A Simple Effective Method for Deadlock Prevention of S³PR, GS³PR, and S⁴PR Petri Net Models of Automated Manufacturing Systems

1. Introduction

Discrete event systems (DES) are widely used systems. They cover different kinds of systems in human society. Most widespread are discrete manufacturing systems usually named automated manufacturing systems (AMS), previously named flexible manufacturing systems (FMS). Railways and road transport are also very widespread. Operations accomplished in them (the activity of rail-switches and semaphores, respectively) can also be understood to be a kind of DES. In such complex systems, deadlocks can occur. In AMS deadlocks can lead to a stoppage of production and succeeding economics damages, while in transport even to dangerous disasters with toll of life. Allocation of resources in AMS is the main source of deadlocks. On this account, as to deadlock prevention in them, resource allocation systems (RAS) are the preferred subject of interest in this paper. However, there exists also many other important kinds of DES inside informatics, business, economy, aeronautics, military, etc. Consequently, the deadlock prevention is very important branch of science. Using Petri nets (PN) for modeling DES is very frequent. It has a great advantage that the PN-based models can also be used in different methods for deadlock prevention. Hence, they will also be applied here, in this paper, for both, modeling DES and the deadlock prevention. We are focused here exclusively on deadlock prevention in AMS. In such systems resources are represented by robots, different kinds of machine-tools, buffers, transport belts, automated guided vehicles, etc. Sharing such devices by multiple production lines represents a dominant source of deadlocks in AMS.

1.1. Petri Nets

In principle, from purely mathematical point of view, PNs are a kind of bipartite directed graphs, i.e., graphs with two types of nodes (places and transitions) and two types of edges (directed arcs from places to transitions and from transitions to places). The places $p_i\in P,i=1,\dots ,n$ (i.e., $\left|P\right|=n$), while transitions $t_j\in T,j=1,\dots ,m$ (i.e., $\left|T\right|=m$). Places model particular operations or activities inside of DES. Their states are expressed by marking $M(p)$ - i.e., by the number of tokens $n_t\in {\mathbb{Z}}_{\ge 0}$ put into them. Transitions represent discrete events which start or finish these operations.

More precisely, a Petri net can be expressed as $N=(P,T,F,W)$ where P and T are finite nonempty sets, $P\cup T\ne \varnothing$ and $P\cap T=\varnothing$; $F=(P\times T)\cup (T\times P)$ is the flow relation in N (consisting of flows from P to T and vice versa); $W:(P\times T)\cup (T\times P)\to \mathbb{N}$ assigns a weight to an arc in such a way that $W(f)>0$ if $f\in F$ and $W(f)=0$ otherwise. Here, $\mathbb{N}=\{0,1,2,\dots \}$ represents the set $\mathbb{Z}$ of integers increased by zero. The net N where $\forall f\in F,W(f)=1$, is named as the ordinary Petri net (OPN) $N=(P,T,F)$. If $\exists f\in F,W(f)>1$, the net $N=(P,T,F,W)$ is named as the generalized net (GPN).

A place $p\in P$ in N is marked at marking M if $M(p)>0$. A place $p\in P$ is insufficiently marked with respect to a transition $t\in p^{•}$ at M if $M(p)<W(p,t)$. Here, $p^{•}$ is the set of output transitions emerging from the place p. If at least one place in a subset $S\subseteq P$ is marked at M then S is marked at M. $M(S)$ denotes the sum of tokens in all places in S i.e., $M(S)={\sum }_{p\in S}M(p)$. If $M(S)=0$, S is said to be empty at M. If $\forall p\in S,\forall t\in p^{•},M(p)<W(p,t)$, S is said to be insufficiently marked at M .

A subset $D\subseteq P$ is marked by M iff (if and only if) at least one place $p\in D$ is marked by M. Thus, $M(D)={\sum }_{p\in D}M(p)$ represents the sum of tokens in all places in D.

In general, the preset and postset of a node $x\in P$ is defined as follows: ${}^{•}x=\{y\in P\cup T|(y,x)\in F\}$ is the preset of a node $x\in P\cup T$, while $x^{•}=\{y\in P\cup T|(x,y)\in F\}$ is the postset of a node $x\in P\cup T$.

Let a marked ordinary Petri net is $(N,M)$. A transition t is enabled at M, expressed by $M[t>$, if $\forall p\in {}^{•}t,M(p)>0$. An enabled transition t at M can be fired. Consequently, a new marking $M_1$ arises. It may be denoted by $M[t>M_1$, where $M_1(p)=M(p)-1,\forall p\in {}^{•}t\setminus t^{•};M_1(p)=M(p)+1,\forall p\in t^{•}\setminus {}^{•}t$; otherwise, $M_1(p)=M(p)$.

PN marking M is an $(n\times 1)$ vector. It is frequently called as the state vector. The evolution of PN marking is performed by means of the following relationship

$$M_{k+1}=M_k+[N].\overrightarrow{{\sigma }_k},k\in \mathbb{N}$$

(1)

where $M_0$ is an initial marking, $[N]$ is the incidence matrix, $\overrightarrow{{\sigma }_k}:T\to \mathbb{N}$ is a vector of non negative integers, called a firing vector. Elements of $\overrightarrow{{\sigma }_k}(t)$ indicate the algebraic sum of all occurrences of t in $\overrightarrow{{\sigma }_k}$. The set of all markings reachable from M by firing any possible sequence of transitions is denoted as $R(N,M)$. $R(N,M_0)$ is the set of reachable markings of a Petri net starting from the initial marking $M_0$.

The equation (1) constitutes the mathematical model of the Petri net N. The matrix $[N]={[Post]}^T-[Pre]$ is the $(n\times m)$ incidence matrix. It presents the matrix form of the set F. It can also be written in the shape $[N](p,t)=W(t,p)-W(p,t)$. Finally, $\overrightarrow{{\sigma }_k}$ is an $(m\times 1)$ vector expressing the firing of transitions. It can also be named as the control vector because it represents the state of transitions. When a transition $t_i$ is enabled (i.e., able to be fired) i-th element of the vector $\overrightarrow{{\sigma }_k}$ is indicated by 1. In the opposite case, when the transition is disabled, it is indicated by 0. Important is to say that the enabled transition may or may not be fired. The marking $M(p)$ of a place p is an integer indicating the number of tokens located in this place. The place p is marked by M iff $M(p)>0$. The mathematical relation ’if and only if’ is expressed by the abbreviation ’iff’ in the whole paper.

A non-empty subset of places $S\subseteq P$ is a siphon iff ${}^{•}S\subseteq S^{•}$. A non-empty subset of places $Q\subseteq P$ is a trap iff $Q^{•}\subseteq {}^{•}Q$. A siphon S is minimal iff there is no siphon contained in it as a proper subset. A minimal siphon S is said to be strict if ${}^{•}S⫋S^{•}$. When a minimal siphon S does not contain a marked trap, it is named to be strict minimal siphon (SMS). Such siphons have to be controlled in order to prevent their emptying.

Siphons are extensively used in a large amount of approaches concerning the deadlock control and liveness-enforcing for both OPN and GPN models of AMSs.

A siphon (trap) S (Q) is said to be empty at marking M iff $M(S)=0$ ($M(Q)=0$).

S or Q is said to be max-marked (min-marked) at a marking M iff $\exists p\in S$ or $\exists p\in Q$ such that $M(p)\ge ma{x}_{p^{•}}(M(p)\ge mi{n}_{p^{•}})$, where $ma{x}_{p^{•}}=ma{x}_{t\in p^{•}}(W(p,t))(mi{n}_{p^{•}}=mi{n}_{t\in p^{•}}(W(p,t)))$.

Any PN is deadlock-free when each reachable marking enables a transition, while it is live when each reachable marking enables an occurrence sequence containing all transitions. A transition t is dead [56] in N if no marking of $[M_0\rangle$ enables t. A deadlock, or dead marking, is a marking enabling no transition. N is deadlock-free if no deadlock belongs to $[M_0\rangle$; otherwise it is deadlockable. The net N is structurally deadlockable if, for every marking M, $(N,M)$ is deadlockable. A transition t is live in N if for every marking M in $[M_0\rangle$, there is a marking $M^{\prime }$ in $[M\rangle$ enabling t. N is live if every transition is live in N. N is structurally live if a marking M exists such that $(N,M)$ is live.

In particular cases, it is possible to prevent deadlocks by a thoughtful change in the system structure. Sometimes even a slight change in the structure of the system itself is sufficient in order to reorganize succession of internal processes in such a way that deadlocks are prevented. However, in systems where the succession of internal processes is strongly prescribed (like in case of technological processes in AMS), supplementing the original structure with an additional structure is necessary. Such a structure is a supervisor, which is designed so that, after connecting to the original system, it is able to guarantee the deadlock prevention in the controlled system.

