By L. Breuer, Dieter Baum

ISBN-10: 1402036302

ISBN-13: 9781402036309

ISBN-10: 1402036310

ISBN-13: 9781402036316

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**Extra info for An introduction to queueing theory and matrix-analytic methods**

**Example text**

Hence the measure ˇ > 0, the measure µ is finite, with total µ := νΛ−1 is stationary for Y. Since λ −1 ˇ mass bounded by λ < ∞. 5) for all j ∈ E yields a stationary distribution for Y. This is unique because ν is unique and the construction of π from ν is reversible. Finally we give two important results for the asymptotic behaviour of a Markov process. 13). We call a Markov process regular if it satisfies the conditions given in the preceding theorem. 6) of the marginal distribution at time t tends to the stationary distribution as t tends to infinity.

Ytπn +s = jn ) for all times t1 < . . < tn with n ∈ N, and states j1 , . . , jn ∈ E. Hence the process Y π is stationary. On the other hand, if π is a stationary distribution, then we necessarily obtain tn n πP (t) = πeG·t = π for all t ≥ 0. As above, this means ∞ n=1 n! πG = 0 for all t ≥ 0, which yields πG = 0 because of the uniqueness of the zero power series. 4) i=j for all j ∈ E. This system can be intepreted as follows. We call the value πi gij stochastic flow from state i to state j in equilibrium.

The balance equations for the Poisson process are given by π0 λ = 0 πi λ = πi−1 λ and for all i ≥ 1. It is immediately evident that these are solvable only by πi = 0 for all i ∈ E, which means that there is no stationary distribution π. 11 Let the underlying Markov chain X in the definition of the Markov process Y be irreducible and positive recurrent. Further assume that ˇ := inf{λi : i ∈ E} > 0. Then there is a unique stationary distribution for λ Y. 18, the transition matrix P of X admits a unique stationary distribution ν with νP = ν.

### An introduction to queueing theory and matrix-analytic methods by L. Breuer, Dieter Baum

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