\over2} . ({2\over3})^{n-2} . {1\over3}~~~~~~n\geq2$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ¢¤ ÷‰µ‰ƒ‰¹‰‚ :\\ \InE{}$\sum_{n=1}^\infty f_{00}^n = {1\over2} + \sum_{n=2}^\infty {1\over2}({2\over3})^{n-2} {1\over3} = {1\over2} + {1\over6}.{1\over1-{2\over3}} = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} •‰Å ø®‰ã‰ƒ‰´ \InE{}$0$\EnE{} “‰‘¥ğ‰È‰µ‰ü ¨‰´.\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} “‰‘¥ğ‰È‰µ‰ü ¨‰´ ğ‰Â ø —‰€‰ú‰‘ ğ‰Â :\\ \InE{}$$\sum_{n=1}^\infty p_{ii}^n = \infty $$\EnE{} \hspace{-8mm} {\siah ÷‰Ø‰µ‰‚ :} “‰‘ —‰›‰‚ “‰‚ ì‰Ì‰ƒ‰‚ ê‰ë õ‰ü—‰ö ™‰‘“‰´ î‰Â¢ û‰Â ğ‰‘ù \InE{}$i \leftrightarrow j $\EnE{} ø \InE{}$i$\EnE{} “‰‘¥ğ‰È‰µ‰ü “‰‘ª‰À:\\ ÷‰Ú‰‘ù \InE{}$j$\EnE{} ÷‰ƒ‰Ã “‰‘¥ğ‰È‰µ‰ü ¨‰´.\\ \hspace{-8mm} {\siah ™‰±‰‘– :}\\ ‰ö \InE{}$i \leftrightarrow j $\EnE{} :\\ \InE{}$ \exists ~r \geq 1~~; ~p_{ij}^r > 0~~~~~~~~~,~~~\exists ~s\geq 1~;~p_{ji}^s > 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰€‰‘“‰Âş‰ß:\\ \InE{}$\sum_{n=0}^\infty p_{jj}^n \geq \sum_{n=0}^\infty p_{jj}^{n+r+s} \geq \sum_{n=0}^\infty p_{ji}^s p_{ij}^r p_{jj}^n = p_{ji}^s p_{ij}^r \sum_{n=0}^\infty p_{jj}^n = \infty$~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ “‰€‰‘“‰Âş‰ß õ‰‘÷‰€‰À ¢ø¤ùı —‰€‰‘ø’, ¡‰‘¬‰ƒ‰´ “‰‘¥ğ‰È‰µ‰ü “‰¢ö ÷‰ƒ‰Ã ş‰× ¡‰‘¬‰ƒ‰´ ¤¢ùı ¨‰´. ş‰ã‰€‰ü ¢¤ û‰Â § û‰İ¤¥ı ş‰‘ û‰Ş‰‚ı ø®‰ã‰ƒ‰´û‰‘ “‰‘¥ğ‰È‰µ‰ü û‰Æ‰µ‰€‰À ø ş‰‘ û‰Ş‰‚ ğ‰Á¤ ¡‰û‰€‰À “‰¢.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ¢¤ ş‰× ¥÷‰¹‰ƒ‰Â ì‰Àô ¥¢ö —‰Ê‰‘¢ê‰ü “‰‘ ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ ä‰À¢ ¬‰½‰ƒ‰¼, ğ‰Â ¢¤ û‰Â ÷‰µ‰Ö‰‘ñ £¤ù “‰‘ Ÿ‰µ‰Ş‰‘ñ \InE{}$p$\EnE{} ş‰× øŸ‰À “‰‚ ¤¨‰´ ø “‰‘ Ÿ‰µ‰Ş‰‘ñ \InE{}$q = 1-p$\EnE{} ş‰× ì‰Àô “‰‚ ‰² “‰Âø¢. ø®‰ã‰ƒ‰´û‰‘ ¤ ¥ ÷‰Ñ‰Â “‰‘¥ğ‰È‰µ‰ü ş‰‘ ğ‰Á¤ “‰¢ö “‰Â¤¨‰ü ‰ƒ‰À\InE{}$(0 0 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$>0$\EnE{} )“‰‘ ª‰Âøá ¥ \InE{}$j$\EnE{} “‰‘„¡‰Âù \InE{}$k$\EnE{} ¤ õ‰…쉑– ‰À, “‰Àøö ş‰€‰Ø‰‚ õ‰¹‰À¢\nasb \InE{}$j$\EnE{} ¤ õ‰…쉑– î‰Â¢ù “‰‘ª‰À( \InE{}$~~\Rightarrow~~\alpha = P$\EnE{}\\ \InE{}$ 1 - F_{jj} \geq \alpha (1-F_{kj}) \geq 0 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ¢¤ ÷‰µ‰ƒ‰¹‰‚ \InE{}$F_{kj} = 1$\EnE{} ø \InE{}$k\rightarrow 0$\EnE{}. ÷‰µ‰ƒ‰¹‰‚: ğ‰Â \InE{}$j \rightarrow k$\EnE{} ş‰‘ \InE{}$j$\EnE{} ø \InE{}$k$\EnE{} ¢¤ ş‰× § û‰Æ‰µ‰€‰À ş‰‘ \InE{}$j$\EnE{} î‰Á¤¨‰´. ş‰ã‰€‰ü ¥ ø®‰ã‰ƒ‰µ‰ú‰‘ı “‰‘¥ğ‰È‰µ‰ü ê‰Ö‰Í “‰‚ “‰‚ Ÿ‰‘󉵉ú‰‘ı “‰‘¥ğ‰È‰µ‰ü û‰Ş‰‘ö § õ‰ü—‰ö ¤ê‰´.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} \InE{}$\{1,2,3\}$\EnE{} : “‰‘¥ğ‰È‰µ‰ü ø \InE{}$\{4,0\}$\EnE{} : ğ‰Á¤\\\\\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} \InE{}$$\bordermatrix{&0&1&2 \cr 0&{1\over2}&{1\over2}&0 \cr 1 &{1\over5}&{4\over5}&0 \cr 2 &{1\over3}&{1\over6}&{1\over2}}$$~~~~\EnE{} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ğ‰Á¤ : \InE{}$\{2\}$ \EnE{} ~~~~~~“‰‘¥ğ‰È‰µ‰ü : \InE{}$\{0,1\}$\EnE{}\\ \InE{}$F_{10} = \sum_{n=1}^\infty f_{10}^n = {1\over5} + ({4\over5} . {1\over5}) + ({4\over5})^2 {1\over5} + \ldots = {1\over5}.{1\over1-{4\over5}} = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$F_{12} = \sum_{n=1}^\infty f_{12}^n = 0 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$F_{21} = \sum_{n=1}^\infty f_{21}^n = {1\over6} + [({1\over3} . {1\over2}) + ({1\over2} . {1\over6})] + [({1\over2})^2 . {1\over6} + {1\over2} . {1\over3} . {1\over2} + {1\over3} .{1\over2}.{1\over2}] + ... $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ + [({1\over2})^{n-1} {1\over6} + ({1\over2})^{n-2} . {1\over3} . {1\over2} + ({1\over2})^{n-3} . {1\over3} . {1\over2} . {1\over2} + ... + ({1\over2})^{n-k} . {1\over3} . ({1\over2})^{k-1} + ... + {1\over3} . ({1\over2})^{n-1}] + ...$\EnE{}\\ \InE{}$ = {1\over6} + [{1\over6} . {1\over2} + {1\over2} . {1\over3}] + ... + [{1\over6} . ({1\over2})^{n-1} + (n-1) {1\over3} . ({1\over2})^{n-1}] ~...$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ = {1\over6}.({1\over1-{1\over2}}) + {1\over3} \sum_{n=1}^\infty n({1\over2})^n = {1\over3} + {1\over2\times3} \sum_{n=1}^\infty n({1\over2})^{n-1} = {1\over3} + {1\over6} . {1-{1\over2} +{1\over2}\over({1\over2})^2} = {1\over3} + {2\over3} = 1$~~~~~~\EnE{}\\ ğ‰Â \InE{}$j$\EnE{} ğ‰Á¤ “‰‘ª‰À \InE{}$\sum_{n} p_{jj}^n = R(j,j) < \infty$\EnE{} ø “‰€‰‘“‰Âş‰ß \InE{}$\lim_{n\to\infty} p_{jj}^n = 0 $\EnE{}\\ \InE{}$~~~~~~~R(i,j) = F_{ij} R(j,j) < \infty$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}~~~~~~~~ø~~~~~~ \InE{}$\lim_{n\to\infty} p_{ij}^n = 0~~~~$\EnE{}\\ øó‰ü ğ‰Â\InE{}$j$ \EnE{} “‰‘¥ğ‰È‰µ‰ü “‰‘ª‰À õ‰Ş‰Ø‰ß ¨‰´ \InE{}$\lim_{n\to\infty} p_{ij}^n$\EnE{} “‰Â“‰Â “‰‘ ¬‰Ô‰Â “‰‘ª‰À ş‰‘ ÷‰±‰‘ª‰À.\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} ó‰Ó( ğ‰Â \InE{}$j$\EnE{} ş‰× ø®‰ã‰ƒ‰´ ğ‰Á¤ “‰‘ª‰À \InE{}$$\lim_{n\to\infty} p_{ij}^n = 0$$\EnE{} ~~~~~~~~~~’( ğ‰Â \InE{}$j$\EnE{} “‰‘¥ğ‰È‰µ‰ü “‰‘ª‰À ø \InE{}$$\lim_{n\to\infty} p_{ij}^n = 0$$\EnE{} ÷‰Â “‰‘“‰‘¥ğ‰È‰µ‰ü •‰œ õ‰ü÷‰‘õ‰ƒ‰İ. ¢¤ ¬‰¤—‰ü  \InE{}$j$\EnE{} “‰‘¥ğ‰È‰µ‰ü 艃‰Â •‰œ ÷‰‘¢ø¤ùı “‰‘ª‰À: \InE{}\\\EnE{} \InE{}$\lim_{n\to\infty} p_{ij}^n = \pi(j) > 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ø\\ \InE{}$\lim_{n\to\infty} p_{ij}^n = F_{ij} \pi(j)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah ÷‰Ø‰µ‰‚ :} ğ‰Â \InE{}$ i \leftrightarrow j $\EnE{} ø \InE{}$j$\EnE{} “‰‘¥ğ‰È‰µ‰ü “‰‘ª‰À, \InE{}$i$\EnE{} ÷‰ƒ‰Ã “‰‘¥ğ‰È‰µ‰ü •‰œ ¡‰û‰À “‰¢.\\ \hspace{-8mm} {\siah ™‰±‰‘– :} “‰‘ —‰›‰‚ “‰‚ ş‰€‰Ø‰‚ \InE{}$ s , r \geq 0 $\EnE{} ø›‰¢ ¢¤¢ “‰‚ ω¤ı : \InE{}$p_{ij}^r>0 , p_{ji}^s >0 $\EnE{}\\ \InE{}$p_{jj}^{n+r+s} \geq p_{ji}^r~p_{ii}^n ~ p_{ij}^s ~~~\Rightarrow~~\pi(j) \geq p_{ji}^n~p_{ij}^s \pi(i)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰€‰‘“‰Âş‰ß ğ‰Â \InE{}$\pi(j) = 0 $\EnE{} ÷‰µ‰ƒ‰¹‰‚ õ‰ü¢û‰À: \InE{}$\pi(i) = 0$\EnE{}.\\ ¢¤ ÷‰µ‰ƒ‰¹‰‚ —‰Ş‰‘ô ä‰Ì‰‘ı ş‰× § “‰‘¥ğ‰È‰µ‰ü •‰œ ş‰‘ “‰‘¥ğ‰È‰µ‰ü õ‰·‰±‰´ ¡‰û‰€‰À “‰¢. ø ş‰× ¥÷‰¹‰ƒ‰Â —‰½‰ş‰Û ÷‰‘•‰Áş‰Â ş‰‘ ğ‰Á¤ ş‰‘ “‰‘¥ğ‰È‰µ‰ü •‰œ ø ş‰‘ “‰‘¥ğ‰È‰µ‰ü õ‰·‰±‰´ ¨‰´.\\ \hspace{-8mm} {\siah ÷‰Ø‰µ‰‚ :} û‰Â ¥÷‰¹‰ƒ‰Â —‰½‰ş‰Û ÷‰‘•‰Áş‰Â “‰‘ —‰ã‰À¢ õ‰µ‰€‰‘û‰ü ø®‰ã‰ƒ‰´, “‰‘¥ğ‰È‰µ‰ü õ‰·‰±‰´ ¨‰´.\\ \hspace{-8mm} {\siah ™‰±‰‘– :} ¢¤ ¬‰¤—‰ü  û‰Ş‰‚ı ø®‰ã‰ƒ‰´û‰‘ ğ‰Á¤ ş‰‘ û‰Ş‰‚ “‰‘¥ğ‰È‰µ‰ü •‰œ “‰‘ª‰€‰À:\\ \InE{}$$\lim_{n\to\infty}p_{ij}^n = 0 ~~~\forall ~j\in ~S$$\EnE{} ø ¢¤ ÷‰µ‰ƒ‰¹‰‚:~~~~~~~~~~~~~~~~~~~~\InE{}$\lim_{n\to\infty}\sum_{j\in S} p_{ij}^n = \sum_{j\in S} \lim_{n\to\infty} p_{ij}^n = 0 $\EnE{}\\ ¢¤ Ÿ‰‘ó‰ü  \InE{}$\sum_{j\in S} p_{ij}^n = 1$\EnE{} “‰‚ ¥ı û‰Â \InE{}$n$\EnE{} ø ş‰ß —‰€‰‘ì‰Ë ¨‰´.\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} ğ‰Â ¥÷‰¹‰ƒ‰Â —‰½‰ş‰Û ÷‰‘•‰Áş‰Â ø ÷‰‘¢ø¤ùı “‰‘ª‰À, ÷‰Ú‰‘ù û‰Ş‰‚ı Ÿ‰‘ó‰´ û‰‘ “‰‘¥ğ‰È‰µ‰ü õ‰·‰±‰´ û‰Æ‰µ‰€‰À, ğ‰Â ø —‰€‰ú‰‘ ğ‰Â ¢¨‰µ‰Ú‰‘ù õ‰ã‰‘¢„– ¡‰Î‰ü \InE{}$$\pi(j) = \sum_{i \in S} \pi(i)p_{ij}$$\EnE{} \InE{}$\sum_{i\in S} \pi(i) = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ¢¤ı ›‰’ \InE{}$\pi = (\pi(1),\pi(2), ...)$\EnE{} “‰‘ª‰À. ğ‰Â ¢¨‰µ‰Ú‰‘ù õ‰ã‰‘¢„– „ ¢¤ı ›‰’ “‰‘ª‰À, ÷‰Ú‰‘ù ›‰’ ‰À\nasb õ‰·‰±‰´ ¨‰´, ›‰’ û‰‘ı ¢ş‰Ú‰Âı ÷‰À¤¢ ø \InE{}$$\pi(j) = \lim_{n\to \infty} p_{ij}^n $$\EnE{} \hspace{-8mm} {\siah ™‰±‰‘– :} ‰‘’û‰‘ı ê‰Âş‰€‰À.\\ “‰‘ ÷‰Ş‰‘¢ õ‰‘—‰Âş‰Æ‰ü ¢¤ş‰İ:\\\\ \InE{}$\pi P = \pi$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ \pi $1$ = $1$ $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\  \InE{}${\rm 1}$\EnE{} “‰Â¢¤ı ¨‰µ‰÷‰ü ¨‰´  —‰Ş‰‘ô ¢¤ş‰‚û‰‘ı ö õ‰Æ‰‘øı ş‰× ¨‰´ ø 쉑“‰Û ®‰Â’ ¢¤ \InE{}$\pi$\EnE{} õ‰ü“‰‘ª‰À.\\ “‰‘ —‰›‰‚ “‰‚ ş‰€‰Ø‰‚ \InE{}$\pi = (\pi(1), ...)$\EnE{} õ‰Ö‰‘¢ş‰Â õ‰·‰±‰´ ¤ ¡‰µ‰ƒ‰‘¤ õ‰ü‰À ø \InE{}$\sum_{i \in S} \pi(i) = 1$\EnE{}, \InE{}$\pi$\EnE{} ş‰× —‰¥ş‰â Ÿ‰µ‰Ş‰‘ñ ¤øı ê‰Ì‰‘ı \InE{}$S$\EnE{} ¨‰´  ÷‰Â —‰¥ş‰â ş‰Æ‰µ‰‘ı\footnote{\InE{}Stationary distribution\EnE{}} ¥÷‰¹‰ƒ‰Â ğ‰ş‰€‰À.\\ —‰›‰ƒ‰‚ ¬‰Î‰… ş‰Æ‰µ‰‘ş‰ü ÷‰‘ª‰ü ¥ ş‰ß øì‰ã‰ƒ‰´ ¨‰´ :\\\\\\ ğ‰Â \InE{}$\pi$\EnE{} —‰¥ş‰â ø󉃉‚ı ¥÷‰¹‰ƒ‰Â “‰‘ª‰À ş‰ã‰€‰ü: \\ \InE{}$P(X_0 = i) = \pi(i)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ÷‰Ú‰‘ù: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\InE{}$P(X_1 = j) = \sum_{j\in S} P(X_1 = j | X_0 = i) P(X_0 = i)$\EnE{}\\ \InE{}$ = \sum_{i\in S}\pi(i)p_{ij} = \pi(j)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ø “‰‚ û‰Ş‰ƒ‰ß —‰Â—‰ƒ‰° “‰‘ —‰›‰‚ “‰‚ ş‰ß ÷‰Ø‰µ‰‚ : \\ \InE{}$ \pi = \pi P = \pi P^2 = \ldots = \pi P^n $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ —‰¥ş‰â Ÿ‰µ‰Ş‰‘ñ \InE{}$X_n$\EnE{} “‰Â“‰Â \InE{}$\pi$\EnE{} ¡‰û‰À “‰¢. ş‰ã‰€‰ü ¢¤ ¬‰¤– ª‰Âøá “‰‘ ş‰ß —‰¥ş‰â ¢¤ —‰Ş‰‘ô ¥÷‰¹‰ƒ‰Â \InE{}$X_n$\EnE{} û‰Ş‰ƒ‰ß —‰¥ş‰â ¤ ¡‰û‰À ¢ª‰´.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘ ê‰Ì‰‘ı Ÿ‰‘ó‰´ \InE{}$\{0,1,2\}$\EnE{} ø õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ \InE{} $P$\EnE{}:\\ $$ P = \left(% \begin{array}{ccc} 0/3 & 0/5 & 0/2 \\ 0/6 & 0 & 0/4 \\ 0 & 0/4 & 0/6 \\ \end{array}% \right) $$ ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À. “‰‚ ¤Ÿ‰µ‰ü õ‰ü—‰ö ÷‰È‰‘ö ¢¢ ¥÷‰¹‰ƒ‰Â ÷‰‘¢ø¤ùı ø —‰½‰ş‰Û ÷‰‘•‰Áş‰Â ¨‰´. Ÿ‰Û õ‰ã‰‘¢ó‰‚ \InE{}$\pi P = \pi$\EnE{} ÷‰Æ‰±‰´ “‰‚ \InE{}$\pi$\EnE{} “‰‚ ¬‰¤– ¥ş‰Â ¨‰´:\\ \InE{}$\pi(0) = 0/3 \pi(0) + 0/6 \pi(1)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$\pi(1) = 0/5 \pi(0) + 0/4 \pi(2)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$\pi(2) = 0/2 \pi(0) + 0/4 \pi(1) + 0/6 \pi(2)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} ‰ö õ‰¹‰Ş‰á ¢¤ş‰‚ û‰‘ı û‰Â ¨‰Î‰Â \InE{}$P$\EnE{} “‰Â“‰Â ş‰× ¨‰´, “‰€‰‘“‰Âş‰ß õ‰ü—‰ö ş‰Ø‰ü ¥ õ‰ã‰‘¢„– ¤ Ÿ‰Áé î‰Â¢. ğ‰Â \InE{}$\pi(0) = \alpha $\EnE{} ¢¤ ÷‰Ñ‰Â ğ‰Âꉵ‰‚ ª‰¢:\\ \InE{}$\pi(1) = {7\over6} \alpha$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\pi(2) = ({7\over6} \alpha - {1\over2} \alpha)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\Rightarrow~~\pi = (\alpha , {7 \over 6} \alpha , {5\over 3} \alpha)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰‘ —‰›‰‚ “‰‚ ş‰€‰Ø‰‚ \InE{}$\sum_{i=0}^2 \pi(i) = 1$\EnE{}, “‰€‰‘“‰Âş‰ß:\\ \InE{}${6+7+10 \over 6} \alpha = 1~~~~\Rightarrow~~\alpha = {6\over23}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\pi = ({6\over23} , {7\over23} , {10\over23})$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰€‰‘“‰Âş‰ß ¥÷‰¹‰ƒ‰Â —‰½‰ş‰Û ÷‰‘•‰Áş‰Â ø “‰‘¥ğ‰È‰µ‰ü õ‰·‰±‰´ “‰‘ —‰¥ş‰â ş‰Æ‰µ‰‘ı \InE{} $({6\over23} , {7\over23} , {10\over23})$\EnE{} ¨‰´.