The local Division of Motor Vehicles (DMV) is concerned with its waiting line system. Currently, the DMV uses a single-server, single-line, single-phase system when processing license renewals. Based on historical evidence, the average number of customers arriving per hour is 9 and is described by a Poisson distribution. The service rate is 12 customers per hour with the service times following an exponential distribution. The customers are patient and come from an infinite population. The manager of the DMV would like you to calculate the operational characteristics of the waiting line system.

(a) What is the average system utilization?

(b) What is the average number of customers in the system?

(c) What is the average number of customers waiting in line?

(d) What is the average time a customer spends in the system?

(e) What is the average time a customer spends waiting in line?

• Before You Begin

The key here is identifying the mean arrival rate and the mean service rate. The average number of customers per hour is given as 9. The server is able to service 12 customers per hour on average. Plug these values into the appropriate formula.

• Solution

(a) Average utilization is 0.75, or 75 percent.

 

(b) Average number of customers in the system is 3.

(c) The average number of customers waiting in line is 2.25.
 

= 9 12 = 0.75