Question

In: Statistics and Probability

A local Division of Motor Vehicles (DMV) is concerned with its waiting line system. Currently, the...

A 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

What is the probability that there are 2 customers in the system.
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Solutions

Expert Solution

Currently, the DMV uses a single-server, single-line, single-phase system

The average number of customers arriving per hour is 9 and is described by a Poisson distribution.

Thus = 9 customers per hour

The service rate is 12 customers per hour with the service times following an exponential distribution

Thus   = 12   customers per hour

Now   p = /                

         p = 9 / 12 = 3 / 4 = 0.75

Thus p = 0.75                               ( here p is traffic intercity )

Probability that there are 2 customers in the system is given by

Prob ( 2 customers in the system ) = ( 1 - p ) * p 2

                                                                      = ( 1 - 0.75 ) * 0.752

Prob ( 2 customers in the system ) = 0.140625

Thus Probability that there are 2 customers in the system is 0.141


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