Important System and Modeling examination quesitons.

SET A

1. Giving examples differentiate between deterministic and stochastic models.
2. 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. 
3. Inventory is withdrawn from a stock of 60 items according to a poisson distribution at the rate of 4 items a day.
• Find the probability that 10 items are withdrawn during the first 2 days. 
• Determine the probability that no items are left at the end of 4 days.
• Determine the average number of items withdrawn over a 4 – day period.
4. 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. 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. Why is simulation important in performance modeling?

SET B

1. List the four desired properties of a random number generator . 
2. 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):-

• Determine the arrival and service rate of the jobs in minutes
• Determine the interarrival and service times for the first six jobs using the first six random numbers in column 1 of table A.1
• Hand simulate the problem to determine the average time each job waits in queue before being processed (assume the system starts at time zero)

SET C

1.  What is a model and why is modeling important?
   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
   • What is the probability a request has to wait in queue before being processed?
   • What is the average service time per transaction?
   • What is the probability there are more than one http request in the system?
2. 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?

SET D

1. Briefly compare the advantages and disadvantages of the analytical modeling and
   the discrete event simulation modeling as applied in queueing system.
2. Using the random numbers 0.6505, 0.3262 and 0.1646, sample from an
   exponential distribution with mean of 6 (6 marks)
   • Find a suitable c for f (x) .
   • Find the probability that the random variable x has a value more than 0.5.
   •  Give an Inverse Transform algorithm for generating x. 

1. 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
2. What is the confidence interval with a 95% confidence level for the mean response time?
3. 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).