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.

The manager is thinking of implementing additional queues to avoid an overloaded system. What is the minimum number of additional queues required? Explain.

How many additional servers are required to ensure the utilization is less than or equal to 50%? Explain.

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.

Answer 1

ANSWER:

From given data,

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.

Average arrival rate, λ = 60 customers per hour
Average service rate, μ = 1 customer per 1.5 minutes = 40 customers per hour
Number of servers, s = 2

The manager is thinking of implementing additional queues to avoid an overloaded system. What is the minimum number of additional queues required? Explain

system utilization, u = λ / (s.μ) = 60 / (2*40) = 75%

A rule of thumb says that more than 60% utilization amounts to overload. So, to avoid overload, utilization should be less than or equal to 60%.

So, λ / (s.μ) ≤ 60%
or, 60 / (s*40) ≤ 0.60
or, s = 2.5 or 3 if we round up.

So, avoiding the overload will require another additional server.

How many additional servers are required to ensure the utilization is less than or equal to 50%? Explain.

If the requirement is utilization ≤ 50%, then,

λ / (s.μ) ≤ 50%
or, 60 / (s*40) ≤ 0.50
or, s = 3

So, with an additional server, it is possible to attain 50% utilization

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.

Now we have one server i.e. s=1

For a stable queue, the requirement is λ ≤ s.μ. or, s ≥ λ/μ
or, 1 ≥ 60/μ
or, μ ≥ 60

So, the minimum service rate has to be 60 per hour or the maximum service has to be 1 minute