This paper provides a detailed study on Discrete Phase Type (DPH) distributions and its acyclic subclass referred to as Acyclic DPH (ADPH). Previously not considered similarities and dierences between DPH and Continuous Phase Type (CPH) distributions are investigated and minimal representations, called canonical forms, for the subclass of ADPH distributions are provided. We investigate the consequences of the recent result about the minimal coefficient of variation of the DPH class [18] and show that below a given order (that is a function of the expected value) the minimal coefficient of variation of the DPH class is always less than the minimal coefficient of variation of the CPH class. Since all the previously introduced Phase Type fitting methods were designed for fitting over the CPH class we provide a DPH fitting method for the first time. The implementation of the DPH fitting algorithm is found to be simple and stable. The algorithm is tested over a benchmark consisting of 10 different continuous distributions. The error resulted when a continuous distribution sampled in discrete points is fitted by a DPH is also considered.

Analytical modeling plays a crucial role in the analysis and design of computer systems. Stochastic Petri Nets represent a powerful paradigm, widely used for such modeling in the context of dependability, performance and performability. Many structural and stochastic extensions have been proposed in recent years to increase their modeling power, or their capability to handle large systems. This paper reviews recent developments by providing the theoretical background and the possible areas of application. Markovian Petri nets are first considered together with very well established extensions known as Generalized Stochastic Petri nets and Stochastic Reward Nets. Key ideas for coping with large state spaces are then discussed. The challenging area of non-Markovian Petri nets is considered, and the related analysis techniques are surveyed together with the detailed elaboration of an example. Finally new models based on Continuous or Fluid Stochastic Petri Nets are briefly discussed.

The analysis of stochastic systems with non-exponential timing requires the development of suitable modeling tools. Recently, some eort has been devoted to generalize the concept of Stochastic Petri nets, by allowing the ring times to be generally distributed. The evolution of the PN in time becomes a stochastic process, for which in general, no analytical solution is available. The paper surveys suitable restrictions of the PN model with generally distributed transition times, that have appeared in the literature, and compares these models from the point of view of the modeling power and the numerical complexity. Key words: Stochastic Petri Nets, Non-exponential Distributions, Phase-type Distributions, Markov and Semimarkov Reward Models, Markov Regenerative Processes, Queueing Systems with Preemption. 1 Introduction The usual denition of Stochastic Petri Net (SPN) implies that all the timed activities associated to the transitions are represented by exponential random ...

The analysis of discrete stochastic models such as generally distributed stochastic Petri nets can be done using state space-based methods. The behavior of the model is described by a Markov chain that can be solved mathematically. The phase-type distributions that are used to describe non-Markovian distributions have to be approximated. An approach for the fast and accurate approximation of discrete phase-type distributions is presented. This can be a step towards a practical state space-based simulation method, whereas formerly this approach often had to be discarded as unfeasible due to high memory and runtime costs. Discrete phases also fit in well with current research on proxel-based simulation. They can represent infinite support distribution functions with considerably fewer Markov chain states than proxels. Our hope is that such a combination of both approaches will lead to a competitive simulation algorithm. 1.

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 Continuous phase-type distributions. Applying the above view, it is shown that there is no qualitative difference between the discrete and the continuous phase-type models.

Analytical modeling is a crucial part in the analysis and design of computer systems. Stochastic Petri Nets represent a powerful tool, widely used for dependability, performance and performability modeling. Many structural and stochastic extensions have been proposed so as to increase their modeling power. In this paper we review the main structural and stochastic extensions of Petri nets, by providing an updated treatment of the theoretical background and the possible areas of application. In particular, we focus our attention on Stochastic and Generalized Stochastic Petri nets (SPN and GSPN), Stochastic Reward Nets (SRN) and on non-Markovian Petri nets such as Markov Regenerative Stochastic Petri Nets (MRSPN) . Fluid Stochastic Petri Nets (FSPN) are discussed. 1 Introduction Analytical evaluation of computer/communication systems is increasingly becoming an integral part of the whole design process. Many diverse model specification techniques have been proposed. Markov chain models...

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For non-trivial (open) queueing networks and also for tandem queueing networks, decomposition often represents the only feasible solution method besides simulation. The network is partitioned into individual nodes which are analyzed in isolation with respect to approximate internal trac representations. The quality of the quickly obtainable results very much depends on the descriptors for the traffic processes within the network. In this paper, the decomposition of tandem networks is based on Markovian arrival processes (MAPs), which allow to capture the correlations in the traffic processes. The correlation structure of network traffic is known to have a considerable impact on performance measures. Moreover, MAP inputs considerably increase the range of applications of the queueing networks with phase type service times and customer losses. Numerical experiments on tandem networks demonstrate the accuracy of the newly proposed approach, which may be extended to general queueing networks with Markovian routing.

Continuous-time Markov chains are used extensively to analyze the performance of various computer networks. However, constructing and solving continuous-time Markov chain is a tedious and error-prone procedure, especially when the studied systems are complex. Stochastic Petri nets and the corresponding software packages provide automated generation and solution to continuous-time Markov chains. This paper gives an overview of stochastic Petri nets. Two examples in ATM networks are presented and studied to illustrate how to use stochastic Petri nets for performance analysis of computer networks. Index Terms: Stochastic Petri Nets, Stochastic Reward Nets, Computer Networks, ATM Networks This research was supported in part by the National Science Foundation under Grant No. EEC9418765. 1 Introduction From ARPAnet to Internet and to Internet 2, from Ethernet to fast Ethernet and to gigabit Ethernet, from packet switching to Asynchronous Transfer Mode (ATM) switching and to label switc...

In this paper two main problems are investigated. The first one is the effect of the goal function of the applied fitting method on the goodness of Phase type fitting. We discuss a numerical method based on a simple numerical optimization procedure that allows us to fit any nonnegative distribution with a Phase type (PH) distribution according to any arbitrary distance measure. By comparing the fitting results obtained by minimizing different distance measures, conclusions are drawn regarding the role of the optimization criteria. The second considered problem is the tail behaviour of Phase type distributions obtained via different fitting methods. To limit the numerical complexity of fitting methods (basically the evaluation of distance measures) the computations (numerical integration) are truncated at some point. Hence the information on the tail behaviour of the distribution is not considered beyond this point. To approximate distributions with heavy tail we propose a complex method that uses different techniques to fit the main part and the tail of the distribution. The proposed method combines the advantages of fitting techniques and this way it overcomes some of their limitations. The goodness of the discussed fitting methods are compared in queuing behaviour as well. The behaviour of the M/G/1 queue is compared with the one of the approximating M/PH/1 queue.