Ergodicity and Polynomial Convergence Rate of Generalized Markov Modulated Poisson Processes

2020 ◽  
pp. 367-381
Author(s):  
Galina Zverkina
Keyword(s):  
Convergence Rate ◽  
Poisson Processes ◽  
Markov Modulated

On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions

1996 ◽  
Vol 33 (3) ◽  
pp. 640-653 ◽  
Author(s):  
Tobias Rydén
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Poisson Processes ◽  
Hitting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Minimum Number ◽  
Markov Modulated ◽  
Two Parameters

An aggregated Markov chain is a Markov chain for which some states cannot be distinguished from each other by the observer. In this paper we consider the identifiability problem for such processes in continuous time, i.e. the problem of determining whether two parameters induce identical laws for the observable process or not. We also study the order of a continuous-time aggregated Markov chain, which is the minimum number of states needed to represent it. In particular, we give a lower bound on the order. As a by-product, we obtain results of this kind also for Markov-modulated Poisson processes, i.e. doubly stochastic Poisson processes whose intensities are directed by continuous-time Markov chains, and phase-type distributions, which are hitting times in finite-state Markov chains.

Analysis of multivariate Markov modulated Poisson processes

1992 ◽  
Vol 12 (1) ◽  
pp. 37-45 ◽  
Author(s):  
Ushio Sumita ◽  
Yasushi Masuda
Keyword(s):  
Poisson Processes ◽  
Markov Modulated

On pair and tuple formation under independent Poisson or renewal arrival processes

2004 ◽  
Vol 41 (4) ◽  
pp. 1138-1144 ◽  
Author(s):  
K. Borovkov
Keyword(s):  
Weak Convergence ◽  
Convergence Rate ◽  
Formation Process ◽  
Short Proof ◽  
Probabilistic Approach ◽  
Convergence Result ◽  
Pair Formation ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Arrival Processes

We present several results refining and extending those of Neuts and Alfa on weak convergence of the pair-formation process when arrivals follow two independent Poisson processes. Our results are obtained using a different, more straightforward, and apparently simpler probabilistic approach. Firstly, we give a very short proof of the fact that the convergence of the pair-formation process to a Poisson process actually holds in total variation (with a bound for convergence rate). Secondly, we extend the result of the theorem to the case of multiple labels: there are d independent arrival Poisson processes, and we are looking at the epochs when d-tuples are formed. Thirdly, we extend the original (weak convergence) result to the case when arrivals follow independent renewal processes (this extension is also valid for the d-tuple formation).

A versatile Markovian point process

1979 ◽  
Vol 16 (4) ◽  
pp. 764-779 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Point Processes ◽  
Probability Distributions ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Probability Models ◽  
Phase Type ◽  
Finite State ◽  
Markov Point Processes ◽  
Arrival Processes ◽  
Markov Modulated

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

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