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Introduction to Matrix Analytic Methods in Queues 1
book

Introduction to Matrix Analytic Methods in Queues 1

by Srinivas R. Chakravarthy
September 2022
Intermediate to advanced content levelIntermediate to advanced
368 pages
8h 4m
English
Wiley-ISTE
Content preview from Introduction to Matrix Analytic Methods in Queues 1

8Matrix-Analytic Methods (Continuous-time)

In this chapter, we will look at the two paradigms as well as the quasi-birth-and-death (QBD) process from the continuous-time point of view. This approach is key when one is interested in studying queueing models from the arbitrary time point of view. This topic is discussed in Volume 2 (see Chakravarthy (2022b)).

8.1. M/G/1-type (scalar case)

Suppose that the generator of the continuous-time Markov chain (CTMC) is of the M/G/1-typein the scalar case. That is, we have a CTMC on the state space {0, 1, …} with the generator of the form:

[8.1] Image

where the (scalars) ck, k ≥ 0, and hk, k ≥ 0, are such that:

[8.2] Image

Define:

[8.3] Image

Letting:

[8.4] Image

we see:

[8.5] Image

which is the transition probability matrix (TPM) of the M/G/1-type for the scalar case as discussed in section 7.1.

Assuming that Q1 is irreducible (which implies that P1 is also irreducible), the conditions for positive recurrence for Q1 are obtained based on the positive recurrence of P

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Introduction to Matrix-Analytic Methods in Queues 2

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Publisher Resources

ISBN: 9781786307323Purchase Link