$\frac{M}{M}\frac{?}{1}$

$P_0=1-,L=\frac{}{-}$

$\frac{M}{M}\frac{?}{s}$

$P_0=\frac{1}{{\displaystyle \underset{n=0}{\overset{s-1}{}}}\frac{(\frac{}{}{)}^n}{n!}+\frac{(\frac{}{}{)}^s}{s!}\frac{1}{1-\frac{}{s}}}$

$L_q=\frac{P_0(\frac{}{}{)}^s}{s!(1-{)}^2},L=(W_q+\frac{1}{})=L_q+\frac{}{}$

$\frac{M}{M}\frac{?}{s}$

$\frac{M}{M}\frac{?}{?}\frac{s}{K}$

$P_0=\frac{1}{{\displaystyle \underset{n=0}{\overset{s}{}}}\frac{(\frac{}{}{)}^n}{n!}+\frac{(\frac{}{}{)}^s}{s!}{\displaystyle \underset{n=s+1}{\overset{K}{}}}(\frac{}{s}{)}^{n-s}}$

For $n=1,2,$dots,$s,$

$P_n=\frac{(\frac{}{}{)}^n}{n!}{P}_0$;

$n=s,s+1,$dots,$K,P_n=\frac{(\frac{}{}{)}^n}{s!s^{n-s}}{P}_0$;

$L_q=\frac{P_0(\frac{}{}{)}^s}{s!(1-{)}^2}[1-{}^{K-s}-(K-s){}^{K-s}(1-)]$

$L={\displaystyle \underset{n=0}{\overset{s-1}{}}}n{P}_n+L_q+s(1-{\displaystyle \underset{n=0}{\overset{s-1}{}}}{P}_n)(20\%)$ Consider a Jackson network with three service facilities having the parameter

values shown below.

$($a$)$ List all the equations that are required to find the total arrival rate at each of the

facilities. $(5\%)$

Suppose that ${}_1=10,{}_2=30,$ and ${}_3=15,$ answer the following $3$ sub$-$questions.

$($b$)$ What is the probability that all the facilities have empty queues? $(5\%)$

$($c$)$ Find the expected total waiting time for a customer. $(5\%)$

$($d$)$ If we hire one more server for facility $3,$ suppose the expected number of customers

in facility $3$ becomes $0\mathrm{.}5.$ Suppose that the salary of each server is $200$ dollars per

hour and the waiting cost is $10$ dollars per hour for each customer. Is it a good idea

to hire one more server for facility $3?(5\%)$