Systems Of Simple Sequential Processes Research Articles

Controllability of Weakly Dependent Siphons under Elementary-Siphon Control

Monitors are added upon problematic siphons to avoid deadlocks. Li and Zhou add monitors to elementary siphons only while controlling the rest dependent siphons to save cost. After failing a Marking Linear Inequality (MLI) test, they perform a Linear Integer Programming (LIP) test (NP-hard). We proposed a new MLI test earlier to avoid the LIP and extended it to S3PGR2 (systems of simple sequential processes with general resource requirements) for strongly dependent siphons (SDS). The control policy for weakly dependent siphons (WDS) is rather conservative due to some negative terms in the MLI. This paper shows that WDS and SDS have the same controllability (i.e., MLI). As a result, the control for WDS needs no longer be that conservative. We also develop an optimization (by redundancy elimination) of the computation required for LIP test to ensure deadlock prevention of S3PR (Systems of Simple Sequential Processes with Resources). A favorable result for this policy is that any n-dependent (n>2) WDS (similar to SDS) needs no monitor and hence the complexity for synthesizing a controller becomes polynomial.

Maximally permissive deadlock prevention via an invariant controlled method

Recently, researchers have proposed a novel and computationally efficient method to design optimal control places and an iteration approach that computes the reachability graph once to obtain a maximally permissive, if any, liveness-enforcing supervisor of flexible manufacturing systems (FMS). The approach solves the set of integer linear inequalities to compute control places. If, given a Petri net model, no solution exists, the optimal control place does not exist for the Petri net model. We discover that a solution always exists for systems of simple sequential processes with resources (S3PR), but not for the case of FMS modelled by generalised Petri nets (GPN). We propose a theory to prove that there are no good states that will be forbidden by the control policy for S3PR, in which live and dead states cannot have the same weighted sum of tokens in the complimentary set of a siphon. For a system of simple sequential processes with general resource requirements (S3PGR2) modelled by GPN, we find the reason why the integer linear programming (ILP) may not have solutions, which is consistent with the fact that optimal supervisor synthesis for GPN remains unknown. We show that live and dead states may have the same weighted sum of tokens in the complimentary set of a siphon in a GPN. These theoretical results are verified by case studies.

Deadlock prevention policy for a class of petri nets based on complementary places and elementary siphons

For a class of Petri nets called Systems of Simple Sequential Processes with Resources $$(\mathrm{{S}}^{3}\mathrm{{PR}})$$ ( S 3 PR ) , this paper proposes a sufficient condition under which there exists a complementary-place supervisor to enforce their liveness. Moreover, an algorithm is proposed to design liveness-enforcing supervisors based on complementary places and elementary siphons. The significance of the proposed policy lies in its design simplicity. A flexible manufacturing system example shows that in some cases, the proposed policy can obtain a structurally simpler supervisor with 99.9 % maximal permissive behavior.

Control policy for a subclass of Petri nets without reachability analysis

Traditional maximally permissive deadlock prevention control for flexible manufacturing systems requires costly reachability analysis. It has been a hot race to synthesise optimal controllers to be maximally permissive with fewest monitors. Previous work shows that among all n -dependent siphons, only one siphon (whose unmarked state follows some token distribution) needs to be controlled. This greatly simplifies the supervisor synthesis as well as minimises the number of monitors required while making the controlled net maximally permissive (i.e. all live states can be reached.). This study further proposes a maximally permissive control policy for a subclass of systems of simple sequential processes with resources (S 3 PR) based on the above theory of token distribution pattern of unmarked siphons.

On the lower bound of monitor solutions of maximally permissive supervisors for a subclass α-S3PR of flexible manufacturing systems

Recently, a novel and computationally efficient method – based on a vector covering approach – to design optimal control places and an iteration approach that computes the reachability graph to obtain a maximally permissive liveness enforcing supervisor for FMS (flexible manufacturing systems) have been reported. However, it is unclear as to the relationship between the structure of the net and the minimal number of monitors required. This paper develops a theory to show that the minimal number of monitors required cannot be less than that of basic siphons in α-S3PR (systems of simple sequential processes with resources). This confirms that two of the three controlled systems by Chen et al. are of a minimal monitor configuration since they belong to α-S3PR and their number in each example equals that of basic siphons.

