Proxel-based Simulation of a Warranty Model

The Proxel method is a state space-based approach to the simulation of discrete stochastic models. It implements the method of supplementary variables in an algorithmic, rather than mathematical way, without using partial differential equations and is able to solve a wide class of stochastic models. The approach has proven to be very competitive with discrete-event simulation for a class of models that is used in reliability modelling. In this paper we report on our experience with the application of the proxel-based method to a problem provided by the DaimlerChrysler Corporation. The model that portrays the problem describes the correlation between system reliability and warranty costs for automobiles. Until recently, this model was analysed using discrete-event simulation, which resulted in very time-consuming computations. By contrast, the proxel-based method achieved comparable accuracy within a few seconds of computation time. INTRODUCTION The analysis of stochastic models is a common task in reliability modelling. The problem of long and expensive simulations is always present and thus also the need for faster and cheaper approaches. Models which are stiff, or whose measures have a large variance, require a large number of replications when Monte Carlo simulations are used; the model considered in this paper required 20 to 30 hours of computation time. Proxel-based methods are a new way of analysing stochastic models, introduced in (Horton 2002). They are a state space approach, using supplementary variables to model the non-Markovian activities. They thus suffer from the state space explosion problem. However, since they are effectively a discretisation of a system of differential equations, they provide a much more controlled convergence towards the solution as the computation progresses. This fact was exploited in this paper to achieve results of comparable accuracy in just a few seconds. In this paper, proxel simulation is very briefly described, followed by the warranty model that was studied. Results of an experiment are given, which illustrate the behaviour of the method and the application of a corresponding software tool to the problem. PROXEL-BASED SIMULATION The proxel-based method (Horton 2002; LazarovaMolnar and Horton 2003, Lazarova-Molnar and Horton 2004) is a state space-based approach which uses supplementary variables to model non-Markovian activities. It is a numerical algorithm which proceeds in discrete time steps of size dt, as opposed to continuoustime approaches such as are described in (German 2000). One important definition in the proxel-based simulation is the one of the term “state” which is the vector composed of the discrete state of the system and the age intensities of the state changes that are possible in those states. The age intensity of a state change is the time that the model has been waiting in a discrete state for that state change to occur. The probability of the state change happening within a time step of length dt can be computed using the instantaneous rate function (IRF) given the age intensity as a parameter. The proxel, as a basic unit of computation, describes a state of the model in a complete and minimal way. A proxel P is defined as the following: P = (S, t, R, p) S = (Discrete State of the System, Age Intensity Vector) R = (State1, State2,..., StateN) where t is the simulation time, and p denotes the probability that the model is in state S at time t, given that it has been reached from the initial state through the sequence of states R. The Age Intensity Vector is a vector containing the age intensities of the enabled state changes in state S. These are needed for a complete definition of the state of the model, because the probabilities of the next state changes are dependent on how long the concurrent activities have been active. A proxel simulator stores a dynamic list of such proxels and proceeds by generating the set of possible successor proxels at a new time step and computing the probability of these new proxels using the IRF of the corresponding activities. Essentially, the method is equivalent to a discrete-time Markov chain approach in which states are generated on-the-fly. Detailed descriptions of the method can be found in (LazarovaMolnar and Horton 2003, Lazarova-Molnar and Horton 2004). DESCRIPTION OF THE WARRANTY MODEL The model that was analysed was used to predict the warranty costs for automobiles using different warranty strategies. These strategies contained a race condition between a time-based and a mileage-based expiration threshold, whereby the simulation time unit t is measured in miles, and physical time is converted into an equivalent number of miles. Figure 1 shows a stochastic Petri net of this model. Figure 1: Petri net of the warranty analysis model The goal of the analysis of this model was to predict the manufacturer's costs for different types of vehicles and analyse different warranty strategies. As shown in Figure 1, the warranty runs out whenever one of the two conditions was satisfied: either the warranty period has run out or the warranty mileage was reached. During the warranty period, the manufacturer incurs costs whenever the vehicle fails and must be repaired. These costs are a decision factor in constructing the warranty strategy. The discrete-event simulation of this model resulted in long computation times, owing to the rarity of the failures. The proxel-based method, as previously mentioned, is less sensitive to the stiffness of models and results in much shorter computation times. The model was analysed using the following parameters: • Y – the number of years under warranty, • X – the mileage under warranty, and • C – average costs per failure and the following distribution functions: • f – failure distribution function, and • g – time to mileage distribution function. In the proxel-based simulation, the number of years was included in the state vector in the same manner as the age intensities are; therefore the model results in only two discrete states, “under warranty”(U) and “out of warranty”(O). The state vector in the general case has the following form: (discrete state, age intensity of the failure transition, number of years passed) with the initial state being: