Queues Applied to Telecoms

10Chapter 2 Exercises

EXERCISE 2.1.– Ergodic

P is a transition matrix for a Markov chain in discrete time since the sum of the coordinates in each row equals 1. The transition graph of the corresponding chain is thus:

Figure 10.1. Transition graph for Exercise 2.1

We found a power of P for which all of the coordinates are strictly positive. The conditions imposed by Proposition 2.4 are satisfied. Thus, the Markov chain is ergodic.

EXERCISE 2.2.– Upper triangular matrix

P is the transition matrix of a Markov chain in discrete time since the sum of the coordinates of each row equals 1.

In algebra, the product of two upper triangular matrices is an upper triangular matrix. Thus, all the powers of matrix P are upper triangular matrices. Consequently, all matrices Pⁿ have at least one coordinate that is zero, which contradicts the condition from Proposition 2.4.

For the stationary distribution, the solution to the equation π = πP gives the following system of equations:

The solution gives us π = (π₀, π₁, π₂) = (0,0,1), so the stationary distribution exists.

We cannot say anything about the ergodicity of the chain. However, what we have found here is that the existence of the stationary ...