CSC 237 - Spring 2000

Michalis Faloutsos

Warm up: Answer the problems from the texbook:

1) 31.1

2) 31.2

3) 31.3

(Hint: the solutions are at the back of the book.
However,

you have to show how they are DERIVED.)

Main course:

4) I have two umbrellas. Everyday, in the morning, I go from home to
school and in the evening from school home. If it rains I will take an
umbrella with me. If there is no umbrella, I get wet. Assume that
the probability of rain in the morning and that of rain in the evening
are different but constant.

a) What is the steady state probability of the system or where
are my umbrellas?

b) What is the probability that it rains and I don't have an umbrella?

5) Find the probability distribution p_n of a birth-death
system with one queue and infinite buffer space and the parameters:

lambda_n = a^n * lambda,
n => 0, 0< a < 1

mu_n = mu,
n => 1

6) Consider the discrete-state Markov chain with continuous time
with n=0,1,2,3 states and transition rates:

P_01 = lambda

P_12 = lambda

P_23 = lambda

P_10 = mu

P_20 = mu

P_30 = mu

and no other transitions (ie P_21 = 0)

a) Is this a birth-death system?

b) Find the transition rate matrix Q

c) Write and solve the equilibrium equations (see class notes)

d) Find the average number of jobs in the system (assuming that state
n has n jobs)