# Lab 5, due Monday, 29 March 04

General Considerations: Your program code should be placed at \$HOME/CZ3102/lab5, as usual. This is the last lab.

### Q1

[This is problem 5 on page 296, modified.] Consider telephone calls on a network with infinitely many lines. That is, every incoming call can be answered. Suppose that the arrival of calls is Poisson distributed with mean rate L, and the terminations of calls are Poisson points with means rate M, and they are independent processes.

(a) If Pn(t) is the probability that exactly n lines are in use at time t, derive the following dynamics

dPn(t)/dt = - (L+n M)Pn(t) + L Pn-1(t) + (n+1) M Pn+1(t),

the index takes values n = 0, 1, 2, 3, ..., where we define Pn(t) = 0 if n < 0.

(b) Show that the mean number of calls, m, satisfies

dm(t)/dt = L - M m(t)

(c) Solve the set of equations Pn(t) numerically. We use L=2, and M=1, and assuming no phone calls at time t = 0. Since n runs from 0 to infinity, we have to cut-off at some finite n, say n = 6. Make a plot of Pn(t) vs t. Check that sum of Pn(t) over n for any t is 1 approximately, but why?

(d) [Difficult] Compute the same Pn(t) by simulation, i.e., generate the random process using random number (Poisson processes) and samples the number of calls. You have to simulate many times (say 10000) to get good statistics.