Reference no: EM132238005

Systems Modeling and Simulation Assignment -

Multiple Choices Questions - Choose the best answer for each of the following 12 questions.

Q1. The acronym GIGO means:

(a) First In First Out

(b) General Input General Output

(c) Geometric Input Geometric Output

(d) Garbage In Garbage Out

(e) Giggles In Giggles Out

Q2. The Coefficient of Variation (CV) is defined as:

(a) E[XY] - E[X][Y]

(b) the ratio of the standard deviation to the mean

(c) the ratio of the median to the mean

(d) the ratio of the variance to the mean

(e) the ratio of the variance to the median

Q3. One of the fundamental conservation laws for queueing models is:

(a) Little's Law

(b) Biggie's Law

(c) Law of Large Numbers

(d) Central Limit Theorem

(e) Newton's Law

Q4. The Kolmogorov-Smirnov (KS) test is best used for:

(a) comparing an empirical pmf with a candidate theoretical distribution

(b) comparing an empirical pdf with a candidate theoretical distribution

(c) comparing an empirical CDF with a candidate theoretical distribution

(d) comparing an empirical CCDF with a candidate theoretical distribution

(e) impressing people at cocktail parties

Q5. The relationship between correlation and independence can be stated as:

(a) the absence of correlation implies independence

(b) the presence of correlation implies independence

(c) independence implies the presence of correlation

(d) independence implies the absence of correlation

(e) they are exactly the same thing

Q6. The only discrete distribution with the memoryless property is the:

(a) Deterministic distribution

(b) Uniform distribution

(c) Geometric distribution

(d) Poisson distribution

(e) Binomial distribution

Q7. The only continuous distribution with the memoryless property is the:

(a) Uniform distribution

(b) Exponential distribution

(c) χ² distribution

(d) Kolmogorov distribution

(e) Weibull distribution

Q8. An example of a heavy-tailed distribution is the:

(a) Deterministic distribution

(b) Uniform distribution

(c) Geometric distribution

(d) Exponential distribution

(e) Pareto distribution

Q9. Consider the execution of a simulation with run length T to determine the value of a performance metric P. In general, as one increases the run length T of this simulation:

(a) the mean of P will decrease with increasing T

(b) the mean of P will increase with increasing T

(c) the variance of P will decrease with increasing T

(d) the variance of P will increase with increasing T

(e) all of the above

Q10. In an M/G/1 queueing model:

(a) the arrival process is Poisson, and service times are Deterministic

(b) the arrival process is Poisson, and service times are Markovian

(c) the arrival process is Poisson, and service times are General

(d) the arrival process is Markovian, and service times are Markovian

(e) the arrival process is Markovian, and service times are Geometric

Q11. In heap-based event list management, the average time for event scheduling is:

(a) O(1) for insertions, and O(n) for deletions

(b) O(n) for insertions, and O(1) for deletions

(c) O(log n) for insertions, and O(log n) for deletions

(d) O(log n) for insertions, and O(1) for deletions

(e) none of the above

Q12. In advanced simulations using a calendar queue, the average event scheduling time is:

(a) O(1) for insertions, and O(n) for deletions

(b) O(n) for insertions, and O(1) for deletions

(c) O(log n) for insertions, and O(log n) for deletions

(d) O(log n) for insertions, and O(1) for deletions

(e) none of the above

Modeling -

Q13. In 1976, George Box made the following famous quote: "All models are wrong; some models are useful."

In this question, we will consider this quote, and how it applies to our simulation model for the ICT elevators in Assignment 4.

(a) Explain in your own words what this quote means, in general.

(b) Using Assignment 4 as the basis for your answer, give three examples of things that were "wrong" about your simulation model, and why they were wrong.

(c) Using Assignment 4 as the basis for your answer, give three examples of things that were "useful" about your simulation model, and why they were useful.

Simulation Concepts and Terminology -

Q14. For each of the following pairs of terms, define each term, and clarify the key difference(s) between the two terms.

(a) "simulation model" and "analytical model"

(b) "transient" and "steady-state"

(c) "confidence interval" and "confidence level"

(d) "χ² test" and "KS test"

(e) "open queueing network" and "closed queueing network"

Simulation Output Analysis -

Q15. In Assignment 2, you implemented a trace-driven simulation model of a single-server queue for a Banff park entry booth. We also used this model in our discussion of simulation output analysis and confidence intervals.