\\ $$ P^\infty = \lim P^n = \left(% \begin{array}{ccc} {6\over23} & {7\over23} & {10\over23} \\ {6\over23} & {7\over23} & {10\over23} \\ {6\over23} & {7\over23} & {10\over23} \\ \end{array}% \right) $$ ¢¤ ¬‰¤—‰ü  ş‰× ¥÷‰¹‰ƒ‰Â ª‰‘õ‰Û ş‰× § “‰‘¥ğ‰È‰µ‰ü õ‰·‰±‰´ ø ş‰× § ğ‰Á¤ “‰‘ª‰À, —‰¥ş‰â õ‰‘÷‰‘ “‰Â¢¤ı ¨‰´  \InE{}$\pi P = \pi$\EnE{} ø \InE{}$\pi $1$ = $1$ $\EnE{} øó‰ü õ‰Ö‰À¤ —‰¥ş‰â ¢¤ ø®‰ã‰ƒ‰´û‰‘ı ğ‰Á¤ “‰Â“‰Â ¬‰Ô‰Â ¨‰´.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ¢¤ ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ¥ş‰Â —‰¥ş‰â õ‰‘÷‰‘ı ¥÷‰¹‰ƒ‰Â ¤ “‰À¨‰´ ø¤ş‰À.\\ \InE{}$$\bordermatrix{&0&1&2 \cr 0 &{1\over3}&{1\over3}&{1\over3} \cr 1 &0&{1\over2}&{1\over2} \cr 2 &0&{2\over3}&{1\over3}}$$\EnE{} § \InE{}$\{1,2\}$\EnE{} “‰‘¥ğ‰È‰µ‰ü ø \InE{}$\{0\}$\EnE{} ğ‰Á¤¨‰´.\\ $$ \pi_{0} = 0~~~~~~~~~(\pi_{1}~~~\pi_{2}) \left[% \begin{array}{cc} {1\over2} & {1\over2} \\ {2\over3} & {1\over3} \\ \end{array}% \right] = [\pi_{1}~~~~\pi_{2}] $$ •‰Å \InE{}$(0 , {4\over7} , {3\over7})$\EnE{} —‰¥ş‰â õ‰‘÷‰‘¨‰´.\\ ‰‚ ÷‰Æ‰±‰µ‰ü ¥ ø®‰ã‰ƒ‰´û‰‘ı ¥÷‰¹‰ƒ‰Â ¢¤ ¢¤¥ õ‰À– õ‰Ö‰À¤ 1 ¡‰û‰À “‰¢? \InE{}${4\over7}$\EnE{}.\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} ğ‰Â \InE{}$j$\EnE{} ş‰× ø®‰ã‰ƒ‰´ “‰‘¥ğ‰È‰µ‰ü ÷‰‘¢ø¤ùı õ‰·‰±‰´ “‰‘ª‰À ø \InE{}$\mu _{j}$\EnE{} õ‰ƒ‰À ¤ş‰‘®‰ü ꉑ¬‰Ü‰‚ ¥õ‰‘÷‰ü “‰ƒ‰ß ¢ø “‰‘¥ğ‰È‰´ “‰‚ \InE{}$j$\EnE{} “‰‘ª‰À. \InE{}$$\pi (j) = \lim_{n\to\infty} p_{ij}^n = {1\over \mu_{j}}$$\EnE{} õ‰·‰\nasb … ğ‰Â õ‰ƒ‰‘÷‰Ú‰ƒ‰ß ꉑ¬‰Ü‰‚ ¥õ‰‘÷‰ü “‰ƒ‰ß ¢ş‰À¤û‰‘ ¥ \InE{}$j$\EnE{} “‰Â“‰Â \InE{}$\mu_{j} = 4$\EnE{} “‰‘ª‰À, ¢¤ ş‰€‰Ê‰¤– “‰‚ ω¤ õ‰µ‰¨‰Í ¢¤ û‰Â 4 ğ‰‘ô, ¥÷‰¹‰ƒ‰Â ş‰Ø‰±‰‘¤ “‰‚ \InE{}$j$\EnE{} õ‰ü¤¨‰À ø Ÿ‰µ‰Ş‰‘ñ Ÿ‰Àı ş‰€‰Ø‰‚ ¥÷‰¹‰ƒ‰Â ¢¤ Ÿ‰‘ó‰´ \InE{}$j$\EnE{} “‰‘ª‰À “‰Â“‰Â ¨‰´ “‰‘ \InE{}$\pi(j) = {1\over4}$\EnE{}.\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} “‰Âı ø®‰ã‰ƒ‰´ “‰‘¥ğ‰È‰µ‰ü ÷‰‘¢ø¤ùı õ‰·‰±‰´ \InE{}$j$\EnE{}, “‰‘ Ÿ‰µ‰Ş‰‘ñ ş‰×, \InE{}$\lim_{n\to\infty} {1\over {n+1}} \sum_{m=0}^n I_j(X_m) = \pi(j)$\EnE{} ¥ ÷‰µ‰‘ş‰¸ ì‰Ì‰ƒ‰‚ “‰‘„ ş‰ß ¨‰´  “‰‘ —‰›‰‚ “‰‚ ş‰€‰Ø‰‚ “‰Âı —‰‘“‰â î‰Â÷‰À¤ \InE{}$f$\EnE{} ¤øı \InE{}$S$\EnE{} ¢¤ş‰İ: \InE{}$\sum_{m=0}^n f(X_m) = \sum_{j\in S} I_j(X_m)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ÷‰µ‰ƒ‰¹‰‚: ¢¤ ş‰× ¥÷‰¹‰ƒ‰Â —‰½‰ş‰Û ÷‰‘•‰Áş‰Â õ‰·‰±‰´ “‰‘ —‰¥ş‰â ş‰Æ‰µ‰‘ı \InE{}$\pi$\EnE{} “‰Âı û‰Â —‰‘“‰â î‰Â÷‰À¤ \InE{}$f$\EnE{} ¤øı \InE{}$S$\EnE{}, “‰‘ Ÿ‰µ‰Ş‰‘ñ ş‰×: \\ \InE{}$\lim_{n\to\infty}{1\over{n+1}} \sum_{m=0}^n f(X_m) = \sum_{j\in S} \pi(j) f(j)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ş‰ß ÷‰µ‰ƒ‰¹‰‚ ¤ “‰Âı õ‰ƒ‰Àû‰‘ı ¤ş‰‘®‰ü ÷‰ƒ‰Ã õ‰ü—‰ö ÷‰È‰‘ö ¢¢.\\ ÷‰µ‰ƒ‰¹‰‚: ¢¤ û‰Â ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘¥ğ‰È‰µ‰ü —‰½‰ş‰Û ÷‰‘•‰Áş‰Â “‰‘ —‰¥ş‰â ş‰Æ‰µ‰‘ı \InE{}$\pi$\EnE{} ø “‰Âı û‰Â —‰‘“‰â î‰Â÷‰À¤ \InE{}$f$\EnE{} ¤øı \InE{}$S$\EnE{} :\\ \InE{}$\lim_{n\to\infty}{1\over{n+1}} \sum_{m=0}^n E_i[f(X_m)] = \sum_{j\in S} \pi(j) f(j)$~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah õ‰ã‰ƒ‰‘¤ı “‰Âı “‰‘¥ğ‰È‰´ •‰Áş‰Âı :}\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :}¢¤ û‰Â ¥÷‰¹‰ƒ‰Â —‰½‰ş‰Û ÷‰‘•‰Áş‰Â “‰‘ ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ \InE{}$\{0,1,2,\ldots\}$\EnE{} ª‰Â¯ „¥ô ø ê‰ü “‰Âı ğ‰Á¤ “‰¢ö ş‰ß ¨‰´  : \InE{}$$\sum_{i,j\in S} p_{ij} y_j = y_i~~~~~(*)$$\EnE{} ¢¤ı ›‰’ î‰Â÷‰À¤ 艃‰Â ™‰‘“‰´ “‰‘ª‰À.\\ ş‰ã‰€‰ü “‰Â¢¤ ¨‰µ‰÷‰ü î‰Â÷‰À¤ 艃‰Â ™‰‘“‰´ \InE{}$y$\EnE{} ø›‰¢ ¢ª‰µ‰‚ “‰‘ª‰À “‰‚ ω¤ı  : \InE{}$$Py = y$$\EnE{} \hspace{-8mm} {\siah õ‰·‰‘ñ :} ¢¤ ¥÷‰¹‰ƒ‰Â ¬‰Ó “‰€‰Àı ğ‰Æ‰Æ‰µ‰‚ “‰‘ õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ¥ş‰Â: $$ P = \left(% \begin{array}{cccc} a_0 & a_1 & a_2 & \ldots \\ 0 & a_0 & a_1 & \ldots \\ 0 & 0 & a_0 & \ldots \\ 0 & 0 & 0 & \ldots \\ \vdots & \vdots & \vdots & \ddots \\ \end{array}% \right) $$ \InE{}$>0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} )\InE{}$k$\EnE{} õ‰È‰µ‰Âı ¢¤ 󉽉щ‚ı \InE{}$n$\EnE{} ô ø¤¢ ¬‰Ó ª‰÷‰À(\InE{}$a_k = P$\EnE{}\\ \InE{}$\sum_{k=0}^\infty a_k = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ÷‰È‰‘ö õ‰ü¢û‰ƒ‰İ ğ‰Â \InE{}$\sum_{k=0}^\infty k a_k > 1$\EnE{} )õ‰µ‰¨‰Í —‰ã‰À¢ õ‰È‰µ‰Âş‰‘÷‰ü  ø¤¢ ¬‰Ó õ‰üª‰÷‰À “‰ƒ‰Ç ¥ ş‰× “‰‘ª‰À( \InE{}$Py = y$\EnE{} ›‰’ î‰Â÷‰À¤ 艃‰Â ™‰‘“‰´ ¢¤¢ ø “‰€‰‘“‰Âş‰ß ¥÷‰¹‰ƒ‰Â ğ‰Á¤¨‰´.\\ \InE{}$y_i = \xi ^i$\EnE{} ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À, “‰‘ —‰›‰‚ “‰‚ ş‰€‰Ø‰‚ \InE{}$p_{ij} = a_{j-i+1}~; ~j\geq i-1$\EnE{} ¢¤ş‰İ :\\ \InE{}$\sum_{j=0}^\infty p_{ij} \xi ^j = \xi ^i ~~~\Rightarrow~~~\sum_{j=i-1}^\infty a_{j-i+1} \xi^j = \xi^i~~~i\geq1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ¢¤ ÷‰µ‰ƒ‰¹‰‚ :\\ \InE{}$\sum_{j=i-1}^\infty a_{j-i+1} \xi^{j-i+1} = \xi$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ş‰‘ õ‰ã‰‘¢ñ ö :\\ \InE{}$\sum_{k=0}^\infty a_k \xi^k = \xi$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ğ‰Â \InE{}$f(\xi) = \sum_{k=0}^\infty a_k \xi^k$\EnE{}, “‰‘ş‰À \InE{}$\xi$\EnE{} ¤ “‰‚ ğ‰÷‰‚ı “‰À¨‰´ ø¤ş‰İ : \InE{}$$f(\xi) = \xi$$\EnE{} “‰‘ª‰À. õ‰ü¢÷‰ƒ‰İ \InE{}$f(1) = 1$\EnE{} ø \InE{}$f(0) = a_0 > 0$\EnE{} ø —‰‘“‰â \InE{}$f(y)$\EnE{} ş‰× ‰€‰À ›‰Ş‰Ü‰‚ı •‰ƒ‰¨‰µ‰‚ ¨‰´. ğ‰Â \InE{}$\sum_{k=0}^\infty ka_k > 1$\EnE{} õ‰Ö‰À¤ \InE{}$\xi_0$\EnE{} “‰ƒ‰ß ¬‰Ô‰Â ø ş‰× ø›‰¢ ¢¤¢, “‰Î‰¤ı  : \InE{}$$f(\xi_0) = \xi_0$$\EnE{} \InE{}$y_i = \xi_0^i~~~i\geq1$~~~~~~~~~~~~~~~~~\EnE{}~ ø \InE{}$y_0 = p_{00}y_0 + \sum_{j=1}^\infty p_{0j}y_j$ \EnE{}~ “‰€‰‘“‰Âş‰ß \InE{}$y_0 = {1\over 1-p_{00}} \sum_{j=1}^\infty p_{0j} y_j$\EnE{}\\ ›‰’ õ‰Î‰Ü‰’ ¤ õ‰ü¢û‰À ø ¥÷‰¹‰ƒ‰Â ğ‰Á¤ ¡‰û‰À “‰¢. „¥ô “‰‚ £î‰Â ¨‰´ ¢¤ ¬‰¤– õ‰ã‰Ü‰ô “‰¢ö \InE{}$ y_i ~~i\geq~$\EnE{}, \InE{}$y_0$\EnE{} “‰À¨‰´ õ‰üş‰À. •‰Å ¢¤ û‰Â ¥÷‰¹‰ƒ‰Â ê‰ü ¨‰´ ø›‰¢ ›‰’ î‰Â÷‰À¤ 艃‰Â ™‰‘“‰´ \InE{}$y_i~~i\geq1$\EnE{} ÷‰È‰‘ö ¢¢ù ª‰¢.\\ ¢¤ Ÿ‰‘󉵉ü  ¥÷‰¹‰ƒ‰Â “‰‘¥ğ‰È‰´ •‰Áş‰Â “‰‘ª‰À õ‰ã‰‘¢ó‰‚ \InE{}$(*)$\EnE{} ›‰’ î‰Â÷‰À¤ 艃‰Â ™‰‘“‰´ ÷‰À¤¢ øó‰ü ¢¤ “‰Æ‰ƒ‰‘¤ı ¥ õ‰¤¢ ÷‰È‰‘ö ¢¢ö ä‰Àô ø›‰¢ ‰€‰ƒ‰ß ›‰“‰ü ¨‰‘ö ÷‰ƒ‰Æ‰´, õ‰Ú‰Â ş‰€‰Ø‰‚ “‰‚ ‰× ÷‰Âôê‰Ã¤û‰‘ı ¤ş‰‘®‰ü ø ¤øª‰ú‰‘ı ä‰À¢ı õ‰¹‰Ş‰ä‰‚ı ›‰“‰ú‰‘ ¤ “‰À¨‰´ ø¤¢ ø ¥ ä‰Àô ø›‰¢ ›‰’ î‰Â÷‰À¤ 艃‰Â ™‰‘“‰´ õ‰Î‰Ş‰ÿ‰ß ª‰À. ì‰Ì‰ƒ‰‚ ¥ş‰Â ¤ø© ¢ş‰Ú‰Âı “‰Âı ÷‰È‰‘ö ¢¢ö “‰‘¥ğ‰È‰´ •‰Áş‰Âı ¢¤ “‰Â¡‰ü õ‰¤¢ ¤ ÷‰È‰‘ö õ‰ü¢û‰À.\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} ¢¤ ş‰× ¥÷‰¹‰ƒ‰Â —‰½‰ş‰Û ÷‰‘•‰Áş‰Â ª‰Â¯ ê‰ü “‰Âı “‰Âğ‰È‰´ •‰Áş‰Âı ş‰ß ¨‰´  ¢÷‰±‰‘ó‰‚ı \InE{}$\{y_i\}$\EnE{} ø›‰¢ ¢ª‰µ‰‚ “‰‘ª‰À  \InE{}$y_i\rightarrow \infty $\EnE{}~ ø \InE{}$$ \sum_{j=0}^\infty p_{ij} y_j \leq y_i$$\EnE{} ş‰‘ “‰Â¢¤ ¨‰µ‰÷‰ü 艃‰Â î‰Â÷‰À¤ \InE{}$y$\EnE{}  \InE{}$Py \leq y$\EnE{} õ‰›‰¢ “‰‘ª‰À.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ¢¤ õ‰·‰‘ñ ¬‰Ó“‰€‰Àı ğ‰Æ‰Æ‰µ‰‚, ğ‰Â “‰‚ ¥ı \InE{}$i \neq 0$\EnE{}, \InE{}$y_i =i $\EnE{} ¢¤ ÷‰Ñ‰Â ğ‰Âꉵ‰‚ ª‰¢, ÷‰Ú‰‘ù “‰Âı \InE{}$i \neq 0$\EnE{}\\ \InE{}$\sum_{j=0}^\infty p_{ij} y_j = \sum_{j=i-1}^\infty j ~a_{j-i+1}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ø ¢¤ ¬‰¤—‰ü  \InE{}$\sum_{k=0}^\infty k~a_{k}\leq 1$\EnE{}, ÷‰Ú‰‘ù:\\ \InE{}$\sum_{j=0}^\infty p_{ij} y_j = \sum_{k=0}^\infty a_k~(k+i-1) = (\sum_{k=0}^\infty k~a_k ) + (i-1) \leq i = y_i$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰€‰‘“‰Âş‰ß ¥÷‰¹‰ƒ‰Â “‰‘¥ğ‰È‰µ‰ü ¨‰´.\\ ¢¤ ¬‰Ó \InE{}$M/G/1$\EnE{} ğ‰Ô‰µ‰ƒ‰İ ê‰Â­ ‰ƒ‰À ø¤ø¢ õ‰È‰µ‰Âş‰‘ö ¢¤ øŸ‰À ¥õ‰‘ö ¢¤ı —‰¥ş‰â •‰¨‰ß “‰‘ª‰À ø \InE{}$X_n$\EnE{} —‰ã‰À¢ õ‰È‰µ‰Âş‰‘ö ¢¤ ¬‰Ó ¢ì‰ƒ‰Ö‰\nasb ‘ “‰ã‰À ¥ ¡‰Âøš \InE{}$n$\EnE{} õ‰ƒ‰ß õ‰È‰µ‰Âı —‰ã‰Âş‰Ó ª‰¢ ø \InE{}$a_k$\EnE{} Ÿ‰µ‰Ş‰‘ñ ø¤ø¢ \InE{}$k$\EnE{} õ‰È‰µ‰Âı “‰ƒ‰ß ¢ø ¡‰Âøš õ‰µ‰ó‰ü “‰‘ª‰À, ÷‰Ú‰‘ù õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ “‰‚ ¬‰¤– ¥ş‰Â ¡‰û‰À “‰¢: $$ P = \left(% \begin{array}{cccc} a_0 & a_1 & a_2 & \ldots \\ a_0 & a_1 & a_2 & \ldots \\ 0 & a_0 & a_1 & \ldots \\ \vdots & \vdots & \vdots & \ddots \\ \end{array}% \right) $$ ø ğ‰Ô‰µ‰ƒ‰İ ¥÷‰¹‰ƒ‰Â “‰‘¥ğ‰È‰µ‰ü ¨‰´ ğ‰Â \InE{}$\sum_{k=0}^\infty k~a_k \leq 1$\EnE{} ø ğ‰Á¤¨‰´ ¢¤ ¬‰¤—‰ü  ş‰ß õ‰Ö‰À¤ “‰Ã¤ğ‰µ‰Â ¥ ş‰× “‰‘ª‰À. õ‰Ö‰À¤ \InE{}$ P = \sum_{k=0}^\infty k~a_k$\EnE{} ¤ ª‰À– —‰Âꉃ‰× ğ‰ş‰€‰À.\\ \hspace{-8mm} {\siah ¨‰ƒ‰Æ‰µ‰İ ¬‰Ó“‰€‰Àı \InE{}$G/M/1$\EnE{} :}\\ ê‰Â­ ‰ƒ‰À ¢¤ ş‰× ¨‰ƒ‰Æ‰µ‰İ ¬‰Ó“‰€‰Àı “‰‘ ş‰× ¨‰Âøş‰Å ¢û‰€‰Àù —‰ã‰À¢ î‰Æ‰‘÷‰ü  ¢¤ øŸ‰À ¥õ‰‘ö ¨‰Âøş‰Å ¢¤ş‰‘ꉴ õ‰ü‰€‰À ¢¤ı —‰¥ş‰â •‰¨‰ß “‰‘ª‰À ø \InE{}$X_n$\EnE{} ¤ —‰ã‰À¢ õ‰È‰µ‰Âş‰‘ö ¢¡‰Û ¬‰Ó ¢¤ 󉽉щ‚ı ø¤ø¢ \InE{}$n$\EnE{} õ‰ƒ‰ß õ‰È‰µ‰Âı “‰Ú‰ƒ‰Âş‰İ ø \InE{}$ q_k$\EnE{} Ÿ‰µ‰Ş‰‘ñ ¡‰Âøš \InE{}$k$\EnE{} õ‰È‰µ‰Âı “‰ƒ‰ß ¢ø ø¤ø¢ “‰‘ª‰À, ¢¤ ş‰ß ¬‰¤– õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ “‰‚ ¬‰¤– $$ P = \left(% \begin{array}{cccc} r_0 & q_0 & 0 & \ldots \\ r_1 & q_1 & q_0 & \ldots \\ \vdots & \vdots & \vdots & \ddots \\ \end{array}% \right) $$ ¡‰û‰À “‰¢. ¢¤ ÷‰µ‰ƒ‰¹‰‚ \InE{}$\sum_{i=0}^\infty q_i = 1$\EnE{}\\ \InE{}$r_0 = 1- q_0 = q_1 + \ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$r_1 = 1- q_0 - q_1 = q_2 + \ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\vdots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$r_n = q_{n+1} + ...$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ğ‰Â \InE{}$r = \sum_{n=1}^\infty n~q_n = r_0 + r_1 + ...