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Simulation &Amp; Modeling Question Paper
Simulation &Amp; Modeling
Course:Bachelor Of Business Information Technology
Institution: Strathmore University question papers
Exam Year:2011
STRATHMORE UNIVERSITY
BACHELOR OF BUSINESS INFORMATION TECHNOLOGY
2011
MAT 4201: SIMULATION & MODELING
DATE: TIME:
INSTRUCTIONS
The Question paper consists of Five questions. Answer Question one and any other
two questions.
Question 1
(a) Giving examples differentiate between deterministic and stochastic models.
(4 marks)
(b) Find a value of ? (arrival rate) for an M/M/1 queue for which the service rate is
10 customers per second, and subject to the requirement that the probability of no
customer in the system is 0.64 (5 marks)
(c) Inventory is withdrawn from a stock of 60 items according to a poisson
distribution at the rate of 4 items a day.
i) Find the probability that 10 items are withdrawn during the first 2
days. (3 marks)
ii) Determine the probability that no items are left at the end of 4 days.
(3 marks)
iii) Determine the average number of items withdrawn over a 4 – day
period. (2 marks)
(d) A single communication server can hold at most five packets. The arrival rate to
the system is 20 packets/sec. The communication sever can serve 40 packets per
sec. Calculate the probability that a packet will have to enter the system.
(5 marks)
(e) Use the linear congruential method of random number generation to generate the
first three random numbers for the case where a = 17, m = 100, c = 43 and the
seed is 27. (6 marks)
(f) Why is simulation important in performance modeling (2 marks)
Total (30 marks)
Question 2
(a) List the four desired properties of a random number generator (4 marks)
(b) Users are connected to a database server through a network. Jobs arrive to the
database server through the network at a time which is exponentially distributed
2
with mean 6 seconds. The server idle time was measured to be 10 seconds during
a one-minute observation interval. The service discipline at the database server is
first come first served (FCFS):-
i.) Determine the arrival and service rate of the jobs in minutes (3 marks)
ii.) Determine the interarrival and service times for the first six jobs using the first six
random numbers in column 1 of table A.1 (8 marks)
iii.) Hand simulate the problem to determine the average time each job waits in queue
before being processed (assume the system starts at time zero) (5 marks)
Total (20 marks)
Question 3
(a) What is a model and why is modeling important (2 marks)
(b) The average time each http request spends in queue before being processed at a
web server is 2 minutes. The system idle time was measured to be 12 seconds
during a one minute observation interval. Use an M/M/1 model for the system to
determine the following
i.) What is the probability a request has to wait in queue before being
processed (3 marks)
ii.) What is the average service time per transaction (4 marks)
iii.) What is the probability there are more than one http request in the system
(2 marks)
iv.) On average, how many requests are waiting in the queue to be processed
(2 marks)
v.) On average, how many requests are in the system (2 marks)
(c) For an M/M/1 queue, how does the mean response time change if we double the
speed of the server? How does mean response time change if we double the speed
of the server and double the arrival rate? (3 marks)
Total (20 marks)
Question 4
(a) Briefly compare the advantages and disadvantages of the analytical modeling and
the discrete event simulation modeling as applied in queueing systems (4 marks)
(b) Using the random numbers 0.6505, 0.3262 and 0.1646, sample from an
exponential distribution with mean of 6 (6 marks)
i.) Find a suitable c for f (x) . (2 marks)
ii.) Find the probability that the random variable x has a value more than 0.5
(4 marks)
3
ii) Give an Inverse Transform algorithm for generating x. (4 marks)
Question 5
(a) A performance analyst simulated a computer system a total of 10 times,
each simulation run independent of all the others. She calculated and
recorded the sample means for system response time from each of the
10 runs, coming up with the following data (measured in seconds):
6,15,17,8,9,7,10,25,5,11
i) What is the confidence interval with a 95% confidence level for the mean
response time? (10 marks)
ii) If measurements from the real system gave a mean figure of 12 seconds
for the response time, Comment on the statistical validity of the
confidence intervals constructed in part (i). (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)
Total (20 marks)
4
5
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
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
?
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