# Question: Consider a single server queueing system with a Poisson input Erlang

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.

(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.

**View Solution:**## Answer to relevant Questions

Consider the E2/M/1 model with λ = 4 and μ = 5. This model can be formulated in terms of transitions that only involve exponential distributions by dividing each interarrival time into two consecutive phases, each having ...Consider the single-server variation of the nonpreemptive priorities model presented in Sec. 17.8. Suppose there are three priority classes, with λ1 = 1, λ2 = 1, and λ3 = 1. The expected service times for priority classes ...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 ...Jim Wells, vice-president for manufacturing of the Northern Airplane Company, is exasperated. His walk through the company’s most important plant this morning has left him in a foul mood. However, he now can vent his ...Consider a two-server queueing system where all service times are independent and identically distributed according to an exponential distribution with a mean of 10 minutes. Service is provided on a first-come-first-served ...Post your question