2. Most Commonly Used Kinds of PN Models of AMS

Mostly used PN models of AMS are Systems of Simple Sequential Processes with Resources (S³PR) and System of Sequential Systems with Shared Process Resources (S⁴PR).

2.1. S³PR Models of AMS

Definition 1 [31,34] A simple sequential process (S²P) is a Petri net $N=(P_A\cup \{p^0\},T,F)$ where:

1) $P_A\ne \varnothing$ is called the set of activity places (here ∅ means the empty set);

2) $p^0\notin P_A$ is called the idle process place;

3) N is a strongly connected state machine; and

4) every circuit of N contains place $p^0$.

Definition 2 [31,34] A simple sequential process with resources (S²PR) is a Petri net $N=(\{p^0\}\cup P_A\cup P_R,T,F)$ such that:

1. The subnet generated by $X=P_A\cup \{p^0\}\cup T$ is an S²P.

2. $P_R\ne \varnothing$ and $(P_A\cup \{p^0)\cap P_R=\varnothing$.

3. $\forall p\in P_A,\forall t\in {}^{•}p,\forall t^{\prime }\in p^{•},\exists r_p\in P_R,{}^{•}t\cap P_R=t^{\prime •}\cap P_R=r_p$ (here ${}^{•}p$ represents all input transitions of the place p, ${}^{•}t$ represents all input places of the transition t, $p^{•}$ represents all output transitions of the place p, $t^{\prime •}$ represents all output places of the transition $t^{\prime }$).

4. The following statements are verified : (a) $\forall r\in P_R,{}^{••}r\cap P_A=r^{••}\cap P_A\ne \varnothing$ and (b) $\forall r\in P_R,{}^{•}r\cap r^{•}=\varnothing$ (here ${}^{••}r$ is the set of all input places of all input transitions of the place r, $r^{••}$ represents the set of all output places of all output transitions of the place r).

5. ${}^{••}(p^0)\cap P_R={(p^0)}^{••}\cap P_R=\varnothing$.

Because resources are represented by places from the set $P_R$, ${}^{•}r$ represents input transitions of the resource r. Then ${}^{••}r={\bigcup }_{t\in {}^{•}r}{}^{•}t$ is the set of all input places of all input transitions of the place r. Similarly, $r^{••}={\bigcup }_{t\in r^{•}}{t}^{•}$ represents the set of all output places of all output transitions of the place r.

Definition 3 [31,34] Let $N=(P_A\cup \{p^0\}\cup P_R,T,F)$ be an S²PR. An initial marking $M_0$ is called an acceptable initial marking for N if:

(1) $M_0(p^0)\ge 1$,

(2) $M_0(p)=0,\forall p\in P_A$, and

(3) $M_0(r)\ge 1,\forall r\in P_R$. An S²PR with such a marking is said to be an acceptably marked one.

Definition 4 [31,34] A system of S²PR, called S³PR for short, is defined recursively as follows :

1. An S²PR is an S³PR.

2. Let $N_i=(P_{A_i}\cup \{p_i^0\}\cup P_{R_i},T_i,F_i),i\in 1,2$, be two S³PR such that $(P_{A_1}\cup \{p_1^0\})\cap (P_{A_2}\cup \{p_2^0\})=\varnothing ,P_{R_1}\cap P_{R_2}=P_C\ne \varnothing$, and $T_1\cap T_2=\varnothing$. Then, the net $N=(P_A\cup P^0\cup P_R,T,F)$ resulting from the composition of $N_1$ and $N_2$ via $P_C$ defined as follows : (a) $P_A=P_{A_1}\cup P_{A_2}$.

(b) $P^0=\{p_1^0\}\cup \{p_2^0\}$.

(c) $P_R=P_{R_1}\cup P_{R_2}$

(d) $T=T_1\cup T_2$.

(e) $F=F_1\cup F_2$ is also an S³PR.

In such a way we can compose S³PR by n S²PR. Then $N=N_1$ if $n=1$ and $N={◯}_{i=1}^n{N}_i$ if $n>1$, or alternatively, $N={◯}_{i=1}^{n-1}\circ N_n$. Transitions in ${(P^0)}^{•}$ are called source transitions representing raw materials.

Definition 5 [31,34] Let N be an S³PR. $(N,M_0)$ is called an acceptably marked S³PR if one of the following statements is true:

1. $(N,M_0)$ is an acceptably marked S²PR. 2. $N=N_1\circ N_2$, where $(N_i,M_{0_i})$ is an acceptably marked S³PR and

(a) $\forall i\in \{1,2\},\forall p\in P_{A_i}\cup \{p_i^0\},M_0(p)=M_{0_i}(p)$.

(b) $\forall i\in \{1,2\},\forall r\in P_{R_i}\setminus P_C,M_0(r)=M_{0_i}(r)$.

(c) $\forall r\in P_C,M_0(r)=max\{M_{0_1}(r),M_{0_2}(r)\}$.

2.2. Generalized S³PR - GS³PR

The difference between S³PR and GS³PR consists in the fact that while S³PR are based on ordinary PN (weights of arcs are only one), GS³PR weights of arcs can be greater than one (i.e., they are based on GPN). Sometimes they are also called [37] weighted S³PR WS³PR. Definitions of them are as follows:

Definition 6 [34,37] A generalized simple sequential process with resources (GS²PR) is a Petri net $N=(P_A\cup \{p^0\}\cup P_R,T,F,W)$ if:

1. The subnet generated by $X=P_A\cup \{p^0\}\cup T$ is an S²P;

2. $P_R\ne \varnothing$ ($r\in P_R$ is called a resource or a resource place in the net formalism) and $(P_A\cup \{p^0\})\cap P_R=\varnothing$;

3. $W=W_A\cup W_R$, where $W_A:((P_A\cup P^0)\times T)\cup (T\times (P_A\cup P^0))\to \{0,1\}$ and $W_R:(P_R\times T)\cup (T\times P_R)\to \mathbb{N}$, where $P^0=\{p^0\}$;

4. $\forall p\in P_A,\forall t\in {}^{•}p,\forall t^{\prime }\in p^{•},{}^{•}t\cap P_R=t^{\prime •}\cap P_R=\{r_p\}$ and $W(r_p,t)=W(t,r_p),\forall r\in P_R,\exists$ a unique minimal P-semiflow $I_r$ s.t. $\left|\right|I_r\left|\right|={}^{••}r\cap P_A\cup \{r\},\left|\right|I_r\left|\right|\cap P_R=\{r\},\left|\right|I_r\left|\right|\cap P^0=\varnothing$, and $I_r(r)=1$. Furthermore, $P_A={\bigcup }_{r\in P_R}(||I_r\left|\right|\setminus P_R)$;

5. The following statements are satisfied : $\forall r\in P_R,{}^{•}r\cap r^{•}=\varnothing$ and ${}^{••}r\cap P_A=r^{••}\cap P_A\ne \varnothing$;

6. ${}^{••}(p^0)\cap P_R={(p^0)}^{••}\cap P_R=\varnothing$.

Definition 7 [34,37] Let $N=(P_A\cup \{p^0\}\cup P_R,T,F,W)$ be a GS²PR. An initial marking $M_0$ is called an acceptable initial marking for N if $\forall p\in P_A,M_0(p)=0;\forall r\in P_R,M_0(r)\ge ma{x}_{p\in \left|\right|I_r\left|\right|}{I}_r(p)$; and $M_0(p^0)\ge 1$.

The couple $(N,M_0)$ with N being a GS²PR and $M_0$ being an acceptable initial marking is called a well-marked GS²PR. An acceptable initial marking guarantees that each transition is potentially fireable.

Definition 8 [34,37] A system of GS²PR, called GS³PR, is defined recursively as follows:

1. A GS²PR is a GS³PR;

2. Let $N_i=(P_{A_i}\cup P_i^0\cup P_{R_i},T_i,F_i,W_i),i\in \{1,2\}$, be two GS³PR such that $(P_{A_1}\cup P_1^0)\cap ((P_{A_2}\cup P_2^0)=\varnothing ,P_{R_1}\cap P_{R_2}=P_C(\ne \varnothing )$, and $T_1\cap T_2=\varnothing$ (in this case we say that $N_1$ and $N_2$ are two composable GS³PR). Then, the net $N=(P_A\cup P^0\cup P_R,T,F,W)$ resulting of the composition of $N_1$ and $N_2$ via PC (denoted as $N=N_1\circ N_2)$ defined as follows :

(a) $P_A=P_{A_1}\cup P_{A_2}$,

(b) $P^0=P_1^0\cup P_2^0$,

(c) $P_R=P_{R_1}\cup P_{R_2}$,

(d) $T=T_1\cup T_2$,

(e) $F=F_1\cup F_2$,

(f) $W=W_1\cup W_2$ is also a GS³PR.

