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]

where the (scalars) c_k, k ≥ 0, and h_k, k ≥ 0, are such that:

[8.2]

Define:

[8.3]

Letting:

[8.4]

we see:

[8.5]

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