On the relation between matrix-geometric and discrete phase-type distributions Sietske Greeuw Kongens Lyngby/Amsterdam 2009 Master thesis Mathematics University of Amsterdam Technical University of Denmark Informatics and Mathematical Modelling Building 321, DK-2800 Kongens Lyngby, Denmark Phone +45 45253351, Fax +45 45882673 [email protected] www.imm.dtu.dk University of Amsterdam Faculty of Science Science Park 404, 1098 XH Amsterdam, The Netherlands Phone +32 20 5257678, Fax +31 20 5257675 [email protected] www.science.uva.nl Summary A discrete phase-type distribution describes the time until absorption in a discrete-time Markov chain with a finite number of transient states and one absorbing state. The density f(n) of a discrete phase-type distribution can be expressed by the initial probability vector α, the transition probability matrix T of the transient states of the Markov chain and the vector t containing the probabilities of entering the absorbing state from the transient states: n−1 f(n) = αT t, n ∈ N. If we take a probability density of the same form, but not necessarily require α,T and t to have the probabilistic Markov-chain interpretation, we obtain the density of a matrix-geometric distribution. Matrix-geometric distributions can equivalently be defined as distributions on the non-negative integers that have a rational probability generating function. In this thesis it is shown that the class of matrix-geometric distributions is strictly larger than the class of discrete phase-type distributions. We give an example of a set of matrix-geometric distributions that are not of discrete phase- type. We also show that there is a possible order reduction when representing a discrete phase-type distribution as a matrix-geometric distribution. The results parallel the continuous case, where the class of matrix-exponential distributions is strictly larger than the class of continuous phase-type distribu- tions, and where there is also a possible order reduction. Keywords: discrete phase-type distributions, phase-type distributions, matrix- exponential distributions, matrix-geometric distributions. ii Resum´e En diskret fasetype fordeling er fordelingen af tiden til absorption i en diskret- tids Markov kæde med et begrænset antal transiente tilstande og ´en absorbe- rende tilstand. Tætheden f(n) af en diskret fasetype fordeling kan udtrykkes ved vektoren med den initielle fordeling α, matricen T, der beskriver de mulige over- gange mellem transiente tilstande i Markov kæden, og vektoren t med sandsyn- ligheder for at springe til den absorberende tilstand fra de transiente tilstande: n−1 f(n) = αT t, n ∈ N. Hvis vi tager en tæthed af samme form, men ikke nødvendigvis kræver, at α,T og t har probabilistisk Markov kæde fortolkning, f˚arvi tætheden af en matrix-geometrisk fordeling. Matrix-geometriske fordelinger kan tilsvarende de- fineres som fordelinger p˚aikke-negative heltal, der har en rationel sandsyn- lighedsgenererende funktion. I denne projekt er det vist, at mængden af matrix-geometriske fordelinger er strengt større end mængden af diskrete fasetype fordelinger. Der gives et ek- sempel p˚aen mængde af matrix-geometriske fordelinger, der ikke er af diskret fasetype. Vi viser ogs˚a,at der er en mulighed for reduktion af størrelse af repræsentationen, n˚arde repræsenterer en diskret fasetype fordeling som en matrix-geometrisk fordeling. Resultaterne svarer til det kontinuerte tilfælde, hvor klassen af matrix- eksponentielle fordelinger er strengt større end klassen af kontinuert fasetype fordelinger, og hvor reduktion af størrelse af repræsentation ogs˚aer muligt. iv Preface This thesis was prepared in partial fulfillment of the requirements for acquiring the master of science degree in Mathematics. It has been prepared during a five month Erasmus-stay at the Danish Technical University, at the institute for Informatics and Mathematical Modelling. I would like to thank my two supervisors, Bo Friis Nielsen and Michel Mandjes, for both being very supportive in my decision of doing my master project abroad. Furthermore I want to thank Bo for his friendly and informative guidance, and Michel for his help with the final parts of my thesis. Finally I want to thank my family and my friends and all other people who have supported me during this process. Amsterdam, March 2009 Sietske Greeuw vi Contents Summary i Resum´e iii Preface v 1 Introduction 1 2 Phase-type distributions 5 2.1 Discrete phase-type distributions . 5 2.2 Continuous phase-type distributions . 19 3 Matrix-exponential distributions 23 3.1 Definition . 24 3.2 Examples of matrix-exponential distributions . 25 4 Matrix-geometric distributions 31 4.1 Definition . 32 4.2 Existence of genuine matrix-geometric distributions . 34 4.3 Order reduction for matrix-geometric distributions . 38 4.4 Properties of matrix-geometric distributions . 