Question

A new full-service, attendant-staffed fuel and car service station on Interstate 95 will service only northbound cars. The manager estimates that customers will arrive every 4 minutes and will require 6 minutes to be served at the pump. How many pumps should be installed if the manager desires a utilization factor of 0.75? What percent of the time will the pumps be idle? Assume that the coefficients of variation of both the inter-arrival times and the service times are equal to 1.

Answer 1

Customer Arrival time = 4 minutes

Customer Service time = 6 minutes

Utilization rate desired = 0.75

Customer Arrival rate = 1/4 X 60 = 15 customers/hr

Customer Service rate = 1/6 = 10 customers/hr

We know that, Utilization = Customer Arrival rate/(Customer Service rate X Number of systems)

Therefore, 0.75 = (15)/(10 X Number of pumps)

Hence, Number of pumps = (15)/(10 X 0.75)

Number of pumps (s) = 15/(7.5) = 2

There should be 2 pumps to have a utilization rate of 0.75.

Percent of time that the pumps will remain idle = ?

We will calculate the Probability that there are zero customers in the system, which will directly mean the probability that the system is idle.

Therefore, Probability that there are no or zero customers in the system =

Po =

Po = [ (0.75)^0/0! + (0.75)^1/1! + ((0.75)^2/2! X (1/(1-0.75))) ]^-1

=> 0.34786

Hence, 34.786 percent of the time the pumps are idle.