CSE 221: Probabilistic Analysis of Computer Systems

Spring 2008

Swapna S. Gokhale

Homework #6

Date:April 21, 2008

1. To determine the probability that a manufacturing plant produces a defective chip, a sample of 100 chips were observed. It was noted that of these 100 chips, 90 chips were functionally correct. Estimate the probability that the manufacturing plant produces a defective chip.

2. The probability that a manufacturing plant produces a defective chip can be better estimated by considering multiple samples. 8 such samples were considered, with 100 chips in each sample. The number of defective chips in each one of the 8 samples is as follows: 5, 6, 16, 7, 2, 3, 10 and 4. Obtain the maximum likelihood estimate of the probability that a chip produced by the plant is defective.

3. Consider a wireless channel, where a single bit is transmitted as many times as it is necessary for it to be received correctly. To estimate the probability of successful transmission, the following experiment was conducted – 10 bits were transmitted and the numbers of retransmissions needed for each bit to be received correctly were recorded. For each bit, the number of retransmissions needed were as follows: 9, 8, 3, 7, 6, 5, 10, 4, 1, 2. Compute the probability of successful transmission.

4. Consider a web server which serves customer requests for static web pages. In a given 24-hour period, the web server receives 400 requests in the peak duration from 9:00 am to 5:00 pm. In the off-peak period, between 5:00 pm to 9:00 am, the web server receives another 400 requests. Assume that the customer requests arrive at the web server according to a Poisson process both during the peak and the off-peak periods, estimate the effective/average arrival rate as viewed by the web server.

5. The time to failure (in days) of 8 workstations are as follows: 2, 0.5, 3, 4, 6, 0.3, 3.5, 4.5. Assume that the time to failure follows an exponential distribution, compute the mean time to failure and the failure rate of the workstations.

6. Consider a server modeled as a three-state homogeneous discrete time Markov chain (DTMC). The states are indexed 1, 2 and 3 and denote the server in busy, idle and failed state respectively. The states of the server are recorded at 21 successive time instants and the recorded sequence is: 1 2 3 3 2 1 1 2 2 3 2 3 1 3 2 3 1 2 3 1 2. Find the maximum likelihood estimates of the transition probability matrix of the DTMC.

7. The average execution of a program is normally distributed with mean and variance sec. The program was tested 49 times and its mean execution time was computed to be 27 sec. Compute the 99% confidence interval for the mean execution time of the program.