Piecewise Affine Dynamical Models of Petri Nets – Application to Emergency Call Centers*

We study timed Petri nets, with preselection and priority routing. We represent the behavior of these systems by piecewise affine dynamical systems. We use tools from the theory of nonexpansive mappings to analyze these systems. We establish an equivalence theorem between priority-free fluid timed Petri nets and semi-Markov decision processes, from which we derive the convergence to a periodic regime and the polynomial-time computability of the throughput. More generally, we develop an approach inspired by tropical geometry, characterizing the congestion phases as the cells of a polyhedral complex. We illustrate these results by a current application to the performance evaluation of emergency call centers in the Paris area. We show that priorities can lead to a paradoxical behavior: in certain regimes, the throughput of the most prioritary task may not be an increasing function of the resources.

Behaviour equivalent max-plus automata for timed petri nets under open-loop race-policy semantics

Timed Petri nets and max-plus automata are well known modelling frameworks for timed discrete-event systems. In this paper we present an iterative procedure that constructs a max-plus automaton from a timed Petri net while retaining the timed behaviour. Regarding the Petri net, we essentially impose three assumptions: (a) the Petri net must be bounded, i.e, the reachability graph must be finite; (b) we interpret the Petri net with single server semantics; and (c) the Petri net operates according to the race policy, i.e., the earliest possible transition will fire and thereby possibly consume tokens required by other competing transitions. Under these assumptions we show that the proposed procedure terminates with a finite deterministic max-plus automaton that realises the same timed behaviour as the Petri net. As a variation of the plain race policy, we also consider that a subsequently designed supervisor may temporarily disable distinguished transitions. Again, we present a terminating procedure that constructs a behaviour equivalent deterministic max-plus automaton. We demonstrate by example how the latter automaton can be utilised as an open-loop model in the context of supervisor control.

Temporal Analysis of Influence of Resource Failures on Cyber-Physical Systems Based on Discrete Timed Petri Nets

Advancement of IoT and ICT provide infrastructure to manage, monitor and control Cyber-Physical Systems (CPS) through timely provision of real-time information from the shop floor. Although real-time information in CPS such as resource failures can be detected based on IoT and ICT, improper response to resource failures may cripple CPS and degrade performance. Effective operations of CPS relies on an effective scheme to evaluate the impact of resource failures, support decision making needed and take proper actions to respond to resource failures. This motivates us to develop a methodology to assess the impact of resource failures on operations of CPS and provide the decision support as needed. The goal of this study is to propose solution algorithms to analyze robustness of CPS with respect to resource failures in terms of the impact on temporal properties. Given CPS modeled by a class of discrete timed Petri nets (DTPNs), we develop theory to analyze robustness of CPS by transforming the models to residual spatial-temporal network (RSTN) models in which capacity loss due to resources is reflected. We formulate an optimization problem to determine the influence of resource failures on CPS based on RSTNs and analyze the feasibility to meet the order deadline. To study the feasibility to solve a real problem, we analyze the computational complexity of the proposed algorithms. We illustrate the proposed method by application scenarios. We conduct experiments to study efficiency and verify computational feasibility of the proposed method to solve a real problem.

A Monitoring Approach Based on Fuzzy Stochastic P-Timed Petri Nets of a Railway Transport Network

This paper proposes a monitoring approach based on stochastic fuzzy Petri nets (SFPNs) for railway transport networks. In railway transport, the time factor is a critical parameter as it includes constraints to avoid overlaps, delays, and collisions between trains. The temporal uncertainties and constraints that may arise on the railway network may degrade the planned schedules and consequently affect the availability of the transportation system. This leads to many problems in the decision and optimization of the railway transport systems. In this context, we propose a new fuzzy stochastic Petri nets for monitoring (SFPNM). The main goal of the proposed supervision approach is to allow an early detection of traffic disturbance to avoid catastrophic scenarios and preserve stability and security of the studied railway networks. Finally, to demonstrate the effectiveness and accuracy of the approach, an application to the case study of the Tunisian railway network is outlined.