Question

Consider two digital fuel pumps A and B that could be used in a single gas station. Pump A has

a mean effective process time of 4 minutes with squared-coefficient of variation of 0.5. Pump B
has a mean effective time of 3 minute with squared-coefficient of variation of 5.
(a) Assume that the arrival rate of cars is 0.2 car per minute with squared-coefficient of variation
of 1. Which pump will have a longer average cycle time? (Hint: the number of machines, m is 1
(c) Again, assume that there could be only one pump in the gas station like Part a. If the arrival
rate increases to 0.24 cars, what will happen to the cycle time? Compare with Part a.
Only Part C please!

Answer 1

In Part C, we are asked to analyze the effect of an increased arrival rate on the cycle time of a single pump in the gas station.

Given:

Arrival rate (λ) = 0.24 cars per minute

Squared-coefficient of variation (CV^2) = 1

Number of machines (m) = 1

To determine the cycle time, we need to calculate the service time and the queue time separately and then sum them up.

Service Time:

For a single pump, the service time follows an exponential distribution with a mean of 4 minutes (from Part A) and CV^2 of 0.5. We can use the formula for the variance of the exponential distribution: Var(service time) = (mean service time)^2.

Var(service time) = (4 minutes)^2 = 16 minutes^2

Queue Time:

To calculate the queue time, we need to consider the arrival rate and the service rate. In this case, the arrival rate is 0.24 cars per minute, and the service rate is 1/mean service time = 1/4 = 0.25 cars per minute.

The formula for the queue time in an M/M/1 queue is given by:

Queue Time = (Squared-coefficient of variation / (2 * (1 - Arrival Rate * Service Time))) * Service Time^2

Plugging in the values:

Queue Time = (1 / (2 * (1 - 0.24 * 4))) * (4 minutes)^2

Now we can calculate the cycle time:

Cycle Time = Service Time + Queue Time

Calculating the queue time and the cycle time for the given values:

Queue Time = (1 / (2 * (1 - 0.24 * 4))) * (4 minutes)^2

Cycle Time = 4 minutes + Queue Time

After performing the calculations, we can compare the new cycle time with the previous cycle time from Part A to see the effect of the increased arrival rate.