Definition 9 [34,37] Let N be a GS³PR. $(N,M_0$) is a well-marked GS³PR if one of the two following statements is true:

1. $(N,M_0)$ is a well-marked GS²PR;

2. $N=N_1\circ N_2$, where $(N_i,M_{i_0}),\forall i\in \{1,2\}$ are well-marked GS²PR;

(a) $\forall i\in \{1,2\},\forall p\in P_{A_i}\cup P_i^0,M_0(p)=M_{i_0}(p)$;

(b) $\forall i\in \{1,2\},\forall r\in P_{R_i}\setminus P_C,M_0(r)=M_{i_0}(r)$;

(c) $\forall r\in P_C,M_0(r)=max\{M_{1_0}(r),M_{2_0}(r)\}$.

Consequently, a GS³PR N composed of k GS²PR $N_1,\dots N_k$ can be denoted by $N={◯}_{i=1}^{k-1}\circ N_k$, $k\in \mathbb{N}\setminus \{0\}$.

2.3. S⁴PR Models of AMS

More powerful modeling ability than S³PR and generalized S³PR (GS³PR) [34,37] has a class of generalized Petri nets named as S⁴R [35]. In the sense of [33] S⁴PR are equivalent to S⁴R.

Definition 10 [33,34] A well-marked S⁴R net $(N,M_0$) is a marked Petri net $N=(P,T,F,W)$ with initial marking $M_0$ such that :

1. $P=P_A\cup P^0\cup P_R$ is a partition such that

(a) $P_A={\bigcup }_{i=1}^n{P}_{A_i}$ is called the set of activity places, where $\forall i,j\in \{1,\dots ,n\},i\ne j,P_{A_i}\ne \varnothing$ and $P_{A_i}\cap P_{A_j}=\varnothing$.

(b) $P^0={\bigcup }_{i=1}^n\{p_i^0\}$ is called the set of idle places.

(c) $P_R={\bigcup }_{i=1}^n\{P_{R_i}\}=\{r_1,r_2,\dots ,r_m\}$ is called the set of resource places;

2. $T={\bigcup }_{j=1}^n{T}_j,T_j\ne \varnothing$. $\forall i\ne j,T_i\cap T_j=\varnothing$;

3. $W=W_A\cup W_R$, where $W_A:((P_A\cup P^0)\times T)\cup (T\times (P_A\cup P^0))\to \{0,1\}$ and $\forall j\ne i,W_A:((P_{A_j}\cup \{p_j^0\}\times T_i)\cup (T_i\times (P_{A_j}\cup \{p_j^0\}))\to \{0\}$, and $W_R:(P_R\times T)\cup (T\times P_R)\to \mathbb{N}$;

4. $N_j$ generated by $P_{A_j}\cup \{p_j^0\}\cup T_j$ is a strongly connected state machine such that every circuit in $N_j$ contains place $p_j^0$;

5. $\forall r\in P_R$, there exists a unique minimal P-semiflow $I_r$ such that $\left|\right|I_r\left|\right|\cap P_R=\{r\},\left|\right|I_r\left|\right|\cap P^0=\varnothing$, and $I_r(r)=1$. Furthermore, we have $PA={\bigcup }_{r\in P_R}(||I_r\left|\right|\setminus P_R)$;

6. N is pure and strongly connected;

7. $\forall p\in P_A,M_0(p)=0;\forall r\in P_R,M_0(r)\ge ma{x}_{p\in \left|\right|I_r\left|\right|}{I}_r(p)$, and $\forall p_j^0\in P^0,M_0(p_j^0)\ge 1$.

2.4. Brief Summary

It can be said that GS³PR is a subclass of S⁴R (or S⁴PR) and a generalized version of a S³PR. A GS³PR is equivalent to WS³PR. An activity place representing a processing stage of a job needs a single unit of a single resource type. The usage of resources in S⁴R is almost arbitrary and requires only conservativeness.

Theorem 1 [38] A Petri net N is (partially) conservative iff there exists an $(m\times 1)$ vector y of positive (nonnegative) integers such that $[N].y=0,y\ne 0$.

3. Using Knowledge of Deadlocks in the Reachability Tree of PN Model to Form a Procedure for Themselves Prevention in that Model

Just as siphons form a kind of a substructure in a Petri net, and similarly P-invariants and T-invariants form another substructures of PN, deadlocks also form a kind of substructure of PN. It results from the reachability tree (RT) the nodes of which are being PN reachable markings (i.e., states of PN). Deadlocks are unambiguously determined as a part (a subset) of RT nodes, because RT is unambiguously determined for any PN. Since, to each PN belongs the only one RT, deadlocks that clearly belong among these nodes of the RT (they are some of PN states at which no transitions can be fired, i.e., they are unwanted terminal states), also have an important informative value about the PN. Let us utilize this as a kind of a hypothetical feedback in the process of deadlock prevention. It may sound indeed paradoxical, but let us use the knowledge of deadlocks obtained from the RT of the PN to prevent deadlocks themselves in this PN. However, the question arises as to how to implement this.

This paper presents an unusual approach to the deadlock prevention in Petri net based models of AMS. Namely, there exist two main approaches to the PN deadlock prevention in the world literature. There are based either on (i) P-invariants; or on (ii) siphons. In both areas a galaxy of literature were written. The invariant-based approach is touched especially in [9,10,12,26,32,41], while the siphon-based approach is presented in [2,3,4,5,6,7,8,14,18,19,20,21,22,23,24,29,30,31,33,34,36,37,40,45,47]. A brief introduction to both approaches was presented in [2].

In this paper we will be based especially on [13,15,27,39,46,48,49,50,51,52,53] as well as on PN theoretical works [16,17,28,38,42,43,56] and on PN practical works [25,26] concerning the GPenSIM editor for Petri nets.

The PN models of AMS contain three groups of places (as it was introduced in Section 2) idle, activity and resource places. The approach that will be presented here, is based on the approach based on P-invariants, but it modifies the choice of elements for the matrix $[L]$ (see below introduced relationships (7) - (15)), which plays a significant role there. It represents prescribed conditions in the inequality (7).

In our approach is characteristic the specific procedure for creating the matrix $[L]$. It is based on a sort of aggregation of activity places occurring in PN deadlock markings (i.e., in RT nodes, being PN state vectors, representing deadlocks) and their incorporating into the matrix $[L]$ as its elements. This operation brings information about deadlocks and their mutual interconnections to the method based on P-invariants. At creation of $[L]$ idle places and resource ones are not used. Namely, it can be said that on the one hand idle places do not affect significantly (the sufficient amount of them is put from outside by the plant operator) the internal structure of deadlocks and, on the other hand, resources are not affected by deadlocks because their amount depends on outside appearances (they are given also from outside). Though a lack of resources can cause deadlocks, but it cannot solve them. Activity (i.e., operation) places present a core of events inside the production processes. Setting the marking of idle places and resource places is an external matter.

3.1. A Short Consideration about Deadlocks

Deadlocks are structural parameters of Petri nets. They are unwanted in real AMS as well as in their PN models. A deadlock occurs there when the system reaches a state where no PN transitions can be fired. The system halts its operation and further continuation of the activity is impossible. When we are trying to find a mutual relations among deadlocks in a Petri net we hit on the answer that the analysis of its reachability tree (RT) is the closest way how to achieve this. Namely, to a PN appertains unique RT. However, the opposite is not true. RT nodes yield particular states of PN, i.e., the whole state space, including deadlocks. It can therefore be assumed that knowledge of deadlocks can be retroactively utilized to influence the structure of the PN. Let us try to use deadlocks for changing the structure of the original PN to remove them, i.e., for finding a supervisor realizing the deadlock prevention in the original PN. It will be performed in further Sections.

3.2. Multisets vs. Sets

It is usual to express PN states by means of multisets in the sense of the definition originally introduced in [40]. A multiset is a generalization of the concept of a set. A multiset $\Omega$, over a non-empty set A, is a mapping $\Omega :A\to \mathbb{N}$, which we represent as a formal sum ${\sum }_{a\in A}\Omega (a).a$. For example, when the PN marking (i.e., state) $M(p)=\{M(p_1)=2,M(p_5)=3,M(p_{11}=1\}$ we will write as $2p_1+3p_5+p_{11}$. For two multisets ${\Omega }_q,{\Omega }_2$ is valid the following: ${\Omega }_1\cup {\Omega }_2={\sum }_{a\in A}max\{{\Omega }_1(a),{\Omega }_2(a)\}.a$ and ${\Omega }_1\cap {\Omega }_2={\sum }_{a\in A}min\{{\Omega }_1(a),{\Omega }_2(a)\}.a$.

