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1 KINGDOM OF SAUDI ARABIA Ministry of Higher Education Al-Imam Muhammad Ibn Saud Islamic University Faculty of Computer Science and Information المملكة العربية السعودية وزارة التعليم العالي جامعة اإلمام محمد بن سعود اإلسالمية كلية علوم الحاسب والمعلومات Mid-Term Exam in Modeling and Simulation (CS-33) Only slides are your own summaries are authorized Summaries must contain only equations. No exercises or solutions are authorized. Duration: 120 minutes اإلسم الثالثي:... المستوى الرقم الجامعي النتيجة: / 0 It is MANDATORY to write the equation. A correct answer will NOT be accepted only if the written equation is correct. If you don t find the correct value in the list, write your own value. يجب إعادة أوراق اإلجابة فقط أوراق األسئلة ال تعاد إلى الدكتور يجب كتابة المعادالت المطلوبة Grading Exercise 1 General Questions 12 points Exercise 2 Continuous Markov Chain 10 points Problem Simulation and Queuing Theory 18 points Total 0 points Dr. Anis Koubâa June 2009

2 Exercise 1. General Questions (12 points) (30 minutes) Question 1 A B C D E F G H I Question 2 A B C D E F G H I Question 3 A B C D E F Question A B C D Exercise 2 (10 points) 1. Draw the Markov Chain of this system. (2 points) (30 minutes) 2. Write the Transition Rate Matrix of this Markov Chain. (2 points) 3. Write the Transition Probability Matrix of this Markov Chain. (2 points) Dr. Anis Koubâa 2/ June 2009

3 . We assume now, that the same restaurant has only ONE counter (and with the same queue) for the handling the service with a service rate 2 cars/mn. Draw the corresponding Markov Chain. (2 points) 5. By applying the result of the birth-death chain to this new system, what is the probability π 0 that there is no car waiting for the service in the system? (2 point) Problem. Simulation and Queuing Theory (18 points) (60 minutes) Queuing Theory Analysis (11 points) 1. What is the type of this network in queuing theory? Explain? (1 points) 2. Write the routing probability matrix P p ij =, for, { 1,2,3, } i j (1 points) Dr. Anis Koubâa 3/ June 2009

4 3. Give the values of the total input traffic at each node 1, 2, 3,. (2 points) Do not forget the unit What is the probability that there is no message in the whole network? (2 points) Compute the response times E S 3 of LAN1 and E S of LAN2? (2 points) E S E S Write the two equations: E S E S [ ] 3 [ ] = sec µ = 10 3 = 7 = 3 3 = sec µ = 10 7 = 3 = 6. Why there is a difference between the response times in both LANs. (1 points) 7. Compute the average number of messages in LAN1. (2 points) Dr. Anis Koubâa / June 2009

5 Simulation (7 points) 8. What are the scenarios that very likely represent the steady state of the network? (2 points) a b c d 9. Justify your choices (2 points) Your Previous Choice of question 8 Write the choice ID: a, b, c or d Its justification Write the ID of the justification in front of it 10. If we want to have a margin of error that does not exceed 0.2 sec for the response time of Router R1, what is the minimum number of messages in one simulation replication that achieves this error bound, assuming that the variance of the response time is equal to 2 σ = 3.5 sec. (2 points) just write the value at the left cell The minimum number of messages is 11. What is the corresponding simulation time (in seconds) that achieves this error bound if the average number of customers that crosses the router is 10 messages/sec? (1 point) The minimum simulation time is Dr. Anis Koubâa 5/ June 2009