Consider a Jackson network with three service facilities having the parameter values shown below.

PROBLEMS

$797$

sj

$$j

aj

$111$

$405030$

$10153$

Facility j

j$1$ j$2$ j$3$

i $1$

pij

i $2$

i $3$

$00\mathrm{.}30\mathrm{.}40\mathrm{.}500\mathrm{.}50\mathrm{.}30\mathrm{.}20$T $($a$)$ Find the total arrival rate at each of the facilities.

$($b$)$ Find the steady$-$state distribution of the number of customers at facility $1,$ facility $2,$ and facility $3.$ Then show the product form solution for the joint distribution of the number at the re$-$

spective facilities.