Performance Evaluation 56 (2004) 121–144 The scale factor: a new degree of freedom in phase-type approximation A. Bobbio a,∗ , A. Horváth b , M. Telek c a Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy b Dipartimento di Informatica, Università di Torino, Turin, Italy c Department of Telecommunications, Technical University of Budapest, Budapest, Hungary Abstract This paper introduces a unified approach to phase-type approximation in which the discrete and the continuous phase-type models form a common model set. The models of this common set are assigned with a non-negative real parameter, the scale factor. The case when the scale factor is strictly positive results in discrete phase-type distributions and the scale factor represents the time elapsed in one step. If the scale factor is 0, the resulting class is the class of CPH distributions. Applying the above view, it is shown that there is no qualitative difference between the discrete and the CPH models. Based on this unified view of phase-type models one can choose the best phase-type approximation of a stochastic model by optimizing the scale factor. © 2003 Elsevier B.V. All rights reserved. Keywords: Discrete and continuous phase-type distributions; Phase-type expansion; Approximate analysis 1. Introduction This paper presents new comparative results on the use of discrete phase-type (DPH) distributions [22] and of continuous phase-type (CPH) distributions [23] in applied stochastic modeling. DPH distributions of order n are defined as the time to absorption in a discrete-state discrete-time Markov chain (DTMC) with n transient states and one absorbing state. CPH distributions of order n are defined, similarly, as the distribution of the time to absorption in a discrete-state continuous-time Markov chain (CTMC) with n transient states and one absorbing state. The above definition implies that the properties of a DPH distribution are computed over the set of the natural numbers while the properties of a CPH distribution are defined as a function of a continuous time variable t . When DPH distributions are used to model timed activities, the set of the natural numbers must be related to a time measure. Hence, a new parameter need to be introduced that represents the time span associated to each step. This new parameter is the scale factor of the DPH distribution, and can be viewed as a new degree of freedom, ∗ Corresponding author. E-mail addresses: andrea.bobbio@mfn.unipmn.it (A. Bobbio), horvath@di.unito.it (A. Horv´ ath), telek@hit.bme.hu (M. Telek). 0166-5316/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.peva.2003.07.003