Question

Fast food franchises considering a drive up window food service operation. assume that cars arrive at the service following a position distribution at a rate of 24 customers per hour. assume that the service distribution is exponential arriving customers place orders on an intercom station at the back of the parking lot and then drive to the service window to pick up their orders and pay for the service.

the situation can be implemented using a single server operation for where an employee who fills the order and the one who takes the money from the customer the average service time for this alternative is 1.45 minutes.

operating characteristics

Probability of no cars in the system

average number of cars waiting for service

average number of cars in the system

average time a car waits for service

average time in the system

probability that an arriving car will have to wait

Answer 1

Arrival rate, = 24 customers per hour = (24/60) per minute = 0.4 customers per minute

Service rate = 1/1.45 customers per minute = 0.6896552 customers per minute

Average utilization of the system, = / = 0.4 / 0.6896552 = 0.58

Using M/M/1 model,

Probability of no cars in the system = 1 - = 1- 0.58 = 0.42

average number of cars waiting for service = / ( - ) = 0.58 * 0.4 / (0.6896552 - 0.4) = 0.80

average number of cars in the system = / ( - ) = 0.4 / (0.6896552 - 0.4) = 1.38

average time a car waits for service = / ( - ) = 0.58/ (0.6896552 - 0.4) = 2.00 minutes

average time in the system = 1 / ( - ) = 1/ (0.6896552 - 0.4) = 3.45 minutes

probability that an arriving car will have to wait = = 0.58