CS 533 / EE 581 Sample Exam 1

All five problems count equally. Do not spend too much time doing arithmetic to simplify your answers.

1. For each of the following, identify the queueing model required. You may use notation like M/M/1/_/_, or you may describe the model in words. Identify any model parameters given in the problem. Do not do any calculations for this problem. You may assume that the times in the problems follow exponential distributions.

a) Five computer users share the use of terminal. No one uses this terminal except these five users. If one wants to use the terminal while the terminal is in use, he or she waits until the terminal is free. If there are several people waiting, they obtain use of the terminal in a first-come, first-served manner.

b) A "modem pool" is available to a large number of personal computer users for making outgoing calls. There are 10 modems in the pool. When a user wants to make a call and a modem is free, the use is randomly assigned to one of the free modems. If all the modems are in use, up to two requests can be held in a queue for the next available modem. If a request is made when this queue is full, the request is rejected.

2. Find the state probabilities for the queueing system M/M/1/2/2 (a single server queue with capacity 2 and population size 2) by constructing the state transition diagram and solving for the state probablities.

3. What is local balance in a queueing system (give a careful definition). Why is local balance an important property for a queueing system to have.

4. For the open queueing network problem on the board, find the average number of customers in the system and the average time in system. You may assume that the interarrival times for external arrivals and the service times follow exponential distributions, and that the queueing network is a product form network.

5. Use the convolution algorithm to find the mean cycle time and the arrival rate with respect to node 1 for the closed queueing network shown on the board.