A Markovian canonical form of second-order matrix-exponential processes

L. Bodrog, A. Heindl, G. Horváth, M. Telek

Research output: Contribution to journalArticle

37 Citations (Scopus)

Abstract

Besides the fact that - by definition - matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that - in two dimensions - the respective sets, MAP(2) and MEP(2), are indeed identical with respect to the stationary behavior. Furthermore, this equivalence extends to acyclic MAPs, i.e., AMAP(2), so that AMAP (2) ≡ MAP (2) ≡ MEP (2). For higher orders, these equivalences do not hold. The second-order equivalence is established via a novel canonical form for the (correlated) processes. An explicit moment/correlation-matching procedure to construct the canonical form from the first three moments of the interarrival time distribution and the lag-1 correlation coefficient shows how these compact processes may conveniently serve as input models for arrival/service processes in applications.

Original languageEnglish
Pages (from-to)459-477
Number of pages19
JournalEuropean Journal of Operational Research
Volume190
Issue number2
DOIs
Publication statusPublished - Oct 16 2008

Fingerprint

Matrix Exponential
Canonical form
Markovian Arrival Process
Equivalence
equivalence
Moment
Correlation coefficient
Notation
Two Dimensions
Higher Order

Keywords

  • Canonical representation
  • Markovian arrival process
  • Matrix-exponential process
  • Moment/correlation matching

ASJC Scopus subject areas

  • Information Systems and Management
  • Management Science and Operations Research
  • Statistics, Probability and Uncertainty
  • Applied Mathematics
  • Modelling and Simulation
  • Transportation

Cite this

A Markovian canonical form of second-order matrix-exponential processes. / Bodrog, L.; Heindl, A.; Horváth, G.; Telek, M.

In: European Journal of Operational Research, Vol. 190, No. 2, 16.10.2008, p. 459-477.

Research output: Contribution to journalArticle

Bodrog, L. ; Heindl, A. ; Horváth, G. ; Telek, M. / A Markovian canonical form of second-order matrix-exponential processes. In: European Journal of Operational Research. 2008 ; Vol. 190, No. 2. pp. 459-477.
@article{793cd877750743eaaddb521bda3be383,
title = "A Markovian canonical form of second-order matrix-exponential processes",
abstract = "Besides the fact that - by definition - matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that - in two dimensions - the respective sets, MAP(2) and MEP(2), are indeed identical with respect to the stationary behavior. Furthermore, this equivalence extends to acyclic MAPs, i.e., AMAP(2), so that AMAP (2) ≡ MAP (2) ≡ MEP (2). For higher orders, these equivalences do not hold. The second-order equivalence is established via a novel canonical form for the (correlated) processes. An explicit moment/correlation-matching procedure to construct the canonical form from the first three moments of the interarrival time distribution and the lag-1 correlation coefficient shows how these compact processes may conveniently serve as input models for arrival/service processes in applications.",
keywords = "Canonical representation, Markovian arrival process, Matrix-exponential process, Moment/correlation matching",
author = "L. Bodrog and A. Heindl and G. Horv{\'a}th and M. Telek",
year = "2008",
month = "10",
day = "16",
doi = "10.1016/j.ejor.2007.06.020",
language = "English",
volume = "190",
pages = "459--477",
journal = "European Journal of Operational Research",
issn = "0377-2217",
publisher = "Elsevier",
number = "2",

}

TY - JOUR

T1 - A Markovian canonical form of second-order matrix-exponential processes

AU - Bodrog, L.

AU - Heindl, A.

AU - Horváth, G.

AU - Telek, M.

PY - 2008/10/16

Y1 - 2008/10/16

N2 - Besides the fact that - by definition - matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that - in two dimensions - the respective sets, MAP(2) and MEP(2), are indeed identical with respect to the stationary behavior. Furthermore, this equivalence extends to acyclic MAPs, i.e., AMAP(2), so that AMAP (2) ≡ MAP (2) ≡ MEP (2). For higher orders, these equivalences do not hold. The second-order equivalence is established via a novel canonical form for the (correlated) processes. An explicit moment/correlation-matching procedure to construct the canonical form from the first three moments of the interarrival time distribution and the lag-1 correlation coefficient shows how these compact processes may conveniently serve as input models for arrival/service processes in applications.

AB - Besides the fact that - by definition - matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that - in two dimensions - the respective sets, MAP(2) and MEP(2), are indeed identical with respect to the stationary behavior. Furthermore, this equivalence extends to acyclic MAPs, i.e., AMAP(2), so that AMAP (2) ≡ MAP (2) ≡ MEP (2). For higher orders, these equivalences do not hold. The second-order equivalence is established via a novel canonical form for the (correlated) processes. An explicit moment/correlation-matching procedure to construct the canonical form from the first three moments of the interarrival time distribution and the lag-1 correlation coefficient shows how these compact processes may conveniently serve as input models for arrival/service processes in applications.

KW - Canonical representation

KW - Markovian arrival process

KW - Matrix-exponential process

KW - Moment/correlation matching

UR - http://www.scopus.com/inward/record.url?scp=41749104465&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41749104465&partnerID=8YFLogxK

U2 - 10.1016/j.ejor.2007.06.020

DO - 10.1016/j.ejor.2007.06.020

M3 - Article

AN - SCOPUS:41749104465

VL - 190

SP - 459

EP - 477

JO - European Journal of Operational Research

JF - European Journal of Operational Research

SN - 0377-2217

IS - 2

ER -