Suppose that you have completed simulation experiments at some fixed load level ρ, and obtained the following results for two different service time models (Exponential and Deterministic) using the method of batch means, with 10 batches:

Service Time Model	Batch	1	2	3	4	5	6	7	8	9	10
Exponential	MeanQ	3.2	3.4	3.0	2.9	3.1	3.1	3.0	2.9	3.3	3.4
Deterministic	MeanQ	2.8	2.4	2.5	2.9	3.0	3.1	2.8	2.7	2.7	2.5

(a) Compute the (sample) mean and standard deviation of the queue size for the simulation results with Exponential service times.

(b) Compute the (sample) mean and standard deviation of the queue size for the simulation results with Deterministic service times.

(c) Compute a 90% confidence interval for the mean queue size for the simulation experiments with Exponential service times. (See handouts for statistical tables.)

(d) Compute a 90% confidence interval for the mean queue size for the simulation experiments with Deterministic service times. (See handouts for statistical tables.)

(e) For this load ρ, is there a statistically significant difference between the mean queue sizes for these two service time distributions at the 90% confidence level? Justify your answer.

Simulation Input Analysis -

Q16. The following table shows the number of journal paper submissions received per month by an ACM journal for the past three years (n = 36 months):

Year	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
2015	7	6	6	8	5	6	6	5	11	13	8	5
2016	7	8	15	4	8	9	4	10	5	4	9	8
2017	10	11	4	7	7	7	8	6	13	9	5	7

The editor of the journal suspects that this data follows a Poisson distribution, but is not sure. Your task is to use a Chi-square test to find out. (See handout materials, if needed.)

(a) Compute a histogram of this data, and sketch a diagram of it. Does it look Poisson? Why or why not?

(b) Compute the mean and variance of the number of paper submissions per month. Do these statistics suggest that the data is from a Poisson distribution? Why or why not?

(c) To the best of your ability, compute (or estimate) the Chi-square statistic for the histogram from this dataset, using χ² = _i=1∑^m(observed[i]-expected[i])²/expected[i]. As a reminder, for the Poisson distribution, Prob[X = k] = (λ^k/k!)e^-λ. Also, expected[i] = nProb[X = i].

(d) Using a Chi-square test at the α = 0.05 level of significance, does this data obey a Poisson distribution? Justify your answer.

Markov Chains -

Q17. In Assignment 4, we built a discrete-event simulation model of the ICT elevators. However, we did not construct an analytical model for this system (thank goodness!).

(a) Draw a simple Markov chain to represent the various states and state transitions for the humans that were modeled in Assignment 4. As a hint, your diagram should have at least 3 states, but no more than 12 states. State any assumptions.

(b) Draw a simple Markov chain to represent the various states and state transitions for a single elevator, as modeled in Assignment 4. As a hint, your diagram should have at least 5 states, but no more than 20 states. State any assumptions.

(c) Suppose that you want to determine the steady-state probability p_i that the (single) elevator is on floor i. With reference to your diagram in (b) above, what additional information would be required in order to solve for the steady-state probabilities, and what would be the (logical) steps for doing so? (Note that you do NOT need to solve the Markov chain to determine the steady-state probabilities; you merely need to indicate HOW one would go about solving it, by listing the main steps involved.)

Queueing Models -

Q18. In Kleinrock's book on Queueing Systems (Volume 1), the well-known PollaczekKhinchin (PK) mean value formula on page 187 states that the average number of customers in an M/G/1 system with FIFO service is:

q¯ = ρ + ρ²((1 + C²)/2(1 - ρ)),

where C² is the squared coefficient of variation (CV) of the service time distribution.

(a) In simple English (at most two sentences), explain what the PK formula indicates about system occupancy.

(b) For exponential service times, show that the PK formula above simplifies to q¯ = ρ/(1-ρ).

(c) For deterministic service times, show that the PK formula simplifies to q¯ = ρ/(1-ρ) - ρ²/(2(1-ρ)).

(d) For the general M/G/1 model at the top of the page, use Little's Law N = λT, where N = q¯, to show the total time T in the system, as a function of ρ. Express your result in the form T = x + W, where x = 1/µ is the mean service time, and W is the waiting time. Note that W will be a (complicated) function of ρ = λ/µ.