$\EnE{}, õ‰ƒ‰À ¤ş‰‘®‰ü —‰ã‰À¢ ¨‰Âøş‰Åû‰‘ş‰ü ¨‰´  ¨‰Âøş‰Å ¢û‰€‰Àù õ‰ü—‰÷‰À “‰ƒ‰ß ¢ø ø¤ø¢ õ‰µ‰ó‰ü ¤‰‚ ¢û‰À.\\ ğ‰Â \InE{}$r\geq 1$\EnE{} “‰‘ª‰À ¨‰Âøş‰Å ¢û‰€‰Àù 쉑¢¤ “‰‚ ›‰“‰Ú‰ş‰ü —‰ã‰À¢ õ‰Â›‰ã‰‚ ‰€‰Àğ‰‘ö û‰Æ‰´ ø ÷‰µ‰Ñ‰‘¤ ¢¤ş‰İ ¬‰Ó “‰‘¥ğ‰È‰µ‰ü “‰‘ª‰À.\\ ğ‰Â \InE{}$r<1$\EnE{} “‰‘ª‰À ¬‰Ó Ÿ‰µ‰Ş‰‘\nasb „ ê‰Ãş‰È‰ü ¨‰´ ø “‰‚ “‰ü ÷‰ú‰‘ş‰´ õ‰ƒ‰Û õ‰ü‰À ø ¥÷‰¹‰ƒ‰Â ğ‰Á¤ ¡‰û‰À “‰¢.\\ ğ‰Â \InE{}$\widetilde{P}$\EnE{} ¤ “‰‚ ¬‰¤– : $$ \widetilde{P} = \left(% \begin{array}{ccccc} q_0 & q_1 & q_2 & q_3 & \ldots \\ q_0 & q_1 & q_2 & q_3 & \ldots \\ 0 & q_0 & q_1 & q_2 & \ldots \\ 0 & 0 & q_0 & q_1 & \ldots \\ \vdots & \vdots & \vdots & \ddots & \ddots \\ \end{array}% \right) $$ —‰ã‰Âş‰Ó ‰ƒ‰İ ş‰× \InE{}$M/G/1$\EnE{} —‰ó‰ƒ‰À õ‰üª‰¢  ÷‰Â ¢øğ‰‘ö \InE{}$G/M/1$\EnE{} ğ‰ş‰€‰À ø ¢¤ øì‰â ¨‰ƒ‰Æ‰µ‰Ş‰ü ¨‰´  ª‰À– ø¤ø¢ ø ¡‰Âøš ö “‰Â“‰Â ª‰À– ¡‰Âøš ø ø¤ø¢ ş‰× \InE{}$M/G/1$\EnE{} ¨‰´. “‰‚ ş‰ß õ‰ã‰€‰‘  õ‰·‰\nasb … ¨‰ƒ‰Æ‰µ‰İ ¢ş‰Ú‰Âı —‰ã‰Âş‰Ó ‰ƒ‰İ  “‰‘ ¡‰Âøš ş‰ß ¨‰ƒ‰Æ‰µ‰İ “‰‚ ¨‰ƒ‰Æ‰µ‰İ “‰ã‰À ø¤¢ ª‰¢ ø ¡‰Âøš ¥ ö “‰‘ û‰Ş‰‘ö —‰¥ş‰â ø¤ø¢ “‰‚ ¨‰ƒ‰Æ‰µ‰İ 쉱‰Û ¬‰¤– ğ‰ƒ‰Â¢. ¢¤ ş‰ß Ÿ‰‘ó‰´ ğ‰Â ¢øğ‰‘ö ›‰Àş‰À ğ‰Á¤ “‰‘ª‰À ¥÷‰¹‰ƒ‰Â \InE{}$M/G/1$\EnE{} “‰‘¥ğ‰È‰µ‰ü ¨‰´ ø ğ‰Â ¥÷‰¹‰ƒ‰Â \InE{}$M/G/1$\EnE{} ğ‰Á¤ “‰‘ª‰À, ¢øğ‰‘ö \InE{}$G/M/1$\EnE{} “‰‘¥ğ‰È‰µ‰ü ¡‰û‰À “‰¢.\\\\\\ \hspace{-8mm} {\siah ¥÷‰¹‰ƒ‰Â ª‰‘¡‰‚ı :}\\ õ‰›‰¢—‰ü ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À  õ‰ü—‰÷‰€‰À õ‰›‰¢– ¢ş‰Ú‰Â ¥ ÷‰á ¡‰¢ —‰ó‰ƒ‰À ‰€‰À, —‰ã‰À¢ õ‰›‰¢– ø󉃉‚ ¤ ÷‰Æ‰Û ø󉃉‚ õ‰ü÷‰‘õ‰ƒ‰İ ø ¥¢ù û‰‘ş‰ü  ¢¤÷‰À ¤ ÷‰Æ‰Û øñ. ş‰€‰ú‰‘ ÷‰ƒ‰Ã “‰‚ ÷‰“‰‚ı ¡‰¢ ¥¢ùû‰‘ş‰ü ¢¤÷‰À  ÷‰Æ‰Û ¢øô ¤ “‰›‰¢ õ‰üø¤÷‰À ø ó‰ü ¡‰Â.\\ \InE{}${}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}~—‰ã‰À¢ ¥¢ùû‰‘ı \InE{}$i$\EnE{} õ‰ƒ‰ß ê‰Â¢ ¥ ÷‰Æ‰Û \InE{}$n$\EnE{} ô : \InE{}$N(n,i)$\EnE{}\\ \InE{}$ \mathop{\rm N(n,i)}_{i \in N}~~~n=1,2,\ldots~~ \stackrel{\mbox {i.i.d}} {\sim} \xi$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} : ê‰Â­ \\ \InE{}$P(\xi = k) = p_k ~~~k=0,1,\ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ğ‰Â \InE{}$X_n$\EnE{} : Ÿ‰¹‰İ ›‰Ş‰ã‰ƒ‰´ ¢¤ ÷‰Æ‰Û \InE{}$n$\EnE{} ô : $$ X_{n+1} = \left\{% \begin{array}{ll} 0 & \hbox{$X_n=0$} \\ N(n,1) + N(n,2) + \ldots + N(n,X_n) & \hbox{$X_n>0$} \\ \end{array}% \right. $$ ê‰Â­ ‰ƒ‰À \InE{}$\{X_n,~n=0,1,2,...\}$\EnE{} ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘ ê‰Ì‰‘ı Ÿ‰‘ó‰´ \InE{}$E = \{0,1,2,\ldots\}$\EnE{} “‰‘ª‰À. “‰‘ —‰›‰‚ “‰‚ —‰ã‰Âş‰Ó \InE{}$0$\EnE{} ş‰× Ÿ‰‘ó‰´ ›‰‘£’ ¨‰´, ğ‰Â \InE{}$$T = min \{n~:~X_n = 0\}$$\EnE{} ¢¤ ş‰ß ¬‰¤– \InE{}$T$\EnE{} ¤ ¥õ‰‘ö ÷‰Ö‰Â­ õ‰ü÷‰‘õ‰€‰À.\\ ğ‰Â \InE{}$T<\infty$\EnE{} “‰‘ª‰À, ÷‰Ú‰‘ù ›‰Ş‰ã‰ƒ‰´ “‰ã‰À ¥ —‰ã‰À¢ õ‰µ‰€‰‘û‰ü ¥ ÷‰Æ‰Ü‰ú‰‘ õ‰€‰Ö‰Â­ õ‰üª‰¢.\\ ğ‰Â \InE{}$p_0 = 0 $\EnE{}, ÷‰Ú‰‘ù “‰‘ ª‰Âøá ¥ \InE{}$i\geq1$\EnE{}\\ \InE{}$X_0 \leq X_1 \leq \ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ø ¢¤ ÷‰µ‰ƒ‰¹‰‚ \InE{}$P(T=\infty) = 1$\EnE{}.\\ ğ‰Â \InE{}$p_0 >0 $\EnE{} ø \InE{}$p_0 + p_1 = 1$\EnE{} ÷‰Ú‰‘ù:\\ \InE{}$X_0 \geq X_1 \geq \ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ø“‰€‰‘“‰Âş‰ß \InE{}$P(T<\infty) = 1$\EnE{}.\\ \InE{}$P_i(T\leq n) = (1-P_i^n)^i ~~~\Rightarrow~~P_i(T<\infty) = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ğ‰Â \InE{}$p_0 > 0$\EnE{} ø \InE{}$p_0 + p_1 <1$\EnE{} ø \InE{}$m = \sum_{k} k~p_k < \infty$\EnE{}, ÷‰Ú‰‘ù \InE{}$P_i(T<\infty) = \eta ^i$\EnE{}  \InE{}$\eta$\EnE{} Ÿ‰µ‰Ş‰‘ñ ÷‰Ö‰Â­ “‰‘ ª‰Âøá ¥ ş‰× õ‰›‰¢ ¨‰´.\\ ğ‰Â \InE{}$m\leq1$\EnE{} ÷‰Ú‰‘ù \InE{}$\eta = 1$\EnE{} ø ğ‰Â \InE{}$m>1$\EnE{} ÷‰Ú‰‘ù \InE{}$0< \eta <1$\EnE{}.\\ \hspace{-8mm} {\siah õ‰Æ‰ÿ‰Ü‰‚ı ®‰Â’ ş‰× § ª‰Àö :}\\ ê‰Â­ ‰ƒ‰À $$ P = \bordermatrix{&0&1&2 \cr 0 &1&0&0 \cr 1 &\alpha&\beta&\gamma \cr 2 &0&0&1} $$\\ ø쉵‰ü  : \InE{}$ ~\alpha > 0~,~\beta > 0~,~\gamma > 0~,~ \alpha + \beta + \gamma = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ğ‰Â ê‰Âş‰€‰À ¢¤ ø®‰ã‰ƒ‰´ 1 “‰‘ª‰À —‰‘ ¥õ‰‘÷‰ü  ø¤¢ ø®‰ã‰ƒ‰´û‰‘ı \InE{}$0$\EnE{} ş‰‘ 2 ÷‰È‰Àù, ¢¤ 1 õ‰üõ‰‘÷‰À øó‰ü “‰‚ õ‰½‰Ë ş‰€‰Ø‰‚ “‰‚ ø®‰ã‰ƒ‰´ \InE{}$0$\EnE{} ¤ê‰´ —‰‘ “‰À ¢¤ ö õ‰üõ‰‘÷‰À )›‰Á’ \InE{}$0$\EnE{} õ‰üª‰¢( ø ğ‰Â “‰‚ 2 û‰İ õ‰€‰µ‰Ö‰Û ª‰¢ ›‰Á’ 2 õ‰üª‰¢. \InE{}$$T = min \{~n \geq 0~;~ X_n = 0~ or ~X_n =2\}$$\EnE{} ş‰ã‰€‰ü ¥õ‰‘ö ›‰Á’ \InE{}$0$\EnE{} ş‰‘ 2 ª‰Àö.\\ õ‰ü¡‰û‰ƒ‰İ “‰±‰ƒ‰€‰ƒ‰İ Ÿ‰µ‰Ş‰‘ñ ›‰Á’ ª‰Àö ‰Ö‰À¤ ¨‰´?\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Ÿ‰µ‰Ş‰‘ñ ş‰€‰Ø‰‚ “‰‘„¡‰Âù ›‰Á’ \InE{}$0$\EnE{} ª‰¢: \InE{}$ u = P[X_T = 0~|~X_0 = 1]$ ~~~~~~~~~~~~~~~~~~\EnE{}\\ õ‰µ‰¨‰Í ¥õ‰‘ö ›‰Á’ ª‰Àö :\InE{}$v = E[T~|~X_0 = 1]$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ u = P(X_T = 0~|~X_0 =1) = \sum_{k=0}^2 P(X_T = 0~|~X_0 = 1,X_1 = k) P(X_1 = k ~|~X_0 = 1) $~~~~~~~~~~\EnE{} \InE{}$= 1(\alpha) + u(\beta) + 0(\gamma)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \InE{}$$ u = \alpha + \beta u $$\EnE{} Ÿ‰µ‰Ş‰‘ñ ›‰Á’ \InE{}$0$\EnE{} ª‰Àö õ‰È‰Âø¯ “‰‚ ş‰€‰Ø‰‚ ¢¤ ø®‰ã‰ƒ‰´ 1 “‰‘ª‰À: \InE{}$\Rightarrow~~u = {\alpha \over {1 - \beta}} = {\alpha \over {\alpha + \gamma}}$~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$$ v = E[T~|~X_0 = 1]$$\EnE{} \InE{}$$ v = \alpha(1) + \beta (1 + v) + \gamma (1)$$\EnE{}\\ \InE{}$ \Rightarrow ~~v = 1 + \beta v~~~\Rightarrow~~~ v = {1 \over {1 = \beta}}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ ¤ù ¢ş‰Ú‰Â :\\ \InE{}$ P(T > k ~| ~ X_0 =1 ) = \beta ^ k ~~~~~~~ k = 0,1,\ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$E[T~|~X_0 =1] = \sum_{k=0}^\infty P(T>k~|X_0 = 1) = \sum_{k=0}^\infty \beta^k = {1 \over {1 - \beta}}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ ş‰ß õ‰·‰‘ñ “‰Æ‰ƒ‰‘¤ ¨‰‘¢ù ω ª‰Àù “‰¢. “‰€‰‘“‰Âş‰ß Ÿ‰µ‰Ş‰‘„– ›‰Á’ “‰‚ ¤Ÿ‰µ‰ü õ‰½‰‘¨‰±‰‚ ª‰Àù÷‰Àøó‰ü û‰Ş‰ƒ‰È‰‚ ş‰€‰Î‰¤ ÷‰ƒ‰Æ‰´.\\ õ‰·‰‘ñ: $$ P = \bordermatrix{&0&1&2&3 \cr 0 &1&0&0&0 \cr 1 &p_{10}&p_{11}&p_{12}&p_{13} \cr 2 &p_{20}&p_{21}&p_{22}&p_{33} \cr 3 &0&0&0&1}$$ “‰€‰‘“‰Âş‰ß ›‰Á’ ¢¤ ø®‰ã‰ƒ‰´ \InE{}$0$\EnE{} ø 3 —‰Ô‰‘ë õ‰üꉵ‰À.\\ ø®‰ã‰ƒ‰´û‰‘ı 1 ø 2 ¤ ğ‰Á¤ \InE{}$(transient)$\EnE{} ğ‰ş‰€‰À. ş‰ã‰€‰ü ø®‰ã‰ƒ‰´û‰‘ş‰ü  “‰‘ ª‰Âøá ¥ ÷‰ú‰‘ õ‰Ø‰‘ö ¢¤¢ ê‰Âş‰€‰À û‰Âğ‰Ã “‰‚ ÷‰ú‰‘ “‰‘¥÷‰Ú‰Â¢¢. \InE{}$$T = min \{~n \geq 0 ~; ~X_n = 0 ~or~ X_n =3\}$$\EnE{} \InE{}$$u_i = P(X_T = 0~|~X_0 = i)~~~i = 1,2$$\EnE{} \InE{}$$v_i = E(T~|~X_0 = i)$$\EnE{} \InE{}$$ u_0 = 1~~,~~u_3 = 0~~,~~v_0 = v_3 = 0$$\EnE{} $$\left\{% \begin{array}{ll} u_1 = p_{10} + p_{11}u_1 + p_{12}u_2 \\ u_2 = p_{20} + p_{21}u_1 + p_{22}u_2 \end{array}% \right.~~~~~~~~~\Rightarrow~~~~~~~~ \left\{% \begin{array}{ll} u_1 = {30 \over 43}~~~\rightarrow ~1-u_1 = {13 \over 43}\\ u_2 = {19 \over 43}~~~\rightarrow ~ 1-u_2 = {24 \over 43} \end{array}% \right. $$\\ \InE{}$v_0 = E[T~|~X_0 = i]$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$v_1 = 1 + p_{11}v_1 + p_{12} v_2$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$v_2 = 1 + p_{21}v_1 + p_{22} v_2$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ $$ \left\{% \begin{array}{ll} v_1 = 1 + 0/3 v_1 + 0/2 v_2 \\ v_2 = 1 + 0/3 v_1 + 0/3 v_2 \end{array}% \right.~~~~~~~~~\Rightarrow~~~~~~~~ \left\{% \begin{array}{ll} v_1 = {90 \over 43}\\ v_2 = {100 \over 43} \end{array}% \right. $$\\ \hspace{-8mm} {\siah ¢¤ Ÿ‰‘ó‰´ î‰Ü‰ü :}\\ ê‰Â­ ‰ƒ‰À \InE{}$\{X_n\}$\EnE{} ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘ª‰À “‰‘ ø®‰ã‰ƒ‰´û‰‘ı õ‰µ‰€‰‘û‰ü \InE{}$\{0 , 1, ... , N\}$\EnE{}.\\ ê‰Â­ ‰ƒ‰À \InE{}$0,1, \ldots , {r-1}$\EnE{} ø®‰ã‰ƒ‰´û‰‘ı ğ‰Á¤ “‰‘ª‰€‰À ø \InE{}$ r , \ldots , N$\EnE{} ø®‰ã‰ƒ‰´û‰‘ı ›‰‘£’ “‰‘ª‰€‰À, ş‰ã‰€‰ü \InE{}$(r\leq i \leq N)$\EnE{}. \InE{}$$ \Rightarrow~~P =~~~ \bordermatrix{&0 \ldots {r-1}& r \ldots N \cr 0 \cr \vdots \cr {r-1} &Q&L \cr r \cr \vdots \cr N &0&K }$$\EnE{} ê‰Âş‰€‰À “‰‘ ª‰Âøá ¥ ş‰× Ÿ‰‘ó‰´ ğ‰Á¤ —‰‘ õ‰À– ‰‘û‰ü ¢¤ ö ø®‰ã‰ƒ‰´û‰‘ ì‰Â¤ õ‰üğ‰ƒ‰Â¢, øó‰ü “‰‘„¡‰Âù ›‰Á’ ş‰Ø‰ü ¥ ø®‰ã‰ƒ‰´û‰‘ı ›‰‘£’ õ‰üª‰¢.\\ Ÿ‰µ‰Ş‰‘ñ ş‰€‰Ø‰‚ “‰‘ ª‰Âøá ¥ Ÿ‰‘ó‰´ ğ‰Á¤ı \InE{}$i$\EnE{} “‰‘„¡‰Âù ›‰Á’ Ÿ‰‘ó‰´ ›‰‘£’ \InE{}$k$\EnE{} ª‰¢~:\InE{}$U_{ik}$~~~~\EnE{}\\ \InE{}$ = p_{ik} + \sum_{\mathop {\rm j=r}_{j \neq k}}^N p_{ij}(0) + \sum_{j=0}^{r-1} p_{ij} U_{jk}~~~~~~~~~i=0,\ldots,{r-1}$\EnE{} )\InE{}$~|~X_0 = i$\EnE{}›‰Á’ \InE{}$k$\EnE{} ª‰Àö ( \InE{}$U_{ik} = P$\EnE{} \InE{}$$U_{ik} = p_{ik} + \sum_{j=0}^{r-1} p_{ij}U_{jk}~~~i=0,\ldots,{r-1}$$\EnE{} \hspace{-8mm} {\siah õ‰·‰‘ñ :} ş‰× õ‰© ¢¤ ¢¡‰Û ş‰× —‰÷‰Û Ÿ‰Â õ‰ü‰À ø “‰‚ —‰‘ëû‰‘ı õ‰¿‰µ‰Ü‰Ó õ‰ü¤ø¢  ª‰Ø‰Û —‰÷‰Û “‰‚ ¬‰¤– ¥ş‰Â ¨‰´: $$ \begin{tabular}{|c|c|c|} \hline % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... 7$(food)$ & 1 & 0 \\ \hline 4 & 3 & 2 \\ \hline 6 & 5 & 8 \InF{}ª‰í \EnF{} \\ \hline \end{tabular} $$\\ ğ‰Â û‰Â —‰‘ë  \InE{}$k$\EnE{} ¤ø¥÷‰‚ ¢ª‰µ‰‚ “‰‘ª‰À, û‰Â ¤ø¥÷‰‚ ¤ “‰‘Ÿ‰µ‰Ş‰‘ñ \InE{}$1/k$\EnE{} ÷‰µ‰¿‰‘’ õ‰ü‰À. Ÿ‰µ‰Ş‰‘ñ ş‰€‰Ø‰‚ è‰Á ¤ 쉱‰Û ¥ ª‰í •‰ƒ‰À ‰À: $$ P = \left(% \begin{array}{ccccccccc} 0 & {1\over2} & {1\over2} & 0 & 0 & 0 & 0 & 0 & 0 \\ {1\over3} & 0 & 0 & {1\over3} & 0 & 0 & 0 & {1\over3} & 0 \\ {1\over3} & 0 & 0 & {1\over3} & 0 & 0 & 0 & 0 & {1\over3} \\ 0 & {1\over4} & {1\over4} & 0 & {1\over4} & {1\over4} & 0 & 0 & 0 \\ 0 & 0 & 0 & {1\over3} & 0 & 0 & {1\over3} & {1\over3} & 0 \\ 0 & 0 & 0 & {1\over3} & 0 & 0 & {1\over3} & 0 & {1\over3} \\ 0 & 0 & 0 & 0 & {1\over2} & {1\over2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array}% \right) $$\\ \InE{}$u_{i7} = P(X_T = 7~|~ X_0 =i)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$u_{07} = {1\over2} u_{17} + {1\over2} u_{27}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$u_{17} = {1\over3} + {1\over3} u_{07} + {1\over3} u_{37}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$u_{27} = {1\over3} u_{07} + {1\over3} u_{37}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$u_{37} = {1\over4} u_{17} + {1\over4} u_{27} + {1\over4} u_{47} + {1\over4} u_{57}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$u_{47} = {1\over3} + {1\over3} u_{37} + {1\over3} u_{67}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$u_{57} = {1\over3} u_{37} + {1\over3} u_{67}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ \Rightarrow~~u_{07} = {1\over2} = u_{67}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$u_{17} = {2\over3} = u_{47}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$u_{27} = {1\over3} = u_{57}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$u_{37} = {1\over 2}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah ê‰Âş‰€‰À •‰¨‰ß :}\\ ş‰× ê‰Âş‰€‰À •‰¨‰ß, ê‰Âş‰€‰Àı ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ \InE{}$\{~X_t~,~ t \in [0,\infty)\}$\EnE{} ¨‰´  \InE{}$X_t$\EnE{} —‰ã‰À¢ —‰Ô‰‘쉑—‰ü ¤ ÷‰È‰‘ö õ‰ü¢û‰À  ¢¤ ꉑ¬‰Ü‰‚ \InE{}$[0,t]$\EnE{} ¤  ¢û‰€‰À. “‰€‰‘“‰Âş‰ß ê‰Âş‰€‰À ª‰Ş‰‘¤ª‰ü ¨‰´. ş‰× õ‰Æ‰ƒ‰Â ÷‰Ş‰÷‰‚ı ¥ ê‰Âş‰€‰À •‰¨‰ß ¤ õ‰ü—‰ö ¢¤ ÷‰Ş‰¢¤ ¥ş‰Â ¢ş‰À.