3.3. A Simple Example of S³PR PN Model of AMS

Let us find a supervisor that has to be added to the original deadlocked system being the S³PR modeled by PN displayed in Figure 1 left, and which will control it in such a way that deadlocks cannot occur. Its RT in this Figure right shows us that there exists only one deadlock.

Let us compare both the siphon based approach to the deadlock prevention and the approach based on activity places contained in deadlocks in RT (i.e., being familiar with all its nodes representing whole state space including deadlocked states).

The state space of the system is given by the RT nodes in the form of row vectors as follows.

$$\begin{array}{c}\hfill \begin{array}{ccccccccccc}\hfill N0=& \hfill (1& \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 0& \hfill 0& \hfill 1)& \hfill \widehat{=}& p_1+p_4+p_5+p_8\hfill \\ \hfill N1=& \hfill (0& \hfill 1& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 1)& \hfill \widehat{=}& p_2+p_5+p_8\hfill \\ \hfill N2=& \hfill (1& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 1& \hfill 0& \hfill 0)& \hfill \widehat{=}& p_1+p_4+p_6\hfill \\ \hfill N3=& \hfill (0& \hfill 0& \hfill 1& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 1)& \hfill \widehat{=}& p_3+p_4+p_8\hfill \\ \hfill N4=& \hfill (0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0)& \hfill \widehat{=}& p_2+p_6\hfill \\ \hfill N5=& \hfill (1& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 1& \hfill 0)& \hfill \widehat{=}& p_1+p_5+p_7\hfill \end{array}\end{array}$$

(2)

The RT shows us that the node $N4$ is the deadlock. The incidence (structural) matrix of the system is the following

$$[N]=\left(\begin{array}{cccccc}\hfill -1& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 1& \hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill -1& \hfill 0& \hfill 0& \hfill 0\\ \hfill -1& \hfill 1& \hfill 0& \hfill 0& \hfill -1& \hfill 1\\ \hfill 0& \hfill -1& \hfill 1& \hfill -1& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill -1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill -1\\ \hfill 0& \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill 1\end{array}\right).$$

(3)

Let us compare the siphon-based approach (see e.g., [2,3,4]) to the deadlock prevention with the deadlock prevention proposed in this paper. It is necessary to show that the RT-based approach, more precisely the deadlock-based approach in our case, is also sound.

3.3.1. Siphon Based Approach

First let us use the elementary siphon based approach to the deadlock prevention. Almost all articles referred to in this paper are interested in the siphon-based approach at the deadlock prevention except [11,27,32,39,41]. There is only one elementary siphon in the system, namely

$$S_1=\left(\begin{array}{cccccccc}\hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 1& \hfill 0& \hfill 1& \hfill 0\end{array}\right).$$

(4)

Hence, the incidence matrix of the supervisor is as follows

$${[N]}_s=S_1.[N]=\left[\begin{array}{cccccc}\hfill -1& \hfill 1& \hfill 0& \hfill -1& \hfill 1& \hfill 0\end{array}\right].$$

(5)

It means that we can use the new Petri net structure containing the place $p_9$ with marking $m(p_9)=1$ being the supervisor. It is connected with the original structure in such a way that interconnections from $p_9$ direct to transitions $t_1$ and $t_4$ while interconnections to $p_9$ direct from transitions $t_2$ and $t_5$. Thus, the new structure is displayed in Figure 2 left while RT is presented in Figure 2 right to document deadlock-freeness of the supervised plant. In this way we can see that no deadlock occurs in the united structure.

3.3.2. Proposed Approach

Note that the deadlock (node $N4$) contains two activity (operation) places $p_2$ and $p_6$. Create the matrix $[L]$ using these places, i.e., $[L]=[01000100]$. Try to put ${[N]}_s=-[L].[N]$. We obtain ${[N]}_s=[-110-110]$. We can see that the supervisor is the same like that in (5).

Such approach was not applied just like haphazardly, of course. It follows from the P-invariant based approach to the deadlock prevention introduced below, where the matrix $[L]$ is composed by means of the activity places contained in the marking representing the deadlock. It was shown here that the activity (operation) places contained in deadlocks (i.e., in RT nodes expressing deadlocks) have an impact on the prevention of themselves. In this simple example even they give the same results as elementary siphons.

Consequently, it can be said that there exists a certain feedback effect of RT on PN itself, although in general it may not be as unambiguous as is the relationship from PN to RT. On the other hand, it should also be added that it cannot be expected that the results of applying this approach will generally be identical to the approach based on siphons.

4. P-invariant Based Approach to Deadlock Prevention in General

Let us first look at the problem of deadlock prevention from the P-invariant point of view, which is the basis of the proposed approach.

Definition 11 [2,11,32,41] A vector I is a place invariant P (shortly, P-invariant) of PN iff $I\ne 0$ and $I^T.[N]=0^T$. Here, $[N]$ is the incidence matrix of N.

Definition 12 [31] A siphon S in an ordinary PN $(N,M_0)$ is invariant controlled by P-invariant I under the initial marking $M_0$ iff $I^T.M_0>0$ and $\forall p\in P\setminus S,I(p)\le 0$, or equivalently, $I^T.M_0>0$ and ${\left|\right|I\left|\right|}^{+}=p\in P|I(p)>0$ is the positive support of I.

For P-invariant I it holds [2,11,32,41] the relation of the state conservation, as follows

$$I^T.M=I^T.M_0$$

(6)

The following inequality representing linear combinations restricting behavior of marking M is important for the deadlock prevention based on P-invariants. Let the following inequality holds

$$[L].M\le b$$

(7)

Here $[L]$ is a $(n_s\times n)$ matrix of integers, M is a $(n\times 1)$ PN marking vector and b is an $(n_s\times 1)$ integer vector. To remove the inequality we can add slack variable s. Thus we have

$$[L].M+s=b,$$

(8)

where s is $(n_s\times 1)$ vector of integers.

When we force $[[L]|{[J]}_s]$ instead of $I^T$ into $I^T.[N]=0^T$ (see the Definition 1), we obtain

$$[L|{[J]}_s].\left[\begin{array}{c}[N]\hfill \\ {[N]}_s\hfill \end{array}\right]=0,$$

(9)

where ${[J]}_s$ is $(n_s\times n_s)$-dimensional identity matrix; $n_s$ is the number of slacks; $[L]$ is a $(n_s\times n)$-dimensional matrix of integers; $[N]$ is the $(n\times m)$ PN incidence matrix; ${[N]}_s$ is the $(n_s\times m)$-dimensional matrix of integers, and m is the number of PN transitions. Thus, after finding ${[N]}_s$ from (9), we obtain the incidence matrix describing the structure of the PN-based model of the supervisor.

Consequently,

$$\begin{array}{ccc}\hfill [L].[N]+{[N]}_s& =& 0\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill {[N]}_s& =& -[L].[N]\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {[N]}_s& =& {[N]}_s^{+}-{[N]}_s^{-},\hfill \end{array}$$

(12)

where ${[N]}_s^{+}$ is ${Post}_s$ and ${[N]}_s^{-}$ is ${Pre}_s$ of ${[N]}_s$.

Marking $M_a$ of the supervised system will consist of marking M of the original (unsupervised) system and marking of the supervisor $M_s$, i.e., $M_a={(M^T{M}_s^T)}^T$. Therefore we can write

$$[L|{[J]}_s].\left[\begin{array}{c}{M}_0\hfill \\ {M_0}_s\hfill \end{array}\right]=b,$$

(13)

Consequently,

$$\begin{array}{c}\hfill [L].M_0+M_{0_s}=b\end{array}$$

(14)

$$\begin{array}{c}\hfill M_{0_s}=b-[L].M_0\end{array}$$

(15)

Thus, there is the only freedom of choice, namely the suitable choice of the structure and elements of the matrix $[L]$ which the only are at the disposal for experiments. The reachability tree (RT) of the net N yields an efficient help at creating $[L]$. RT nodes are state vectors (markings) of PN. These states include among themselves also PN deadlocks (dead states). The proposed method just uses the places of deadlock markings that correspond to activity (operation) places of the PN. Namely, just these places represent production processes inside AMS.

In [47] the RT nodes are divided (as to marking) into four categories: good, dangerous, bad, and deadlock markings. At formulation of our approach, the partial knowledge from [12,13,45,46] was utilized, especially those concerning the isomorphism of graphs and isomorphism of PNs. The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic.

