A new sandwich shop is nearing completion and a parking lot needs to be designed. The storeowners anticipate that, on the average 12-hour day, 360 vehicles will visit the sandwich shop. The owners also anticipate that the average vehicle will remain parked for 10 minutes. How many parking spaces need to be provided in order to guarantee that no more than 1 vehicle in 50 will be unable to find a parking space?
[Solution Shown Below]
First, we need to determine the traffic load. The incoming flow rate is calculated as shown below.
Q = 360 vehicles/12 hours
The average parking duration is 10 minutes or 0.167 hours. The traffic load is calculated as shown below.
A = 30 vehicles/hour * 0.167 hours
The maximum probability of rejection is 1 in 50, or 0.02. Using the probability of rejection equation, we can solve for the number of spaces required.
P = (AM/M!)/(1 + A + A2/2 + . . . + AM/M!)
Solving the equation for M yields a value of 10. The parking lot at the sandwich shop must have at least 10 spaces, in order to meet the owners expectations.
Note that we have used average parking rates in this analysis. The sandwich shops particular situation could dictate that more spaces are required. For example, say that the shop serves 80% of its customers between 11 A.M and 2 P.M. The majority of the customers are arriving during a much shorter time frame than the 12 hours that we used to find the incoming flow rate. In this case, more parking spaces would be required.