Question

Discuss the relationship between the M/M/1 queue and the situation described in problem 9 of homework 3.

What similarities are there between arrival processes in these two examples?

What about similarities in service-time distribution?

Compute the stationary distribution of the Markov chain obtained in problem 9 of homework 3 under the assumption that p < q. Explain the significance of this assumption.

problem 9 of homework 3

Suppose customers can arrive to a service station at times n = 0, 1, 2, .... In any given period, independent of everything else, there is one arrival with probability p, and there is no arrival with probability 1 − p. Suppose customers are served one-at-a-time on a first-come-first-served basis. If at the time of an arrival, there are no customers present, then the arriving customer immediately enters service. Otherwise, the arrival joins the back of the queue.

In a time period n, events happen in the following order: (i) arrivals, if any, occur; (ii) service completions, if any, occur; (iii) service begins on a new customer if there has been an arrival to an empty queue or a service has just finished and there is another customer present.

Assume that service times are i.i.d. geometric random variables (each with parameter q) that are independent of the arrival process. Note that a customer who enters service in time t can complete service, at the earliest, in time t+1 (in which case his service time is 1).

Let Xn be the number of customers at the station at the end of time period n; i.e., after the time-n arrivals and services. Note that Xn includes both customers waiting as well as any customer being served.

Answer 1