Course Documents Authorized Duration: 120 minutes. Exercise 1. Monte Carlo Simulation and Confidence Interval (5 points)

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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 اإلسم الثالثي:... المستوى الرقم الجامعي النتيجة: 40/ Final Exam Modeling and Simulation (CS-433) Course Documents Authorized Duration: 120 minutes Exercise 1. Monte Carlo Simulation and Confidence Interval (5 points) We have used a Monte Carlo simulation method to estimate the area of a circle, with radius equal to 1, using 800 random numbers for generating 400 points (X, Y). We have found that 308 random points are inside the circle. a. Compute the estimation of the circle area and the estimation of. (1 points) b. Compute the 90% confidence interval for this estimated value assuming that the standard deviation is equal to (1 points) c. Compute the 99% confidence interval for this estimated for the same standard deviation (1 points) Dr. Anis Koubâa 1/6

2 d. Interpret ( (فس ر the previous result. (2 points) Exercise 2. Simulation and Point Estimators (8 points) A network engineer has made a simulation study on the network of the University. He wants to evaluate the average delay to transmit packets from the queue of a router. The engineer makes ONE simulation run that sends N = 100 packets according to a Poisson distribution λ 2 / and computes the delay of each packet. He finds that the average delay obtained from this simulation is 102. a. Does this is value 102 give a precise estimation of the average delay? Explain. (1 points) b. Can you have any confidence on this average delay? On which parameters does the observed average delay depend to have a confidence on it? (2 points) c. The engineer computes the standard deviation of the average delay from the output sample and finds that σ 4. Determine the margin of error for this point estimator obtained from this simulation. (1 points) d. What is the confidence level on the error that found in the previous question? (1 points) Dr. Anis Koubâa 2/6 January 2008

3 Using a computer program, the engineer makes 500 simulation runs and in each simulation he computes the average value where What will be the distribution of the average value? Give the expressions of the mean and the variance of. (3 points) e. What is the distribution of the inter-arrival time of packets? (1 points) Exercise 3. Experimental Analysis and Sample Size (10 points) We consider the following ping application on the website We have obtained the following output: ping -n 10 Pinging [ ] with 32 bytes of data: Reply from : bytes=32 time=263ms TTL=52 Reply from : bytes=32 time=286ms TTL=52 Reply from : bytes=32 time=308ms TTL=52 Reply from : bytes=32 time=229ms TTL=52 Reply from : bytes=32 time=252ms TTL=52 Reply from : bytes=32 time=276ms TTL=52 Reply from : bytes=32 time=299ms TTL=52 Reply from : bytes=32 time=220ms TTL=52 Reply from : bytes=32 time=243ms TTL=52 Reply from : bytes=32 time=265ms TTL=52 We would like to study the round trip time from a computer to the Yahoo! Website. Observe column time in the figure (in grey color). As you can see, time is a random number. 1. Compute the mean (average) round trip time? (1 point) 2. Compute the standard deviation? (1 point) 3. Compute the margin of error for 95% confidence interval? (1 point) 4. Compute the minimum sample size if we want that the estimate error be less than 5 ms. Use the value of the standard deviation that you have found in Question 2. (2 point) Dr. Anis Koubâa 3/6 January 2008

4 Now, we would like to compare the round trip time of Yahoo! Website with Google website. Thus, we send ping packets to the website and we get the following output. We find that the mean (average) round trip time to Google website is 260 ms and the standard deviation is ms obtained for n=30 observed packets. 5. If we assume that the average round trip time is normally distributed, give an estimatee of the difference of the mean delays between Yahoo! and Google websites with 90% confidence interval. (1 point) 6. Do you think that the confidence on the difference of mean delays is high? Explain the reasons. (2 points). 7. Is Yahoo! Website faster to respond than Google website or not? Explain. (2 points) Problem: Aircraft Simulation (17 Points) Consider the aircraft simulation problem as shown in the following figure. Figure 1. Aircraft Simulation Model We assume that aircrafts arrive according to Poisson process. Each aircraft that arrives must wait for the others that already came before. When it gets its turn for landing, it spends a landing time exponentially distributed. We denote by Ai the i th aircraft. Consider the following simulation events: Dr. Anis Koubâa 4/6 January 2008

5 Aircraft Arrival Time Landing Duration Parking Duration A A A This indicates that the first aircraft arrives at time t=1. It spend 5 time units to land (so, it will be landed a time t=6). It then spends 8 time units in the parking (so the departure will be on time t=6+8=14). The same interpretation holds for the other aircraft. 1. Complete the following simulation table. (5 points) Event Time Event Name InTheAir OnTheGround RunwayFree 0 Nothing 0 0 TRUE 1 A1 Arrived 1 0 FALSE 4 A2 Arrived 2. We define the delay and the waiting time before landing plus the landing duration. Delay = waiting time + landing duration. From the previous table, compute the delay of each aircraft. (3 point) Delay Aircraft A1 = Delay Aircraft A2 = Delay Aircraft A3 = Average Delay = Standard Variation = 3. In this simulation, we have used a random number generator RNG that generates a random number u uniformly distributed between 0 and 1. However, the inter-arrival time between two consecutive aircraft arrivals follows an exponential distribution with a mean µ= 30 minutes. Dr. Anis Koubâa 5/6 January 2008

6 Explain how can we generate a random number exponentially distributed with mean µ= 30 using the RNG (0,1)? (2 points) 4. We have generated 100 aircraft arrivals using this random number generator. The observed and theoretical frequencies of the number of arrivals in a time unit are presented in the following table. Note that the expected frequency corresponds to a Poisson distribution. Complete the following table (2 points). Number of Arrivals Observed Frequency Oi Expected Frequency Ei ² Total Total ² 5. Make the chi-square test to decide if the two distributions are equivalent or not, assuming that χ²(0.05, 6) = χ² critical = (2 points) 6. Look at the table and determine what the mean value of number of arrival per time unit is. (2 points) Dr. Anis Koubâa 6/6 January 2008