13.10. Consider a one-pump gas station that satisfies the assumptions for the M/M/1 model. It is estimated that, on average, customers arrive to buy gas when their tanks are one-eighth full. The mean time to service a customer is 4 minutes, and the arrival rate is six customers per hour.

a. Determine the expected length of the queue and the expected time in the system.

b. Suppose that customers perceive a gas shortage (when there is none) and respond by changing the fill-up criterion to more than one-eighth full on average. Assuming that changes in λ are inversely proportional to changes in the fill-up criterion, compare results when the fill-up criterion is one-quarter full with the results in part a.

c. Making the same assumption as in part

b, compare the results obtained if the fill-up criterion is one-half full.

Do we have the makings of a behaviorally induced gasoline panic?

d. It is reasonable to assume that the time to service a customer will decrease as the fill-up criterion increases.

Under “normal” conditions, it takes, on average, 2 minutes to pump the gasoline and 2 minutes to clean the windshield, check the oil, and collect the money. Rework parts b and c if the time to pump the gasoline changes proportionally to changes in the fill-up criterion.