A Framework for Performability Modelling Using Proxels

Performability modelling and analysis is used as a measure of both performance and dependability of, mainly, fault-tolerant systems. Proxels are a paradigm that provides deterministic analysis of discrete stochastic models which contain general types of distribution functions, which is not otherwise possible using the standard approaches, such as discrete-event simulation or Markov reward modelling. Because of the flexible definition of the proxels, they can easily be extended to provide a widely-applicable approach to performability modelling. Proxels work in such a way that they track the complete behaviour of the system and correspondingly distribute the probability, depending on the distribution functions and the time spent in each discrete state without anything happening (denoted as age intensity of the state change). Therefore the term state with respect to the proxels has been extended to include the age intensities of the relevant state changes. In this paper we establish the framework for proxel-based performability modelling and analysis, as well as for general reward modelling. Two types of rewards are considered, impulse (associated with transitions) and rate rewards (associated with discrete states). The advantage of the proxel-based simulation is that both types of rewards can be modelled as functions of any other timedependent quantity of the model, and models with general types of distribution functions can be analysed in a deterministic way. For the purpose of demonstrating the method, we will include an example which will show in detail how the method works. Discussion and comparisons with some of the existing approaches will be included too, which will provide more insight into the advantages of the proxel-based method. 1. GOALS AND INTRODUCTION The goal of this paper is to show how the basic proxel-based method can be adapted for doing performability analysis of discrete stochastic models. In that sense, establishing the formalities that are behind the adaptation process are the subject of the paper. The motivation for this new application field of the proxelbased method is a recent practical experience with the method for analysing a warranty model, a project which we carried out for DaimlerChrysler, as described in [1]. The model that we needed to analyse involved manipulating impulse rewards in order to analyse the costs. It turned out to be a straightforward adaptation of the basic proxel method, without imposing any additional significant complications with respect to the computational and memory complexity of the implementation. Based on that experience we realised that the method can be useful for carrying out a more general performability analysis which also includes reward rates and that is the topic that we treat here. As a beginning this section will give a short overview on the performability theory and the proxel-based method, as well as present and discuss some of the more popular existing approaches. Further we will describe and explain why the proxels are a good option for carrying out performability analysis of discrete stochastic systems and describe how it is to be achieved. To aid the comprehension and show how it works in practice, in the experiments’ section we introduce an example model which we than use to test two different cases of performability modelling. Finally, in our last section we give a brief summary and an outlook of our previous and future research with respect to the topic of the paper. 1.1. Performability Modelling Performability modelling [2] is a modelling approach to simultaneously evaluating both the performance of a stochastic system and its dependability. This measure is of an especial importance when analysing fault-tolerant systems whose performance depends on many components which fail and get repaired, i.e. behave, in a stochastic manner. Performance alone is a measure of how efficient one system is (can be measured as throughput, response time, etc.), whereas dependability is a measure of its ability to function correctly over a specific period of time i.e. how reliable it is. Reliability (or dependability) of a discrete and stochastic system S can be expressed mathematically in the following form: R(t) = Pr(S operates correctly in [0, t)). (1) If we now denote the lifetime of a system by L, and F is the distribution function of L, then the reliability of the system at time t can be computed as R(t) = Pr(L > t) = 1− F (t). (2) In common words, performability modelling tries to evaluate and answer the following question: How much work will be done (lost) in a given interval by a given system including the effects of its failures and repairs? and find the function that describes it. The work is then the accumulated performance over the given time interval, given that it changes over time depending on the operativeness of the separate components of the system i.e. the performability. The performance of the system is measured using a reward function P (DS, τ) which evaluates its efficiency in each of its