Closed form solutions for mapping general distributions to quasi-minimal PH distributions

https://doi.org/10.1016/j.peva.2005.06.002Get rights and content

Abstract

Approximating general distributions by phase-type (PH) distributions is a popular technique in stochastic analysis, since the Markovian property of PH distributions often allows analytical tractability. This paper proposes an algorithm for mapping a general distribution, G, to a PH distribution, which matches the first three moments of G. Efficiency of our algorithm hinges on narrowing the search space to a particular subset of the PH distributions, which we refer to as Erlang–Coxian (EC) distributions. The class of EC distributions has a small number of parameters, and we provide closed form solutions for these. Our solution applies to any distribution whose first three moments can be matched by a PH distribution. Also, our resulting EC distribution requires a nearly minimal number of phases, within one of the minimal number of phases required by any acyclic PH distribution.

Section snippets

Motivation

There is a large body of literature on the topic of approximating general distributions by phase-type (PH) distributions, whose Markovian properties make them far more analytically tractable. Much of this research has focused on the specific problem of finding an algorithm which maps a general distribution, G, to a PH distribution, P, where P and G agree on the first three moments. Throughout this paper we say that G is well represented by P if P and G agree on their first three moments. We

Overview of key ideas and definitions

We start with some definitions that we use throughout the paper.

Definition 1

A PH distribution is the distribution of the absorption time in a continuous time Markov chain. A PH distribution, F, is specified by a generator matrix, TF, and an initial probability vector, τF.

Fig. 1 shows a three-phase PH distribution, F, with τF=(τ1,τ2,τ3) andTF=(λ12+λ13+λ14)λ12λ13λ21(λ21+λ23+λ24)λ23λ31λ32(λ31+λ32+λ34).There are n=3 internal states. With probability τi we start in the i th state. The absorption time is

Characterizing phase type distributions

Set S(n) is characterized by Theorem 2: S(n)T(n)S(n+1)for all n2. In this section we prove the first part of the theorem.

Lemma 1

For all integers n2, S(n)T(n).

The second part, T(n)S(n+1), follows immediately from the construction of the Complete solution (see Corollary 3 in Section 6).

We begin by defining the ratio of the normalized moments.

Definition 10

The ratio of the normalized moments of a distribution F, rF, is defined as rF=m3F/m2F and is also referred to as the r-value of F.

A nice property of the r

EC distribution

The purpose of this section is two-fold: to provide a detailed characterization of the EC distribution, and to discuss a narrowed-down subset of the EC distributions with only five free parameters (λY is fixed) which we will use in our moment matching algorithms. Both results are summarized in Theorem 3.

To motivate the theorem in this section, suppose one is trying to match the first three moments of a given distribution, G, to a distribution, P, which is the convolution of exponential

Simple closed form solution

The Simplesolution is the simplest among our three closed form solutions, and the Complete and Positive solutions will be built upon the Simple solution. Before, presenting the Simple solution, we first classify the input distributions. This classification is used, in particular, to determine the number of phases used in the Simple solution.

Complete closed form solution

The Complete solution improves upon the Simple solution in the sense that it is defined for all the input distributions GPH3. Fig. 6 shows an implementation of the Complete solution. Below, we elaborate on the Complete solution, and prove an upper bound on the number of phases used in the Complete solution.

Positive closed form solution

The Simple and Complete solutions can have mass probability at zero (i.e. p<1). In some applications, mass probability at zero is not an issue. Such applications include approximating busy period distributions in the analysis of multiserver systems [17] and approximating shortfall distributions in inventory system analysis [26], [27]. However, there are also applications where a mass probability at zero increases the computational complexity or even makes the analysis intractable. For example,

Conclusion

In this paper, we propose a closed form solution for the parameters of a PH distribution, P, that well represents a given distribution G. Our solution is the first that achieves all of the following goals: (i) the first three moments of G and P agree, (ii) any distribution G that is well represented by a PH distribution (i.e., GPH3) can be well represented by some P, (iii) the number of phases used in P is at most OPT(G)+1, (iv) the solution is expressed in closed form.

The key idea is the

Acknowledgements

This paper combines two papers [18], [19] which appeared in TOOLS 2003, and also includes some new results beyond what was included in [18], [19]. In particular, the Positivesolution proposed in Section 7 of this paper is entirely new. The Positive solution is motivated by a discussion with Miklos Telek and Armin Heindl at the TOOLS conference. Also, Section 3 and Appendix A in [19] have been replaced by a simpler proof of the same result which is contained in Section 3 of this paper.

Takayuki Osogamiis a Ph.D. candidate at the Department of Computer Science, Carnegie Mellon University. He received a B.Eng. degree in electronic engineering from the University of Tokyo, Japan, in 1998. In 1998–2001, he was at IBM Tokyo Research Laboratory, where the principal project was development of optimization algorithms. His current research interest includes performance analysis of resource allocation policies for multi-server systems.

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    Takayuki Osogamiis a Ph.D. candidate at the Department of Computer Science, Carnegie Mellon University. He received a B.Eng. degree in electronic engineering from the University of Tokyo, Japan, in 1998. In 1998–2001, he was at IBM Tokyo Research Laboratory, where the principal project was development of optimization algorithms. His current research interest includes performance analysis of resource allocation policies for multi-server systems.

    Mor Harchol-Balter is an associate professor of Computer Science at Carnegie Mellon University. She received her doctorate from the Computer Science Department at the University of California at Berkeley under the direction of Manuel Blum. She is a recipient of the McCandless Chair, the NSF CAREER award, the NSF Postdoctoral Fellowship in the Mathematical Sciences, multiple best paper awards, and several teaching awards, including the Herbert A. Simon Award for Teaching Excellence. She is heavily involved in the ACM SIGMETRICS research community. Her work focuses on designing new scheduling/resource allocation policies for various distributed computer systems including Web servers, distributed supercomputing servers, networks of workstations, and database systems. Her work spans both queueing analysis and implementation and emphasizes integrating measured workload distributions into the problem solution.

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