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)