#### Date of Award

8-2016

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Legacy Department

Mathematical Science

#### Committee Member

Dr. Brian Fralix, Committee Chair

#### Committee Member

Dr. Peter Kiessler

#### Committee Member

Dr. Xin Liu

#### Committee Member

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.

#### Recommended Citation

Joyner, Jason, "A New Look at Matrix Analytic Methods" (2016). *All Dissertations*. 1724.

https://tigerprints.clemson.edu/all_dissertations/1724