Analysis of transportation system with the use of Petri nets

The paper considers problem of city transportation system performance. Reliability analysis of such a complex system is complicated by several factors. One of th em is the possibility of logistic support elements unreliability defined as unavailability of spare elements when de sired, what in result may lead to decrease of perfo rmance of the system being supported. Thus, both systems must be considered in a single model. However, the simultaneous setting of all structural parameters ( .g. redundancy, repair shop capacity) and control variables (e.g. spare part inventory levels, maintenance poli cy parameters, time resource) is mathematically a h ard problem. This paper investigates Petri net model of the system with the use of Monte Carlo simulation as a solution technique. Comparison of the simulation re sults with characteristics of real-life system is g iven. Kowalski Marcin, Magott Jan, Nowakowski Tomasz, Wer bińska-Wojciechowska Sylwia Analysis of transportation system with the use of P etri nets 266 maintenance parameters (or cost of maintenance) on one side, and the performance level on the other. The interest in development and investigation of maintenance problems has been extensively discussed in the literature since the early 1960s. The basic review in the area of maintenance modelling i s prepared by Pierskalla & Voelker [20], where authors investigated discrete time vs. continuous time maintenance models, later updated by ValdezFlores & Feldman [22]. For other surveys see e.g. [3], [14], [15], [1], [19], [21], [23], [24]. However, most of the maintenance models investigated in the literature on reliability theor y assume, that all the necessary logistic support resources, which include maintenance resources, support personnel, logistic information and data, spares and repair parts, and facilities, are immediately provided when it is desired. In practic e, the repair capacity is not infinite, and logistic information may be unreliable. Moreover, the influence of a spare provisioning policy on the maintenance policy also cannot be ignored, since spares are ordered and carried in the limited quantity, and the procurement lead time is not negligible. The problem of providing an adequate and efficient supply of spare parts, in support of maintenance an d repair of operational systems, has been researched for many decades. Recent overview of these models is made by Nowakowski & Werbi ńskaWojciechowska in [17]. Consequently, reliability of complex systems (e.g. transportation systems, aircraft systems) can be difficult to analyze for several reasons. First, bo th systems, logistic and being supported are integrate d and thus should be considered in a single model. However, growing body of existing literature in the investigated research area treats maintenance, replacement and inventory decisions separately [24] . Second, commonly used analytical techniques for reliability evaluation are applied probability theo ry, renewal reward processes, Markov decision theory, and Fault Trees. Each of these techniques has advantages and disadvantages and the choice depends on the system being modelled. All of them require simplifying assumptions about time to failure behaviour of the system components. Moreover, Markov method analyses the system by identifying all the different states in which the system can reside and is able to produce accurate system reliability measures by assigning rates of transition between these states. However, the Markov method has its own drawbacks in its application for a relatively large system to establ ish the state transition model is an intractable task. Traditional Fault Tree Analysis (FTA) [6] is probabilistic approach to safety, reliability, and risk analysis. Traditional fault trees contain Boolean gates to represent how component failures combine to produce system failures. These fault trees are n ow called static. In papers [4], [5], dynamic fault tr ees (DFTs) are presented. Gates of DFTs can express the following features: • dynamic replacement of failed components from pools of spares, • failures can occur only in a predefined order. In paper [2], DFTs have been extended by repair boxes. These boxes can express a repair time of components. In this paper, transformations of different gates and repair boxes into Stochastic Hi gh Level Colored Petri Net are given. In calculation of probabilistic characteristics of systems using DFTs, the following formalisms are used: Markov models [4], [5], Petri nets [2], Bayesian networks [13]. When analyzing the transportation system, we have to analyze not only repair and lead time, but time consuming replacement process and time resource as well. Hence, DFTs with repair boxes are not sufficient to represent the transportation system. Moreover, Fault Tree with Time Dependencies provides an interesting solution for nondeterministic models [11], [12]. In these models, time parameters are described by minimal and maximal values, but without probabilistic characteristics. The FTTD technique has been proposed as a convenient approach to describe the values of the delay times of system of systems task performance on the ESREL conference in 2008 [10]. The investigated problem has regarded to simple logistic support model performance. Later, there is proposed method application example presented in [9]. However, this method cannot be applied when time between failures and repair times are expresse d probabilistically. In contrast to the analytic approaches Monte Carlo simulation can be broadly used. However, Monte Carlo simulation is time-consuming because of the intensive computations. This is because an extremely large number of simulated samples may be needed to estimate the reliability parameters at a high level of confidence. Following this consideration, in the paper, Petri n ets are used to support the reliability analysis of complex real-life system performance. The primary contribution of this research is to propose a Stochastic High-level Petri Net model for presented below transportation system. This model is based on standard of High-level Petri Net [7] an d on generalized stochastic Petri nets [1]. For this model, simulation experiments have been