By L. Breuer, Dieter Baum

ISBN-10: 1402036302

ISBN-13: 9781402036309

The textbook includes the documents of a two-semester direction on queueing conception, together with an advent to matrix-analytic tools. The direction is directed to final yr undergraduate and primary yr graduate scholars of utilized chance and computing device technological know-how, who've already accomplished an advent to likelihood idea. Its objective is to provide fabric that's shut adequate to concrete queueing types and their purposes, whereas offering a legitimate mathematical beginning for his or her research. A in demand a part of the ebook may be dedicated to matrix-analytic equipment. it is a number of ways which expand the applicability of Markov renewal the right way to queueing concept via introducing a finite variety of auxiliary states. For the embedded Markov chains this ends up in transition matrices in block shape similar to the constitution of classical versions. Matrix-analytic tools became rather renowned in queueing conception over the last 20 years. The goal to incorporate those in a scholars' advent to queueing concept has been the most motivation for the authors to write down the current publication. Its goal is a presentation of an important matrix-analytic suggestions like phase-type distributions, Markovian arrival strategies, the GI/PH/1 and BMAP/G/1 queues in addition to QBDs and discrete time techniques.

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**Additional resources for An Introduction to Queueing Theory: and Matrix-Analytic Methods **

**Example text**

An exhaustive introduction to Markov chains on general state spaces and conditions for their positive recurrence is given in Meyn and Tweedie [59]. 1 Let (X Xn : n ∈ N0 ) be a family of iid random variables with discrete state space. Show that X = (X Xn : n ∈ N0 ) is a homogeneous Markov chain. 2 Let (X Xn : n ∈ N0 ) be iid random variables on N0 with probabilities ai := P(X Xn = i) for all n, i ∈ N0 . The event Xn > max(X0 , . . , Xn−1 ) for n ≥ 1 is called a record at time n. e. T0 := 0 and Ti+1 := min{n ∈ N : Xn > XTi } for all i ∈ N0 .

Proof: Assume that i ∈ E is transient and i ↔ j. Then there are numbers m, n ∈ N with 0 < P m (i, j) ≤ 1 and 0 < P n (j, i) ≤ 1. 7). 15 this means that j is transient, too. If j is recurrent, then the same inequalities lead to rii ≥ P m (i, j)P n (j, i)rjj = ∞ which signifies that i is recurrent, too. Since the above arguments are symmetric in i and j, the proof of the first statement is complete. For the second statement assume that i ∈ E belongs to a communication class C ⊂ E and pij > 0 for some state j ∈ E \ C.

N ≤ ∆t · ˆ · t)n (2λ n! which in the limit n → ∞ yields 0 ≤ P (t) − P˜ (t) ≤ 0 and consequently P (t) = P˜ (t). As t has been chosen arbitrarily, the statement is proven. Hence the generator of a Markov process uniquely determines all its transition matrices. This can also be seen from the definition, if we agree (without loss 46 AN INTRODUCTION TO QUEUEING THEORY of generality) upon the convention pii = 0 for all ∈ E. Then the parameters for the definition of the Markov process can be recovered by λi = −gii and pij = gij −gii for all i = j ∈ E.

### An Introduction to Queueing Theory: and Matrix-Analytic Methods by L. Breuer, Dieter Baum

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