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Simulation &Amp; Modelling Question Paper
Simulation &Amp; Modelling
Course:Bachelor Of Business Information Technology
Institution: Strathmore University question papers
Exam Year:2007
STRATHMORE UNIVERSITY
FACULTY OF INFORMATION TECHNOLOGY
Bachelor Of Business Information Technology
END OF SEMESTER EXAMINATION
STA 4201: SIMULATION & MODELLING
DATE: 16th March 2007 TIME: 2 Hours
INSTRUCTIONS
THE QUESTION PAPER CONSISTS OF FIVE QUESTIONS. ANSWER QUESTION
ONE AND ANY OTHER TWO QUESTIONS.
Question 1
(a) What is the difference between a discrete and a continuous system? (2 marks)
(b) Find a value of µ (service rate) for an M/M/1 queue for which the arrival rate is
8 customers per second, and subject to the requirement that the probability of
more than one customer in the system is 0.64 (4 marks)
(c) The simulation model-building (or simulation life cycle) can be broken into four
phases. Explain briefly the main tasks of each of these phases? (8 marks)
(d) What is meant by the “System State” in a simulation? What can change the
system state in a single server queuing system? (3 marks)
(e) Describe five key components of a Discrete Event simulation (5 marks)
(f) Use congruential method of random number generation to generate the first three
random numbers for the case where a = 10, m = 7, and the seed is 10. (7 marks)
(g) How can the width of a confidence interval be reduced? (1 mark)
Total (30 marks)
Question 2
(a) Consider a simple queuing network where customers enter the system with
exponential inter-arrival times with expectation 1 minute. One server then serves
the incoming people with a service time uniform between 0.3 and 0.5 minutes.
After that service people leave the system with probability 80% whereas with
probability 20% they have to join the queue again to wait for another service. The
simulation should start with an empty system and last for 4 hours.
i.) What are the entities and what are the resources and what are the events
for this simple network? (6 marks)
ii.) What are two variables you can use as state variable for that system
(2 marks)
iii.) Is the system transient or steady state? Explain (2 marks)
(b) The average response time for http requests at a web server is 2 minutes. The
system busy time was measured to be 50 seconds during a one minute observation
interval. Use an M/M/1 model for the system to determine the following.
2
i.) What is the average service time per transaction (4 marks)
ii.) What is the probability there are more than one http request in the system
(2 marks)
iii.) On average, how many requests are in the system (2 marks)
iv.) What is the average time a request spends in the queue? (2 marks)
Total (20 marks)
Question 3
(a) What is an event list (2 marks)
(b) A certain faculty printer is shared by staff and students. Suppose that the rate of
generating requests to the printer by staff is twice that of students, but that the
average time to print a student printout is the same as average time for a faculty
printout. If the utilization of the printer by the students is 25%, the utilization of
the printer by the faculty is 50%, the overall average service time is 1 minute,
what is the average time the students spend waiting for their printout (time spend
in system)? (8 marks)
(c) Twenty observations were made over a 90-second period on the inter-arrival
times of customers arriving at the drinks stall at the Frontier canteen. The times
(in seconds) were:
4 3 6 2 1 3 4 6 8 10
1 4 15 5 7 1 2 1 4 3
i.) Prepare a histogram and hypothesize the distribution of the
transaction times (4 marks)
ii.) Using the Kolmogorov-Smirnov test, test the hypothesis that the
inter-arrival times are exponentially distributed. Use a level of
significance of a = 0.05 (6 Marks)
Total (20 marks)
Question 4
(a) The following data are arrival times (in minutes counting from 0) and service
times (in minutes) for the first thirteen customers arriving at a dental clinic with
one dentist on duty. Upon arrival, a customer either enters service if the dentist is
free or joins the waiting line. When the dentist has finished work on a customer,
the next one in line (i.e., the one who has been waiting longest) enters service
Arrival
times
12 31 63 95 99 154 178 221 304 345 411 455 537
Service
times
40 32 55 48 18 50 23 12 72 30 54 46 13
i.) Hand simulate the problem and determine the departure times of the
thirteen customers ( 5 marks)
ii.) How much spare time, if any, does the dentist have? ( 1 mark)
iii.) What is the average time a customer waits in line to be served (1 mark)
3
iv.) Draw a curve for number of customers in the queue, Q(t), depending on
variable t. (3 marks)
(b) Let a random variable X represent the lifetime in months of a certain brand of
light bulb. The pdf of X (life time) is given by
( 1), 0 2
( )
0,
k x x
f x
otherwise
? + < <
=??
i) Determine the value of k (3marks)
ii) Find the probability that a bulb will last less than 1 month (1 mark)
iii) Find the probability that a bulb will last between half a month and three
quarter of a month (2marks)
iv) Apply the inverse transformation method to generate a variate from the pdf.
(4 marks)
Total (20 marks)
Question 5
(a) Validation and verification are important in simulation. Distinguish between these
two processes. (4 marks)
(b) Describe the differences between a steady-state and transient-state system
(4 marks)
(c) Define confidence interval. What do we mean by saying “the 90% confidence
interval for the mean response time of a computer is 12 to 15 seconds”?(3 marks)
(d) Assume you are supervising a research student who is working on simulation. His
aim was to determine the overhead of his new algorithm in the system. He runs
four statistically independent replications and these are his results (in
milliseconds):
233.7 225.8 226.4 241.1
i.) Is this the proper way to present the results? How would you advise him to
present his results, with a confidence of say 90%. (2 marks)
ii.) You of course tell him to run more replications, and he runs 6 more
replications. If he wants to achieve an error of at most 5 milliseconds, at
90% confidence, was his number of additional replications enough?
(7 marks)
Total (20 marks)
4
Some useful formulae
Average time in the queue =
µ (µ ? )
?
-
, Average time in the system =
µ -?
1
Expected queue length =
( )
2
µ µ ?
?
-
, Average number of units in the system =
µ ?
?
-
Probability that there are n units in the system at a particular time is
n P = (1- ? )? n = (
µ
?
1- ) ( )n
µ
?
Student's t Distribution
Degrees
of
freedom
0.100 0.05 0.025 0.010 0.005 0.001
1 3.078 6.314 12.706 31.821 63.657 318.31
2 1.886 2.920 4.303 6.965 9.925 22.326
3 1.638 2.353 3.182 4.541 5.841 10.213
4 1.533 2.132 2.776 3.747 4.604 7.173
5 1.476 2.015 2.571 3.365 4.032 5.893
6 1.440 1.943 2.447 3.143 3.707 5.208
7 1.415 1.895 2.365 2.998 3.499 4.785
8 1.397 1.860 2.306 2.896 3.355 4.501
9 1.383 1.833 2.262 2.821 3.250 4.297
10 1.372 1.812 2.228 2.764 3.169 4.144
11 1.363 1.796 2.201 2.718 3.106 4.025
12 1.356 1.782 2.179 2.681 3.055 3.930
13 1.350 1.771 2.160 2.650 3.102 3.852
14 1.345 1.760 2.145 2.624 2.977 3.787
15 1.341 1.753 2.131 2.602 2.947 3.733
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