José Manuel Colom

Linear Algebraic and Linear Programming Techniques for the Analysis of Place or Transition Net Systems

The structure theory of Place/Transition net systems is surveyed — incorporating new contributions — in a tutorial style, mainly from a linear algebraic perspective. Topics included are: state equation based analysis of safety properties (e.g., boundedness, mutual exclusion, deadlock-freeness, etc.), linear invariants, siphons and traps, implicit places and their application to improve the accuracy of the state equation, and rank theorems (structural conditions for liveness and boundedness based on the rank of the incidence matrix).

A Banker's solution for deadlock avoidance in FMS with flexible routing and multiresource states

Bankers-like approaches to deadlock avoidance are based on a decision procedure to grant active processes resources using information about the maximum needs of resources that a process can request in order to ensure termination. The paper presents an extension of the classical Bankers algorithm to a class of flexible manufacturing systems modeled by means of Petri nets. These systems have two interesting characteristics from the application point of view. First, flexible routing of parts is allowed, and second, a multiset of resources is allowed to be used at each processing step. The decision procedure introduced is polynomial in the Petri net model size.

A Petri Net Structure– Based Deadlock Prevention Solution for Sequential Resource Allocation Systems

A new method for the deadlock prevention problem in concurrent systems where a set of processes share a set of common resources in a conservative way is proposed. It can be applied to flexible manufacturing systems, modeled with Petri nets. In this paper, we present a set of important results related to the deadlock prevention problem in S4PR nets. First, a liveness characterization is introduced, establishing how deadlocks can be studied in terms of circular waits. Second, we show how a circular wait situation corresponds to a particular marking related to a siphon of the Petri net model. Finally, this last characterization is used to obtain an iterative method that successively forbids deadlock related states, synthesizing the control necessary to ensure a final live behavior. The method can be implemented by means of the solutions of a set of integer linear programming problems.

On Weighted T-Systems

Structure theory is a branch of net theory devoted to investigate the relationship between the structure and the behaviour of net system models. Many of its powerful results have been derived for some subclasses of ordinary net systems. The aim of this contribution is to draw a general perspective of the structure theory for a subclass with Marked Graph-like underlying graph but allowing weights: weighted T-graphs (WTG). Weights are convenient to properly model systems with bulk services and arrivals. Properties of WTG and the corresponding weighted T-systems (WTS) are presented at three different levels: purely structural (e.g. in consistent WTG conservativeness is equivalent to strong connectedness), inter-relationships between the structure and the behaviour (e.g. structural liveness and boundedness is equivalent to consistency and strong connectedness) and liveness and reachability characterizations (e.g. deciding liveness is linear wrt. the 1-norm of the unique minimal T-semiflow of a consistent, even unbounded, WTS). Classical results for Marked Graphs can be derived as corollaries. Nevertheless, even in live and consistent WTS, important properties of Marked Graphs do not hold (e.g. P-semiflows based characterization of reachability).

Properties and performance bounds for timed marked graphs

A class of synchronized queueing networks with deterministic routing is identified to be equivalent to a subclass of timed Petri nets called marked graphs. Some structural and behavioral properties of marked graphs are used to show interesting properties of this class of performance models. In particular, ergodicity is derived from the boundedness and liveness of the underlying Petri net representation. In the case of unbounded (i.e., nonstrongly connected) marked graphs, ergodicity is computed as a function of the average transition firing delays. For steady-state performance, linear programming problems defined on the incidence matrix of the underlying Petri nets are used to compute tight (i.e., attainable) bounds for the throughput of transitions for marked graphs with deterministic or stochastic time associated with transitions. These bounds depend on the initial marking and the mean values of the delays but not on the probability distortion functions. The benefits of interleaving qualitative and quantitative analysis of marked graph models are shown. >

Choice-free Petri nets: a model for deterministic concurrent systems with bulk services and arrivals

Among discrete event systems, those exhibiting concurrency are especially challenging, requiring the use of formal methods to deal with them. Petri nets are a well-established such formalism. The structure theory aims at overcoming the state space explosion problem, inherent to the analysis of concurrent systems, by bridging structural and behavioral properties. To date, this has been successfully achieved mainly for some subclasses of ordinary nets. However weights are a modeling convenience in many situations. In this paper we study a formal model for a subclass of concurrent systems with bulk services and arrivals which structurally avoids conflicts. Structural results and techniques for dealing with them are introduced. These include structural conditions on properties of correct behavior and a unified framework for checking general behavioral properties by reasoning solely on the structure.

The resource allocation problem in flexible manufacturing systems

The analysis of resource allocation related aspects is a precondition for the design and control of Flexible Manufacturing Systems. The formulation of this application-driven problem in terms of Petri nets leads to a class of models, with a specific structure-based characterization, which we explore in this presentation. We will concentrate our efforts on the characterization of the liveness of such models. We will also discuss the structural causes of the non-liveness (deadlock of some manufacturing processes) that will allow to state the foundations to introduce control elements which eliminate all the bad states.

Approximate throughput computation of stochastic marked graphs

A general iterative technique for approximate throughput computation of stochastic strongly connected marked graphs is presented. It generalizes a previous technique based on net decomposition through a single input-single output cut, allowing the split of the model through any cut. The approach has two basic foundations. First, a deep understanding of the qualitative behavior of marked graphs leads to a general decomposition technique. Second, after the decomposition phase, an iterative response time approximation method is applied for the computation of the throughput. Experimental results on several examples generally have an error of less than 3%. The state space is usually reduced by more than one order of magnitude; therefore, the analysis of otherwise intractable systems is possible. >

Improving the linearly based characterization of P/T nets

The state equation is a linear description of the reachable markings and firing count vectors of a P/T net. It has the disadvantage that its solution space, in general, includes additional integer unreachable or/and unfirable vectors. As a result, the analysis of properties using this linear characterization, usually leads to necessary or sufficient conditions for satisfying it, but not both. The appearance of these spurious solutions is due to the fact that the state equation does not take into account the order in which transitions fire.

Operational analysis of timed Petri nets and application to the computation of performance bounds

Operational analysis techniques are used to partially characterize the behavior of timed Petri nets under very weak assumptions on their timing semantics. New operational inequalities are derived that are typical of the presence of synchronization and that were therefore not considered in queuing network models. An interesting application of the operational laws to the statement and the efficient solution of problems related to the estimation of performance bounds insensitive to the timing probability distributions is shown. The results obtained generalize and improve in a clear setting results that were derived in the last few years for several different subclasses of timed Petri nets. In particular, the extension to well-formed colored nets appears straightforward and allows an efficient exploitation of model symmetries.<<ETX>>