(a) Generalize Exercise 7.26 to the case in which there are K types of customers, each with independent Poisson arrivals and each with independent exponen- tial service times. Let λk and μk be the arrival rate and service rate respectively for the kth user type, 1 ≤ k ≤ K. Let ρk = λk/μk and ρ = ρ1+ρ2+·· ·+ρK . In particular, show, as before, that the probability of n customers in the system is Qn = p(0, ... , 0)ρn/n! for 0 ≤ n ≤ m.
(b) View the customers in (a) as a single type of customer with Poisson arrivals of rate λ = J, λk and with a service density J, (λk/λ)μk exp(-μkx). Show that the expected service time is ρ/λ. Note that what you have shown is that, if a service distribution can be represented as a weighted sum of exponentials, then the distribution of customers in the system is the same as for the M/M/m/m queue with equal mean service time.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.