Robustness of deadlock control for a class of Petri nets with unreliable resources

A variety of deadlock control policies based on Petri nets have been proposed for automated manufacturing systems (AMSs). Most of them prevent deadlocks by adding monitors for emptiable siphons that, without an appropriate control policy, can cause deadlocks, where the resources in a system under consideration are assumed to be reliable. When resources are unreliable, it is infeasible or impossible to apply the existing control strategies. For systems of simple sequential processes with resources (S3PR), this paper bridges the gap between a divide-and-conquer deadlock control strategy and its application to real-world systems with unreliable resources. Recovery subnets and monitors are designed for unreliable resources and strict minimal siphons that may be emptied, respectively. Normal and inhibitor arcs are used to connect monitors with recovery subnets in case of necessity. Then reanalysis of the original Petri net is avoided and a robust liveness-enforcing supervisor is derived. Examples are presented to illustrate the proposed methodology.

Recursive solution of number of reachable states of a simple subclass of FMS

This paper aims to compute the number of reachable (forbidden, live and deadlock) states for flexible manufacturing systems (FMS) without the construction of reachability graph. The problem is nontrivial and takes, in general, an exponential amount of time to solve. Hence, this paper focusses on a simple version of Systems of Simple Sequential Processes with Resources (S3PR), called kth-order system, where each resource place holds one token to be shared between two processes. The exact number of reachable (forbidden, live and deadlock) states can be computed recursively.

Transition-Based Deadlock Detection and Recovery Policy for FMSs Using Graph Technique

A transition-controlled deadlock detection and recovery prevention policy is presented for a subclass of Petri nets used to model flexible manufacturing systems. The subclass is called systems of simple sequential processes with resources (S 3 PR). The proposed policy is different from the standard deadlock prevention policies. Instead of adding control places, this policy adds a controlled transition to solve a group of deadlocked markings that have the same graph-based property. Finally, the results of our study indicate that the proposed policy appears to be more permissive than those existing ones that add control places.

The Liveness of WS<sup>3</sup>PR: Complexity and Decision

Petri nets are a kind of formal language that are widely applied in concurrent systems associated with resource allocation due to their abilities of the natural description on resource allocation and the precise characterization on deadlock. Weighted System of Simple Sequential Processes with Resources (WS3PR) is an important subclass of Petri nets that can model many resource allocation systems in which 1) multiple processes may run in parallel and 2) each execution step of each process may use multiple units from a single resource type but cannot use multiple resource types. We first prove that the liveness problem of WS3PR is co-NP-hard on the basis of the partition problem. Furthermore, we present a necessary and sufficient condition for the liveness of WS3PR based on two new concepts called Structurally Circular Wait (SCW) and Blocking Marking (BM), i.e., a WS3PR is live iff each SCW has no BM. A sufficient condition is also proposed to guarantee that an SCW has no BM. Additionally, we show some advantages of using SCW to analyze the deadlock problem compared to other siphon-based ones, and discuss the relation between SCW and siphon. These results are valuable to the further research on the deadlock prevention or avoidance for WS3PR. key words: concurrent systems, Petri nets, liveness, deadlock, resource allocation

Design of Liveness-Enforcing Supervisors for S3PR Based on Complementary Places

In this article, an algorithm is proposed to design liveness-enforcing supervisors for systems of simple sequential processes with resources (S 3 PR) based on complementary places. Firstly, a mixed integer programming (MIP) based deadlock detection method is used to find unmarked strict minimal siphons from an infinite-capacity net. Next, the finite-capacity net, in which liveness can be enforced, is obtained by adding capacity function to the infinite-capacity net. Finally, complementary-place transformation is used to transform the finite-capacity net into an infinite-capacity net. This article focuses on adding a complementary place to each operation place that is related to unmarked siphons, deals with the deadlock problem from a new view point, and hence advances the deadlock control theory. Compared with the existing methods, the new policy is easier to implement for real industrial systems. More importantly, design of a complementary-place supervisor is very easy. Finally, in some cases, the new policy can obtain a structurally simpler supervisor with more permissive behavior than the existing methods do. A flexible manufacturing systems (FMS) example is used to compare the proposed policy with some other methods.