\\\\\\ —‰ã‰À¢ —‰Ô‰‘쉑– ¢¤ ꉑ¬‰Ü‰‚ \InE{}$(t,t+s]$\EnE{} ş‰ã‰€‰ü \InE{}$X_{t+s} - X_t$\EnE{} —‰€‰ú‰‘ ø“‰Æ‰µ‰‚ “‰‚ \InE{}$s$\EnE{} ¨‰´ ø ÷‰‚ “‰‚ \InE{}$t$\EnE{}. “‰ã‰…øù ş‰ß —‰ã‰À¢ “‰‚ \InE{}$X_t$\EnE{} ş‰ã‰€‰ü —‰ã‰À¢ —‰Ô‰‘쉑– —‰‘ 󉽉щ‚ \InE{}$t$\EnE{} “‰Æ‰µ‰Ú‰ü ÷‰À¤¢ ø ¢¤ û‰Â 󉽉щ‚ Ÿ‰À‰Â ›‰ú‰È‰ü “‰‚ ÷‰À¥ùı øŸ‰À ¤  õ‰ü¢û‰À.\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} ê‰Âş‰€‰À \InE{}$\{~X_t ~,~t\geq 0\}$\EnE{} ¤ ê‰Âş‰€‰À •‰¨‰ß ğ‰ş‰€‰À, û‰Â ğ‰‘ù: \\ \InE{}$ w.p.1~~ X_0 = 0~(i$\EnE{}\\ \InE{}$(ii$\EnE{} “‰‘ Ÿ‰µ‰Ş‰‘ñ ş‰× û‰Â ›‰ú‰Ç \InE{}$t \rightarrow X_t$\EnE{} “‰‚ ÷‰À¥ùı øŸ‰À “‰‘ª‰À.) ¢¤ ê‰Âş‰€‰À —‰¹‰Àş‰À ğ‰Â \InE{}$X_1 > 0$\EnE{} “‰‘ Ÿ‰µ‰Ş‰‘ñ 1, ÷‰Ú‰‘ù û‰Â ›‰ú‰Ç “‰‘ Ÿ‰µ‰Ş‰‘ñ 1 “‰‚ ÷‰À¥ù øŸ‰À ¨‰´.(\\ \InE{}$ X_{t+s} - X_t(iii$\EnE{} “‰‚ ¥ı û‰Â \InE{}$s$\EnE{} ø \InE{}$t$\EnE{} ê‰Ö‰Í “‰‚ \InE{}$s$\EnE{} “‰Æ‰µ‰Ú‰ü ¢¤¢.\\ \InE{}$(iv$\EnE{} “‰‚ ¥ı û‰Â \InE{}$t~,~s\geq0$\EnE{}, \InE{}$X_{t+s} - X_t$\EnE{} õ‰Æ‰µ‰Ö‰Û ¥ \InE{}$\{~X_u~,~u\leq t\}$\EnE{} ¨‰´.\\ “‰‘ —‰›‰‚ “‰‚ )’(:\\ “‰Âı \InE{}$t_1,\ldots,t_n \leq t$\EnE{}, \InE{}$X_{t+s} -X_t$\EnE{} õ‰Æ‰µ‰Ö‰Û ¥ \InE{}$X_{t_1} , \ldots,X_{t_n}$\EnE{} ¨‰´.\\ “‰€‰‘“‰Âş‰ß \InE{}$X_{t+s} - X_t$\EnE{} õ‰Æ‰µ‰Ö‰Û ¥ \InE{}$X_{t_1} , \ldots , X_{t_{n-1}} - X_{t_{n-2}} , X_{t_n} - X_{t_{n-1}} $\EnE{} ¨‰´.\\ “‰‘ ş‰ß ¨‰µ‰À„ñ ÷‰µ‰ƒ‰¹‰‚ õ‰üª‰¢ ş‰ß ê‰Âş‰€‰À ¢¤ı ÷‰Ş‰û‰‘ı õ‰Æ‰µ‰Ö‰Û ¨‰´, ş‰ã‰€‰ü:\\ \InE{}$ X_{t_1} , X_{t_2} - X_{t_1} , \ldots$\EnE{} ¥ û‰İ õ‰Æ‰µ‰Ö‰Û û‰Æ‰µ‰€‰À. ş‰ã‰€‰ü —‰ã‰À¢ —‰Ô‰‘쉑– ¢¤ ꉬ‰Û ¥õ‰‘÷‰ü õ‰¹‰Ã ¥ û‰İ õ‰Æ‰µ‰Ö‰Ü‰€‰À.\\ “‰‘ —‰›‰‚ “‰‚ ‰ú‰‘¤ ¡‰‘¬‰ƒ‰´ ê‰ë õ‰ü—‰ö ÷‰È‰‘ö ¢¢ : \InE{}$$\forall ~t>0~~~X_t \sim P(\lambda t)$$\EnE{}  \InE{}$\lambda \geq 0$\EnE{}, õ‰µ‰¨‰Í —‰ã‰À¢ —‰Ô‰‘쉑– ¢¤ ş‰× øŸ‰À ¥õ‰‘ö ¨‰´.\\\\ \InE{}$ \Rightarrow~~~E(X_t) = \lambda t ~~~,~~\forall (X_t) = \lambda t$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\siah ÷‰µ‰ƒ‰¹‰‚ :} ¢¤ ê‰Âş‰€‰À •‰¨‰ß \InE{}$\{X_{\theta}~,~ \theta \geq 0 \}$\EnE{} “‰‘ ÷‰Â  \InE{}$\lambda$\EnE{}.\\\\ \InE{}$P(X_{t+s} - X_t = k~|~ X_u~,~u \leq \theta) = P(X_{t+s} - X_t = k) = {{e^{-s \lambda} (s \lambda)^k }\over {k!}}~~~~k=0,\ldots$~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \hspace{-8mm} {\siah õ‰·‰‘ñ :} ¢¤ ê‰Âş‰€‰À •‰¨‰ß \InE{}$\{X_t~,~t\geq 0\}$\EnE{} “‰‘ ÷‰Â  \InE{}$\lambda = 8$\EnE{} ¢¤ş‰İ:\\\\ \InE{}$P(X_{2/5} = 17 , X_{3/7} = 22 , X_{4/3} = 36) $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$= P(X_{2/5} = 17 , X_{3/7} - X_{2/5} = 5 , X_{4/3} - X_{3/7} = 14) $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$= P(X_{2/5} = 17) P(X_{1/2} = 5) P(X_{0/6} = 14) = {e^{-8(2/5)} (8(2/5))^17 \over 17!} \times {e^{-8(1/2)} (8(1/2))^5 \over 5!} \times {e^{-8(0/6)} (8(0/6))^14 \over 14!}$\EnE{}\\\\ ê‰Â­ ‰ƒ‰À:\\ \InE{}$T_1$\EnE{}: ¥õ‰‘ö øó‰ƒ‰ß ¤  ¢¢ ¢¤ ş‰× ê‰Âş‰€‰À •‰¨‰ß \\ \InE{}$T_2$\EnE{}: ¥õ‰‘ö ¢øõ‰ƒ‰ß ¤  ¢¢ ¢¤ ş‰× ê‰Âş‰€‰À •‰¨‰ß\\ \InE{}$\vdots$\EnE{}\\ \InE{}$T_n$\EnE{}: ¥õ‰‘ö \InE{}$n$\EnE{} õ‰ƒ‰ß ¤  ¢¢ ¢¤ ş‰× ê‰Âş‰€‰À •‰¨‰ß “‰‘ ÷‰Â  \InE{}$\lambda$\EnE{} “‰‘ª‰À\\\\ \InE{}$P(T_{n+1} - T_n > t ~ | ~ T_0 , \ldots , T_n) = P(N_t = 0) = e^{-t \lambda}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$t \geq 0$\EnE{} ş‰ã‰€‰ü \InE{}$T_{n+1} - T_n$\EnE{} ş‰× õ‰µ‰ç‰ƒ‰Â ÷‰Ş‰‘ş‰ü “‰‘ ÷‰Â  \InE{}$\lambda $\EnE{} ¨‰´, •‰Å :\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} ğ‰Â \InE{}$\{X_t ~ ,~ t\geq 0\} = X$\EnE{} ê‰Âş‰€‰Àı “‰‘ª‰À  —‰ã‰À¢ ¤  ¢¢û‰‘ ¤ —‰‘ 󉽉щ‚ \InE{}$t$\EnE{} ÷‰È‰‘ö õ‰ü¢û‰À ø \InE{}$T_1, T_2, \ldots $\EnE{} ¥õ‰‘öû‰‘ı “‰ƒ‰ß ¤  ¢¢û‰‘ı õ‰µ‰ó‰ü “‰‘ª‰À, ÷‰Ú‰‘ù \InE{}$X$\EnE{} ş‰× ê‰Âş‰€‰À •‰¨‰ß “‰‘ ÷‰Â  \InE{}$\lambda$\EnE{} ¨‰´ ğ‰Â ø ê‰Ö‰Í ğ‰Â \InE{}$T_1,T_2 - T_1, \ldots $\EnE{} õ‰µ‰ç‰ƒ‰Âû‰‘ı õ‰Æ‰µ‰Ö‰Û “‰‘ —‰¥ş‰â ÷‰Ş‰‘ş‰ü “‰‘ •‰‘¤õ‰µ‰Â \InE{}$\lambda$\EnE{} “‰‘ª‰À.\\ \hspace{-8mm} {\siah ÷‰Ø‰µ‰‚ :} û‰Â ğ‰‘ù \InE{}$\{X_t~,~t\geq 0\}$\EnE{} ø \InE{}$\{Y_t~,~ t\geq 0\}$\EnE{} ¢ø ê‰Âş‰€‰À •‰¨‰ß “‰‚ —‰Â—‰ƒ‰° “‰‘ ÷‰Â û‰‘ı \InE{}$\lambda$\EnE{} ø \InE{}$\mu$\EnE{} “‰‘ª‰€‰À ÷‰Ú‰‘ù: \InE{}$$N_t = X_\theta + Y_t~~~~t\geq 0$$\EnE{} ÷‰ƒ‰Ã ş‰× ê‰Âş‰€‰À •‰¨‰ß “‰‘ ÷‰Â  \InE{}$\lambda + \mu$\EnE{} ¨‰´.\\ \hspace{-8mm} {\siah —‰Ş‰Âş‰ß :}\\ ó‰Ó( ê‰Âş‰€‰À •‰¨‰ß \InE{}$\{X_t~,~ t\geq 0\}$\EnE{} “‰‘ ÷‰Â  \InE{}$\lambda = 3 $\EnE{} ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À. ş‰× õ‰Æ‰ƒ‰Â ÷‰Ş‰÷‰‚ı —‰‘ 󉽉щ‚ı \InE{}$t =100$\EnE{} ¤ ª‰±‰ƒ‰‚¨‰‘¥ı ‰ƒ‰À.\\ ’( ê‰Âş‰€‰À •‰¨‰ß \InE{}$\{Y_t~,~ t\geq 0\}$\EnE{} “‰‘ ÷‰Â  \InE{}$\mu = 2$\EnE{} ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À ğ‰Â\\ \InE{}$$M_t = (X_t - Y_t)^t = max\{0, X_t - Y_t\}$$\EnE{} ş‰× õ‰Æ‰ƒ‰Â ÷‰Ş‰÷‰‚ı “‰‚ ωñ \InE{}$100$\EnE{} ¥ ş‰ß ê‰Âş‰€‰À ¤ ª‰±‰ƒ‰‚¨‰‘¥ı ‰ƒ‰À.\\ ‰€‰ƒ‰ß ê‰Âş‰€‰Àı ¤ \InE{}$M/M/1$\EnE{} ğ‰ş‰€‰À ø ωñ ¬‰Ó ¤ ¢¤ û‰Â 󉽉щ‚ ÷‰È‰‘ö õ‰ü¢û‰À. ¢¤ ¬‰¤—‰ü  —‰ã‰À¢ ø¤ø¢ õ‰È‰µ‰Âı “‰‚ ¬‰Ó •‰¨‰ß ø ¥õ‰‘ö ¨‰Âøş‰Å¢û‰ü ÷‰ƒ‰Ã ÷‰Ş‰‘ş‰ü \InE{}$\mu = 2$\EnE{} “‰‘ª‰À.\\ \hspace{-8mm} {\siah ÷‰Ñ‰Âş‰‚ —‰¹‰Àş‰À :}\\ ¢¤ ê‰Âş‰€‰À •‰¨‰ß ¢ş‰Àş‰İ ¥õ‰‘öû‰‘ı “‰ƒ‰ß ¢ø •‰ƒ‰È‰‘õ‰À õ‰Æ‰µ‰Ö‰Û ø û‰İ—‰¥ş‰â ÷‰Ş‰‘ş‰ü ¨‰´.\\ ê‰Â­ ‰ƒ‰À \InE{}$X_1,X_2, \ldots \stackrel {\rm i.i.d}{\sim} F$\EnE{} ø \InE{}$F(0) = P(X_n = 0) < 1$\EnE{}\\ \InE{}$X_n$\EnE{} ¤ ¥õ‰‘ö “‰ƒ‰ß •‰ƒ‰È‰‘õ‰À \InE{}$n$\EnE{} ô ø \InE{}$n+1$\EnE{} ô ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À ø \\ \InE{}$s_0 = 0 ~~,~~s_n = \sum_{i=1}^n X_i~~~n \geq 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\  \InE{}$s_n$\EnE{} ¥õ‰‘ö \InE{}$n$\EnE{} õ‰ƒ‰ß •‰ƒ‰È‰‘õ‰À “‰‘ª‰À\\ \InE{}$ = N(t) = max \{n~;~s_n \leq t\}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} —‰ã‰À¢ •‰ƒ‰È‰‘õ‰Àû‰‘ —‰‘ ¥õ‰‘ö \InE{}$t$\EnE{}\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} ê‰Âş‰€‰À ª‰Ş‰‘¤ª‰ü \InE{}$\{ N(t)~;~t\geq 0\}$\EnE{} ¤ ş‰× ê‰Âş‰€‰À —‰¹‰Àş‰À õ‰ü÷‰‘õ‰€‰À. ‰ö ¥õ‰‘öû‰‘ı “‰ƒ‰ß ¢ø •‰ƒ‰È‰‘õ‰À õ‰Æ‰µ‰Ö‰Û ø û‰İ—‰¥ş‰â û‰Æ‰µ‰€‰À, ¢¤ û‰Â —‰¹‰Àş‰À, ê‰Âş‰€‰À ¥ 󉽉‘à Ÿ‰µ‰Ş‰‘„—‰ü ¥ ÷‰ ª‰Âøá õ‰üª‰¢.\\\\ \InE{}$P(N(t) = n) = P(N(t) \geq n) - P(N(t) \geq {n+1}) $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = P(s_n \leq t) - P(s_{n+1} \leq t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = F_n^{(n)}(t) - F_{n+1}^{(n+1)}(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\  ‰ö \InE{}$F^{(n)} , s_n = \sum_{i=1}^n X_i$\EnE{} •‰ƒ‰»‰Ç \InE{}$n$\EnE{} ğ‰‘÷‰‚ı \InE{}$F$\EnE{} “‰‘ ¡‰¢© ¨‰´.\\ —‰‘“‰â \InE{}$m(t) = E(N(t))$\EnE{} ş‰ã‰€‰ü õ‰ƒ‰À ¤ş‰‘®‰ü —‰ã‰À¢ —‰¹‰Àş‰Àû‰‘ —‰‘ 󉽉щ‚ı \InE{}$t$\EnE{} ô ¤ —‰‘“‰â —‰¹‰Àş‰À õ‰ü÷‰‘õ‰€‰À.\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} \InE{}$$m(t) = \sum_{n=1}^\infty F^{(n)}(t)$$\EnE{} \hspace{-8mm} {\siah ™‰±‰‘– :} \\ $$ I_n(t) = \left\{% \begin{array}{ll} 1~~~ & \mbox {\InF{} ¤  ¢û‰À\EnF{}$[0,t]$\InF{} õ‰ƒ‰ß —‰¹‰Àş‰À ¢¤ \EnF{}$n$\InF{}ğ‰Â \EnF{}}\\ 0 ~~~& \mbox{\InF{}¢¤ 艃‰Â ş‰€‰Ê‰¤–\EnF{}} \\ \end{array}% \right. ~~~~~~\Rightarrow~~N(t) = \sum_{n=1}^\infty I_n(t) $$\\ “‰€‰‘“‰Âş‰ß \\ \InE{}$E[N(t)] = E[\sum_{n=1}^\infty I_n(t)]$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$= \sum_{n=1}^\infty E[I_n(t)]$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = \sum_{n=1}^\infty P(I_n(t) = 1)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = \sum_{n=1}^\infty P(s_n \leq t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = \sum_{n=1}^\infty F^{(n)}(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ş‰× ›‰\hamze à ¥ ş‰× ¢¨‰µ‰Ú‰‘ù ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À  û‰Ş‰¤ù õ‰¤¢ ¨‰µ‰Ô‰‘¢ù ì‰Â¤ õ‰üğ‰ƒ‰Â¢ ø 쉑“‰Û —‰ã‰ş‰Ë ¨‰´ ğ‰Â \InE{}$X$\EnE{} ¤ ωñ 䉪‰Â ş‰ß ›‰\hamze à ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À “‰‘ —‰¥ş‰â \InE{}$F$\EnE{}. û‰Â “‰‘¤  ş‰ß ›‰\hamze à ¡‰Â’ õ‰üª‰¢ ÷‰Â “‰‘ ş‰× ÷‰ 䉭 õ‰ü‰€‰À. û‰Â —‰ã‰ş‰Ë ş‰× —‰¹‰Àş‰À ¨‰´ ø —‰ã‰À¢ —‰¹‰Àş‰Àû‰‘ ê‰Âş‰€‰À —‰¹‰Àş‰À ¡‰û‰À “‰¢.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ğ‰Â ¢¤ ş‰× ê‰Âş‰€‰À —‰¹‰Àş‰À ¥õ‰‘ö “‰ƒ‰ß ¢ø —‰¹‰Àş‰À ¢¤ı —‰¥ş‰â ş‰Ø‰€‰¡‰´ \InE{}$(0,1)$\EnE{} “‰‘ª‰À, ÷‰Ú‰‘ù:\\\\ \InE{}$m(t) = \sum_{n=1}^\infty F^n(t)~~~~~0\leq t \leq 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$X_1, \ldots , X_n \stackrel {\rm i.i.d}{\sim} U(0,1)~~~\Rightarrow~~X_1 + \ldots + X_n \sim F^n~~~~~F^n (t) = {t^n \over n!}~~~~~n=1,2,\dots ~~~~~0\leq t \leq 1$\EnE{}\\\\ \InE{}$\Rightarrow~~~~m(t) = sum_{n=1}^\infty {t^n \over n!} = e^t -1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\siah —‰Ş‰Âş‰ß :} ÷‰È‰‘ö ¢û‰ƒ‰À: \InE{}$F^n(t) = {t^n \over n!}$~~~~~~~~~~\EnE{}\\ ™‰‘“‰´ õ‰üª‰¢  —‰‘“‰â —‰¹‰Àş‰À “‰Âı \InE{}$ 0 \leq t < \infty$\EnE{} “‰‚ ¬‰¤– õ‰€‰½‰Ê‰Â “‰Ô‰Â¢ —‰¥ş‰â “‰ƒ‰ß •‰ƒ‰È‰‘õ‰Àû‰‘ \InE{}$F$\EnE{} ¤ õ‰È‰¿‰É õ‰ü‰À. “‰Âı õ‰·‰‘ñ \InE{}$m(t) = t \lambda $\EnE{} õ‰µ‰€‰‘҉ ¨‰´ “‰‘ —‰¥ş‰â ÷‰Ş‰‘ş‰ü “‰‘ õ‰ƒ‰‘÷‰Ú‰ƒ‰ß \InE{}${1 \over \lambda}$\EnE{}.