**CDA6530: ****Performance Models of Computers and Networks**

**Fall 2012**

**Homework 1: Probability and Random Variables
**

(assigned 08/30; due: 09/09 midnight, submitted via Webcourse)

**1. **Bill and George go target shooting together.
Both shoot at a target at the same time. Suppose Bill hits the
target with prob. 0.7, George, independently, hits the target
with prob. 0.4.

a). Now both of them shoot once and exactly one shot hits the
target. What is the prob. that it was George's shot?

b). Now both of them shoot once and the target is hit. What is
the prob. that George hit it?

2. A gambler has in his pocket a fair coin and a
two-headed coin. He selects one of the coins at a random, and
when he flips it, it shows heads.

a). What is the probability that it is a fair coin?

b). Suppose that he flips the same coin a second time and again
it shows heads. Now what is the probability that it is a fair
coin?

3. An airline knows that 5% of the people making
reservations on a certain flight will not show up. Consequently,
their policy is to sell 52 tickets for a flight that can only
hold 50 passengers. What is the probability that there will be a
seat available for every passenger who show up?

4. Consider three trails (they may not be
independent), each of which is either a success or not. Let X
denote the number of successes. Suppose E[X]=1.8

a). What is the largest possible value of P(X=3)?

b). What is the smallest possible value of P(X=3)?

5. Consider the networks shown in the figure
below. Assume computers in the institutional network send out 13
requests per second to the "origin servers" shown in the figure.
Since each request is very small, we assume there is no delay in
sending out these requests and they take 1 second to arrive at
the "orginal servers" and web response data take another 1
second to come back at the Internet-side router on the "access
link". Each web response object average size is 100kbits.

Now congestion could happen at the access link when web response objects come back (we assume there is no delay in the institutional network LAN). By using M/M/1 queue to model the access delay in the 1.5Mbps access link, the average response time (it is the time for a web response object arrives at the access link until it enters into institutional LAN) is E[T]=1/(m-n), where n is the arrival rate of objects to the access link (i.e., how many webpage contents go to the link per second) and m is the service rate of the access link (i.e., how many webpage contents can be served to pass the link per second).

a). Find the total average response time when no institutional cache is used. The response time is the time from a browser sends out a request until it receives the web response object. (Hint: total delay includes Internet delay and access link delay)

b). Now suppose the
institutional cache is used. The hit rate for the cache is 0.6. Find
the total average response time.

6. The advantage of packet switching vs. circuit
switching. The following figure shows that many users are
sharing a 1Mbps access link to the Internet. Suppose each user
is either in active status that required data access rate of
100kbps, or in silence status that the user requires no
data. Each user is active only 10% of the time , and users
are independent with each other in their activities.

a). If circuit switching is used, how many user can this access link support?

b). If packet switching is used and there are 10 users. How often is this access link idle? How often does this link serve exactly two users?

b). If packet switching is used and there are 35 users, how often does the access link be congested? (Hint: the link is congested if and only if there are more than 10 users be active at the same time).