Definition 13 [46]: Let $G_1=(V_1,A_1)$ and $G_2=(V_2,A_2)$ be two directed graph, where $\{V_1,V_2\}$ and $\{A_1,A_2\}$ indicate the vertex and edge sets, respectively. Suppose that there is an isomorphism between $G_1$ and $G_2$, which is denoted as $G_1\cong G_2$, such that there exist two functions $f:V_1\leftrightarrow V_2$ and $g:A_1\leftrightarrow A_2$.

Definition 14 [53] Given two graphs $G=(V_1,E_1)$ and $H=(V_2,E_2)$ an isomorphism between G and H is a bijection $\phi :V_1\to V_2$ satisfying for every pair of nodes $u,v\in V_1$, $(u,v)\in E_1\iff (\phi (u),\phi (v))\in E_2$.

And even automorphisms in $G=(V_1,E_1)$ is the set of bijections from $V_1$ onto itself satisfying $(u,v)\in E_1\iff (\phi (u),\phi (v))\in E_1$.

These definitions can be applied to PN model, which is also a class of directed graphs (namely, bipartite directed graph). In [46] were proved the next Theorems, where authors lean on the algorithm ’Think-globally-act-locally’, presented in [45]:

Theorem 2 [46]: Given two PN models $N_1(P_1,T_1,F_1,W_1)$ and $N_2(P_2,T_2,F_2,W_2)$. Let their vertex sets be $\{P_1\cup T_1,P_2\cup T_2\}$ and edge sets be $\{F_1,F_2\}$. If there are two bijective functions $f_p:P_1\cup T_1\leftrightarrow P_2\cup T_2$ and $g_p:F_1\leftrightarrow F_2$, it is verified that $N_1\cong N_2$.

Theorem 3 [46]: Given two isomorphic PN models $N_1\cong N_2$ and with the same initial marking $M_0$. It is said that the related reachability graphs $R_1(N_1,M_0)$ and $R_2(N_2,M_0)$ also have isomorphism relationship, where $R_1\cong R_2$.

Already more then 60 years ago in [48] a heuristic algorithm was proposed for testing pairs of directed line graphs for isomorphism. However, till now it is not known the problem to be solvable in polynomial time, however nor to be NP-complete, and therefore may be in the computational complexity class called as NP-intermediate. Relatively new research in this area was published in [49]. In [50] a category of nested-unit PN (NUPN) were defined in order to study PN isomorphism. A quasi-polynomial ($(exp(O(\sqrt{n.logn}))$ where n is th number or graph vertices) algorithm for isomorphism of graphs was published in [51]. Isomorphism in connection with parallel processing of calculations, or rather the number of parallel processors required, was addressed in the work [53].

4.1. Creating the Matrix $[L]$

In order to obtain elements of the integer matrix $[L]$ an aggregate of markings of activity places has to be created. We start from RT nodes corresponding to deadlocks (occurring among PN state vectors). Such an aggregate, the set S, consists of the union of subsets $S_i,i=1,n_d$, where $n_d$ is the number of deadlocks

$$S={\displaystyle \bigcup _{i=1}^{n_d}}{S}_i$$

(16)

Suppose that there are activity places $p_a,p_b,\dots ,p_k$ in the PN. Of course, these places don’t have to be consecutive. In particular deadlocks $N_{d_i},i\in \{1,n_d\}$ they occur together with their markings $c_{a_i},c_{b_i},\dots ,c_{k_i}$, being greater or equal to zero. Then,

$$S_i=\{c_{a_i}{p}_a,c_{b_i}{p}_b,\dots ,c_{k_i}{p}_k\},i=1,2,\dots ,n_d$$

(17)

corresponds to the deadlock $N_{d_i}$. Namely, in different deadlocks these places are presented with different markings. Elements of the set S are just derived from markings of corresponding activity places. Namely,

$$S=\{({\displaystyle \underset{i=1}{\overset{n_d}{max}}}{c}_{a_i})p_a,({\displaystyle \underset{i=1}{\overset{n_d}{max}}}{c}_{b_i})p_b,\dots ,({\displaystyle \underset{i=1}{\overset{n_d}{max}}}{c}_{k_i})p_k\}.$$

(18)

Thus, such aggregated markings of activity places are then becoming the corresponding elements of the matrix $[L]$. It means that to the $(1\times n)$ zero matrix $[L]$ (n is the number of PN places) we insert the aggregates of markings which correspond to activity places and remaining elements of $[L]$ stay zeros, because we do not consider them. Thus, the elements of $[L]$:

$$L_i\in \{{\mathbf{0}}_{idle},\{{\displaystyle \underset{i=1}{\overset{n_d}{max}}}{c}_{a_i},{\displaystyle \underset{i=1}{\overset{n_d}{max}}}{c}_{b_i},\dots ,{\displaystyle \underset{i=1}{\overset{n_d}{max}}}{c}_{k_i}\},{\mathbf{0}}_{res}\},i\in \{1,2,\dots ,n\},$$

(19)

where ${\mathbf{0}}_{idle}$ and ${\mathbf{0}}_{resource}$ are, respectively, the set of zero elements in positions of idle places and that in positions of resource ones. However, it should be emphasized here, that the arrangement of these elements in the matrix $[L]$ depends on the numbering of places in the investigated PN. Therefore, the elements of the matrix $[L]$ have to be aligned in the same order, i.e., succession of idle, activity and resource places need not be consecutive (i.e., can be mixed), because the order of elements in deadlocks also depends on the numbering of places in the investigated PN. The important thing is that all elements of the matrix $[L]$, except those corresponding to activity places, are zero. This is also presented in several examples (even, may be said the whole series) introduced in next Sections of this paper.

Finally, we find (in the sense of (11) the incidence matrix of the supervisor as follows

$${[N]}_s=-[L].[N]$$

(20)

In next Sections are introduced examples verifying correctness of the proposed approach to the deadlock prevention.

5. Verification of Proposed Approach

In this section several case studies of the kind of S³PR, GS³PR and S⁴PR PN models of AMS.

5.1. AMS Modeled by Generalized Petri Nets

Consider the S⁴PR PN model of a simple manufacturing system in Figure 3 consisting of two subsystems (subnets) sharing two resources.

The RT nodes are represented by the state vectors as follows

$$\begin{array}{c}\hfill \begin{array}{ccccccccccc}\hfill N0=& \hfill (1& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 2& \hfill 2)& \\ \hfill N1=& \hfill (0& \hfill 1& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 2& \hfill 1)& \\ \hfill N2=& \hfill (1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 1& \hfill 2)& \\ \hfill N3=& \hfill (0& \hfill 0& \hfill 1& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 1)& \\ \hfill N4=& \hfill (0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 1& \hfill 1)& \hfill \widehat{=}{p}_2+p_6\\ \hfill N5=& \hfill (1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 2& \hfill 0)& \\ \hfill N6=& \hfill (0& \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 0& \hfill 0& \hfill 2& \hfill 0)& \\ \hfill N7=& \hfill (0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 1& \hfill 0& \hfill 1& \hfill 0)\end{array}\end{array}$$

(21)

$$[N]=\left(\begin{array}{ccccccc}\hfill -1& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 1& \hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill -1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill -1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill -1\\ \hfill 0& \hfill -2& \hfill 2& \hfill 0& \hfill -1& \hfill 1& \hfill 0\\ \hfill -1& \hfill 0& \hfill -1& \hfill 2& \hfill 0& \hfill -2& \hfill 2\end{array}\right).$$

(22)

The supervised system, where the supervisor computed by another method for generalized PN (see e.g., that in [4]), is presented in Figure 4 left while its RT is right.

When in Figure 4 we put marking $m(p_{10})=2$ and when we assign the weight of the arc $p_{10}\to t_5$ as $W(p_{10},t_5)=2$ we obtain also the resulting deadlock-free system, only RT will be displayed in Figure 5.

5.1.1. Proposed Aproach

The RT node

$N4=\{p_2+p_6+p_8+p_9\}$

is the deadlock. It contains two activity places $p_2$ and $p_6$. After their projection into the matrix $[L]$ we have the matrix

$[L]=[010001000]$.

Consequently, we obtain the incidence matrix of the supervisor

${[N]}_s=-[L].[N]=[-1100-110]$.

With respect to (12) negative elements belong to $Pr{e}_s$ while the positive ones belong to $Pos{t}_s$. Hence, PN arcs from the supervisor lead to the transitions $t_1$ and $t_5$ while arcs from transitions $t_2$ and $t_6$ arrive to the supervisor. The supervised system is displayed in Figure 6 left while its RT documenting the deadlock-freeness is right.

Comparing the RT in Figure 6 with that in Figure 4 we can see that the RT is very similar, but it has for one branch more.

