Question

 Problem 3: Assume that a single-server queueing system has a Poisson interarrival process with a rate of 10 customers per hour. Also, assume that the service time is exponential with at a rate

 of 12 customers per hour. Answer the following questions to 3 significant digits:

 a) What is the expected utilization of the server?

 b) What is the log-run time average of number of customers in the system?

 c) Using Little's law, use the answer from part (b) to calculate the average waiting time in system.

 Assume that the system in the question changes in the following way: Instead of an exponential service time, suppose service is now a general random variable with a rate of 12 customers per

 hour and a variance of 1. Answer the following questions to 3 significant digits:

 d) What is the expected number of customers in the system?

 e) What is the expected amount of time that a customer will spend in the system?

Answer 1

Answer 2

Answer 3

Mean Arrival rate = 10 customers per hour (Poisson)

Service rate = 12 customer per hour (Exponential)

a.Expected utilization of the server p = Mean arrival rate / service rate = 10/12 = 83.333%

b. Average number of the customers in the system = Mean arrival rate/ (service rate - mean arrival rate)

= 10/ (12-10) = 10/2 = 5 customers

c. Average waiting time in the system= (Expected utilization of the server) * (1/(service rate - mean arrival rate))

= 83.33% *(1/(12-10)) = 83.333% * (1/2) = 0.417 hours or 25 minutes.