CS/ECpE 5516 Spring 1998

Homeworks 4 and 5

(Distributed 9 April, due 21 April 1998 at start of class)

1. Suppose that the arrivals of packets to a switch forms a Poisson distribution with rate 1000 packets/second. Suppose we observe the arrival stream for 10 seconds, which we designate t=0 to t=10. (Hint: one of (a), (b), and (c) is a "trick" question with a trivial solution.)

(a) If we count the number of arrivals in the observation period, what is the probability that exactly 12 packets arrive?

(b) (i) What is the probability that the first arrival in the observation period occurs by t=0.001? (ii) What is the probability that the first two arrivals in the observation period occur by t=1 (the first need not occur by t=0.001)?

2. Suppose that the interarrival times of packets to a local area network is uniformly distributed between 0 and 20 seconds. Suppose we start observing the network when a packet arrives (call this t=0). Suppose we observe the network for 10 seconds and see no packet arrival. (Thus there are no arrivals in interval [0,10].) Derive the expected time at which the next packet will arrive. (Hint: Because the weather is nice outside, and I'm sure you want to soak up a few rays in the sun rather than slaving over 5516 homework, I'll give you the answer: "15." Your job is merely to formally demonstrate that the answer is 15. And don't forget that P[A|B]=P[A«B]/P[B].)

3. Do [BG] 3.13(a). Hint: This problem at first glance appears to require a state representation with 2 components, corresponding to a two dimensional Markov chain. However, we only go over 1 dimensional chains in the class. So you need to find a clever way to map the 2 dimensions onto one dimension. I found the mapping when I tried to solve the problem by thinking hard about all possible 2 dimensional states, and then saw that certain states can never occur.

4. Do [BG] 3.9 (b), both (1) and (2). The answers in the text correspond to TDM. In the textbook's problem statement, change "where five of the sessions transmit" to "where five of the sessions each transmit," and "1.038 sec." to "1.04 sec." Note that "Kbits" means "1000 bits."

5. Do [BG] 4.3 (a) and (b).

Hints:

a) This is straightforward. Just review the equations of the two curves for Figure 4.4 and use the values of q_a and q_r given.

b) This is also straightforward, using the formula for Psucc. (On an exam, I might ask you to calculate the probability that a transmission will succeed, given that all nodes are backlogged -- a very interesting piece of information to know! This quantity is P_succ.)

Last modified on 9 April 1998 by abrams@vt.edu.