scholarly journals Renewal characterization of Markov modulated Poisson processes

1989 ◽  
Vol 2 (1) ◽  
pp. 53-70 ◽  
Author(s):  
Marcel F. Neuts ◽  
Ushio Sumita ◽  
Yoshitaka Takahashi
Keyword(s):  
Markov Chain ◽  
Poisson Process ◽  
Renewal Process ◽  
Jump Process ◽  
Probability Vector ◽  
Markov Modulated Poisson Process ◽  
Markov Modulated ◽  
Bivariate Markov Chain ◽  
Characterization Theorems

A Markov Modulated Poisson Process (MMPP) M(t) defined on a Markov chain J(t) is a pure jump process where jumps of M(t) occur according to a Poisson process with intensity λi whenever the Markov chain J(t) is in state i. M(t) is called strongly renewal (SR) if M(t) is a renewal process for an arbitrary initial probability vector of J(t) with full support on P={i:λi>0}. M(t) is called weakly renewal (WR) if there exists an initial probability vector of J(t) such that the resulting MMPP is a renewal process. The purpose of this paper is to develop general characterization theorems for the class SR and some sufficiency theorems for the class WR in terms of the first passage times of the bivariate Markov chain [J(t),M(t)]. Relevance to the lumpability of J(t) is also studied.

scholarly journals A bivariate two-state Markov modulated Poisson process for failure modeling

2021 ◽  
Vol 208 ◽  
pp. 107318
Author(s):  
Yoel G. Yera ◽  
Rosa E. Lillo ◽  
Bo F. Nielsen ◽  
Pepa Ramírez-Cobo ◽  
Fabrizio Ruggeri
Keyword(s):  
Poisson Process ◽  
Failure Modeling ◽  
Markov Modulated Poisson Process ◽  
Markov Modulated

A characterization of the Poisson process

1974 ◽  
Vol 11 (1) ◽  
pp. 72-85 ◽  
Author(s):  
S. M. Samuels
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complete Proof ◽  
Necessary And Sufficient

Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.

scholarly journals Estimating the parameters of a seasonal Markov-modulated Poisson process

2015 ◽  
Vol 26 ◽  
pp. 103-123 ◽  
Author(s):  
Armelle Guillou ◽  
Stéphane Loisel ◽  
Gilles Stupfler
Keyword(s):  
Poisson Process ◽  
Markov Modulated Poisson Process ◽  
Markov Modulated

A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

1995 ◽  
Vol 32 (4) ◽  
pp. 1103-1111 ◽  
Author(s):  
Qing Du
Keyword(s):  
Poisson Process ◽  
Arrival Rate ◽  
Loss Probability ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Markov Modulated Poisson Process ◽  
Server Queue ◽  
Monotonicity Result ◽  
Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i. The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i. In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

A characterization of the Poisson process

1986 ◽  
Vol 23 (01) ◽  
pp. 233-235 ◽  
Author(s):  
Pushpa Lata Gupta ◽  
Ramesh C. Gupta
Keyword(s):  
Poisson Process ◽  
Real Number ◽  
Renewal Process ◽  
Residual Life ◽  
Mild Conditions

Denoting by v(t) the residual life of a component in a renewal process, Çinlar and Jagers (1973) and Holmes (1974) have shown that if E(v(t)) is independent of t for all t, then the process is Poisson. In this note we prove, under mild conditions, that if E(G(v(t))) is constant, then the process is Poisson. In particular if E((v(t))r) for some specific real number r ≧ 1 is independent of t, then the process is Poisson.

A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

1995 ◽  
Vol 32 (04) ◽  
pp. 1103-1111 ◽  
Author(s):  
Qing Du
Keyword(s):  
Poisson Process ◽  
Arrival Rate ◽  
Loss Probability ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Markov Modulated Poisson Process ◽  
Server Queue ◽  
Monotonicity Result ◽  
Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i . The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i . In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

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