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.

Stochastic inequalities for an overflow model

1987 ◽  
Vol 24 (3) ◽  
pp. 696-708 ◽  
Author(s):  
Arie Hordijk ◽  
Ad Ridder
Keyword(s):  
Failure Rate ◽  
Lower Bounds ◽  
Approximation Method ◽  
Queueing Networks ◽  
Upper And Lower Bounds ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Decreasing Failure Rate ◽  
General Method

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.

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.

Stochastic inequalities for an overflow model

1987 ◽  
Vol 24 (03) ◽  
pp. 696-708 ◽  
Author(s):  
Arie Hordijk ◽  
Ad Ridder
Keyword(s):  
Failure Rate ◽  
Lower Bounds ◽  
Approximation Method ◽  
Queueing Networks ◽  
Upper And Lower Bounds ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Decreasing Failure Rate ◽  
General Method

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.

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.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

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.

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.

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