# CSC2281 Quiz 1Spring 2013

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This is quiz 1 to be taken by Friday, 8 February 2013. It should take you between half an hour and 2 hours to answer the following questions. You should be sure to read Chapter 1 (pp 1 -- 35) of H. J. Bernstein, M. Goldstein, "Network Design and Implementation Lecture Notes Spring 1984" at least through the section on Queuing Theory.

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Please answer the following questions on this form (or on a paper copy of this form).

2. What is a Poisson distribution? Explain in detail.

3. What is an M/M/1 queue. Explain in detail.

4. Give the expected waiting time in an M/M/1 queue in terms of customer arrival rate and service rate. Explain what happens when the customer arrival rate approaches the service rate.

5. What is Little's formula? Explain each term of the formula and give a specific example.

6. Apply Liitle's formula to give a formula for the average delay in a set of M/M/1 queues.

7. Give a reasonable approximation to the expected queue length for a queuing system with multiple servers assuming Poisson distributions.

8. Explain when, in practical terms it is reasonable to use a Poisson distribution instead of a binomial distribution.

9. Prove that r*t is the expected number of events in a time interval, t = n*t0, in which event arrival is governed by a binomial distribution with short-term rate of event arrival r over minimal time-step t0.

10. A. K. Erlang studied the behavior of telephone systems, and developed the basic formulae used for allocation of sufficient trunk capacity in telephone systems. Derive the Erlang formulae giving the probability of finding all lines busy in a telephone exchange having m outgoing lines, a single queue of incoming lines, and Poisson distributed traffic, under the assumption that all traffic for unavailable lines is lost, instead of queued.

11. What is the expected delay time for a queuing system composed of 100 smaller queuing systems, fifty of which have a customer arrival rate of 10 per second and a expected delay time of 5 seconds each, and 50 of which have a customer arrival rate of 20 per second and expected queue lengths of 1 each?

12. Suppose an M/M/1 queuing system has a customer arrival rate of 5 per second and an expected queue length of 3. What is the service rate?

13. Assume an airport can safely land an average of 30 planes per hour. Suppose the average time from arrival of a plane in the airport's controlled space and landing is 6 minutes. On average, how planes are in landing patterns or landing at the airport at any one time.

14. Briefly describe your proposed project for this course.

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Revised 20 January 2013