A Merging Method for Siphon-Based FMS Maximally Permissive Control with Simple Structures

It has been a hot race to design optimal controllers to be maximally permissive with fewest monitors in the shortest amount of time. Recent maximally permissive deadlock prevention controls for S3PR (Systems of Simple Sequential Processes with Resources) reduce the computation burden by considering only a small portion of all forbidding markings and employs much fewer monitors by a linear integer programming method. However, it still requires costly reachability analysis. This paper proposes a method (the first of its kind) to merge several monitors into a single one while not losing states. It achieves the same best results in the literature while avoiding the time-consuming reachability analysis which does not scale well with the size of the nets.

Optimal controllability of 2-composed siphons in a class of Petri nets

Proposesd is a controllability condition of a siphon that is composed of two elementary siphons in a class of Petri nets, Systems of Simple Sequential Processes with Resources (S3PR). It is shown that, under the condition, an S3PR admits a maximally permissive liveness-enforcing supervisor that is expressed by a set of monitors if every dependent siphon is 2-composed, which can be decided by an algorithm with polynomial complexity.

A new optimal control policy for a well-known S3PR (systems of simple sequential processes with resources)

There are many studies reported in the literature comparing the effectiveness of new control policies by testing them against a well-known S3PR (systems of simple sequential processes with resources) model. We propose a new approach that recovers the system from empty-siphon states to former live states. It therefore attains the same number of states as the original uncontrolled model by adding monitors (and control arcs) similar to the prevention approach. There is no need to perform a reachability analysis. INA (integrated net analyser) analysis indicates that the resulting controlled model is live and reaches all 26,750 states (in the uncontrolled model), more than the maximally permissive 21,581 states. Only seven monitors are employed, fewer than in most other approaches. This arises from the fact that no new problematic siphons are generated due to the added monitors. We discuss the disadvantages (a variant of the scheme to overcome the disadvantages is also discussed) and physical meaning of the policy. We further propose a lossless approach by colouring some arcs. This not only avoids material loss, but also tackles the livelock problem.

An algorithm to find the minimal initial markings of resource places ensuring liveness of finite-capacity S3PR

Deadlock prevention can be achieved by configuring proper initial markings to obtain a live Petri net model. For a class of Petri nets called finite-capacity Systems of Simple Sequential Processes with Resources (S3PR), an algorithm is proposed to find the minimal initial markings of resource places ensuring its liveness. First, conditions on initial markings of resource places are computed under which the strict minimal siphons (SMS) can never be emptied. Then, the minimal initial markings of resource places are computed to ensure liveness of the net. By the proposed algorithm, a live net can be obtained without changing the structure of the original model. The main practical implication of our work is that it can lower the facility cost. A simple example is used to illustrate the application of the algorithm.

An extraction algorithm for a set of elementary siphons based on mixed-integer programming

Elementary siphons are useful in the development of a deadlock prevention policy for a discrete event system modeled with Petri nets. This paper proposes an algorithm to iteratively extract a set of elementary siphons in a class of Petri nets, called system of simple sequential processes with resources (S3PR). At each iteration, by a mixed-integer programming (MIP) method, the proposed algorithm finds a maximal unmarked siphon, classifies the places in it, extracts an elementary siphon from the classified places, and adds a new constraint in order to extract the next elementary siphon. This algorithm iteratively executes until no new unmarked siphons can be found. It finally obtains a unique set of elementary siphons and avoids a complete siphon enumeration. A theoretical analysis and examples are given to demonstrate its efficiency and practical potentials.

A novel liveness condition for S3PGR2

Systems of simple sequential processes with resources (S3PR), modeled by ordinary Petri nets (OPNs) are live if and only if no siphons (a set of places) can ever become empty of tokens. Systems of simple sequential processes with general resources requirement (S3PGR2), modeled by general Petri nets (GPNs), are a generalization of S3PR. It has been a hot research topic to find the sufficient and necessary condition of liveness for S3PGR2. When an OPN (respectively, GPN) is deadlocked, the set of all unmarked (respectively, non-max-marked) places forms a siphon, which is said to be deadly marked. However, when an S3PGR2 is livelocked but not deadlocked, e.g., both live and dead transitions exist, the set of non-max-marked places, denoted by [Formula: see text], may not form a siphon. Thus, it is limited using deadly marked or max-controlled siphons to characterize liveness of S3PGR2. The first author proposed earlier max’ -controlled siphons to characterize liveness of S3PGR2. However, the presence of a set of non-max’ -marked places is not a sufficient condition for an S3PGR2 to be non-live and cannot be relied on to devise mixed integer programming (MIP) testing. In this paper, we present the sufficient and necessary liveness condition of S3PGR2. We define pivot markings, under which we prove that [Formula: see text] forms a siphon. To detect livelocks, the number of disabled arcs must be maximal under the pivot marking. This result is useful to reorient the MIP test of S3PR for S3PGR2.