When in Figure 6 we put marking $m(p_{10})=2$ and to the arc $p_{10}\to t_5$ we assign the weight $W(p_{10},t_5)=2$ we obtain also the resulting deadlock- free system, only RT will be displayed in Figure 7.

5.2. Another GS³PR PN Model of AMS

Consider the GS³PR model in Figure 8.

The PN model has the following incidence matrix

$$[N]=\left(\begin{array}{cccccc}\hfill -1& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 1& \hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill -1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill -1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill -1\\ \hfill -1& \hfill 1& \hfill 0& \hfill 0& \hfill -2& \hfill 2\\ \hfill 0& \hfill -2& \hfill 2& \hfill -1& \hfill 1& \hfill 0\end{array}\right).$$

(23)

5.2.1. Siphon Based Approach to Deadlock Prevention

There is only only one siphon in the PN model. It is the following

$$S_1=\left(\begin{array}{cccccccc}\hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 1\end{array}\right).$$

(24)

The supervisor incidence matrix is the following

$${[N]}_s=-[L].[N]=\left(\begin{array}{cccccc}\hfill -1& \hfill 0& \hfill 1& \hfill -1& \hfill 0& \hfill 1\end{array}\right).$$

(25)

The supervised system is in Figure 9 left, with the supervisor depicted in red, while the RT documenting the deadlock-freeness is right.

5.2.2. Proposed Approach Resolving the Problem

As we can see in RT of the PN model in Figure 8 (see also the list of its nodes below) there is the deadlock $N12=(02002000)$.

$$\begin{array}{c}\hfill \begin{array}{ccccccccc}\hfill N0=(2& \hfill 0& \hfill 0& \hfill 2& \hfill 0& \hfill 0& \hfill 2& \hfill 2)& \\ \hfill N1=(1& \hfill 1& \hfill 0& \hfill 2& \hfill 0& \hfill 0& \hfill 1& \hfill 2)& \\ \hfill N2=(2& \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 0& \hfill 2& \hfill 1)& \\ \hfill N3=(0& \hfill 2& \hfill 0& \hfill 2& \hfill 0& \hfill 0& \hfill 0& \hfill 2)& \\ \hfill N4=(1& \hfill 0& \hfill 1& \hfill 2& \hfill 0& \hfill 0& \hfill 2& \hfill 0)& \\ \hfill N5=(1& \hfill 1& \hfill 0& \hfill 1& \hfill 1& \hfill 0& \hfill 1& \hfill 1)& \\ \hfill N6=(2& \hfill 0& \hfill 0& \hfill 0& \hfill 2& \hfill 0& \hfill 2& \hfill 0)& \\ \hfill N7=(2& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 1& \hfill 0& \hfill 2)& \\ \hfill N8=(0& \hfill 1& \hfill 1& \hfill 2& \hfill 0& \hfill 0& \hfill 1& \hfill 0)& \\ \hfill N9=(0& \hfill 2& \hfill 0& \hfill 1& \hfill 1& \hfill 0& \hfill 0& \hfill 1)& \\ \hfill N10=(1& \hfill 1& \hfill 0& \hfill 0& \hfill 2& \hfill 0& \hfill 1& \hfill 0)& \\ \hfill N11=(2& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 0& \hfill 1)& \\ \hfill N12=(0& \hfill 2& \hfill 0& \hfill 0& \hfill 2& \hfill 0& \hfill 0& \hfill 0)& \hfill \widehat{=}2p_2+2p_5\end{array}\end{array}$$

(26)

Respecting activity places only (finally, this deadlock does not even contain other places), N12$\widehat{=}\{2p_2,2p_5\}$. Consequently we can put

$[L]=[02002000]$.

The incidence matrix of the supervisor is

$N_s=-[L].[N]=[-220-220]$.

Then the supervised system is in Figure 10 left together with its RT right. The supervisor is depicted in red. RT shows that no deadlock is in the supervised system.

5.3. Slightly Larger S³PR and GS³PR PN Models

In this case as well as in further cases, we will not already introduced the siphon-based approach in order not to unnecessarily increase the scope of the article.

5.3.1. S³PR PN Model with Minimal Resources

Consider S³PR model of a simple AMS displayed in Figure 11. Similar model was solved in [4] using the siphon-based approach.

The transpose of its incidence matrix (${[N]}^T$) is expressed in the form of the table in Figure 12

The RT of this PN model is depicted in Figure 13. It documents existence of two deadlocks in the PN model - nodes $N13,N14$.

As we can see in the RT there are two deadlocks in the uncontrolled PN model:

Consider deadlocks

$M_{13}=p_1+p_2+p_3+p_5+2p_8$

$M_{14}=2p_1+p_2+p_5+p_6+p_8$.

Hence, we can see that we have two sets of activity places $\{p_2,p_3,p_5\}$ and $\{p_2,p_5,p_6\}$. After their union $\{p_2,p_3,p_5\}\cup \{p_2,p_5,p_6\}$ we have the set $\{p_2,p_3,p_5,p_6\}$. Hence, we can create the matrix $[L]$ in the form

$$[L]=\left(\begin{array}{ccccccccccc}0& 1& 1& 0& 1& 1& 0& 0& 0& 0& 0\end{array}\right).$$

(27)

and applying (12) we have the incidence matrix of the supervisor

$${[N]}_s=-[L].[N]=\left(\begin{array}{cccccccc}-1& 0& 1& 0& -1& 0& 1& 0\end{array}\right).$$

(28)

The supervised system is displayed in Figure 14. Its RT is in Figure 15. The RT documents that there are no deadlocks in the supervised model.

Let us compare figures Figure 14 and Figure 15 on the one hand with results (presented in [4]) displayed in Figure 16 on the other hand. There are 3 supervisors synthesized by the method based on siphons in Figure 16 on the left and the corresponding RT of the controlled model on the right which has 14 nodes. It is clear that we have achieved here, in this paper, the RT (in Figure 15) which on the one hand has indeed only 10 nodes and is a little bit less branched, but on the other hand these results were achieved by the single supervisor only.

5.3.2. A GS³PR Model described by Generalized PN (GPN)

Consider the GPN model displayed in Figure 17.

Its RT has 38 nodes (consequently, it cannot be introduced here) and it points out two deadlocks

$N34=p_1+2p_2+p_3+2p_5+2p_8$ and

$N35=2p_1+2p_2+2p_5+p_6+p_8$.

Their unifying gives us $2p_2+p_3+2p_5+p_6$, hereof

$[L]=[02102100000]$ and

${[N]}_s=-[L].[N]=[-2110-2110]$.

The supervisor structure is displayed in Figure 18 left. The RT of the supervised GPN model is displayed in the same figure right. It can be seen that there is no deadlock in the RT.

5.4. Simple S⁴PR GPN Model

Let us use the S⁴PR GPN model displayed in Figure 19.

This PN model of AMS has RT with 21 nodes. Hence, it cannot be introduced here. There is one deadlock in this model - the node

N9 = $(10200010)$.

Consequently,

$[L]=(10200000)$.

The supervisor

${[N]}_s=-[L].[N]=(-110-220)$.

The supervised system has RT with 16 nodes. The supervised system is displayed in Figure 20 up, while the RT documenting deadlock-freeness of the controlled model is displayed down.

5.5. Further S⁴PR PN Model

Consider the S⁴PR PN model and its RT in Figure 21. The RT points out to two deadlocks

$N7=(00100100100)$ and

$N8=(01000010001)$

what leads to two subset of activity places $\{p_3,p_6\}$ and $\{p_2,p_7\}$. Hence,

$[L]=[01100110000]$.

Consequently,

${[N]}_s=-[L].[N]=[-1010-1010]$.

The supervised system is in Figure 22 (left) with the supervisor $p_{12}$ displayed in red, and its RT (right) documenting deadlock-freeness.

The alternative siphon based approach leads to 6 siphons what complicates the supervisor synthesis.

5.6. Another S⁴PR PN Model

Consider the deadlocked S⁴PR PN model with the structure displayed in Figure 23 (left) together with its RT (right).

As we can see there are not only two deadlocks but also a dwarfism of the state space. The transpose of the incidence matrix $[N]$ of the S⁴PR PN model is given in Figure 24.

The RT points out to two following deadlocks:

Node 3: $(9010100000)$

Node 4: $(910091000)$

Ignoring idle places $p_1,p_5$ we can see that in the former node there is the activity (operation) place $p_3$ while in the latter node the operation places $p_2,p_6$. When we unify them into one set $\{p_2,p_3,p_6\}$ we can construct the matrix

$[L]=[011001000]$.

Consequently, we can find the incidence matrix of the supervisor as follows

${[N]}_s=-[L].[N]=[-1010-110]$.

Applying the supervisor into the original PN model we obtain the controlled structure and RT documented its deadlock-freeness in Figure 25.

