Operator-Geometric Stationary Distributions for Markov Chains, with Application to Queueing Models.

Abstract

This paper considers a class of Markov chains on a bivariate state space (N,E), whose transition probabilities have a particular 'block-partitioned' structure. Examples of such chains include those studied by Neuts (1978) who took E to be finite: they also include chains studied in queueing theory, such as (Nn, Sn) where Nn is the number of customers in a GI/G/1 queue immediately before, and Sn the remaining service time immediately after, the nth arrival. We show that the stationary distribution for these chains has an 'operator-geometric' nature where the operator S is the mininal solution of a non-linear operator equation. Necessary and sufficient conditions for Pi to exist are also found. In the case of the GI/G/1 queueing chain above these are exactly the usual stability conditions.

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Document Details

Document Type
Technical Report
Publication Date
Dec 01, 1980
Accession Number
ADA096030

Entities

People

  • R. L. Tweedie

Organizations

  • University of Delaware

Tags

Communities of Interest

  • Materials and Manufacturing Processes

DTIC Thesaurus Topics

  • Air Force
  • Applied Mathematics
  • Convergence
  • Distribution Functions
  • Equations
  • Geometric Forms
  • Markov Chains
  • Mathematics
  • Probability
  • Queueing Theory
  • Random Variables
  • Random Walk
  • Sequences
  • Stationary
  • Stochastic Processes
  • Theorems
  • Transitions

Fields of Study

  • Mathematics

Readers

  • Mathematical Modeling and Probability Theory.

Technology Areas

  • Space