Q$2.$ The jobs are first processed on server $1$ with an exponential rate of ${}_1$ and then on server $2$ with an exponential rate of ${}_2.$

a$)$ Draw a part of the transition diagram to show the transitions emanating from a typical state.

b$)$ Draw a part of the transition diagram to show the transitions ending in a typical state.

c$)$

$($i$)$ Give the steady$-$state balance equation for a typical state.

$($ii$)$ Give formulations of the following steady$-$state performance measures in terms of the steady$-$state probabilities $P_{(n_f,n_1,n_2)}.$

expected number of customers at server $1($including the customers waiting in line to be processed at server $1)$

expected number of customers at server $2($including the customers waiting in line to be processed at server $2)$

expected number of backorders

expected number of idle fixtures

expected percent utilization of the fixtures

e$)$ In order to answer the numerical questions in parts $($f$),($g$)$ and $($h$),$ truncate the state space assuming that $n_{f}B.$ In this case, the maximum number of backorders allowed in the system is equal to $|B|.$ A job that arrives when there are $|B|$ backorders in the system would be lost.

$($i$)$ Give the truncated state space and revise your answer for part $($c$).$

$($ii$)$ What is the ergodic structure of the chain when the number of backorders is limited Does the steady$-$state exist for this ergodic structure?

f$)$ Let $K=2,{}_1={}_2=3.$ Answer the questions below for combinations of the following two different backorder limits and two different job arrival rates: $B=-20,B=-50$ and $=1,=1\mathrm{.}8.$

$($i$)$ Calculate the steady$-$state probabilities. $($Make sure that the code you submit gives the steady$-$state probabilities for any given $K,B,,{}_1$ and ${}_2.)$

Hint. You may use a matrix inversion method or an LP solver to compute the steady$-$state probabilities. Note that the matrix inversion method is suggested here.

$($ii$)$ Is there a significant difference between steady$-$state probabilities obtained with different backorder limits$?$ Explain. How do the probabilities change as the maximum number of backorders increases?

$($iii$)$ Is there a significant difference between steady$-$state probabilities obtained with different $$ values? Explain.