5.7. GS³PR PN Model with a More Complex Structure

Let us consider the PN model of an automated manufacturing system in Figure 26. Its transposed incidence matrix ${[N]}^T$ is displayed in Figure 27 in the form of a table.

The RT of the GS³PR PN model has 247 nodes. Consequently, it cannot be displayed here. There are five deadlocks in it as follows:

$N15:(10000073000101)N48:(63010100000000)N50:(7201091000000)N52:(8101082000000)N58:(9001073000000)$

Taking only operation places into account in particular deadlocks we have following subsets of such places; $S_1=\{3p_7\},S_2=\{3p_2,p_4\},S_3=\{2p_2,p_4,p_7\},S_4=\{p_2,p_4,2p_7\},S_5=\{p_4,3p_7\}$. The union of these subsets $S_1\cup S_2\cup S_3\cup S_4\cup S_5$ has the form $\{3p_2,p_4,3p_7\}$. Taking into account corresponding markings of these places in particular deadlocks, we obtain the matrix:

$[L]=[0301003000000]$

and the supervisor

${[N]}_s=-[L].[N]=[-331-10-330000]$

displayed in Figure 28 where the supervisor consists of the place $p_{14}$ with corresponding directed arcs (with corresponding weights) to transitions $t_1,t_3,t_6$ of the original PN model and directed arcs (with corresponding weights) from transitions $t_2,t_4,t_7$ back to the supervisor. The marking of the place $p_{14}$ is $M(p_{14})=4$.

The RT of the supervised PN model has 136 nodes. No of them is a deadlock. It was verified by the GPenSIM tool [14,26] being joined to Matlab.

5.8. Another S⁴PR PN Model

Consider another S⁴PR PN model displayed in Figure 29. Its RT is in Figure 30 where one deadlock occurs:

$N12=(31002110200)$.

Corresponding activity places are $p_2,p_6,p_7$, hence

$[L]=[01000110000]$

and the supervisor

${[N]}_s=-[L].[N]=[-1100-1010]$.

The supervisor is displayed in Figure 31 left, while the RT documenting the deadlock-freeness of the supervised system is right.

The supervisor is displayed in Figure 31 left in red, while the RT of the supervised system is right. The RT shows that the supervised model is without deadlocks.

6. Large-Scale S³PR PN Model

Consider the S³PR PN model displayed in Figure 32. There can be seen that the model is relatively large-scale because it consists of 4 cooperating production lines. Correct coordination of them is not easy, and less so the deadlock prevention. Nevertheless, the approach presented here, offers effective solution.

Here, the places $p_1,p_6,p_{11},p_{16}$ are idle places and $p_{21},p_{22},p_{23}$ are resources that all 4 production lines compete (or bid) for. RT of this model has 1144 nodes but 6 of them are deadlocks:

$$\begin{array}{c}\hfill \begin{array}{ccccccccccccccccccccccc}\hfill N60:(0& 0& 1\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0)\hfill \\ \hfill N63:(0& 1& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0)\hfill \\ \hfill N66:(0& 1& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1)\hfill \\ \hfill N67:(0& 1& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0)\hfill \\ \hfill N189:(0& 0& 1\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0)\hfill \\ \hfill N221:(0& 1& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0)\hfill \end{array}\end{array}$$

Here we have the following sets of activity (operation) places: $\{p_3,p_7,p_{12}\}$, $\{p_2,p_8,p_{12}\}$, $\{p_2,p_7,p_{13}\}$, $\{p_2,p_7,p_{12},p_{17}\}$, $\{p_3,p_7,p_{12},p_{18}\}$, $\{p_2,p_8,p_{12},p_{18}\}$. When we apply the union of these sets of operation places (i.e., without idle and resource places) from these deadlocks we obtain the set

$\{p_2,p_3,p_7,p_8,p_{12},p_{13},p_{17},p_{18}\}$,

and so we can create the matrix

$$\begin{array}{ccccccccccccccccccccccc}[L]=[0& 1& 1& 0& 0& 0& 1& 1& 0& 0& 0& 1& 1& 0& 0& 0& 1& 1& 0& 0& 0& 0& 0]\end{array}$$

(29)

and consequently the incidence matrix of supervisor

$${[N]}_s=\begin{array}{cccccccccccccccccccc}[-1& 0& 1& 0& 0& -1& 0& 1& 0& 0& -1& 0& 1& 0& 0& -1& 0& 1& 0& 0]\end{array}$$

(30)

The supervisor $p_{24}$ is depicted in Figure 33. After applying this supervisor to the PN model we obtain the system with 653 states but without deadlocks. This was verified by means of the above mentioned GPenSIM tool [14,26] in Matlab. Authentic behavior of the supervised model can be verified by simulation in any respectable PN editor.

7. S⁴PR PN Models with More Complicated Structure

Here, two relatively complex S⁴PR PN models are introduced. In the second model will be shown that in case when we are not satisfied with the behavior of the supervised system we can apportion the matrix $[L]$ with one row into the matrix $[L]$ with two rows.

7.1. Example 1

Consider S⁴PR PN model of AMS in Figure 34.

RT of this model has 1097 nodes (computed by GPenSIM tool [14,26]) including two deadlocks - nodes:

$N391:(5230082000000)$

$N461:(7210072010000)$

Eliminating idle places we have two subsets of places in particular deadlocks are $S_1=\{2p_2,3p_3,2p_7\}$ and $S_2=\{2p_2,p_3,2p_7,p_9\}$. After using the union operation on them we can find $S=S_1\cup S_2=\{2p_2,3p_3,2p_7,p_9\}$ and consequently, the matrix

$[L]=[0230002010000]$.

Then, the incidence matrix of the supervisor is

${[N]}_s=-[L].[N]=[-2-1300-22-110]$.

The supervisor is displayed in Figure 35. The supervised system has RT with 305 states and it has no deadlocks. The authentic behavior of the controlled model can be verified in any serious PN editor.

7.2. Example 2

Consider another S⁴PR PN model with significantly complicated structure displayed in Figure 36.

This example was resolved by the siphon-based approach in [21], more precisely said by so called augmented elementary siphons. There, the supervisor consists of 7 elements.

Here, we have 16 142 nodes in the RT (computed by GPenSIM tool [14,26]) of the uncontrolled model in Figure 36, including 10 deadlocks as follows:

N2466: (5 1 3 0 1 0 8 2 0 0 0 0 0 0 0 0)

N3166: (5 2 2 1 0 0 7 2 0 1 0 0 0 0 0 0)

N4845: (5 2 2 0 1 0 7 2 0 0 1 0 0 0 0 0)

N5741: (5 3 1 1 0 0 6 2 0 1 1 0 0 0 0 0)

N6621: (6 1 3 0 0 0 5 2 0 3 0 0 0 0 0 0)

N7639: (5 3 1 0 1 0 6 2 0 0 2 0 0 0 0 0)

N8555: (5 4 0 1 0 0 5 2 0 1 2 0 0 0 0 0)

N9466: (6 2 2 0 0 0 4 2 0 3 1 0 0 0 0 0)

N11909:(6 3 1 0 0 0 3 2 0 3 2 0 0 0 0 0)

N13792:(6 4 0 0 0 0 2 2 0 3 3 0 0 0 0 0)

and consequently we have following set of activity places (i.e., exclusive of idle places $p_1,p_7$ and resources) as follows: $\{p_2,3p_3,p_5,2p_8\}$, $\{2p_2,2p_3,p_4,2p_8,p_{10}\}$, $\{2p_2,2p_3,p_5,2p_8,p_{11}\}$, $\{3p_2,p_3,p_4,2p_8,p_{10},p_{11}\}$, $\{p_2,3p_3,2p_8,3p_{10}\}$, $\{3p_2,p_3,p_5,2p_8,2p_{11}\}$, $\{4p_2,p_4,2p_8,p_{10},2p_{11}\}$,

$\{2p_2,2p_3,2p_8,3p_{10},p_{11}\}$, $\{3p_2,p_3,2p_8,3p_{10},2p_{11}\}$, $\{4p_2,2p_8,3p_{10},3p_{11}\}$.

Using union operation on these subsets we can find the set

$\{4p_2,3p_3,p_4,p_5,2p_8,3p_{10},3p_{11}\}$,

and consequently the matrix

$[L]=[0431100203300000]$.

After performing the operation ${[N]}_s=-[L].[N]$, we obtain the supervisor incidence matrix in the form

${[N]}_s=[-412010-22-3030]$.

The supervisor is displayed in Figure 37.

The supervised system has RT with 47 states and it is without deadlocks.

