Many problems in computing, service, and manufacturing systems can be modeled via infinite repeating Markov chains with an infinite number of levels and a finite number of phases. Many such chains are quasi-birth-death processes (QBDs) with transitions that are skip-free in level, in that one can only transition between consecutive levels, and unidirectional in phase, in that one can only transition from lower-numbered phases to higher-numbered phases. We present a procedure, which we call Clearing Analysis on Phases (CAP), for determining the limiting probabilities of such Markov chains exactly. The CAP method yields the limiting probability of each state in the repeating portion of the chain as a linear combination of scalar bases raised to a power corresponding to the level of the state. The weights in these linear combinations can be determined by solving a finite system of linear equations.

Clearing Analysis on Phases: Exact Limiting Probabilities for Skip-free, Unidirectional, Quasi-birth-death Processes

Analyzing M/G/1 paradigmsthrough QBDs: the role of the block structure in computing the matrixG

A conditional probability approach to M/G/1-like queues

A variety of problems in computing, service, and manufacturing systems can be modeled via infinite repeating Markov chains with an infinite number of levels and a finite number of phases. Many such chains are quasi-birth-death processes with transitions that are skip-free in level, in that one can only transition between consecutive levels, and unidirectional in phase, in that one can only transition from lower-numbered phases to higher-numbered phases. We present a procedure, which we call Clearing Analysis on Phases (CAP), for determining the limiting probabilities of such Markov chains exactly. The CAP method yields the limiting probability of each state in the repeating portion of the chain as a linear combination of scalar bases raised to a power corresponding to the level of the state. The weights in these linear combinations can be determined by solving a finite system of linear equations.

/pdf/clearing-analysis-on-phases-exact-limiting-probabilities-for-5742q5dihd.pdf

Clearing Analysis on Phases: Exact Limiting Probabilities for Skip-Free, Unidirectional, Quasi-Birth-Death Processes

We describe the ETAQA (efficient technique for the solution of quasi birth-death processes) approach for the exact analysis of M/G/1 and GI/M/1-type processes, and their intersection, i.e., quasi birth-death processes. ETAQA exploits the repetitive structure of the infinite portion of the chain and derives a finite system of linear equations. In contrast to the classic techniques for solution of such systems, the solution of this finite linear system does not provide the entire probability distribution of the state space, but simply allows calculation of the aggregate probability of a finite set of classes of states from the state space, appropriately defined. Nonetheless, these aggregate probabilities allow for computation of a rich set of measures of interest such as the system queue length or any of its higher moments. The proposed solution approach is exact and, for the case of M/G/1-type processes, compares favorably to the classic methods as shown by detailed time and space complexity analysis. Detailed experimentation further corroborates that ETAQA provides significantly less expensive solutions when compared to the classic methods.

/pdf/etaqa-solutions-for-infinite-markov-processes-with-5uay71q51x.pdf

ETAQA Solutions for Infinite Markov Processes with Repetitive Structure

http://www.cs.wm.edu/~andreas/publications/newEtaqa.pdf

Zhili Hua

Papers

Bridging ETAQA and Ramaswami's formula for the solution of M/G/1-type processes