Question

A queuing system with a Poisson arrival rate and exponential service time has a single queue, two servers, an average arrival rate of 60 customers per hour, and an average service time of 1.5 minutes per customer.

Answer the following questions. Show ALL formulas and calculations used in your response.

1. The manager is thinking of implementing additional queues to avoid an overloaded system. What is the minimum number of additional queues required? Explain.
2. How many additional servers are required to ensure the utilization is less than or equal to 50%? Explain.
3. If the manager loses a server, what service time would be necessary to ensure that the queue length is not at risk of approaching infinity? Explain.

Instructions:

• Answers should include FULL explanations, including formulas

Answer 1

Given

No.of servers,s = 2

customers per hour

Average service time = 1.5 min per customer = 40 customers per hour

System utilization , u =

u = 60 / (2*40) = 0.75 = 75%

Here

The utilization should be less than or equal to 60%

servers

The no.of additional servers required to ensure the utilization is less than or equal to 50%

servers

If the manager loses a server, what service time would be necessary to ensure that the queue length is not at risk of approaching infinity

s = 2-1 =1

For stable queue,

The minimum service rate should be 60 per hour or the maximum service time should be 1 minute.