$-$A bank has established two counters one for commercial banking and the second for personal

banking. Arrival and service rates at the commercial counter are $6$ and $12$ per hour, respectively.

The corresponding numbers at the personal banking counter are $12$ and $24,$ respectively. Assume

that arrivals occur in Poisson processes and service times have exponential distributions

$($a$)$ Assuming that the two counters operate independently of each other, determine the ex$-$

pected number of waiting customers and their mean waiting time at each counter $.$

$($b$)$ What is the effect of operating the two queues as a two$-$server queue with arrival rate

$18/$hour and service rate $18/$hour$?$ What conclusion can you draw from this operation $($i$.$e$.,$

which one is more efficient the two$-$server queue operation or two single$-$server operations$)?$