\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} )¥õ‰‘ö —‰ì‰Ó \InE{}$stopping ~time$\EnE{}( õ‰µ‰ç‰ƒ‰Â —‰Ê‰‘¢ê‰ü \InE{}$N$\EnE{} ¤ ş‰× ¥õ‰‘ö —‰ì‰Ó “‰Âı ¢÷‰±‰‘ó‰‚ \InE{}$X_1, X_2, \ldots$\EnE{} ğ‰ş‰ƒ‰İ ğ‰Â “‰‚ ¥ı û‰Â \InE{}$n =1, 2, \ldots$\EnE{}, •‰ƒ‰È‰‘õ‰À \InE{}$\{N=n\}$\EnE{} õ‰Æ‰µ‰Ö‰Û ¥ \InE{}$X_{n+1}, X_{n+2}, \ldots$\EnE{} “‰‘ª‰À.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ê‰Â­ ‰ƒ‰À \InE{}$X_1, \ldots \stackrel {\rm i.i.d}{\sim} B(1,{1\over2})$\EnE{} ø \InE{}$$N = min \{n~;~X_1 + \ldots + X_n = 10\}$$\EnE{}\\ ÷‰Ú‰‘ù \InE{}$N$\EnE{} ş‰× ¥õ‰‘ö —‰ì‰Ó ¨‰´. \InE{}$N$\EnE{} ¤ ¥õ‰‘ö —‰ì‰Ó ¥õ‰‘ş‰Ç “‰À÷‰ƒ‰İ  ›‰Ş‰â •‰ƒ‰Âø¥ıû‰‘ “‰‚ \InE{}$10$\EnE{} “‰Â¨‰À.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} \InE{}$X_1, X_2, \ldots i.i.d$~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚)õ‰ã‰‘¢ó‰‚ øó‰À( :}\\ ğ‰Â \InE{}$X, X_1, X_2, \ldots $\EnE{} õ‰µ‰ç‰ƒ‰Âû‰‘ı õ‰Æ‰µ‰Ö‰Û ø û‰İ—‰¥ş‰â “‰‘ õ‰ƒ‰À ¤ş‰‘®‰ü õ‰µ‰€‰‘û‰ü “‰‘ª‰€‰À ø \InE{}$N$\EnE{} ş‰× ¥õ‰‘ö —‰ì‰Ó “‰Âı \InE{}$X_1, X_2, \ldots$\EnE{} “‰‘ \InE{}$E(N) < \infty $\EnE{} , ÷‰Ú‰‘ù: \InE{}$$E[\sum_{n=1}^N X_n] = E[N] E[X]$$\EnE{} \hspace{-8mm} {\siah ™‰±‰‘– :}\\ \InE{}$\sum_{n=1}^N X_n = \sum_{n=1}^\infty X_n . I_n$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ø쉵‰ü : $$ I_n = \left\{% \begin{array}{ll} 1 & \hbox{$N \geq n$}\\ 0 & \hbox{$Nt$~~~~~~~~~~~~~~~~\EnE{}\\\\ •‰Å \InE{}$\{N(t) + 1 = n\}$\EnE{} ê‰Ö‰Í “‰‚ \InE{}$X_1, \ldots, X_n$\EnE{} “‰Æ‰µ‰Ú‰ü ¢¤¢ ø õ‰Æ‰µ‰Ö‰Û ¥ \InE{}$X_{n+1}, \ldots $\EnE{} ¨‰´. “‰€‰‘“‰Âş‰ß \InE{}$N(t)+1$\EnE{} ş‰× ¥õ‰‘ö —‰ì‰Ó ¨‰´.\\ \InE{}$ \Rightarrow~E[X_1 + \ldots + X_{N(t)+1}] = E[X] E[N(t) + 1]$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\siah ÷‰µ‰ƒ‰¹‰‚ :} ğ‰Â \InE{}$E[X] = \mu < \infty $\EnE{} ÷‰Ú‰‘ù: \InE{}$$E[s_{N(t)+1}] = \mu (m(t) + 1)$$\EnE{} \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :})ì‰Ì‰ƒ‰‚ õ‰Ö‰Àõ‰‘—‰ü —‰¹‰Àş‰À( \InE{}$${m(t) \over t} ~\rightarrow~{1 \over \mu}~~,~~t\rightarrow \infty$$\EnE{} \hspace{-8mm} {\siah ™‰±‰‘– :} ¢¤ øì‰â “‰‚ ş‰ß õ‰ã‰€‰‘¨‰´  ÷‰Æ‰±‰´ õ‰µ‰¨‰Í —‰ã‰À¢ —‰¹‰Àş‰À ¢¤ øŸ‰À ¥õ‰‘ö “‰‚ \InE{}${1 \over \mu}$\EnE{} ş‰ã‰€‰ü ä‰Ø‰Å õ‰ƒ‰À û‰Â —‰¹‰Àş‰À õ‰ƒ‰Û õ‰ü‰À. “‰µ‰À ê‰Â­ ‰ƒ‰À \InE{}$\mu < \infty$\EnE{}\\ \InE{}$s_{N(t)+1} > t~~\mu(m(t)+1) > t~~\Rightarrow~~{m(t) \over t} > {1 \over \mu} - {1 \over t}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$\Rightarrow~~\lim_{t \to \infty} inf {m(t) \over t} \geq {1 \over \mu}~~~~~~(*)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ “‰Âı ›‰ú‰´ ¢ş‰Ú‰Â õ‰Ö‰À¤ ™‰‘“‰´ \InE{}$M$\EnE{} ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À ø ê‰Âş‰€‰À —‰¹‰Àş‰À \InE{}$\{\bar{X}_n~,~n=1,2,\ldots\}$\EnE{} ¤ “‰‚ ¬‰¤– ¥ş‰Â ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À: $$ \bar{X}_n = \left\{% \begin{array}{ll} X_n & \hbox{$X_n \leq M$}\\ M & \hbox{$X_n>M$}\\ \end{array}% \right. ~~~~~~~~~~~~~~~~~~n=1,2,\ldots $$\\ \InE{}$= X_n I_{[X_n\leq M]} + MI_{[X_n>M]}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$\bar{s}_n = \sum_{i=1}^n \bar{X}_i$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$\bar{N}(t) = sup\{n~;~\bar{s}_n \leq t\}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow~~\bar{s}_{N(t)+1} \leq {t+M}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow~~(\bar{m}(t)+1)\mu_M \leq {t+M}~~~~(\mu_M = E(\bar{X}_n))$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$\Rightarrow~~{\bar{m}(t) \over t} \leq {1 \over \mu_M} + \frac{M}{t \mu_M} -{1 \over t}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$\Rightarrow~~{m(t) \over t} \leq {1 \over \mu_M} + {M \over t \mu_M} -{1 \over t}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ ¢¤ ¬‰¤—‰ü  \InE{}$\mu<\infty$\EnE{} “‰‘ª‰À, ™‰±‰‘– “‰Æ‰ƒ‰‘¤ ¨‰‘¢ù—‰Â ¨‰´: \InE{}$ \Rightarrow~~\lim_{t \to \infty} sup{m(t) \over t} \leq {1 \over \mu}~~~~(**)$~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ E[s_{N(t)}] \leq t~~\Rightarrow~~~\mu_{m(t)} \leq t $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$\lim_{t \to \infty} sup{m(t) \over t} \leq {1 \over \mu}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ ¤ù Ÿ‰Û “‰‘„ “‰Âı \InE{}$\mu = \infty$\EnE{} ›‰±‰‘¤ı ¨‰´. \InE{}$$(*) , (**) ~~\Rightarrow~~~\lim_{t \to \infty} {m(t) \over t} = {1 \over \mu}$$\EnE{}\\ ğ‰Â \InE{}$\mu = \infty $\EnE{} “‰‘¥ û‰İ ê‰Âş‰€‰À “‰Âş‰Àù ¤ ¢¤ ÷‰Ñ‰Â õ‰üğ‰ƒ‰Âş‰İ. ‰ö \InE{}$\mu \rightarrow \infty$\EnE{} ÷‰µ‰ƒ‰¹‰‚ õ‰ü¢û‰À \InE{}$\mu_M \rightarrow \infty$\EnE{} ø ì‰Ì‰ƒ‰‚ ™‰‘“‰´ õ‰üª‰¢.\\ ¢¤ ê‰Âş‰€‰À ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ \InE{}$\{X_t~,~t\geq 0\}$\EnE{} ÷‰Ş‰ ê‰Âş‰€‰À õ‰€‰Ñ‰¤ \InE{}$X_{t_1} - X_{t_0}$\EnE{} ¨‰´.\\ ş‰× ê‰Âş‰€‰À ¤ “‰‘ ÷‰Ş‰û‰‘ı õ‰Æ‰µ‰Ö‰Û ğ‰ş‰€‰À ğ‰Â “‰Âı ¥õ‰‘öû‰‘ı “‰Àøö ª‰µ‰Âí ÷‰Ş‰û‰‘ õ‰Æ‰µ‰Ö‰Û “‰‘ª‰€‰À. ş‰ã‰€‰ü \InE{}$X_{t_n} - X_{t_{n-1}} , \ldots, X_{t_1} - X_{t_0}$\EnE{} õ‰Æ‰µ‰Ö‰Û “‰‘ª‰€‰À.\\ ş‰× ê‰Âş‰€‰À ¤ ş‰Æ‰µ‰‘ ğ‰ş‰€‰À ğ‰Â “‰‚ ¥ı û‰Â \InE{}$(X_{t_0}, \ldots, X_{t_n})$\EnE{} ¢ª‰µ‰‚ “‰‘ª‰ƒ‰İ: \InE{}$$(X_{t_0}, \ldots, X_{t_n}) \stackrel {\rm d}{=} (X_{t_) +h}, \ldots, X_{t_n + h})~~~\forall h>0 $$\EnE{} ø ê‰Âş‰€‰À ¤ ¢¤ı ÷‰Ş‰û‰‘ı ş‰Æ‰µ‰‘ ğ‰ş‰€‰À ğ‰Â \InE{}$$(X_{t_1} - X_{t_0}, \ldots, X_{\theta_n} - X_{t_{n-1}}) \stackrel{\rm d}{=} (X_{t_1+h} - X_{t_0}, \ldots, X_{t_n+h} - X_{t_{n-1}+h})$$ \EnE{} \hspace{-8mm} {\siah ÷‰Ø‰µ‰‚ :} ğ‰Â ê‰Âş‰€‰À ¢¤ı ÷‰Ş‰ ş‰Æ‰µ‰‘ “‰‘ª‰À ø \InE{}$E(X_t) < \infty$\EnE{} ÷‰Ú‰‘ù \\ \InE{}$E(X_t) = m_0 + m_1 t$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\  \InE{}$m_0 = E(X_0)$\EnE{} ø \InE{}$m_1 = E(X_1) - E(X_0)$\EnE{}.\\ \hspace{-8mm} {\siah ™‰±‰‘– :}\\ \InE{}$f(t) \stackrel{\rm def}{=} E(X_t) - E(X_0)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰‘ —‰›‰‚ “‰‚ ş‰€‰Ø‰‚ \InE{}$X_{t+s} - X_s \stackrel {\rm d}{=} X_t - X_0$\EnE{} “‰€‰‘“‰Âş‰ß \\ \InE{}$E(X_{t+s}) - E(X_s) = f(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$X_{t+s} - X_0 = X_{t+s} - X_t + X_t - X_0~~\Rightarrow~~~f(t+s) = f(t) + f(s)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ğ‰Â\\ \InE{}$f^ \prime(t+s) = f^\prime(t) = f^\prime(s) ~~\forall s,t$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$f^\prime(t) = m_1~~\Rightarrow~~f(t) = ct + d~~~f(0) = d = 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$\Rightarrow~~f(t) = ct~~\Rightarrow~~~f(1) = c = E(X_1) - E(X_0) = m_1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$E(X_t) = E(X_0) + ct = m_0 + m_1t$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ê‰Âş‰€‰À •‰¨‰ß ş‰× ê‰Âş‰€‰À “‰‘ ÷‰Ş‰û‰‘ı õ‰Æ‰µ‰Ö‰Û ø ş‰Æ‰µ‰‘¨‰´: \InE{}$$E(X_0) = 0 ~~~,~~~E(X_t) = \lambda t$$\EnE{} ê‰Âş‰€‰À —‰¹‰Àş‰À ó‰Ãøõ‰\nasb ‘ ¢¤ı ÷‰Ş‰û‰‘ı ÷‰Æ‰µ‰Ö‰Û ÷‰ƒ‰Æ‰´ ø ó‰Ãøõ‰\nasb ‘ ÷‰Ş‰ ş‰Æ‰µ‰‘ ÷‰À¤¢. õ‰ü—‰ö ÷‰È‰‘ö ¢¢ —‰€‰ú‰‘ ê‰Â÷‰ƒ‰À —‰¹‰Àş‰Àı  û‰İ ÷‰Ş‰û‰‘ı õ‰Æ‰µ‰Ö‰Û ¢¤¢ ø û‰İ ÷‰Ş‰û‰‘ı ş‰Æ‰µ‰‘, ê‰Âş‰€‰À •‰¨‰ß ¨‰´.\\ —‰Ş‰Âş‰€‰ú‰‘ş‰ü  õ‰Ô‰ƒ‰À ¨‰´ Ÿ‰Û ‰ƒ‰À:\\ \InE{}$cinlar$\EnE{} ¬‰Ô‰½‰‚ı \InE{}$180$\EnE{}:\\ \InE{}$10.4 , 8.4 , 7.4 ,6.4 , 5.4 , 5.4 , 4.4 , 3.4 , 2.4 , 1.4 $\EnE{}\\ ¬‰Ô‰½‰‚ı \InE{}$244$\EnE{}:\\ \InE{}$ 23.8 , 9.8 , 5.8 , 4.8 , 2.8 , 1.8$\EnE{}\\ ¬‰Ô‰½‰‚ı \InE{}$131$\EnE{} :\\ \InE{}$7.8 , 6.8 , 5.8 , 4.8 , 3.8 , 2.8 , 1.8$\EnE{}\\ ÷‰¿‰Æ‰µ‰ƒ‰ß ¢¤§ ¢¤ ê‰Â÷‰Àû‰‘ı —‰Ê‰‘¢ê‰ü:\\ ¬‰Ô‰½‰‚ı \InE{}$ 1:~84$\EnE{}\\ ¬‰Ô‰½‰‚ı \InE{}$126$\EnE{}.\\\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ê‰Â­ ‰ƒ‰À \InE{}$\{Y_n~,~n=1,2,\ldots\}$\EnE{} ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘ ê‰Ì‰‘ı Ÿ‰‘ó‰´ ğ‰Æ‰Æ‰µ‰‚ ø \InE{}$j$\EnE{} Ÿ‰‘󉵉ü ™‰‘“‰´ “‰‘ª‰À. ê‰Â­ ‰ƒ‰À \InE{}$s_1,s_2, \ldots$\EnE{} —‰ã‰À¢ õ‰ÂŸ‰Û õ‰µ‰ó‰ü —‰‘ ¢ş‰À¤ \InE{}$j$\EnE{} “‰‘ª‰€‰À. ğ‰Â Ÿ‰‘ó‰´ ø󉃉‚ \InE{}$j$\EnE{} “‰‘ª‰À ¥õ‰‘ö “‰ƒ‰ß “‰‘¥ğ‰È‰´û‰‘ \InE{}$\underbrace{s_1}_{X_1}~, \underbrace{s_2 - s_1}_{X_2}~, \ldots$\EnE{}õ‰Æ‰µ‰Ö‰Û ø û‰Ş‰µ‰¥ş‰â÷‰À. •‰Å ğ‰Â \InE{}$Y_0 = j$\EnE{} ÷‰Ú‰‘ù \InE{}$\{X_i~,~i=1,2,\ldots\}$\EnE{} ş‰× ê‰Âş‰€‰À —‰¹‰Àş‰À ¨‰´.\\ \InE{}$N(t) = \sum_{n=0}^\infty I_[0,t](s_n) $\EnE{}\\\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} ê‰Â­ ‰ƒ‰À ¥õ‰‘öû‰‘ı “‰ƒ‰ß ¢ø —‰¹‰Àş‰À \InE{}$(X_i)$\EnE{} ¢¤ı õ‰ƒ‰‘÷‰Ú‰ƒ‰ß ø ø¤ş‰‘÷‰Å õ‰µ‰€‰‘û‰ü \InE{}$\mu$\EnE{} ø \InE{}$\sigma^2$\EnE{} “‰‘ª‰€‰À. ¢¤ ş‰ß ¬‰¤–:\\\\ \InE{}$P(\frac{N(t) - \frac{t}{\mu}}{\sigma \sqrt{\frac{t}{\mu^3}}} < y) = P(N(t)< \underbrace{\frac{t}{\mu} + y \sigma \sqrt{\frac{t}{\mu^3}}}_{r_t}~) $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$= P(s_{r_t} > t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = P(\frac{s_{r_t} - r_t \mu}{\sigma \sqrt(r_t)} > \frac{t - r_t \mu}{\sigma \sqrt(r_t)})$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \approx P(Z > {-y(1+\frac{y \sigma}{t \mu})^{-({1 \over 2})}})$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$\approx P(Z > -y) = P(Zt$}\\ 1+E(N(t-x))~~ & \hbox{$x \leq t$}\\ \end{array}% \right. $$ $$ ~~~~~~~~~~~~~~~~~~= \left\{% \begin{array}{ll} 0~~ & \hbox{$x>t$}\\ 1+m(t-x)~~ & \hbox{$x \leq t$}\\ \end{array}% \right. $$ “‰€‰‘“‰Âş‰ß \\ \InE{}$m(t) = E[N(t)]$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\ = \int_0^t \!