Consider a queueing system with two servers, where the customers arrive from two different sources. From source 1, the customers always arrive 2 at a time, where the time between consecutive arrivals of pairs of customers has an exponential distribution with a mean of 20 minutes. Source 2 is itself a two-server queueing system, which has a Poisson input process with a mean rate of 7 customers per hour, and the service time from each of these two servers has an exponential distribution with a mean of 15 minutes. When a customer completes service at source 2, he or she immediately enters the queueing system under consideration for another type of service. In the latter queueing system, the queue discipline is preemptive priority where customers from source 1 always have preemptive priority over customers from source 2. However, service times are independent and identically distributed for both types of customers according to an exponential distribution with a mean of 6 minutes.
(a) First focus on the problem of deriving the steady-state distribution of only the number of source 1 customers in the queueing system under consideration. Define the states and construct the rate diagram for most efficiently deriving this distribution (but do not actually derive it).
(b) Now focus on the problem of deriving the steady-state distribution of the total number of customers of both types in the queueing system under consideration. Define the states and construct the rate diagram for most efficiently deriving this distribution (but do not actually derive it).
(c) Now focus on the problem of deriving the steady-state joint distribution of the number of customers of each type in the queueing system under consideration. Define the states and construct the rate diagram for deriving this distribution (but do not actually derive it).