A Method to Compute Strict Minimal Siphons in a Class of Petri Nets Based on Loop Resource Subsets

Strict minimal siphons (SMS) play an important role in the development of deadlock control policies for flexible manufacturing systems (FMS). For a class of Petri nets called Systems of Simple Sequential Processes with Resources (S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> PR), the resource circuit-based method is an effective way to compute SMS. In this paper, a more effective one to compute SMS is proposed. First, the concepts of loop resource subsets and their characteristic resource subnets are proposed. Next, sufficient and necessary conditions for loop resource subsets to generate SMS are established. Finally, an algorithm is given to find all the SMS based on loop resource subsets. Since the number of loop resource subsets is much less than that of resource circuits and their combinations, the computational efficiency of the SMS enumeration task is significantly improved by the proposed method. An FMS example is used to illustrate the application of the proposed method, and computational time comparisons are provided on several S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> PRs to show its superior efficiency.

On Intrinsically Live Structure of a Class of Generalized Petri Nets Modeling FMS

The deadlock control and liveness-enforcement are mostly implemented based on siphons in most existing work using structural theory. Our previous work has revealed a kind of intrinsically live structures in a class of generalized Petri nets called systems of simple sequential processes with weighted resources allocation (WS3PR). The liveness-enforcement is achieved by analyzing the structure including weight information and reconfiguring the initial marking of resource places without considering the initial token counts in idle process places. This work extends it to a more general class than WS3PR. Instead of siphons, a new concept of structural objects is defined to carry weight information and reflect the circular waits of resources. The numerical relationship between initial markings of resource places and weights of arcs is investigated to yield an intrinsically live structure-based liveness-enforcing method that requires no external monitors. The work provides an insight into structures of generalized Petri nets and a new revenue to achieve the desired deadlock control. Several examples are used to illustrate the proposed method.

A Polynomial Algorithm for Computing Elementary Siphons in a Class of Petri Nets

A set of elementary siphons plays a key role in the development of deadlock prevention policies for automated manufacturing systems. This paper addresses the computation problem for elementary siphons in a subclass of Petri nets which are basic systems of simple sequential processes with resources (BS3PR) and can model many automated manufacturing systems. An algorithm for enumerating elementary siphons is established by the one-to-one relationship between maximal perfect resource-transition circuits (MPCs) and strict minimal siphons. A set of MPCs is first computed, followed by a set of elementary siphons in a BS3PR. The presented algorithm is proved to have polynomial-time complexity. An example is used to illustrate the algorithm.

Segment theory to compute elementary siphons in Petri nets for deadlock control

Unlike other techniques, Li and Zhou add control nodes and arcs for only elementary siphons greatly reducing the number of control nodes and arcs (implemented by costly hardware of I/O devices and memory) required for deadlock control in Petri net supervisors. Li and Zhou propose that the number of elementary siphons is linear to the size of the net. An elementary siphon can be synthesized from a resource circuit consisting of a set of connected segments. We show that the total number of elementary siphons, |ПE|, is upper bounded by the total number of resource places |PR | lower than that min(|P|, |T|) by Li and Zhou where |P| (|T|) is the number of places (transitions) in the net. Also, we claim that the number of elementary siphons |ПE| equals that of independent segments (simple paths) in the resource subnet of an S3PR (systems of simple sequential processes with resources). Resource circuits for the elementary siphons can be traced out based on a graph-traversal algorithm. During the traversal process, we can also identify independent segments (i.e. their characteristic T-vectors are independent) along with those segments for elementary siphons. This offers us an alternative and yet deeper understanding of the computation of elementary siphons. Also, it allows us to adapt the algorithm to compute elementary siphons in [2] for a subclass of S3PR (called S4PR) to more complicated S3PR that contains weakly dependent siphons.