(1 + m(t-x)) \ dF(x)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ = F(t) + \int_0^t \! m(t-x) \ dF(x)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\\\ \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚ :} û‰Â ğ‰‘ù \InE{}$a$\EnE{} ş‰× —‰‘“‰â õ‰ã‰Ü‰ô “‰‘ª‰À ¢¤ ş‰€‰Ê‰¤– ›‰’ õ‰ã‰‘¢ó‰‚ \InE{}$$A(t) = a(t) + A*F(t)$$\EnE{} 䉱‰‘¤– ¨‰´ ¥: \InE{}$$A(t) = a(t) + m*a(t)$$\EnE{}  \InE{}$m$\EnE{} —‰‘“‰â —‰¹‰À¢ş‰À \InE{}$F$\EnE{} ¨‰´: \InE{}$$m(t) = \sum_{n=1}^\infty F^{(n)}(t)$$\EnE{} ş‰ã‰€‰ü “‰‚ ¥ı û‰Â \InE{}$A$\EnE{} î‰Â÷‰À¤ \InE{}$$A*F = m*a$$\EnE{} \hspace{-8mm} {\siah ™‰±‰‘– :}\\ \InE{}$A*F(t) = a*F(t) + m*a*F(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = a*F(t) + a*(m(t) - F(t))$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = a*F(t) + a*m(t) - a*F(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = a*m(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ¢¤ ê‰Âş‰€‰À —‰¹‰Àş‰À \InE{}$ = Y(t)$\EnE{} ¥õ‰‘ö “‰‘쉃‰Ş‰‘÷‰Àù —‰‘ —‰¹‰Àş‰À “‰ã‰À ¢¤ 󉽉щ‚ \InE{}$t$\EnE{} ¢¤ ¬‰¤—‰ü  õ‰µ‰ç‰ƒ‰Âû‰‘ı \InE{}$X_i$\EnE{} ¥õ‰‘öû‰‘ı “‰ƒ‰ß —‰¹‰Àş‰À ¢¤ı —‰¥ş‰â ÷‰Ş‰‘ş‰ü \InE{}$E(\lambda)$\EnE{} “‰‘ª‰À: \InE{}$$g(t) = E[Y(t)] = ?$$\EnE{} \InE{}$g(t) = E(E(Y(t)|X_1))$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = \int_0^\infty \! E(Y(t)|X_1=x) \ ,dF(x)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} $$ E(Y(t)|X_1 =x) = \left\{% \begin{array}{ll} x-t & \hbox{$x>t$}\\ g(t-x) & \hbox{$x0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$$ \Rightarrow~~~Y(t) \sim E(\lambda)$$\EnE{}\\ \hspace{-8mm} {\siah ê‰Âş‰€‰À “‰‘¥•‰ƒ‰Àş‰ü :}\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} ş‰× ê‰Â÷‰ƒ‰À \InE{}$\{X(t)~;~t\geq 0\}$\EnE{} “‰‘ ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ \InE{}$\{0,1,\ldots\}$\EnE{} ¤ ê‰Âş‰€‰À “‰‘¥•‰ƒ‰Àş‰ü ğ‰ş‰€‰À û‰Â ğ‰‘ù ¥õ‰‘ö \InE{}$s_1$\EnE{} ø›‰¢ ¢ª‰µ‰‚ “‰‘ª‰À “‰Î‰¤ı  ¢õ‰‚ ê‰Âş‰€‰À “‰ã‰À ¥ ¥õ‰‘ö \InE{}$s_1$\EnE{} ¥ ÷‰Ñ‰Â Ÿ‰µ‰Ş‰‘ñ —‰Ø‰Â¤ õ‰¹‰À¢ ê‰Âş‰€‰À ¥ 󉽉щ‚ \InE{}$0$\EnE{} “‰‘ª‰À.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ğ‰Â \InE{}$\{X(t)~;~t\geq 0\}$\EnE{} —‰ã‰À¢ õ‰È‰µ‰Âş‰‘ö ¢¤ ş‰× ¨‰ƒ‰Æ‰µ‰İ \InE{}$M/G/1$\EnE{} ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ “‰‘ª‰À ø ¢¤ 󉽉щ‚ \InE{}$0$\EnE{} ¬‰Ó ¡‰‘ó‰ü “‰‘ª‰À, ÷‰Ú‰‘ù \InE{}$\{X(t)~;~t\geq 0\}$\EnE{} ş‰× ê‰Âş‰€‰À “‰‘¥•‰ƒ‰Àş‰ü “‰‘ ê‰Ì‰‘ı ø®‰ã‰ƒ‰´ \InE{}$\{0,1,\ldots\}$\EnE{} ¨‰´ ø ÷‰Ö‰‘¯ “‰‘¥•‰ƒ‰Àş‰ü ÷‰Ö‰‘ωü ¨‰´  ¬‰Ó ¡‰‘ó‰ü õ‰üª‰¢.\\\\ \hspace{-8mm} {\siah ¥÷‰¹‰ƒ‰Âû‰‘ı õ‰‘¤î‰Ó ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ :}\\ ê‰Â­ ‰ƒ‰À ê‰Âş‰€‰À \InE{}$\{X(t)~;~0\leq t <\infty\}$\EnE{} ê‰Âş‰€‰Àı “‰‘ª‰À  ä‰À¢ ¬‰½‰ƒ‰¼ ÷‰‘õ‰€‰Ô‰ü ¤ ¡‰µ‰ƒ‰‘¤ õ‰ü‰À ø ş‰× ê‰Âş‰€‰À õ‰‘¤î‰Ó “‰‘ Ÿ‰µ‰Ş‰‘„– ÷‰µ‰Ö‰‘ñ ş‰Æ‰µ‰‘ “‰‘ª‰À.\\ \InE{}$$p_{ij}(t) = P\{X(t+u) = j | X(u) = i\}~~~~i,j = 1,2,\ldots$$\EnE{} ¥ \InE{}$ u \geq 0$\EnE{} õ‰Æ‰µ‰Ö‰Û ¨‰´.\\\\ \hspace{-8mm} {\siah ¡‰¬‰ü ¥ ê‰Âş‰€‰À •‰¨‰ß :}\\ 1( \InE{}$P\{X(t+h) - X(t) = 1 | X(t) = x\} = \lambda h + o(h)~~~~x=0,1,2,\ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ 2(\InE{}$P\{X(t+h) - X(t) = 0|X(t) = x\} = 1 - \lambda h + o(h)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ 3(\InE{}$X(0) = 0~~~~~~~~~~~$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\siah ¥÷‰¹‰ƒ‰Âû‰‘ı õ‰‘¤î‰Ó ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ :}\\ ê‰Âş‰€‰À —‰Ê‰‘¢ê‰ü ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ \InE{}$\{X(t) ; t\geq0\}$\EnE{} ¤ “‰‘ ø®‰ã‰ƒ‰´û‰‘ı ğ‰Æ‰Æ‰µ‰‚ ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ¥õ‰‘ö ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ ¨‰´ ğ‰Â :\\ \InE{}$P(X_{t+s} = j | X_s = i , X_u = x_u , ~~0 \leq u \leq s) = P(X_{t+s} = j | X_s = i)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ø ¢¤ ¬‰¤—‰ü  ÷‰Ş‰û‰‘ı ş‰Æ‰µ‰‘ ¢ª‰µ‰‚ “‰‘ª‰À\\ \InE{}$P(X_{t+s} = j | X_s = i) = p_{ij}(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ : T_i$\EnE{} õ‰À– ¥õ‰‘÷‰ü  ê‰Âş‰€‰À ¢¤ ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} õ‰üõ‰‘÷‰À 쉱‰Û ¥ ÷‰µ‰Ö‰‘ñ “‰‚ ş‰× ø®‰ã‰ƒ‰´ õ‰µ‰Ô‰‘ø– ¢¤ı ¡‰‘¬‰ƒ‰´ ä‰Àô Ÿ‰‘ê‰Ñ‰‚ ¨‰´.\\ \InE{}$T_i \sim exponential$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah ÷‰Ø‰µ‰‚ :} ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ “‰‘ ÷‰Ş‰ ş‰Æ‰µ‰‘ ¢¤ı ş‰ß ¡‰‘¬‰ƒ‰´ ¨‰´  õ‰À– ¥õ‰‘÷‰ü  ê‰Âş‰€‰À ¢¤ ø®‰ã‰ƒ‰´ ™‰‘“‰´ \InE{}$i$\EnE{} õ‰üõ‰‘÷‰À ¢¤ı —‰¥ş‰â ÷‰Ş‰‘ş‰ü ¨‰´ “‰‘ ÷‰Â  õ‰·‰\nasb … \InE{}$\nu_i$\EnE{} ¨‰´. ¢¤ ¬‰¤—‰ü  \InE{}$\nu_i = 0$\EnE{} “‰‘ª‰À ÷‰Ú‰‘ù \InE{}$i$\EnE{} ¤ ş‰× ø®‰ã‰ƒ‰´ ›‰‘£’ ğ‰ş‰€‰À. “‰€‰‘“‰Âş‰ß \InE{}$\nu_i$\EnE{} ÷‰Â  —‰ç‰ƒ‰ƒ‰Â ¥ ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} ¨‰´. ¢¤ 󉽉щ‚ —‰ç‰ƒ‰ƒ‰Â ø®‰ã‰ƒ‰´ “‰‘ Ÿ‰µ‰Ş‰‘ñ \InE{}$p_{ij}$\EnE{} “‰‚ ø®‰ã‰ƒ‰´ \InE{}$j$\EnE{} õ‰ü¤ø¢.\\ \InE{}$E(T_i) = \frac{1}{\nu_i}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} ÷‰Â  —‰ç‰ƒ‰ƒ‰Â ¥ \InE{}$i$\EnE{} “‰‚ \InE{}$j$\EnE{} : \InE{}$q_{ij} = \nu_i p_{ij}~~~~i \neq j$\EnE{}\\\\ \hspace{-8mm} {\siah ê‰Âş‰€‰À —‰ó‰À õ‰½‰Ë :}\\ ş‰× —‰ã‰Ş‰ƒ‰İ ¥ ê‰Âş‰€‰À •‰¨‰ß ş‰ß ¨‰´  ê‰Â­ ‰ƒ‰İ ª‰‘÷‰Å øì‰á ş‰× •‰ƒ‰È‰‘õ‰À ¢¤ 󉽉щ‚ ¥ ¥õ‰‘ö “‰‚ —‰ã‰À¢ •‰ƒ‰È‰‘õ‰Àû‰‘ş‰ü  쉱‰\nasb … ¤  ¢¢ù “‰Æ‰µ‰Ú‰ü ¢ª‰µ‰‚ “‰‘ª‰À. õ‰·‰‘ó‰ü ¥ ş‰ß ş‰ß Ÿ‰‘ó‰´ õ‰Â“‰¯ “‰‚ —‰ó‰ƒ‰À õ‰·‰Û ¤ğ‰‘÷‰ƒ‰Æ‰İ ¥÷‰Àù —‰½‰´ ª‰Âş‰Í è‰Áı ê‰ü, õ‰Âï ø õ‰ƒ‰Â ø õ‰ú‰‘›‰Â– ø Ÿ‰µ‰Ş‰‘ñ ş‰× —‰ó‰À “‰‚ ÷‰À¥ùı ›‰Ş‰ã‰ƒ‰´ “‰Æ‰µ‰Ú‰ü ¢¤¢.\\ “‰Â ¨‰‘§ ÷‰Â  —‰ç‰ƒ‰ƒ‰Â– õ‰ü—‰ö ÷‰Â  —‰ç‰ƒ‰ƒ‰Â ¢¤ û‰Â ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} ø Ÿ‰µ‰Ş‰‘ñ —‰ç‰ƒ‰ƒ‰Â “‰‚ ş‰× ø®‰ã‰ƒ‰´ \InE{}$j$\EnE{} ¤ “‰À¨‰´ ø¤¢: \InE{}$$\sum_{j \neq i} p_{ij} = 1$$\EnE{} \InE{}$$ \nu_i = \sum_{j \neq i} q_{ij}~~~~~~~p_{ij} = \frac{q_{ij}}{\sum_{j \neq i} q_{ij}}$$\EnE{} \begin{flushleft} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ õ‰À– ¥õ‰‘÷‰ü  ê‰Âş‰€‰À ¢¤ ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} õ‰üõ‰‘÷‰À 쉱‰Û ¥ õ‰…쉑– \InE{}$ T_{ij} = ( j \neq i)~j$\EnE{}\\ \end{flushleft} \InE{}$T_i = \mathop {\rm min_{j}} \{ T_{ij} ~,~j \neq i\}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$T_{ij} = E(q_{ij})$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} )¬‰Ó \InE{}$M/M/1$\EnE{}(\\ ¢¤ ş‰ß Ÿ‰‘ó‰´ : \InE{}$ \sim E(\mu)$\EnE{} ¥õ‰‘ö ¨‰Âøş‰Å ¢û‰ü ø ÷‰ƒ‰Ã ÷‰Â  ø¤ø¢ “‰Â“‰Â \InE{}$\lambda$\EnE{} ¨‰´.\\ \InE{}$q_{i {i+1}} = \lambda ~~~~~~i = 0, 1, \ldots $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$q_{i {i-1}} = \mu ~~~~~~i = 1, 2, \ldots $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$ \Rightarrow~~~ \nu_i = \lambda + \mu ~~~~~i = 1, 2, \ldots $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\nu_0 = \lambda$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\\\\\\\\\\\ \hspace{-8mm} {\siah ê‰Âş‰€‰Àû‰‘ı ¥¢ ø õ‰Âï \InE{}$ Birth ~and ~Death ~Process$\EnE{} :}\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} “‰‚ ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ “‰‘ ø®‰ã‰ƒ‰´û‰‘ı \InE{}$ 0 , 1 , \ldots$\EnE{}  \InE{}$$q_{ij} = 0 ~~~~|i - j|>1$$\EnE{} ê‰Âş‰€‰À ¥¢ ø õ‰Âï ğ‰ş‰€‰À.\\ \InE{}$q_{i {i+1}} = \lambda_i~~~~~~i = 0,1, \ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$q_{i {i-1}} = \mu_i~~~~~~i = 1,2, \ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$p_{i{i+1}} = \frac{\lambda_i}{\lambda_i + \mu_i} = 1 - p_{i{i-1}}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\lambda_i$\EnE{} ÷‰Â  —‰ó‰À ¢¤ ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} ø \InE{}$\mu_i$\EnE{} ÷‰Â  õ‰Âï ¢¤ ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} ¨‰´. \begin{flushleft} \InE{}$\sim E(\lambda_i)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} õ‰À– ¥õ‰‘ö õ‰‘÷‰Àğ‰‘¤ı ¢¤ \InE{}$i$\EnE{} —‰‘ ş‰× —‰ó‰À: \InE{}$T_{i{i+1}}$\EnE{} \end{flushleft} \InE{}$ \sim E(\mu_i)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\EnE{} õ‰À– ¥õ‰‘ö õ‰‘÷‰Àğ‰‘¤ı ¢¤ \InE{}$i$\EnE{} —‰‘ ş‰× õ‰Âï: \InE{}$T_{i{i-1}}$\EnE{}\\ \InE{}$ \sim E(\lambda_i + \mu_i)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} õ‰À– ¥õ‰‘ö õ‰‘÷‰Àğ‰‘¤ı ¢¤ \InE{}$i$\EnE{} —‰‘ ş‰× —‰ç‰ƒ‰ƒ‰Â: \InE{}$T_i$\EnE{}\\ \hspace{-8mm} {\siah õ‰·‰‘ñ : )¬‰Ó “‰€‰Àı \InE{}$M/M/C$\EnE{} (}\\ ¢¤ ş‰ß Ÿ‰‘ó‰´ ø¤ø¢ õ‰È‰µ‰Âş‰‘ö “‰‘ ÷‰Â  \InE{}$\lambda $\EnE{} ø \InE{}$\sim E(\mu)$\EnE{} ¥õ‰‘ö ¨‰Âøş‰Å¢û‰ü ¨‰´ ø ¨‰Âøş‰Å¢û‰ü —‰¨‰Í \InE{}$C$\EnE{} ¨‰Âøş‰Å¢û‰€‰Àùı õ‰Æ‰µ‰Ö‰Û “‰‘ ÷‰Â  \InE{}$\mu$\EnE{} ÷‰¹‰‘ô õ‰üğ‰ƒ‰Â¢.\\\\ \InE{}$\lambda_i = \lambda~~~~~~i = 0,1, \ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \mu_i = \left\{ \begin{array}{ll} i\mu ~~& \hbox{$1 \leq i \leq C $} \cr C \mu ~~& \hbox{${C+1} \leq i $} \cr \end{array} \right.$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} ş‰× ê‰Âş‰€‰À ¤ õ‰Âï õ‰½‰Ë õ‰üğ‰ş‰€‰À, ğ‰Â: \InE{}$ \forall ~i~~~~~ \lambda_i = 0$\EnE{}\\ ø —‰ó‰À õ‰½‰Ë ğ‰ş‰€‰À, ğ‰Â: \InE{}$\forall~i~~~~~ \mu_i = 0$\EnE{}\\ \hspace{-8mm} {\siah —‰ã‰Âş‰Ó :} ş‰× ê‰Âş‰€‰À —‰ó‰À õ‰½‰Ë ¤ ê‰Âş‰€‰À ş‰ñ õ‰ü÷‰‘õ‰€‰À “‰‘ ÷‰Â  \InE{}$\lambda$\EnE{} ğ‰Â: \InE{}$ \mathop{\rm \lambda_i}_{i = 1,2 , \ldots} = i \lambda$\EnE{}. ş‰ß ê‰Âş‰€‰À ¥ ›‰Ş‰ã‰ƒ‰µ‰ü  û‰Â ä‰Ì‰ —‰ó‰ƒ‰À õ‰·‰Û õ‰Æ‰µ‰Ö‰Û “‰‘ ÷‰Â  \InE{}$\lambda$\EnE{} ¢¤¢ ø û‰ƒ‰º î‰Å ÷‰Ş‰üõ‰ƒ‰Â¢. \InE{}$X(t)$\EnE{} ›‰Ş‰ã‰ƒ‰´ ¢¤ 󉽉щ‚ı \InE{}$t$\EnE{} ¨‰´.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ê‰Â­ ‰ƒ‰À \InE{}$\{X(t)~,~ t \geq 0\}$\EnE{} ş‰× ê‰Âş‰€‰À ş‰ñ “‰‘ ÷‰Â  \InE{}$\lambda$\EnE{} “‰‘ª‰À: \InE{}$$p_{ij}(t) = ?