$13\mathrm{.}10$ Busy Period in M$/$M$/14$

A busy period, B$,$ is the time from when the system has $1$ job until the system is first empty. $($Obviously$,$ the number of jobs may go up and down a lot in going from state $1$ to state $0.)$ Your goal is to derive E $[$B$]$ for an M$/$M$/1.$

$($a$)$ Draw the CTMC for the M$/$M$/1$ with arrival rate $\backslash$lambda and service rate $\backslash$mu $.$

$($b$)$ Write a recurrence relation for E $[$B$]$ by conditioning on the next state that

the CTMC moves to after leaving state $1.$

$($c$)$ SolvetherecurrencerelationforE$[$B$].$WhatdoestheexpressionforE$[$B$]$

remind you of$?$