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