~~~~~ 1 \leq i \leq j$$\EnE{} \InE{}$T_i \sim E(i \lambda)$\EnE{}: ¥õ‰‘ö õ‰‘÷‰Àğ‰‘¤ı ¢¤ \InE{}$i$\EnE{} —‰‘ —‰ó‰À “‰ã‰À ş‰‘ ¢¤ øì‰â ¥õ‰‘ö “‰ƒ‰ß ¢ø —‰ó‰À\\\\ \InE{}$P(T_1 + \cdots + T_j \leq t) = P(X(t) \geq {j+1}~|~X(0) = 1)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow~~~p_{ij}(t) = P(X(t) \geq j ~|~X(0) = 1) - P(X(t) \geq {j+1}|X(0) = 1)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = P(T_1 + \cdots + T_{j-1} \leq t) - P(T_1 + \cdots + T_j \leq t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ õ‰ü—‰ö ÷‰È‰‘ö ¢¢\\ \InE{}$ = (1 - e^{- \lambda t})^{j-1} - (1 - e^{ - \lambda t})^j$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = e^{ - \lambda t} (1 - e^{ - \lambda t})^{j-1}~~~~~~ j=1,2,\ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ —‰Æ‰‘øı ¨‰ô ¤ “‰‚ 䉀‰ö —‰Ş‰Âş‰ß ÷‰È‰‘ö ¢û‰ƒ‰À. )‰‘’ ¤§ ¬‰Ô‰½‰‚ 441(\\\\ \InE{}$X(t)|X(0) = 1 \sim Ge(e^{-\lambda t})$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$X(t)|X(0) = i \sim NB(i,e^{-\lambda t})$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$p_{ij}(t) = \left (\begin{array}{c} {j-1} \\ {i-1} \end{array} \right) e^{ - i \lambda t} (1 - e^{- \lambda t})^{j-i}~~~~~j \geq i \geq 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \begin{flushleft} \InE{}Kolmogorov~Diffenential~Equations~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \end{flushleft} \InE{}$p_{ij} = P(X_{(t+s)} = j | X_{(s)} = i)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah ó‰İ :} ª‰Ø‰Û ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ı —‰Æ‰‘øı ‰³‰Ş‰ß î‰Ü‰Ş‰ğ‰Âøé:\\ \InE{}$p_{ij}(t+s) = \sum_{k \in S} p_{ik}(t) p_{kj}(s)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah ™‰±‰‘– :}\\\\ \InE{}$P(X_{()t+s} = j | X_0 = i) = \sum_{k \in S} P(X_{(t+s)} = j , X(t) = k | X_0 = i)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = \sum_{k \in S} P(X_{(t+s)} = j | X_t = k , X_0 = i ) P(X_t = k | X_0 = i)$~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = \sum_{k \in S} p_{kj}(s) p_{ik}(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ õ‰‘—‰Âş‰Å \InE{}$P(t)$\EnE{} ¤ “‰‚ ¬‰¤– ¥ş‰Â ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À: $$ P(t) = \left( \begin{array}{ccc} p_{00}(t) & p_{01}(t) & \ldots \\ p_{10}(t) & p_{11}(t) & \ldots \\ \vdots & \vdots & \vdots \\ \end{array} \right) $$ \InE{}$$ \Rightarrow~~~~P(t+s) = P(t) P(s)$$\EnE{}\\ \hspace{-8mm} {\siah ó‰İ :} ¢¤ ş‰× ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó “‰‘ ÷‰Â  —‰ç‰ƒ‰ƒ‰Â ø®‰ã‰ƒ‰´ \InE{}$(q_{ij})$\EnE{}\\\\ ó‰Ó( \InE{}$ \lim_{h \to 0} \frac{ 1 - p_{ii}(h)}{h} = \nu_i$\EnE{}  \InE{}$\nu_i = \sum_{j} q_{ij}$\EnE{}\\ ’( \InE{}$ \lim_{h \to 0}\frac{p_{ij}(h)}{h} = q_{ij}~~~~ i \neq j $\EnE{}\\\\ \hspace{-8mm} {\siah ™‰±‰‘– :} \begin{flushleft} \InE{}$ = \nu_i h + o(h)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} )¤  ¢¢ö ş‰× —‰ç‰ƒ‰ƒ‰Â ¥ \InE{}$i$\EnE{} ¢¤ ꉑ¬‰Ü‰‚\InE{}$h$\EnE{} ( \InE{} $1 - p_{ii}(h) = P$\EnE{} \end{flushleft} \begin{flushleft} \InE{}$ = q_{ij}h + o(h)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} ) ¤  ¢¢ö ş‰× —‰ç‰ƒ‰ƒ‰Â ¥ \InE{}$i$\EnE{} “‰‚ \InE{}$j$\EnE{} ¢¤ ꉑ¬‰Ü‰‚ı \InE{}$h$\EnE{}( \InE{}$p_{ij}(h) = P$\EnE{}\\ \end{flushleft} \hspace{-8mm} {\siah ì‰Ì‰ƒ‰‚:} \InE{}$$p_{ij}\prime (t) = \sum_{k \neq i} q_{ik} p_{kj}(t) - \nu_i p_{ij}(t)~~~~~t \geq 0$$\EnE{} \hspace{-8mm} {\siah ™‰±‰‘– :}\\\\ \InE{}$p_{ij}(t+h) = \sum_{k \in S} p_{ik}(h)p_{kj}(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow ~~~p_{ij}(t+h) - p_{ij}(t) = \sum_{k \neq i} p_{ik}(h)p_{kj}(t) - [1 - p_{ii}(h)] p_{ij}(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \begin{flushleft} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¤“‰Î‰‚ ™‰‘“‰´ õ‰üª‰¢ \InE{}$\Rightarrow$\EnE{} \end{flushleft} \InE{}$q_{ij}$\EnE{} ¤ “‰‚ ¬‰¤– ¥ş‰Â ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À:\\ \InE{}$q_{ii} = - \sum_{j \neq i} q_{ij} = - \nu_i$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ø õ‰‘—‰Âş‰Å \InE{}$Q$\EnE{} ¤ “‰‚ ¬‰¤– ¥ş‰Â ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À:\\ $$ Q = \left(% \begin{array}{cccc} -\nu_0 & q_{01} & q_{02} & \ldots \\ q_{10} & -\nu_1 & q_{12} & \ldots \\ \vdots & \ddots & \ddots & \ddots \\ \end{array} \right)% $$\\ \InE{}$ \Rightarrow~~~p \prime _{ij} (t) = \sum_{k} q_{ik}p_{kj}(t)~~~~~~~~~~backward~equation$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰‚ û‰Ş‰ƒ‰ß ª‰Ø‰Û õ‰ü—‰ö ÷‰È‰‘ö ¢¢\\ \InE{}$ p\prime_{ij}(t) = \sum_{k} p_{ik}(t) q_{kj}~~~~~~~~~~~~forward ~ equation$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ş‰ã‰€‰ü \\ \InE{}$P \prime (t) = Q P(t) = P(t) Q$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ õ‰ü—‰ö ÷‰È‰‘ö ¢¢ —‰€‰ú‰‘ Ÿ‰Û õ‰ã‰‘¢ó‰‚ õ‰‘—‰Âş‰Æ‰ü “‰‘„ “‰‚ ê‰Âô ¥ş‰Â ¨‰´:\\ $$ P(0) = \left(% \begin{array}{ccc} 1 & \ldots & 0\\ \vdots & \ddots & \vdots \\ 0 &\ldots & 1 \\ \end{array}% \right) $$\\ \InE{}$P(t) = e^{Qt} = \sum_{h=0}^\infty \frac{(Qt)^h}{h!} = \sum_{h=0}^\infty \frac{Q^h t^h}{h!}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \hspace{-8mm} {\siah õ‰·‰‘ñ)1(:} ê‰Â­ ‰ƒ‰À ş‰× ¢¨‰µ‰Ú‰‘ù “‰‚ ω¤ õ‰µ‰€‰‘ø’ ¢¤ ¢ø ø®‰ã‰ƒ‰´ \InE{}$ON$\EnE{} ø \InE{}$OFF$\EnE{} ì‰Â¤ ¢¤¢. ¥õ‰‘öû‰‘ı \InE{}$ON$\EnE{} ¥ û‰İ õ‰Æ‰µ‰Ö‰Û ø \InE{}$iid$\EnE{} “‰‘ —‰¥ş‰â \InE{}$E(\lambda)$\EnE{} ø ¥õ‰‘öû‰‘ı \InE{}$OFF$\EnE{} õ‰Æ‰µ‰Ö‰Û \InE{}$E(\mu)$\EnE{} “‰‘ª‰€‰À ø \\ \InE{}$ X(t) = \left\{ \begin{array}{ll} 1 & \hbox{$on$} \cr 0 & \hbox {$off$} \cr \end{array} \right.$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$p_{00}(t)$\EnE{} ¤ “‰À¨‰´ ø¤ş‰À.\\\\ \InE{}$ Q = \left[ \begin{array}{cc} -\mu & \mu \cr \lambda & \lambda \cr \end{array} \right] $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$P\prime(t) = P(t)Q ~~\Rightarrow~~P\prime(t) = \left[ \begin{array}{cc} p_{00}(t) & p_{01}(t) \cr p_{10}(t) & p_{11}(t) \cr \end{array} \right] \left[ \begin{array}{cc} -\mu & \mu \cr \lambda & -\lambda \end{array} \right]$~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow ~~p\prime_{00}(t) = - \mu p_{00}(t) + \lambda p_{01}(t) $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = -\mu p_{00}(t) + \lambda (1 - p_{00}(t))$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = \lambda - (\mu + \lambda) p_{00}(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow~~p\prime_{00}(t) + (\mu + \lambda) p_{00}(t) = \lambda$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow~~e^{(\mu + \lambda)t} [p\prime_{00}(t) + (\mu + \lambda)p_{00}(t)] = \lambda e^{(\mu + \lambda)t} $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow~~ e^{(\mu + \lambda)t}p_{00}(t) = \frac{\lambda}{{\mu + \lambda}} e^{(\mu + \lambda)t} + c$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$p_{00}(0) = 1 ~~\Rightarrow~~\frac{\lambda}{\mu + \lambda} + c = 1 ~~\Rightarrow~~c = 1- \frac{\lambda}{\mu + \lambda} = \frac{\mu}{\mu + \lambda}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow~~p_{00}(t) = \frac{\lambda}{\mu + \lambda} + \frac{\mu}{\mu + \lambda} e^{-(\mu + \lambda)t}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰‚ û‰Ş‰ƒ‰ß —‰Â—‰ƒ‰° õ‰ü—‰ö ÷‰È‰‘ö ¢¢\\\\ \InE{}$p_{11}(t) = \frac{\mu}{\mu + \lambda} + \frac{\lambda}{\mu + \lambda} e^{- (\mu + \lambda)t}~~~~~~~~~t\geq 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\siah õ‰·‰‘ñ)2(:} õ‰ã‰‘¢„– •‰ƒ‰È‰Âø î‰Ü‰Ş‰ğ‰Âøé “‰Âı ê‰Âş‰€‰À ¥¢ ø õ‰Âï “‰‚ ¬‰¤– ¥ş‰Â ¨‰´ \\ $$ \left(% \begin{array}{cccccc} -\lambda_0 & \lambda_0 & 0 & 0 & \ldots & 0 \\ \mu_1 & (\mu_1+\lambda_1) & \lambda_1& 0 & \ldots & 0 \\ 0 & \mu_2 & (\mu_2 + \lambda_2) & \lambda_2 & 0 & \ldots \\ \vdots & \ddots& \ddots & \ddots & \ddots & \ddots \\ \end{array}% \right) $$\\ \InE{}$ \Rightarrow~~p\prime_{i0}(t) = -\lambda_0 p_{i0}(t) + \mu_1 p_{i1}(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ ø ÷‰ƒ‰Ã \\ \InE{}$ p\prime_{ij}(t) = \lambda_{j-1}p_{i{j-1}}(t) + \mu_{j+1} p_{i {j+1}}(t) - (\lambda_j + \mu_j) p_{ij}(t)~~~~~~~~~~~~ j \neq 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \hspace{-8mm} {\siah õ‰·‰‘ñ)3(:} õ‰ã‰‘¢„– •‰ƒ‰È‰Âø î‰Ü‰Ş‰ğ‰Âøé “‰Âı ê‰Âş‰€‰À ¥¢ õ‰½‰Ë “‰‚ ¬‰¤– ¥ş‰Â ¨‰´ \\ )1( \InE{}$$ Q = \left( \begin{array}{ccccc} -\lambda_1 & \lambda_1 & 0 & \ldots & 0 \cr 0 & -\lambda_2 & \lambda_2 & 0 & \ldots \cr \vdots & \ddots & \ddots & \ddots & \ddots \cr \end{array} \right)$$\EnE{} \InE{}$$p \prime_{ij}(t) = p_{ij}(t) = 0 ~~~j < i ~~~~,~~~~p\prime_{ii}(t) = -\lambda_i p_{ii}(t)$$\EnE{} )2( \InE{}$$ p\prime_{ij}(t) = \lambda_{j-1} p_{i {j-1}}(t) - \lambda_j p_{ij}(t)~~~~~j>i$$\EnE{}\\ “‰‘ —‰›‰‚ “‰‚ ş‰€‰Ø‰‚ \InE{}$p_{ii}(0) = 1$\EnE{} ¢¤ş‰İ: \InE{}$(1)~~\Rightarrow~~p_{ii}(t) = e^{-\lambda_i t}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$(2)~~\Rightarrow~~ \lambda_{j-1}p_{i{j-1}}(t) = p\prime_{ij}(t) + \lambda_j p_{ij}(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow~~e^{\lambda_j t} \lambda_{j-1}p_{i{j-1}}(t) = e^{\lambda_j t} (p\prime_{ij}(t) + \lambda_j p_{ij}(t)) = \frac{d}{d\theta} (e^{\lambda_j t} p_{ij}(t))~~~~~~~~~~~~~j>i$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$\Rightarrow ~~p_{ij}(t) = e^{\lambda_j t} \lambda_{j-1} \int_0^t \! e^{\lambda_j s} p_{i {j-1}(s)} \ ds$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\  “‰Âı ê‰Âş‰€‰À ş‰ñ \InE{}$\lambda_j = j\lambda$\EnE{} ÷‰µ‰ƒ‰¹‰‚ı  쉱‰\nasb … “‰À¨‰´ ø¤¢ş‰İ “‰À¨‰´ õ‰üş‰À. \InE{}$$ p_{ij}(t) = \left( \begin{array}{c} j-1 \cr i-1 \cr \end{array} \right) e^{- \lambda t i} (1 - e^{- \lambda t})^{j-i}~~~~~~j \geq i \geq 1$$\EnE{}\\ \hspace{-8mm} {\siah ş‰Ø‰€‰¡‰´¨‰‘¥ı:}\\ ş‰Ø‰ü ¥ “‰Â¢û‰‘ı ş‰ß ¤ø© “‰Âı õ‰½‰‘¨‰±‰‚ı \InE{}$p_{ij}(t)$\EnE{} ¨‰´. ¥õ‰‘÷‰ü  ÷‰Â û‰‘ı ÷‰µ‰Ö‰‘ñ î‰Â÷‰À¤ ¨‰´, ş‰× ê‰Âş‰€‰À õ‰‘¤î‰Ó ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ \InE{}$ \{X(t)~,~t\geq 0\}$\EnE{} ¤ “‰‘ \InE{}$( \nu_i ~,~p_{ij}, j \neq i)$\EnE{} ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À.