Suppose that customers arrive to a service system according to a Poisson process with rate 6 arrivals per hour. Suppose furthermore that service can potentially consist of two tasks,AandB, to be done in order. More specifically, for 10 percent of arriving customers, service consists of taskAonly; for the remaining 90 percent of arriving customers, service consists of first completing taskAand then completing taskB. Assume that the time it takes to complete taskAis exponentially distributed with rate 12 per hour and that the time it takes to complete taskBis exponentially distributed with rate 15 per hour. Finally, assume that all random variables (i.e., interarrival times, task times, and whether individual customers need to complete one or both tasks) are independent.

Reminder:IfXis an exponential random variable with rate >0, thenE[X] = 1/andE[X²] = 2/².

a. (8 points) Suppose that there is a single server who serves customers in the order that they arrive (if a customer requires both tasksAandB, the server will do taskAfirst and then taskBbefore helping the next customer). It is not difficult to see that this system is stable. Determine the long-run average system (sojourn) timeWper customer in the system (including both waiting time in queue and service time). Explain.

b. (7 points) Suppose that there are two servers, namely one for taskAand another one for taskB. Thus customers may have to wait before service starts for taskAand also before service starts for taskB(for customers who require taskB). It is not difficult to see that this system is stable. Determine the long-run average system (sojourn) timeWper customer in the system (including waiting times and service times). Explain (note that you may want to model this situation as an open Jackson network).