All Dissertations

8-2016

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Committee Member

Dr. Brian Fralix, Committee Chair

Committee Member

Dr. Peter Kiessler

Dr. Xin Liu

Dr. Robert Lund

Abstract

In the past several decades, matrix analytic methods have proven eﬀective at studying two important sub-classes of block-structured Markov processes: G/M/1-type Markov processes and M/G/1-type Markov processes. These processes are often used to model many types of random phenomena due to their underlying primitives having phase-type distributions. When studying block-structured Markov processes and its sub-classes, two key quantities are the “rate matrix” R and a matrix of probabilities typically denoted G. In [30], Neuts shows that the stationary distribution of a Markov process of G/M/1-type, when it exists, possess a matrix-geometric relationship with R. Ramaswami’s formula [32] shows that the stationary distribution of an M/G/1-type Markov process satisﬁes a recursion involving a well-deﬁned matrix of probabilities, typically denoted as G. The ﬁrst result we present is a new derivation of the stationary distribution for Markov processes of G/M/1-type using the random-product theory found in Buckingham and Fralix [9]. This method can also be modiﬁed to derive the Laplace transform of each transition function associated with a G/M/1-type Markov process. Next, we study the time-dependent behavior of block-structured Markov processes. In [15], Grassmann and Heyman show that the stationary distribution of block-structured Markov processes can be expressed in terms of inﬁnitely many R and G matrices. We show that the Laplace transforms of the transition functions associated with block-structured Markov processes satisﬁes a recursion involving an inﬁnite collection of R matrices. The R matrices are shown to be able to be expressed in terms of an inﬁnite collection of G matrices, which are solutions to ﬁxed-point equations and can be computed iteratively. Our ﬁnal result uses the random-product theory to a study an M/M/1 queueing model in a two state random environment. Though such a model is a block-structured Markov process, we avoid computing any R or G matrices and instead show that the stationary distribution can be written exactly as a linear combination of scalars that can be determined recursively.

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