\\ ê‰Â­ ‰ƒ‰À  \InE{}$ \nu_i \leq \nu ~~\forall i$\EnE{}. ê‰Âş‰€‰À ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ õ‰‘¤î‰Ó ›‰Àş‰Àı ¤ “‰‘ ÷‰Â  \InE{}$\nu$\EnE{} “‰‚ ¥ı û‰Â \InE{}$i$\EnE{} ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À. “‰Àş‰ß ¬‰¤–  ¥õ‰‘÷‰ü  ÷‰µ‰Ö‰‘ñ ¤  ¢û‰À “‰‘ ÷‰Â  \InE{}$\nu$\EnE{} —‰€‰ú‰‘ \InE{}$\frac{\nu_i}{\nu}$\EnE{} ¥ ş‰ß —‰ç‰ƒ‰ƒ‰Â– —‰ç‰ƒ‰ƒ‰Â “‰‚ ¡‰‘¤š ¥ \InE{}$i$\EnE{} “‰‘ª‰À ø “‰Ö‰ƒ‰‚ —‰ç‰ƒ‰ƒ‰Â– “‰‚ ø®‰ã‰ƒ‰´û‰‘ı ¢ş‰Ú‰Âı “‰‘ª‰€‰À  “‰‚ ›‰‘ı ÷‰ú‰‘ û‰Ş‰‘ö \InE{}$i$\EnE{} ş‰‘¢¢ª‰´ ª‰Àù ¨‰´. “‰€‰‘“‰Âş‰ß ê‰Âş‰€‰À õ‰‘¤î‰Ó ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ø󉃉‚ õ‰ã‰‘¢ñ ¨‰´ “‰‘ ş‰ß ê‰Âş‰€‰À ›‰Àş‰À “‰‘ \InE{}$(\nu , R_{ij})$\EnE{}  \\ $$ R_{ij} = \left\{% \begin{array}{ll} \frac{\nu_i}{\nu} p_{ij} & \hbox{$ j \neq i$}\\ 1 - \frac{\nu_i}{\nu} & \hbox{$ j = i$} \\ \end{array} \right. $$\\ ğ‰Â \InE{}$N(t)$\EnE{} ¤ —‰ã‰À¢ —‰ç‰ƒ‰ƒ‰Â– —‰‘ ¥õ‰‘ö \InE{}$t$\EnE{} ¢¤ ê‰Âş‰€‰À õ‰‘¤î‰Ó ›‰Àş‰À —‰ã‰Âş‰Ó ‰ƒ‰İ \InE{}$\{N(t) ~,~ t\geq 0\}$\EnE{} ş‰× ê‰Âş‰€‰À •‰¨‰ß “‰‘ ÷‰Â  \InE{}$\nu$\EnE{} ¡‰û‰À “‰¢. “‰€‰‘“‰Âş‰ß \\\\ \InE{}$p_{ij}(t) = P\{X(t) = j~|~ X(0) = i\}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = \sum_{n=0}^\infty P\{X(t) = j ~|~ X(0) = j ~,~ N(t) = n\}P\{N(t) = n ~|~ X(0) = i\}$~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = \sum_{n=0}^\infty R^{(n)}_{ij} \frac{e^{-\nu t } (\nu t)^n}{n!}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$R^{(n)}_{ij}$\EnE{} Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ωü \InE{}$n$\EnE{} ì‰Àô “‰‘ \InE{}$R^0_{ii} = 1$ \EnE{} ¨‰´. õ‰ã‰‘¢ó‰‚ “‰‘„ “‰Âı õ‰½‰‘¨‰±‰‚ \InE{}$p_{ij}(t)$\EnE{} ¨‰‘¢ù—‰Â ¥ õ‰ã‰‘¢„– ¢ş‰Ô‰Â÷‰Æ‰ƒ‰Û î‰Ü‰Ş‰ğ‰Âøé ¨‰´.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ )4(:} õ‰·‰‘ñ 1 ¤ õ‰¹‰À¢\nasb ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À\\ \InE{}$$ \nu_0 = \mu ~~~,~~~\nu_1 = \lambda ~~~,~~~p_{01} = p_{10} = 1$$\EnE{} ¢¤ ê‰Âş‰€‰À õ‰‘¤î‰Ó ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ı ›‰Àş‰À ¡‰û‰ƒ‰İ ¢ª‰´:\\ \InE{}$ \nu = \lambda + \mu$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$R_{00} = \frac{\lambda}{\lambda + \mu} = 1 - R_{01}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$R_{10} = \frac{\lambda}{\lambda + \mu} = 1 - R_{11}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$$\Rightarrow ~~~R= \left( \begin{array}{cc} \frac{\lambda}{\lambda + \mu} & \frac{\mu}{\lambda + \mu} \cr \frac{\lambda}{\lambda + \mu} & \frac{\mu}{\lambda + \mu}\cr \end{array} \right) = \frac{1}{\nu} Q + I$$\EnE{}\\ “‰€‰‘“‰Âş‰ß \\ \InE{}$R^{(n)}_{i0} = \frac{\lambda}{\lambda + \mu}~~~n \geq 1 ~;~i = 0, 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$p_{00}(t) = \sum_{n=0}^ \infty R^{(n)}_{00} \frac{ e^{-(\lambda + \mu)t} ((\lambda + \mu)t)^n}{}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = e^{-(\lambda + \mu)t} + \frac{\lambda}{\lambda + \mu} e^{-(\lambda + \mu)t} \sum_{n=1}^\infty \frac{[(\lambda + \mu)t]^n}{n!}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = \frac{\lambda}{\lambda + \mu} + \frac{\mu}{\lambda + \mu} e^{-(\lambda + \mu)t}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ = 1 - p_{01}(t)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$R$\EnE{} ¢¤ øì‰â õ‰‘—‰Âş‰Å Ÿ‰µ‰Ş‰‘ñ ÷‰µ‰Ö‰‘ñ ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ¥õ‰‘ö ğ‰Æ‰Æ‰µ‰‚ ø“‰Æ‰µ‰‚ “‰‚ ¥÷‰¹‰ƒ‰Â õ‰‘¤î‰Ó ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ “‰‘ øŸ‰À ¥õ‰‘ö \InE{}$\nu$\EnE{} ¨‰´.\\ \hspace{-8mm} {\siah Ÿ‰µ‰Ş‰‘„– Ÿ‰Àı ¢¤ ê‰Âş‰€‰Àû‰‘ı õ‰‘¤î‰Ó ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ :}\\ ¢¤ ş‰ß ì‰Æ‰Ş‰´ ä‰…ì‰Ş‰€‰Àş‰İ “‰‚ “‰Â¤¨‰ü \InE{}$\lim_{ t \to \infty} p_{ij}(t)$\EnE{}. ş‰ß Ÿ‰µ‰Ş‰‘„– “‰‚ ¤Ÿ‰µ‰ü 쉑“‰Û õ‰½‰‘¨‰±‰‚ û‰Æ‰µ‰€‰À. õ‰ü—‰ö ÷‰È‰‘ö ¢¢ ¢¤ ¬‰¤—‰ü  Ÿ‰À ø›‰¢ ¢ª‰µ‰‚ “‰‘ª‰À “‰‚ ÷‰Ö‰Î‰‚ ª‰Âøá “‰Æ‰µ‰Ú‰ü ÷‰À¤¢: \InE{}$$\lim_{t \to \infty} p_{ij}(t) = p_j$$\EnE{} “‰‘ —‰›‰‚ “‰‚ ş‰€‰Ø‰‚ \InE{}$ P\prime(t) = P(t) Q$\EnE{} ø ş‰€‰Ø‰‚ ¢¤ ¬‰¤—‰ü  \InE{}$ \lim_{t \to \infty} p_{ij}(t)$\EnE{} õ‰›‰¢ “‰‘ª‰À \InE{}$\lim_{t \to \infty} p\prime_{ij}(t) = 0$\EnE{}. “‰€‰‘“‰Âş‰ß :\\ $$ \left( \begin{array}{ccc} 0 & \ldots & 0 \\ 0 & \ldots & 0 \\ \vdots &\ldots & \vdots \\ \end{array} \right) = \left( \begin{array}{ccc} p_1 & p_2 & \ldots \\ p_1 & p_2 & \ldots \\ \vdots & \vdots & \vdots \\ \end{array} \right) Q ~~~~~\Rightarrow~~~ ~\underbrace{ [p_1 ~ p_2 ~ \ldots ]}_P Q = 0 ~~~~~(1) $$ ¥ ωÂê‰ü \\ \InE{}$\sum_{j}p_{ij}(t) = 1 ~~~\Rightarrow~~~ \sum_{j}p_j = 1 ~~~~~~~~~~~~~(2)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ “‰‘ ¨‰µ‰Ô‰‘¢ù ¥ ¤“‰Î‰‚ \InE{}$(1)$\EnE{} ø \InE{}$(2)$\EnE{} õ‰ü—‰ö \InE{}$p_1,p_2, \ldots$\EnE{} ¤ “‰À¨‰´ ø¤¢. ¢¤ ¬‰¤—‰ü  ›‰’ ş‰Ø‰µ‰‘ı õ‰·‰±‰´ ø›‰¢ ¢ª‰µ‰‚ “‰‘ª‰À ÷‰Â —‰¥ş‰â ş‰Æ‰µ‰‘ı ê‰Âş‰€‰À ğ‰ş‰€‰À ø ê‰Âş‰€‰À ¤ ¤ğ‰¢ş‰× õ‰ü÷‰‘õ‰€‰À. ¢¤ 艃‰Â ş‰€‰Ê‰¤– ê‰Âş‰€‰À ğ‰Á¤ ş‰‘ “‰‘¥ğ‰È‰µ‰ü õ‰·‰±‰´ ¡‰û‰À “‰¢.\\ \hspace{-8mm} {\siah õ‰·‰‘ñ :} ê‰Âş‰€‰À ¥¢ ø õ‰Âï ¤ ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À:\\ $$ Q = \left( \begin{array}{cccc} -\lambda_1 & \lambda_1 & 0 & \ldots\\ \mu_2 & -(\mu_2 + \lambda_2) & \lambda_2 & \ldots\\ \ddots & \ddots & \ddots & \ddots\\ \end{array} \right) $$\\ \InE{}$ PQ = 0 ~~~\Rightarrow~~~-p_1 \lambda_1 + p_2 \mu_2 = 0 $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$p_1\lambda_1 - (\mu_2 + \lambda_2)p_2 + p_3 \mu_3 = 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\vdots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$p_{n-1} \lambda_{n-1} - (\mu_n + \lambda_n) p_n + \mu_{n+1} p_{n+1} = 0$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$p_1 \lambda_1 = p_2 \mu_2 ~~~~~~~~~~~~\Rightarrow ~~~p_2 = \frac{\lambda_1}{\mu_2} p_1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$p_2 \lambda_2 = p_3 \mu_3 ~~~~~~~~~~~\Rightarrow~~~ p_3 = \frac{\lambda_2}{\mu_3} p_2 = \frac{\lambda_1 \lambda_2}{\mu_2 \mu_3} p_1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$\vdots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\ \InE{}$p_n \lambda_n = p_{n+1} \mu_{n+1}~~~~\Rightarrow~~~ p_n = \frac{\lambda_1 \lambda_2 \ldots \lambda_{n-1}}{\mu_2 \mu_3 \ldots \mu_n} p_1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ ‰ö \InE{}$\sum_{i=1}^\infty p_i = 1$\EnE{} “‰€‰‘“‰Âş‰ß \\ \InE{}$1 = \sum_{i = 1}^\infty p_i = p_1 ( 1 + \frac{\lambda_1}{\mu_2} + \frac{\lambda_1 \lambda_2}{\mu_2 \mu_3} + \ldots )$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$\Rightarrow ~~~p_n = \frac{\lambda_1 \ldots \lambda_{n-1}}{\mu_2 \mu_3 \ldots \mu_n}(1 + \frac{\lambda_1}{\mu_2} + \frac{\lambda_1 \lambda_2}{\mu_2 \mu_3} + \ldots)^{-1}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} õ‰È‰Âø¯ “‰‚ ş‰€‰Ø‰‚ \InE{}$ (1 + \frac{\lambda_1}{\mu_2} + \ldots) < \infty$\EnE{}  ¢¤ øì‰â ª‰Â¯ ¤ğ‰¢ş‰× “‰¢ö ê‰Âş‰€‰À ¨‰´.\\ õ‰ã‰‘¢„– \InE{}$P Q = 0$\EnE{} —‰Ô‰Æ‰ƒ‰Â ›‰‘󉱉ü ¢¤¢. ¢¤ øì‰â ê‰Âş‰€‰À ¢¤ ¬‰¤—‰ü —‰ã‰‘¢ñ ¢¤¢  ÷‰Â  ¡‰Âøš ¥ û‰Â ø®‰ã‰ƒ‰´ “‰Â“‰Â ÷‰Â  ø¤ø¢ “‰‚ ö ø®‰ã‰ƒ‰´ “‰‘ª‰À. \begin{flushleft} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ÷‰Â  ø¤ø¢ \InE{}$ = p_1 \lambda_1=p_2 \mu_2 =$\EnE{} ÷‰Â  ¡‰Âøš~~~~~~~ : ø®‰ã‰ƒ‰´ 1 \end{flushleft} \begin{flushleft} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \InE{}$(\mu_n + \lambda_n) p_n = p_{n-1} \lambda_{n-1} + p_{n+1} \mu_{n+1}$\EnE{}~~~~~~: ø®‰ã‰ƒ‰´ \InE{}$n$\EnE{} \end{flushleft} ¢¤ ¬‰¤—‰ü  õ‰ã‰‘¢„– —‰ã‰‘¢ñ “‰‚ ¤Ÿ‰µ‰ü 쉑“‰Û Ÿ‰Û ÷‰±‰‘ª‰€‰À ¨‰µ‰Ô‰‘¢ù ¥ —‰‘“‰â õ‰ó‰À õ‰ü—‰÷‰À õ‰Ô‰ƒ‰À “‰‘ª‰À. “‰Àş‰ß ¬‰¤–  —‰‘“‰â õ‰ó‰À \InE{}$\{p_0,p_1,p_2, \ldots\}$\EnE{} ¤ “‰À¨‰´ õ‰üø¤ş‰İ \InE{}$$\phi (z) = \sum_{i=0}^\infty z^i p_i$$\EnE{} ø —‰›‰‚ \InE{}$$ \phi (1) = \lim_{z \to 1} \phi (z) = 1$$\EnE{} \hspace{-8mm} {\siah õ‰·‰‘ñ :} ¢¤ ¬‰Ó \InE{}$M/M/1$\EnE{} \\ $$ Q = \bordermatrix{&0&1&2&\ldots \cr 0 &-\lambda&\lambda&0&\ldots \cr 1 &\mu&-(\mu + \lambda)& \lambda& \ldots \cr 2 &0&\mu&-(\mu + \lambda)&\ldots \cr \vdots &\vdots&~\vdots&~\vdots&~\vdots }\\ $$\\ \begin{eqnarray*} \Rightarrow~~~~~~~~~~~~~~ \lambda p_n & = & \mu p_{n+1} ~~~~~n=0,1,\ldots\\\\ \Rightarrow~~~~ \lambda \sum_{n=0}^\infty z^n p_n & = & \mu \sum_{n=0}^\infty z^n p_{n+1} = \frac{\mu}{z} \sum_{n=0}^\infty z^{n+1} p_{n+1}\\\\ \Rightarrow ~~~~~~~~~~~ \lambda \phi (z) & =& \frac{\mu}{z} (\phi (z) - p_0)\\\\ \Rightarrow~~~~~~~~~~~~~ \phi(z) &=& \frac{p_0}{1-pz}~~~~~\rho = \frac{\lambda}{\mu} \\ \end{eqnarray*} \InE{}$\phi (1) = 1 ~~~\Rightarrow~~~p_0 = 1-p$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow~~~\phi (z) = \frac{1-p}{1-\rho z} = (1 - \rho) \sum_{n=0}^\infty (\rho z)^n$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{}\\\\ \InE{}$ \Rightarrow ~~~p_n = (1 - \rho) \rho^n ~~~~~n=0,1, \ldots$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} \begin{flushleft} \InE{}$\sum_{n=1}^\infty n p_n = (1 - \rho) \sum_{n=1}^\infty n \rho^n = \phi\prime (1) = \frac{\rho}{1-\rho}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\EnE{} õ‰µ‰¨‰Í —‰ã‰À¢ õ‰È‰µ‰Âş‰‘ö ¢¤ ¬‰Ó \\ \end{flushleft} \hspace{-8mm} {\siah ê‰Âş‰€‰À •‰¨‰ß õ‰€‰Ö‰Î‰â \InE{}$( Intrrupted ~Poisson)$\EnE{} :}\\ ê‰Â­ ‰ƒ‰À ¢ø Ÿ‰‘ó‰´ ø¤ø¢ \InE{}$(ON)$\EnE{} ş‰‘ )1( ø \InE{}$(OFF)$\EnE{} ş‰‘ \InE{}$(0)$\EnE{} ¢¤ş‰İ ¢¤ Ÿ‰‘󉵉ü  ¢¤ ¥õ‰‘÷‰ü  ¢¤ ø®‰ã‰ƒ‰´ \InE{}$(ON)$\EnE{} ì‰Â¤ ¢¤¢ õ‰Æ‰µ‰Ö‰Û ¥ ¥õ‰‘ö \InE{}$OFF$\EnE{} “‰‘ —‰¥ş‰â ÷‰Ş‰‘ş‰ü \InE{}$\alpha$\EnE{} ø \InE{}$\beta$\EnE{} ¨‰´. ¢¤ ø®‰ã‰ƒ‰´ \InE{}$ON$\EnE{} •‰ƒ‰È‰‘õ‰Àû‰‘ş‰ü ¤  õ‰ü¢û‰À “‰Â ¨‰‘§ ê‰Âş‰€‰À •‰¨‰ß “‰‘ ÷‰Â  \InE{}$\lambda$\EnE{} ø ¢¤ Ÿ‰‘ó‰´ \InE{}$OFF$\EnE{} “‰‘ û‰ƒ‰º •‰ƒ‰È‰‘õ‰Àı ¤  ÷‰Ş‰ü¢û‰À. ş‰ß ê‰Âş‰€‰À ª‰Ş‰‘¤ª‰ü ¤ ê‰Âş‰€‰À •‰¨‰ß õ‰€‰Ö‰Î‰â ğ‰ş‰€‰À.\\\\ \hspace{-8mm} {\siah ê‰Âş‰€‰À 󉽉‘ë •‰¨‰ß ø õ‰‘¤î‰Ó \InE{} $(Markov ~ modulated~ Poisson) $\EnE{}:}\\ ê‰Â­ ‰ƒ‰À \InE{}$m$\EnE{} Ÿ‰‘ó‰´ \InE{}$1,2,\ldots,m$\EnE{} “‰Â ¨‰‘§ ş‰× ê‰Âş‰€‰À õ‰‘¤î‰Ó ¥õ‰‘ö •‰ƒ‰¨‰µ‰‚ “‰‘ ÷‰Â û‰‘ı ÷‰µ‰Ö‰‘ñ \InE{}$(q_{ij} ~,~i,j = 1, \ldots , m ~;~j \neq i)$\EnE{} ¢¤ş‰İ. ¢¤ û‰Â ø®‰ã‰ƒ‰´ \InE{}$i$\EnE{} —‰Ô‰‘쉑—‰ü ¤  õ‰ü¢û‰À “‰Â ¨‰‘§ ê‰Âş‰€‰Àı •‰¨‰ß “‰‘ ÷‰Â  \InE{}$\lambda_i$\EnE{} ¤  õ‰ü¢û‰À. ê‰Âş‰€‰À ª‰Ş‰‘¤ª‰ü  —‰ã‰À¢ —‰Ô‰‘쉑– ¢¤ û‰Â 󉽉щ‚ ¤ õ‰üª‰Ş‰‘¤¢ ¤ ş‰× ê‰Âş‰€‰À 󉽉‘ë •‰¨‰ß õ‰‘¤î‰Ó ğ‰ş‰€‰À “‰‘ \InE{}$(q_{ij},\lambda_i)$\EnE{}. \end{document}