45 viii CONTENTS Chapter 1 Introduction In this thesis the relation between matrix-geometric and discrete phase-type distributions will be explored. Matrix-geometric distributions (MG) are distri- butions on the non-negative integers that possess a density of the form n−1 f(n) = αT t, n ∈ N (1.1) together with f(0) = ∆, the point mass in zero. The parameters (α,T, t) are a row vector, matrix and column vector respectively, that satisfy the necessary conditions for f(n) to be a density. Hence αT n−1t ≥ 0 for all n ∈ N and P∞ −1 n=0 f(n) = ∆ + α(I − T ) t = 1. The class MG is a generalization of the class of discrete phase-type distributions (DPH). A discrete phase-type distribution describes the time until absorption in a finite-state discrete-time Markov chain. A discrete phase-type density has the same form as (1.1) but requires the parameters to have the interpretation as initial probability vector (α), sub-transition probability matrix (T ) and exit probability vector (t) of the underlying Markov chain. The analogous setup in continuous time is given by matrix-exponential distri- butions (ME) that are a generalization of continuous phase-type distributions (PH). The latter describe the time until absorption in a continuous-time finite- state Markov chain. The name phase-type refers to the states of the Markov chain. Each visit to a 2 Introduction state can be seen as a phase with a geometric (in continuous time exponential) distribution, and the resulting Markovian structure gives rise to a discrete (con- tinuous) phase-type distribution. The method of phases has been introduced by A.K. Erlang in the beginning of the 20th century and has later been generalized by M.F. Neuts. A general probability distribution can be approximated arbi- trarily closely by a phase-type distribution. This makes phase-type distributions a powerful tool in the modelling of e.g. queueing systems. Another nice feature of phase-type distributions is that they often give rise to closed form solutions. The main reference on discrete phase-type distributions is ‘Probability distribu- tions of phase type’ by M.F. Neuts, published in 1975 [12]. This article gives a thorough introduction to discrete phase-type distributions, their main proper- ties and their use in the theory of queues. In his 1981 book ‘Matrix-geometric Solutions in Stochastic Models’ [13], M.F. Neuts gives again an introduction to phase-type distributions, this time focussing on the continuous case. Another clear introduction of continuous phase-type distributions can be found in the book by G. Latouche and V. Ramaswami [10]. A good survey on the class ME is given in the entry on matrix-exponential distributions in the Encyclopedia of Statistical Sciences by Asmussen and O’Cinneide [4]. In this article the rela- tion of the class ME to the class PH is discussed, some properties of ME are given and the use of these distributions in applied probability is explained. The authors mention the class of matrix-geometric distributions as the analogous discrete counterpart of ME. However, no explicit examples are given. The main question addressed in this thesis is what the relation is between the classes DPH and MG. From their definition it is clear that DPH ⊆ MG. It is however not immediately clear, whether there exists distributions that are in MG but do not have a discrete phase-type representation. The second question is on the order of the distribution. This order is defined as the minimal dimension of the matrix T in (1.1) needed to represent the distribution. It is of interest to know if there exist discrete phase-type distributions that have a lower order representation when they are represented as a matrix-geometric distribution. Such a reduced-order representation can be much more convenient to work with, for example in solving high-dimensional queueing systems. In the continuous case it is known that PH ( ME, that is, PH is a strict subset of ME [3, 4]. A standard example of a distribution that is in ME but not in PH is a distribution represented by a density which has a zero on (0, ∞). Such a distribution cannot be in PH, as a phase-type density 3 is strictly positive on (0, ∞) [13]. There is no equivalent result in discrete time, as discrete phase-type densities can be zero with a certain periodicity. A sec- ond way of showing that ME is larger than PH is by using the characterization of phase-type distributions stated by O’Cinneide in [14] which is based on the absolute value of the poles of the Laplace transform. In [15] O’Cinneide gives a lower bound on the order of a phase-type represen- tation. He shows that there is possible order reduction when using a matrix- exponential representation rather than a phase-type representation. We will use the characterization of discrete phase-type distributions by O’Cinneide [14], to give a set of distributions that is in MG but not in DPH.