### Abstract

A multivariate Markov modulated Poisson process M(t) = [M_{1}(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 |
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Pages (from-to) | 37-45 |

Number of pages | 9 |

Journal | Operations Research Letters |

Volume | 12 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1992 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Management Science and Operations Research
- Statistics, Probability and Uncertainty
- Discrete Mathematics and Combinatorics
- Modelling and Simulation

### Cite this

*Operations Research Letters*,

*12*(1), 37-45. https://doi.org/10.1016/0167-6377(92)90020-4

**Analysis of multivariate Markov modulated Poisson processes.** / Sumita, Ushio; Masuda, Yasushi.

Research output: Contribution to journal › Article

*Operations Research Letters*, vol. 12, no. 1, pp. 37-45. https://doi.org/10.1016/0167-6377(92)90020-4

}

TY - JOUR

T1 - Analysis of multivariate Markov modulated Poisson processes

AU - Sumita, Ushio

AU - Masuda, Yasushi

PY - 1992

Y1 - 1992

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

UR - http://www.scopus.com/inward/record.url?scp=0026895404&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026895404&partnerID=8YFLogxK

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

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

M3 - Article

AN - SCOPUS:0026895404

VL - 12

SP - 37

EP - 45

JO - Operations Research Letters

JF - Operations Research Letters

SN - 0167-6377

IS - 1

ER -