Applying this supervisor into the unsupervised system in Figure 36 we indeed achieve the deadlock-free system, however its behavior is not satisfied. The right part (right subsystem) of the system is developed intensively while the left one very modestly, almost at all. The controlled system development is too asymmetrical. It is undesirable situation.

Fortunately, we found the interesting way how to resolved such a problem, namely by a decomposition of the supervisor into two ones as follows. The decomposition consists in splitting the matrix $[L]$ into two portion and create the new matrix with two rows. In our case into the first row of the new matrix we put first eight elements of $[L]$ and other eight elements we put zeros. Into the second row of the matrix we put first eight elements zeros while into the second eight elements the second eight elements of original matrix $[L]$. In this way we have obtained the following $(2\times 16)$ matrix. Let us name it repeatedly $[L]$.

$$[L]=\left[\begin{array}{cccccccccccccccc}\hfill 0& \hfill 4& \hfill 3& \hfill 1& \hfill 1& \hfill 0& \hfill 0& \hfill 2& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 3& \hfill 3& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]$$

(31)

Consequently,

$${[N]}_s=-[L].[N]=\left[\begin{array}{cccccccccccc}\hfill -4& \hfill 1& \hfill 2& \hfill 0& \hfill 1& \hfill 0& \hfill -2& \hfill 2& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -3& \hfill 0& \hfill 3& \hfill 0\end{array}\right]$$

(32)

Thus, we have two supervisors given in Figure 38. Applying them to the original unsupervised system we obtain deadlock-free system with the RT having 1028 states (computed by GPenSIM tool [14,26]). These two supervisors cover the same transitions like the original single supervisor. The second supervisor $p_{18}$ is quasi wrested out of the original supervisor in Figure 37. It should be noticed that after some fictitious optical identification (reverse composition or reassembling) of both supervisors in Figure 38 (i.e., amalgamating supervisors $p_{17}$ and $p_{18}$ into one) we get formally the original single supervisor in Figure 37. However, while the single supervisor yields only 47 states of the supervised system, applying two supervisors yields 1028 states and more satisfying behavior. Simulation on any serious PN editor shows that while in the former case the behavior of the supervised system is not satisfied, in the latter case it fully meets the needs. But for calculating RT, the GPenSIM tool [14,26] is suitable, especially in case of huge amount of nodes. It offers results relatively quickly - all RT states including deadlocks in the form of multisets (see Section 3.2) and integers of nodes being deadlocks.

8. Large-Scale S⁴PR PN Model of AMS

Consider large-scale S⁴PR PN Model of AMS in Figure 39. It was resolved in [46] by another approach.

It has 23 places and 18 transitions. Transpose of its incidence matrix is given in Figure 40 in the form of the table.

According to our calculations, the RT model has 19,299 nodes. Among them there are 25 deadlocks. The list of these deadlocks is introduced there, in [46]. Here, they are introduced in the Table 1.

Let us apply our approach to resolve that problem, i.e., to synthesize a supervisor realizing the deadlock prevention for that PN model. Utilize the list of deadlocks to create subsets $S_i,i=1,25$ of activity places corresponding to those deadlocks. They are introduced in Table 2.

Here, $p_1,p_5,p_{13}$ are idle places and $p_{19},p_{20},p_{21},p_{22},p_{23}$ are resources. Applying the union relation on the subsets introduced in Table 2 we have

${\bigcup }_{i=1}^{25}{S}_i=\{2p_2,p_3,2p_6,2p_7,2p_8,2p_{10},2p_{14},2p_{15},2p_{16}\}$

Thus, we can write the matrix:

$[L]=[02100222020002220000000]$

and to obtain the incidence matrix of the supervisor

${[N]}_s=-[L].[N]=[-2110-200200200-20020]$.

Applying such supervisor into the original uncontrolled system we obtain system without deadlocks. The supervisor is displayed in Figure 41. Number of RT nodes of the controlled system is 299.

Verifying behavior of the system controlled by this supervisor on a PN editor shows the balanced deadlock-free development of its subsystems. In [46], where the problem was resolved by another way, another supervisor was obtained. The RT of the supervised system in that case has 270 nodes and the system is deadlock-free.

9. Results

The method for deadlock prevention based on P-invariants but with the specific construction of the matrix $[L]$ incident in relationships (7)-(15) was proposed and verified on several examples of S³PR, GS³PR, and S⁴PR. In SubsubSection 5.1.1, in SubsubSection 5.2.1, and in SubsubSection 5.2.2 the proposed approach was tested on relatively simple GPN models. In the SubsubSection 5.3.1 we tested the S³PR PN model with strictly minimal resources. Obtained results were compared with results obtained at solving the same model resolved in [4] by the siphon-based approach. A GS³PR PN model was presented and resolved in SubsubSection 5.3.2. In the SubSection 5.4, in the SubSection 5.5, and in the SubSection 5.6 S⁴PR PN models were introduced and resolved as to deadlock prevention. In the SubSection 5.7 a GS³PR PN model with more complex structure was resolved. In the SubSection 5.8 the S⁴PR PN model was resolved.

In the Section 6 the large-scale S³PR PN model was resolved and finally, in the Section 7 two S⁴PR PN models with more complicated structures were resolved in the Example 1 in SubSection 7.1 and in Example 2 in SubSection 7.2, respectively. In the latter example the matrix $[L]$ were rebuilt from the one-row matrix into the matrix with two rows in order to ensure not only the deadlock-freeness but also a symmetry of development of both subsystems (left and right) of the supervised model.

Thus, an important result was achieved. Namely, when the single supervisor based on the one-row matrix $[L]$ accomplished deadlock-freenes but it does not satisfy the expected quality of the controlled system behavior, it can be decomposed into two (in general more) supervisors (or better said we can create the two-row or, in general multi-row, matrix $[L]$).

In such a way built supervisors achieve together not only deadlock-freeness but also better behavior of the controlled system. The research in this direction seems to be promising. However, such research may be time-consuming for experiments, especially in case of many deadlocks.

In the Section 8 our proposed approach is applied on the large-scale S⁴PR PN model presented in [46]. The model is displayed in Figure 39 and the transpose of incidence matrix $[N]$ in the model is displayed in Figure 40. This S⁴PR PN model has RT with 19 299 nodes inclusive of 25 deadlocks. After finding the supervisor displayed in Figure 41 and its application on the original model we obtain deadlock-free system having RT with 299 nodes. The supervisor found in [46] is different. After its applying on the original model the supervised system has 270 states and it is deadlock-free too. Both of supervised models were verified by simulation in a PN editor.

10. Discussion

In this paper we present the simple effective method for deadlock prevention of S³PR, GS³PR, and S⁴PR Petri net models of AMS. It present a novelty at the construction of the matrix $[L]$ prescribing the set of constraints in the P-invariant based method of deadlock prevention for PN models of AMS presented in Section 4. As elements of the integer matrix $[L]$ an aggregation of activity places , occurring among PN state vectors, being nodes of RT corresponding to deadlocks, are inserted. More details are introduced in SubSection 4.1.

The proposed method was successfully examined on a number of varied kinds and different sizes of PN models of AMS. Compared to other methods, this method appears to be effective and relatively simple to be promising. From a formal point of view, this method is a form of a feedback between the reachability tree of a deadlocked Petri net and the structure of the net itself. The result of such feedback yields the supervisor ensuring the deadlock-freeness of the net controlled by it. It may seem to be paradoxical, but knowledge of deadlocks (namely marking of their activity places) obtained from the deadlocked RT nodes of the Petri net is, contrarily, used at computing the supervisor ensuring the deadlock-freeness of the net. 253 PN to prevent deadlocks themselves in this PN.

11. Conclusions

The deadlock prevention of AMS represents very wide area of research. Accordingly, there exist many directions of development of methods in both in theory and in applications. Two directions leave the most distinctive and noticeable mark on this way. They are approaches to deadlock prevention based on P-invariants and those based on siphons. This paper was devoted to a contribution to former direction. It brings an approach to creating the integer matrix $[L]$ which represents a key element (core) of the approach based on P-invariants. Elements of $[L]$ are derived from deadlocks themselves obtained from the reachability tree of the Petri net, which is just to be investigated as to deadlock prevention. Namely, markings of activity places occurring in particular deadlocks are unified into an aggregate and consequently create $[L]$. After multiplying the PN incidence matrix $[N]$ by the matrix $[L]$ from the left we obtain the supervisor ${[N]}_s$. Applying such supervisor on the PN ensures the deadlock-freenes of the PN. Normally we make do with the single-row matrix $[L]$. However, in case of too structurally complex AMS, as it was shown in the Example 2 in the Section 7, when we are not gratified with the behavior of the supervised PN we can decompose the matrix $[L]$ into a double-row matrix. This topic seems to be interesting as to further investigation.