Queueing question

Question

A local country garage has one petrol pump. On average, customers are served at a rate of 6 per minute. On normal days, customers arrive at the rate of 30 every 10 minutes. Arrivals follow a Poisson distribution and service times follow an exponential distribution.

a. What is the average number of customers in the queue?

Answer 1

Make sure the rates are in the same "units" -- in this case, let's scale both arrival times and service times to be "per minute"

 

For the service time, 6 customers/minute: λ=6

For arrival times: 30 customers/10 minutes = 3 customers/minute; µ=3

 

Let Lq = Average number of customers in the queue.

Using the formula for Lq for an M/M/1 proces: 

 

Lq = λ² / [µ ( µ - λ)] = 3² / [6*(6-3)] = 9/18 =0.5 

 

So the average number of customers in the queue is: 0.5 customers every minute or 1 customer every 2 minutes.