## Download e-book for iPad: An introduction to queueing theory and matrix-analytic by L. Breuer, Dieter Baum

By L. Breuer, Dieter Baum

ISBN-10: 1402036302

ISBN-13: 9781402036309

ISBN-10: 1402036310

ISBN-13: 9781402036316

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

**Example text**

Hence πP = π implies first π0 · (1 − p) = π0 ⇒ π0 = 0 since 0 < p < 1. Assume that πn = 0 for any n ∈ N0 . This and the condition πP = π further imply for πn+1 πn · p + πn+1 · (1 − p) = πn+1 ⇒ πn+1 = 0 22 AN INTRODUCTION TO QUEUEING THEORY which completes an induction argument proving πn = 0 for all n ∈ N0 . Hence the Bernoulli process does not have a stationary distribution. 21 The solution of πP = π and P = j∈E πj = 1 is unique for 1−p p p 1−p with 0 < p < 1. Thus there are transition matrices which have exactly one stationary distribution.

1) is given by ✛✘ ✛✘ ✛ ✛ ✛✘ λ λ ✲ ✲ ✲ ✚✙ ✚✙ ✚✙ ✚✙ ✚✙ ... 2. 6 The transition probabilities Pij (t) of a Markov process satisfy the systems dP Pij (t) = Pik (t)gkj = gik Pkj (t) dt k∈E k∈E of differential equations. These are called the Kolmogorov forward and backward equations. 4, it follows by induction on the number of jumps that all restricted probabilities P (n) (t) are Lebesgue inte(n) grable with respect to t over finite intervals. Since the sum of all Pij (t) is a probability and thus bounded, we conclude by majorized convergence that also P (t) is Lebesgue integrable with respect to t over finite intervals.

The (M +1) (M ) probability for {ττj − τj = ∞} is 1 − fij for M = 0 and 1 − fjj for M > 0. Once more the strong Markov property is the reason for independence of the subevents. Now multiplication of the probabilities leads to the formulae in the statement. e. depending on fjj there are almost certainly infinitely many visits to a state j ∈ E. This result gives rise to the following definitions: A state j ∈ E is called recurrent if fjj = 1 and transient otherwise. Let us further define the potential matrix R = (rij )i,j∈E of the Markov chain by its entries rij := E(N Nj |X0 = i) for all i, j ∈ E.

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

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