CS 533 /  EE 581                                                  Sample  Exam 2

 

All five problems count equally.  Do not spend too much time doing arithmetic to simplify your answers.  When you are finished, staple this question sheet and your formula sheet to your answer sheets.

 

 

1. In an M/M/1  queueing system, suppose we were give a choice of service rate.  We could obtain a server of rate µ for a cost of c * µ per hour.  We also have a waiting cost (which applies to customers in the queue or in service) of d per customer per hour.  Find (in terms of the constants c and d ) the optimal service rate µ (to minimize the overall system cost).  Assume that the arrival rate is 10 jobs per hour.  (If you don’t know how to find the optimal service rate, at least find an expression for the total system cost based on the constants c and d.)

 

 

2.   We want to model a buffer as an M/M/ queueing system.  The buffer has the capacity to hold two jobs, but since we feel that it will be very rare for the buffer to be full, we decide to model the buffer as an M/M/1 queue (thus ignoring the finite capacity).  The arrival rate is 5 jobs per hour and the service rate is 20 jobs per hour.

 

a) In estimating average response time, what's the percentage error in using an M/M/1 model instead of the M/M/1/2 model.  (The formulas for the M/M/1/c queue are on the board.)

 

b) Using the M/M/1/c model, what proportion of arriving jobs will be rejected because the buffer is full?

 

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.  In studying the performance of local area networks, we examined the performance of Aloha and slotted Aloha networkss.  For each, we derived a formula for the throughput, S, in terms of the offered load, G, of the form

 

                                                S = G ´ X

 

where the fomula for X depended on the particular network we were studying.  The formulas for X were:

 

Aloha:  X = e-2G

 

Slotted Aloha: X = e-G

 

Explain why the formulas for X are different for Aloha and slotted Aloha.  That is, explain specifically what it is about these two protocols that leads to these two different formulas.