Operator-geometric stationary distributions for markov chains, with application to queueing models

Richard L. Tweedie

98 citations0/0 files storedAdvances in Applied Probability

No readable copy stored yet — this record is metadata only.

No open-access location is known for this record yet.

Literature note Cite in a project Citation map

Advanced Queuing Theory Analysis Probability and Statistical Research Stochastic processes and statistical mechanics

Abstract

This paper considers a class of Markov chains on a bivariate state space , whose transition probabilities have a particular ‘block-partitioned' structure. Examples of such chains include those studied by Neuts [8] 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 Πfor these chains has an ‘operator-geometric' nature, with , where the operator S is the minimal solution of a non-linear operator equation. Necessary and sufficient conditions for Πto exist are also found. In the case of the GI/G/1 queueing chain above these are exactly the usual stability conditions. G1/G/1 QUEUE; PHASE-TYPE; INVARIANT MEASURE; FOSTER'S CONDITIONS

Paper brief

Loading…

Cite this work

Citation unavailable for this work.

BibTeX RIS CSL-JSON

No public copy is stored for this record, so only the metadata is cited.

Citations in this library

Citation map →

Stored only when both papers are in the corpus — 98 citations exist upstream.

References

1978Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector

Cited by

1996Some properties of the rate perators in level dependent uasi-birth-and-death processes with countable number of phases 2007Stability of Parallel Queueing Systems with Coupled Service Rates 1984Matrix-analytic methods in queuing theory 1986A general class of Markov processes with explicit matrix-geometric solutions 2004Transient Analysis of Fluid Flow Models via Stochastic Coupling to a Queue 2009Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks 2004Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process 2010Markovian arrivals in stochastic modelling: a survey and some new results

Explore the map →

Papers that cite the same references — the relational fallback while embeddings are still backfilling.

Imported from OpenAlex (CC0) — nothing here is generated.