Analysis of multivariate Markov modulated Poisson processes

Ushio Sumita; Yasushi Masuda

doi:10.1016/0167-6377(92)90020-4

Analysis of multivariate Markov modulated Poisson processes

Ushio Sumita, Yasushi Masuda

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

A multivariate Markov modulated Poisson process M(t) = [M₁(t),...,M_K(t)] governed by a Markov chain {J(t):t ≥ 0} on N = {0, 1,...,N} is introduced where jumps of M_k(t) occur according to a Poisson process with intensity λ(k, i) whenever the Markov chain J(t) is in state i, 1 ≤ k ≤ K, 0 ≤ i ≤ N. Of interest to the paper is the time-dependent joint distribution of the multivariate process [M(t), J(t)]. In particular, the Laplace transform generating function is explicitly derived and its probabilistic interpretation is given. Asymptotic expansions of the cross moments and covariance functions of M(t) are also discussed.

Original language	English
Pages (from-to)	37-45
Number of pages	9
Journal	Operations Research Letters
Volume	12
Issue number	1
DOIs	https://doi.org/10.1016/0167-6377(92)90020-4
Publication status	Published - 1992 Jul
Externally published	Yes

Keywords

asymptotic analysis
covariance functions
multivariate Markov modulated Poisson processes
probability generating function
time-dependent joint distribution

ASJC Scopus subject areas

Software
Management Science and Operations Research
Industrial and Manufacturing Engineering
Applied Mathematics

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10.1016/0167-6377(92)90020-4

Cite this

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title = "Analysis of multivariate Markov modulated Poisson processes",

abstract = "A multivariate Markov modulated Poisson process M(t) = [M1(t),...,MK(t)] governed by a Markov chain {J(t):t ≥ 0} on N = {0, 1,...,N} is introduced where jumps of Mk(t) occur according to a Poisson process with intensity λ(k, i) whenever the Markov chain J(t) is in state i, 1 ≤ k ≤ K, 0 ≤ i ≤ N. Of interest to the paper is the time-dependent joint distribution of the multivariate process [M(t), J(t)]. In particular, the Laplace transform generating function is explicitly derived and its probabilistic interpretation is given. Asymptotic expansions of the cross moments and covariance functions of M(t) are also discussed.",

keywords = "asymptotic analysis, covariance functions, multivariate Markov modulated Poisson processes, probability generating function, time-dependent joint distribution",

author = "Ushio Sumita and Yasushi Masuda",

note = "Funding Information: * This work has been partially supported by Nippon Telegraph and Telephone Corporation Research Fund.",

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doi = "10.1016/0167-6377(92)90020-4",

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journal = "Operations Research Letters",

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TY - JOUR

T1 - Analysis of multivariate Markov modulated Poisson processes

AU - Sumita, Ushio

AU - Masuda, Yasushi

N1 - Funding Information: * This work has been partially supported by Nippon Telegraph and Telephone Corporation Research Fund.

PY - 1992/7

Y1 - 1992/7

N2 - A multivariate Markov modulated Poisson process M(t) = [M1(t),...,MK(t)] governed by a Markov chain {J(t):t ≥ 0} on N = {0, 1,...,N} is introduced where jumps of Mk(t) occur according to a Poisson process with intensity λ(k, i) whenever the Markov chain J(t) is in state i, 1 ≤ k ≤ K, 0 ≤ i ≤ N. Of interest to the paper is the time-dependent joint distribution of the multivariate process [M(t), J(t)]. In particular, the Laplace transform generating function is explicitly derived and its probabilistic interpretation is given. Asymptotic expansions of the cross moments and covariance functions of M(t) are also discussed.

AB - A multivariate Markov modulated Poisson process M(t) = [M1(t),...,MK(t)] governed by a Markov chain {J(t):t ≥ 0} on N = {0, 1,...,N} is introduced where jumps of Mk(t) occur according to a Poisson process with intensity λ(k, i) whenever the Markov chain J(t) is in state i, 1 ≤ k ≤ K, 0 ≤ i ≤ N. Of interest to the paper is the time-dependent joint distribution of the multivariate process [M(t), J(t)]. In particular, the Laplace transform generating function is explicitly derived and its probabilistic interpretation is given. Asymptotic expansions of the cross moments and covariance functions of M(t) are also discussed.

KW - asymptotic analysis

KW - covariance functions

KW - multivariate Markov modulated Poisson processes

KW - probability generating function

KW - time-dependent joint distribution

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U2 - 10.1016/0167-6377(92)90020-4

DO - 10.1016/0167-6377(92)90020-4

M3 - Article

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SN - 0167-6377

VL - 12

SP - 37

EP - 45

JO - Operations Research Letters

JF - Operations Research Letters

IS - 1

ER -

Analysis of multivariate Markov modulated Poisson processes

Abstract

Keywords

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