The Design of an Online Petri Net Based Token (Ludo) Game

The behavior of Petri nets with exponentially distributed firing times can be represented by labeled directed "slate" graphs in which labels describe the probabilities of transition firings (displacement of tokens) between vertices of the graph. The interactive firing of transitions in subsequent markings is known as token game. This development is analogous to that of games of strategy, in which each player is assigned a set of possible strategies (actions or moves) and each possible combination of strategies, one for each player, produces an outcome. That is, player's fortunes are intertwined and determined by chance events. The design of an interactive Petri net-based token (ludo) game is presented in which transition firing is determined by dice cast. Programming is considered and developed in Visual Basic 6.0 programming language, which is an object-oriented, event-driven and visual programming environment

M-timed Petri nets, priorities, preemptions, and performance evaluation of systems

In M-timed Petri nets, firing times are exponentially distributed random variables associated with transitions of a net. Several classes of M-timed Petri nets are discussed in this paper to show increasing “modelling power” of different nets. Conflict-free nets can model M- and E k -type queueing systems. Free-choice nets can also represent H k -type systems. Systems with several classes of users and with service priorities assigned to user classes require nets with inhibitor arcs. Preemption of service can be represented by extended nets with escape (or generalized inhibitor) arcs. Finally, to provide flexible modelling of scheduling and decision strategies, enhanced Petri nets are introduced with two classes of transitions, immediate and timed ones, and with (exponentially distributed) firing times associated with the timed transitions. It is shown that the behavior of bounded M-timed Petri nets can be represented by finite “state” graphs which are finite-state continuous-time homogeneous Markov processes. Stationary probabilities of states can thus be obtained by standard techniques used for analysis of Markov chains, and then operational analysis can be applied for performance evaluation. Simple models of interactive systems are used as an illustration of modelling

Simulation and numerical solution of stochastic Petri nets with discrete and continuous timing

We introduce a novel stochastic Petri net formalism where discrete and continuous phase-type firing delays can appear in the same model. By capturing deterministic and generally random behavior in discrete or continuous time, as appropriate, the formalism affords higher modeling fidelity and efficiencies to use in practice. We formally specify the underlying stochastic process as a general state space Markov chain and show that it is regenerative, thus amenable to renewal theory techniques to obtain steady-state solutions. We present two steady-state analysis methods depending on the class of problem: one using exact numerical techniques, the other using simulation. Although regenerative structures that ease steady-state analysis exist in general, a noteworthy problem class arises when discrete-time transitions are synchronized. In this case, the underlying process is semi-regenerative and we can employ Markov renewal theory to formulate exact and efficient numerical solutions for the stationary distribution. We propose a solution method that shows promise in terms of time and space efficiency. Also noteworthy are the computational tradeoffs when analyzing the embedded versus the subordinate Markov chains that are hidden within the original process. In the absence of simplifying assumptions, we propose an efficient regenerative simulation method that identifies hidden regenerative structures within continuous state spaces. The new formalism and solution methods are demonstrated with two applications

Symbolic Analysis of Timed Petri Nets

In timed Petri nets temporal properties are associated with transitions as transition firing times (or occurrence times). Specific properties of timed nets, such as boundedness or absence of deadlocks, can depend upon temporal properties and sometimes even a small change of these properties has a significant effect on the net’s behavior (e.g., a bounded net becomes unbounded or vice versa). The objective of symbolic analysis of timed nets is to provide information about the net’s behavior which is independent of specific temporal properties, i.e., which describes preperties of the whole class of timed nets with the same structure

Optimal trajectory generation for Petri nets

Recently, the increasing complexity of IT systems requires the early verification and validation of the system design in order to avoid the costly redesign. Furthermore, the efficiency of system operation can be improved by solving system optimization problems (like resource allocation and scheduling problems). Such combined optimization and validation, verification problems can be typically expressed as reachability problems with quantitative or qualitative measurements. The current paper proposes a solution to compute the optimal trajectories for Petri net-based reachability problems with cost parameters. This is an improved variant of the basic integrated verification and optimization method introduced in [11] combining the efficiency of Process Network Synthesis optimization algorithms with the modeling power of Petri nets

Timed petri net simulation of flexible manufacturing systems

Standard Petri nets have been used to model and analyze Flexible Manufacturing Systems. The timed Petri net, which can incorporate the time delay associated with manufacturing events, provides additional information about real time behavior of practical systems. The Timed Petri Net Simulation Tool, a highly interactive graphical tool, is applied to simulate the modeled flexible manufacturing systems. Timed Petri net models are experimented with. Machine utilization data and throughput are obtained. Analysis of the results shows that the system performance can be optimized by choosing proper parameters

M-timed Petri nets and Markov chains in modelling of computer systems

It is shown that the behavior of enhanced free-choice bounded M-timed Petri nets, i.e., Petri nets with two classes of transitions, immediate and timed ones, and with exponentially distributed firing times of timed transitions, can be represented by state-transition graphs which are finite continuous-time homogeneous Markov chains. Moreover, for each finite continuous-time homogeneous Markov chain there exists an enhanced free-choice bounded M-timed Petri net with the state-transition graph isomorphic to this chain. These two classes of models are thus equivalent. A straight-forward application of (enhanced) M-timed Petri nets is modelling and performance evaluation of Markovian queueing systems, and in particular closed-network models of computer systems. Simple models of interactive systems are used as an illustration of modelling. as an illustration of modelling

Timed Petri nets and performance evaluation of systems

Several simple applications of timed Petri nets to modeling and performance analysis of concurrent systems are presented as an illustration of a uniform approach to analysis of a wide class of discrete-event systems. Such a unified approach is used in a graduate course on performance evaluation of systems at Memorial University