Question

SHOW ALL WORK!

The Quick Snap photo machine at the Lemon County bus station takes snapshots in exactly 100 seconds. Customers arrive at the machine according to a Poisson distribution at the mean rate of 25 per hour. On the basis of this information, determine the following:

a. the average number of customers waiting to use the photo machine
b. the average time a customer spends in the system
c. the probability an arriving customer must wait for service

Answer 1

a. 1.92 customer

b. 400 seconds.

c. 0.6944

Explanation:

The system described is a waiting line type (M / M / 1) with 100s service time, that is, a service rate mu = 36 customers / hour; and a lambda arrival rate = 25 customers / hour. In this way,

a. The number of customers waiting to use the machine is: lambda ^ 2 / [mu (mu - lambda)] = (25^2) / [36 (36 - 25)] = (25^2) / (36*9) = 1.92 customers .

b. The average time a client spends in the system is: 1 / (mu - lambda) = 1 / (36 - 25) = 1/9 = 0.11 hours = 400 seconds.

c. The probability that a arriving customer should wait for the service is given by: lambda / mu = 25/36 = 0.6944.