dolution

Consider a single-server queueing system with a Poisson input, Erlang service times, and a finite queue. In particular, suppose that k = 2, the mean arrival rate is 2 customers per hour, the expected service time is 0.25 hour, and the maximum permissible number of customers in the system is 2. This system can be formulated in terms of transitions that only involve exponential distributions by dividing each service time into two consecutive phases, each having an exponential distribution with a mean of 0.125 hour, and then defining the state of the system as (n, p), where n is the number of customers in the system (n = 0, 1, 2), and p indicates the phase of the customer being served (p = 0, 1, 2, where p = 0 means that no customer is being served).

(a) Construct the corresponding rate diagram. Write the balance equations, and then use these equations to solve for the steadystate distribution of the state of this queueing system.

(b) Use the steady-state distribution obtained in part (a) to identify the steady-state distribution of the number of customers in the system (P₀, P₁, P₂) and the steady-state expected number of customers in the system (L).

(c) Compare the results from part (b) with the corresponding results when the service-time distribution is exponential.