Presentation of phase-type distributions as proper mixtures

1985 ◽  
Vol 22 (01) ◽  
pp. 247-250 ◽  
Author(s):  
David Assaf ◽  
Naftali A. Langberg
Keyword(s):  
Phase Type Distribution ◽  
Type Distribution ◽  
Phase Type ◽  
Phase Type Distributions

It is shown that any phase-type distribution can be represented as a proper mixture of two distinct phase-type distributions. Using different terms, it is shown that the class of phase-type distributions does not include any extreme ones. A similar result holds for the subclass of upper-triangular phase-type distributions.

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.

Phase-type distributions and invariant polytopes

1991 ◽  
Vol 23 (03) ◽  
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.

Presentation of phase-type distributions as proper mixtures

1985 ◽  
Vol 22 (1) ◽  
pp. 247-250 ◽  
Author(s):  
David Assaf ◽  
Naftali A. Langberg
Keyword(s):  
Phase Type Distribution ◽  
Type Distribution ◽  
Phase Type ◽  
Phase Type Distributions

It is shown that any phase-type distribution can be represented as a proper mixture of two distinct phase-type distributions. Using different terms, it is shown that the class of phase-type distributions does not include any extreme ones. A similar result holds for the subclass of upper-triangular phase-type distributions.

RELIABILITY OF A k-OUT-OF-n SYSTEM WITH REPAIR BY A SERVICE STATION ATTENDING A QUEUE WITH POSTPONED WORK

2007 ◽  
Vol 14 (04) ◽  
pp. 379-398 ◽  
Author(s):  
A. KRISHNAMOORTHY ◽  
VISWANATH C. NARAYANAN ◽  
T. G. DEEPAK
Keyword(s):  
Cost Function ◽  
System State ◽  
Phase Type Distribution ◽  
State Distribution ◽  
Service Facility ◽  
Type Distribution ◽  
Service Completion ◽  
Phase Type ◽  
Phase Type Distributions ◽  
External Customer

In this paper the reliability of a repairable k-out-of-n system is studied. Repair times of components follow a phase type distribution. In addition, the service facility offers service to external customers which arrive according to a MAP. An external customer, who finds an idle server on its arrival, is immediately selected for service. Otherwise, the external customer joins the queue in a pool of postponed work of infinite capacity with probability 1 if the number of failed components in the system is < M (M ≤ n - k + 1) and if the number of failed components ≥ M it joins the pool with probability γ or leaves the system forever. Repair times of components of the system and that of the external customers have independent phase type distributions. At a service completion epoch if the buffer has less than L customers, a pooled customer is taken for service with probability p, 0 < p < 1 If at a service completion epoch no component of the system is waiting for repair, a pooled customer, if any waiting, is immediately taken for service. We obtain the system state distribution under the condition of stability. A number of performance characteristics are derived. A cost function involving L, M, γ and p is constructed and its behaviour investigated numerically.

A Note on Characterizing Service Interruptions with Phase-Type Distribution

2013 ◽  
Vol 31 (4) ◽  
pp. 671-683 ◽  
Author(s):  
A. Krishnamoorthy ◽  
P. K. Pramod ◽  
S. R. Chakravarthy
Keyword(s):  
Phase Type Distribution ◽  
Type Distribution ◽  
Service Interruptions ◽  
Phase Type

scholarly journals A generalized class of correlated run shock models

2018 ◽  
Vol 6 (1) ◽  
pp. 131-138 ◽  
Author(s):  
Femin Yalcin ◽  
Serkan Eryilmaz ◽  
Ali Riza Bozbulut
Keyword(s):  
Stopping Time ◽  
Random Variables ◽  
Random Variable ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Shock Models ◽  
Phase Type ◽  
Over Time

AbstractIn this paper, a generalized class of run shock models associated with a bivariate sequence {(Xi, Yi)}i≥1 of correlated random variables is defined and studied. For a system that is subject to shocks of random magnitudes X1, X2, ... over time, let the random variables Y1, Y2, ... denote times between arrivals of successive shocks. The lifetime of the system under this class is defined through a compound random variable T = ∑Nt=1 Yt , where N is a stopping time for the sequence {Xi}i≤1 and represents the number of shocks that causes failure of the system. Another random variable of interest is the maximum shock size up to N, i.e. M = max {Xi, 1≤i≤ N}. Distributions of T and M are investigated when N has a phase-type distribution.

A closure characterisation of phase-type distributions

1992 ◽  
Vol 29 (01) ◽  
pp. 92-103 ◽  
Author(s):  
Robert S. Maier ◽  
Colm Art O'Cinneide
Keyword(s):  
Analogous Result ◽  
Finite Automata ◽  
Automata Theory ◽  
Phase Type Distribution ◽  
Exponential Distributions ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase ◽  
Rational Sequence ◽  
Family Of Distributions

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class. We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.

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