$H5\mathrm{.}2$ Consider $an$ open Jackson network consisting $ofK$ first$-$come, first$-$serve, single$-$server queues. External

arrivals $to$ queue i are according $toan$ independent Poisson arrival process $of$ rate $r_i,1iK{r}_i=0is$

possible, $in$ which case the queue has $no$ external arrivals, but $we$ require $r_i>0$ for $at$ least one queue $in$ the

systemi are exponentially distributed with mean $\frac{1}{{}_i}.$ Once a job completes service

$at$ queue $i,it$ proceeds $to$ join queue $j$ with probability $P_{ij},1jK,or$ leaves the system altogether with

probability $P_{i,\text{o}\text{u}\text{t}}=1-{\displaystyle \underset{j=1}{\overset{K}{}}}{P}_{ij}.$

The one difference with a standard open Jackson network $is$ that any external job arriving when there are $M$

customers $in$ the network $is$ blocked from entering and lost. $In$ other words, there can never $be$ more than a total

$ofM$ jobs $in$ the entire system. Derive the stationary distribution for the state $of$ the system.

Hint: Consider transforming the system into $an$ "equivalent" closed Jackson network with $M$ jobs and $an$ addi$-$

tional $(K+1{)}^{st}$ queue. The goal $isto$ structure this additional queue $to$ emulate the arrival rates $to$ each queue

$in$ the open system.