An invariant of representations of phase-type distributions and some applications

1996 ◽  
Vol 33 (2) ◽  
pp. 368-381 ◽  
Author(s):  
C. Commault ◽  
J. P. Chemla
Keyword(s):  
Markov Chains ◽  
Laplace Transform ◽  
Rational Functions ◽  
Absorbing State ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Finite State ◽  
The Laplace Transform ◽  
The Difference ◽  
Real Poles

In this paper we consider phase-type distributions, their Laplace transforms which are rational functions and their representations which are finite-state Markov chains with an absorbing state. We first prove that, in any representation, the minimal number of states which are visited before absorption is equal to the difference of degree between denominator and numerator in the Laplace transform of the distribution. As an application, we prove that when the Laplace transform has a denominator with n real poles and a numerator of degree less than or equal to one the distribution has order n. We show that, in general, this result can be extended neither to the case where the numerator has degree two nor to the case of non-real poles.

An invariant of representations of phase-type distributions and some applications

1996 ◽  
Vol 33 (02) ◽  
pp. 368-381 ◽  
Author(s):  
C. Commault ◽  
J. P. Chemla
Keyword(s):  
Markov Chains ◽  
Laplace Transform ◽  
Rational Functions ◽  
Absorbing State ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Finite State ◽  
The Laplace Transform ◽  
The Difference ◽  
Real Poles

In this paper we consider phase-type distributions, their Laplace transforms which are rational functions and their representations which are finite-state Markov chains with an absorbing state. We first prove that, in any representation, the minimal number of states which are visited before absorption is equal to the difference of degree between denominator and numerator in the Laplace transform of the distribution. As an application, we prove that when the Laplace transform has a denominator with n real poles and a numerator of degree less than or equal to one the distribution has order n. We show that, in general, this result can be extended neither to the case where the numerator has degree two nor to the case of non-real poles.

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.

Phase-type distributions and invariant polytopes

1991 ◽  
Vol 23 (3) ◽  
pp. 515-535 ◽  
Author(s):  
Colm Art O'Cinneide
Keyword(s):  
Lower Bounds ◽  
Stieltjes Transform ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Natural Approach ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Real Poles

The notion of an invariant polytope played a central role in the proof of the characterization of phase-type distributions. The purpose of this paper is to develop invariant polytope techniques further. We derive lower bounds on the number of states needed to represent a phase-type distribution based on poles of its Laplace–Stieltjes transform. We prove that every phase-type distribution whose transform has only real poles has a bidiagonal representation. We close with three short applications of the invariant polytope idea. Taken together, the results of this paper show that invariant polytopes provide a natural approach to many questions about phase-type distributions.

scholarly journals On Maxima and Ladder Processes for a Dense Class of Lévy Process

2006 ◽  
Vol 43 (01) ◽  
pp. 208-220 ◽  
Author(s):  
Martijn Pistorius
Keyword(s):  
Laplace Transform ◽  
Error Estimates ◽  
Lévy Process ◽  
Iterative Procedure ◽  
Levy Process ◽  
Distribution Of The Maximum ◽  
Phase Type ◽  
The Laplace Transform ◽  
The Law ◽  
Dense Class

In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.

Diffraction by a wave-guide of finite length

1952 ◽  
Vol 48 (1) ◽  
pp. 118-134 ◽  
Author(s):  
D. S. Jones
Keyword(s):  
Electromagnetic Wave ◽  
Approximate Solution ◽  
Laplace Transform ◽  
Integral Equations ◽  
Finite Length ◽  
Wave Guide ◽  
The Laplace Transform ◽  
The Difference

AbstractWhen the electric intensities on two parallel planes, of which the two perfectly conducting sides of a wave-guide of finite length and infinite width are portions, are taken as unknowns, the problem of the diffraction of a plane harmonic electromagnetic wave polarized parallel to the edges of the guide leads to two integral equations. By means of the Laplace transform these equations are converted into others suitable for solution by successive substitutions. The series thus obtained is too complex for practical purposes, and so an approximate solution is found for the case when the length of the guide is large compared with the wavelength. Finally, there is a brief discussion of the difference between the distant fields when l is large and when l is infinite.

scholarly journals On Maxima and Ladder Processes for a Dense Class of Lévy Process

2006 ◽  
Vol 43 (1) ◽  
pp. 208-220 ◽  
Author(s):  
Martijn Pistorius
Keyword(s):  
Laplace Transform ◽  
Error Estimates ◽  
Lévy Process ◽  
Iterative Procedure ◽  
Levy Process ◽  
Distribution Of The Maximum ◽  
Phase Type ◽  
The Laplace Transform ◽  
The Law ◽  
Dense Class

In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.

scholarly journals Explicit results for the distribution of the number of customers served during a busy period for $M^X/PH/1$ queue

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Veena Goswami ◽  
M. L. Chaudhry
Keyword(s):  
Functional Equation ◽  
Busy Period ◽  
Stochastic Modelling ◽  
Lagrange Inversion ◽  
Special Cases ◽  
Service Distribution ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Laplace Transform ◽  
Number Of Customers

<p style='text-indent:20px;'>We give analytically explicit solutions for the distribution of the number of customers served during a busy period for the <inline-formula><tex-math id="M1">\begin{document}$ M^X/PH/1 $\end{document}</tex-math></inline-formula> queues when initiated with <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> customers. When customers arrive in batches, we present the functional equation for the Laplace transform of the number of customers served during a busy period. Applying the Lagrange inversion theorem, we provide a refined result to this functional equation. From a phase-type service distribution, we obtain the distribution of the number of customers served during a busy period for various special cases such as exponential, Erlang-k, generalized Erlang, hyperexponential, Coxian, and interrupted Poisson process. The results are exact, rapid and vigorous, owing to the clarity of the expressions. Moreover, we also consider computational results for several service-time distributions using our method. Phase-type distributions can approximate any non-negative valued distribution arbitrarily close, making them a useful practical stochastic modelling tool. These distributions have eloquent properties which make them beneficial in the computation of performance models.</p>

scholarly journals The Limit Theorems for Function of Markov Chains in the Environment of Single Infinite Markovian Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Zhanfeng Li ◽  
Min Huang ◽  
Xiaohua Meng ◽  
Xiangyu Ge
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Strong Convergence ◽  
Sufficient Conditions ◽  
Weighted Sums ◽  
Strong Law ◽  
Large Numbers ◽  
Finite State ◽  
The Difference ◽  
Markovian Systems

This paper is intended to study the limit theorem of Markov chain function in the environment of single infinite Markovian systems. Moreover, the problem of the strong law of large numbers in the infinite environment is presented by means of constructing martingale differential sequence for the measurement under some different sufficient conditions. If the sequence of even functions gnx,n≥0 satisfies different conditions when the value ranges of x are different, we have obtained SLLN for function of Markov chain in the environment of single infinite Markovian systems. In addition, the paper studies the strong convergence of the weighted sums of function for finite state Markov Chains in single infinitely Markovian environments. Although the similar conclusions have been carried out, the difference results performed by previous scholars are that we give weaker different sufficient conditions of the strong convergence of weighted sums compared with the previous conclusions.

Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes

1991 ◽  
Vol 28 (04) ◽  
pp. 822-832 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Time Interval ◽  
Reliability Model ◽  
Finite State ◽  
The Laplace Transform ◽  
Semi Markov Process ◽  
Number Of Visits ◽  
Semi Markov Processes ◽  
Finite State Space

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB (t) stand for the number of visits of Y to B during the time interval [0, t], t &gt; 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB (t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB (t) as t →∞.

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