$13\mathrm{.}14$ Threshold Queue

We define a threshold queue with parameter $T$ as follows: When the number

of jobs is $>>TT=2E[N]TT=0T=0=\frac{}{}T=2,$ then jobs arrive according $to$ a Poisson process with rate $$ and

their service time $is$ Exponentially distributed with rate $,$ where $>(i.e.,$

the queue $is$ running $in$ overload$).$ However, when the number $of$ jobs $is>T,$

then jobs arrive with Exponential rate $$ and are served with Exponential rate

$.$

Figure $13\mathrm{.}10$ shows the $CTMC$ for the case $ofT=2.$ Compute $E[N],$ the

mean number $of$ jobs $in$ the system, $as$ a function $ofT.As$ a check, evaluate

your answer when $T=0.$ Note that when $T=0,we$ have $=\frac{}{}.$

Figure $13\mathrm{.}10.$ Threshold queue with $T=2.$