Input Modeling with Phase-Type Distributions and Markov Models

Author(s):  
Peter Buchholz ◽  
Jan Kriege ◽  
Iryna Felko
Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1404
Author(s):  
Alessio Angius ◽  
András Horváth ◽  
Marcello Urgo

The application of theoretical scheduling approaches to the real world quite often crashes into the need to cope with uncertain events and incomplete information. Stochastic scheduling approaches exploiting Markov models have been proposed for this class of problems with the limitation to exponential durations. Phase-type approximations provide a tool to overcome this limitation. This paper proposes a general approach for using phase-type distributions to model the execution of a network of activities with generally distributed durations through a Markov chain. An analytical representation of the infinitesimal generator of the Markov chain in terms of Kronecker algebra is proposed, providing a general formulation for this class of problems and supporting more efficient computation methods. This entails the capability to address stochastic scheduling in terms of the estimation of the distribution of common objective functions (i.e., makespan, lateness), enabling the use of risk measures to address robustness.


2014 ◽  
Vol 30 (4) ◽  
pp. 576-597 ◽  
Author(s):  
V. Ramaswami ◽  
N. C. Viswanath

2004 ◽  
Vol 36 (1) ◽  
pp. 116-138 ◽  
Author(s):  
Yonit Barron ◽  
Esther Frostig ◽  
Benny Levikson

An R-out-of-N repairable system, consisting of N independent components, is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R-1. A failed component is sent to a repair facility. After a failed component has been repaired it is as good as new. Formulae for the availability of the system using Markov renewal and semi-regenerative processes are derived. We assume that either the repair times of the components are generally distributed and the components' lifetimes are phase-type distributed or vice versa. Some duality results between the two systems are obtained. Numerical examples are given for several distributions of lifetimes and of repair times.


2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.


2002 ◽  
Vol 30 (3) ◽  
pp. 6-8 ◽  
Author(s):  
Alma Riska ◽  
Vesselin Diev ◽  
Evgenia Smirni

1987 ◽  
Vol 24 (3) ◽  
pp. 696-708 ◽  
Author(s):  
Arie Hordijk ◽  
Ad Ridder

A general method to obtain insensitive upper and lower bounds for the stationary distribution of queueing networks is sketched. It is applied to an overflow model. The bounds are shown to be valid for service distributions with decreasing failure rate. A characterization of phase-type distributions with decreasing failure rate is given. An approximation method is proposed. The methods are illustrated with numerical results.


2020 ◽  
Vol 23 (5) ◽  
pp. 1431-1451 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Martin Bladt ◽  
Mogens Bladt

Abstract We extend the Kulkarni class of multivariate phase–type distributions in a natural time–fractional way to construct a new class of multivariate distributions with heavy-tailed Mittag-Leffler(ML)-distributed marginals. The approach relies on assigning rewards to a non–Markovian jump process with ML sojourn times. This new class complements an earlier multivariate ML construction [2] and in contrast to the former also allows for tail dependence. We derive properties and characterizations of this class, and work out some special cases that lead to explicit density representations.


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