CS533 Sample Mid-Term Exam 1

All five problems count equally. In the mathematical problems, show all your work, and do not spend too much time simplifying your answers.

1. Suppose a random variable X has probability density function

f(x) = (3/8) x², for 0 £ x £ 2.

(Note that this says that the only possible values of x are between 0 and 2.) Find

a) the probability that X will be between 1 and 2.

b) the mean of X.

2. a) State the Central Limit Theorem.

b) Explain how (by showing calculations or by writing an explanation) the Central Limit Theorem allows us to construct confidence intervals.

3. a) What is a benchmark program? (Give a definition.)

b) Discuss the shortcomings of the classic benchmark programs, such as Whetstone, Dhrystone, and describe how these shortcomings have been overcome in modern benchmark suites, such as SPEC and TPC.

4. Suppose you have written a simulation program to simulate the operation of an on-line transaction processing system. Your program produces as one of its outputs the average response time per transaction. How could you use your simulation program to construct a confidence interval for the mean response time. Describe both

a) the method of independent replications, and

b) the method of batch means.

5. 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?