IE 581 Introduction to Stochastic Simulation. Three pages of hand-written notes, front and back. Closed book. 120 minutes.

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1 Three pages of hand-written notes, front and back. Closed book. 120 minutes. 1. SIMLIB. True or false. (2 points each. If you wish, write an explanation of your thinking.) (a) T F The time of the next event and the type of the next event are stored as "attributes" in a "list". (b) T F The average size, over time, of every list is collected automatically. (c) T F The average size of a list is an observational statistic, maintained through the routine "sampst". (d) T F The variance of the size of a list is a time-based statistic. (e) T F When simulating a system using SIMLIB, the state cannot change continuously through time. (f) T F When simulating a system using SIMLIB, the next-event list is ordered based on the scheduled time of the event. (g) T F When simulating a system using SIMLIB, the "main" routine is user written. (h) T F When simulating a system using SIMLIB, reducing variance by using control variates is not possible. (i) T F When simulating a system using SIMLIB, reducing variance by using antithetic variates is not possible. (j) T F When simulating a system using SIMLIB, an "arrival" event must always be scheduled to occur at time zero. (k) T F In SIMLIB, every event occurs at a point in time, not over an interval of time. Schmeiser Page 1 of 6 Final Exam, Spring 2011 (May 6)

2 2. True or false. (2 points each. If you wish, write an explanation of your thinking.) (a) T F The only reason to use multiple random-number streams is to induce correlation in the hope of improving the efficiency of the simulation experiment. (b) T F If an experiment with sample size n = produces a point estimator θˆ with variance var(θˆ)=100, then the "ones" digit is meaningful. (c) T F If two uniform (0,1) random numbers U and V are independent, then the fractional part of their sum is also uniform (0,1). (d) T F The average number of jobs in a system is a time integral divided by the elapsed time of simulation. (e) T F The average time of a customer in the system is a time integral divided by the time of simulation. (f) T F For a stationary set of data Y 1, Y 2,..., Y n, the variance of Y i can be estimated by the usual estimator S 2 = n Σ i=1 (Y i Y ) 2 n 1 where Y is the sample average. (That is, S 2 goes to var(y i ) as the simulation run length goes to infinity.) (g) T F If the length of a simulation run is quadrupled, then the standard error of the point estimators is approximately halved, whether or not the output observations are autocorrelated. (h) T F The "loss factor" in linear control variates arises because linearity is only an approximation. Schmeiser Page 2 of 6 Final Exam, Spring 2011 (May 6)

3 3. (4 points) In a simulation experiment to estimate a performance measure θ, an ideal control variate C is one that is (1) inexpensive to compute, (2) is highly correlated with the point estimator θˆ, and (3) has known mean E(C ). Since we are free to choose any random variable as the control, why not chooseθˆ itself; that is, why not choose C =θˆ? 4. (2 points each) Indicate whether the statistic is a "time average" (T) or "observational" (O). (a) T O Variance of the number of customers waiting for service. (b) T O Average number of customers waiting for service. (c) T O Fraction of customers who wait for service. 5. (2 points each) Suppose you rerun a simulation experiment, keeping everything the same except the random-number seeds. Which are constants, which are random, and which are undefined. (Circle one.) (a) θ constant random undefined (b) S 2 constant random undefined (c) s.e.(θˆ) constant random undefined Schmeiser Page 3 of 6 Final Exam, Spring 2011 (May 6)

4 6. Suppose that we are keeping statistics about the time spent in the system, say T i for customer i. We wish to estimate the steady-state mean E(T ), variance Var(T ), and P(T < 10 minutes). (a) (5 points) Some statistical accumulators need to be maintained. Choose names and write the logic to update them for one new observation, say t i. (b) (5 points) Write the logic to compute the point estimators at the end of the simulation. Schmeiser Page 4 of 6 Final Exam, Spring 2011 (May 6)

5 7. Let L (t ) denote the number of customers in the system at time t in a steady-state simulation. We wish to estimate the long-term average number of customers in the system, θ=lim t E(L (t )). Suppose that we observe L (t ) from time t = 0 until the simulation ends at time t end. (a) (5 points) Write a reasonable point estimatorθˆ. (b) (5 points) Write the definition of the steady-state variance of L (t ). (c) (5 points) Write a reasonable point estimator for your answer to Part (b). Schmeiser Page 5 of 6 Final Exam, Spring 2011 (May 6)

6 8. (2 points each) Why is the "initial transient" a concern? That is, why do we sometimes discard data from the beginning of a simulation replication? Circle all correct answers. Reduce the bias ofθˆ. Reduce the variance ofθˆ. Get a better estimator of the standard error ofθˆ. 9. (5 points) In the thinning algorithm for generating the next arrival time t i given the last arrival time t i 1 from a Poisson process with rate λ(t ), suppose a potential arrival has arrived at time t* via the Poisson process with rate λ*. What is the probability that the potential arrival enters the system? 10. (2 points each) The variance-reduction techniques of common random numbers, antithetic variates, and (external) control variates require "correlation induction". (a) T F The correlation being induced is (i) a property of the simulation experiment and (ii) not a property of the model being analyzed. (b) T F The induced correlation should always be positive. (c) T F The use of NORTA (normal to anything) is an example of such correlation induction. Schmeiser Page 6 of 6 Final Exam, Spring 2011 (May 6)