Queueing Theory
Consider a post office with two counters. Customers arrive according to a Poisson process of rate 1 customer every 5 minutes. The service time for each customer is exponentially distributed with an average of 2 minutes, independent of other customers' service time.
(a) (15 pts) If there are two separate queues: one for each counter. Incoming customers choose a queue at random so each queue has a Poisson arrive rate of 1 customer every 10 minutes. (Hint: Consider the system as two independent M/M/1 queues.)
- (5 pts) What is the average number of customers in the post office?
- (5 pts) What is the probability that an incoming customer has to wait in queue?
- (5 pts) What is the percentage of customers who will spend more than a total of 4 minutes in the post office?
(b) (15 pts) If customers wait in a combined line. When a customer completes service at either counter, the customer at the front of the line goes into service. (Hint: Consider the system as one M/M/2 queue.)
- (5 pts) What is the average number of customers in the store?
- (5 pts) What is the probability that an incoming customer has to wait in queue?
- (5 pts) What is the percentage of customers who will spend more than 1 minute waiting in queue?