Deadlock Prevention Policy for Flexible Manufacturing Systems: Petri Net-Based Approach Utilizing Iterative Synthesis and Places Invariant ^†

Abstract

An iterative method was developed in this study within a Petri net system (PNS) for flexible manufacturing systems (FMSs) to eliminate deadlocks. The algorithm employs variant tokens, ranging from a few to many, in idle places to identify subnet deadlocks through reachability states. Additionally, it utilizes the place invariant (PI) to resolve deadlocks. An algorithm with a simple example was developed and applied using a complex scenario. The developed algorithm is user-friendly and effectively eliminates deadlocks in the PNS of FMSs, producing optimal results.

1. Introduction

Deadlocks occur from simultaneous attempts to access and share a resource, creating an undesired situation in a Petri net of flexible manufacturing systems. The sole method to resolve deadlocks is to reassign the workflow of the shared resource. Three solutions are used to eliminate deadlocks: deadlock avoidance, deadlock detection and recovery, and deadlock prevention [1,2].

Deadlock avoidance involves implementing an online control method to identify the correct evolution of the Petri net system (PNS). The primary objective is to ensure that the PNS avoids deadlocks and operates smoothly. Consequently, this method does not eliminate all deadlocks in the PNS [3,4,5]. The second method is deadlock detection and recovery. When deadlocks happen, a deadlock-free state is detected, and the deadlock is omitted by relocating resources. Generally, this approach demands a large amount of data to guide the algorithm, making it complex, especially with shared resources [6,7,8,9,10]. The third method is the deadlock prevention policy. This method aims to prevent deadlocks statically for optimal reachability states. No dead markings are reached after implementing the deadlock prevention policy. We used the deadlock prevention policy to address deadlocks in the PNS to obtain optimal results [1,2,6,11]. We employed variant token numbers in specific locations to detect deadlocks and applied the PI to eliminate them through iterative computation. The algorithm yielded results consistent with those in Ref. [6].

The remainder of this article is organized as follows. The algorithm for the deadlock prevention policy is presented in Section 2. Section 3 provides two illustrative examples demonstrating the detailed application of the algorithm. Finally, the conclusion is presented in Section 4.

2. Iterative Synthesis and PI

We introduced the deadlock prevention policy in this study [12]. We employed iterative syntheses as a proactive approach to eliminate deadlocks within the PNS. This strategy aligns with the findings and recommendations presented in the referenced report, contributing to the enhancement of deadlock resilience in our system.

For two idle places (p₁, p₄), two resource places (p₇, p₈), and operation places (active places), the saturated line is (p₁, p₄) = (2, 2) [13] and the maximal reachability states are 8. In this case, the PNS has one deadlock. The PI is obtained by using the control place (CP) equation ${(D}_{\mathrm{c}}=-L{·D}_p, u_c=b- u_p$) where D_c is the PNS controller, L is an n_c × n matrix, D_p is the PNS of the process, u_c is the marking of the controller places which is an n_c × 1 vector, b is an n_c × 1 vector, and u_p is the marking vector of PNS modeling the process [12,14,15,16]. n_c is the number of constraints by the u_i + u_j ≤ 1 where u_i and u_j are the marking places of p_i and p_j by the process PNS. One can refer to [12] for a more in-depth description.

The iterative number token of the idle places from the small to the saturated line [13] was used to search the deadlocks. Then, the PI was used to obtain CPs (D_c) and select dead marking minus 1. The algorithm is presented in Figure 1. The maximal reachability states (RSs) are 8, and the saturated line is (p₁, p₄) = (2, 2). When (p₁, p₄) is equal to (1, 1), a dead marking (M₄ = p₂ + p₅) is obtained. The operation places are omitted. According to the dead marking, D_p is obtained using Equation (1). D_c ($D_{\mathrm{c}}=-L{·D}_p$) when D_c = t₂ + t₅ − t₁ − t₄. The dead marking coefficient sum (DMCS) is 2 in this case. By subtracting 1 from the DMCS, the marking of the controller places (u_c = 1) is obtained.

Figure 2 shows the result of the CP in time PNA (TINA) [17].

$${\mathrm{D}}_{\mathrm{p}}=\begin{array}{c}\begin{array}{c}\\ {\mathrm{p}}_2\end{array}\\ {\mathrm{p}}_5\end{array}\begin{array}{c}\begin{array}{cc}{\mathrm{t}}_1& \begin{array}{cc}{\mathrm{t}}_2& \begin{array}{cc}{\mathrm{t}}_4& {\mathrm{t}}_5\end{array}\end{array}\end{array}\\ \left[\begin{array}{cc}-1& \begin{array}{cc}1& \begin{array}{cc}0& 0\end{array}\end{array}\\ 0& \begin{array}{cc}0& \begin{array}{cc}-1& 1\end{array}\end{array}\end{array}\right]\end{array}$$

(1)

In the above algorithm, the following parameters are defined.

• Input: PNM prone to deadlocks
• Output: PNM without deadlocks
• Saturated line for PNS
• Subtracting 1 from the saturated line as the token number in idle places
• Searching the dead markings
• $D_{\mathrm{c}}=-L{·D}_p, u_c=b- u_p$ to obtain CPs
• Zero deadlocks
• Obtaining the live PNS
• End

3. FMS

An FMS of the factory is shown in Figure 3a and its sequence production is illustrated in Figure 3b [6,11]. The PNS is illustrated in Figure 4. In the figure, p₂₁ and p₂₂ are I/O (input and output), p₁₂, p₁₃, and p₁₅ are machines 1–3, p_11, and p₁₄ are robots 1–2. The initial reachability states are 380; this PNS has a live lock, so it is not live and not reversible.

In the developed method, (p₂₁, p₂₂) = (1, 2) and the dead marking was (p₁₁ + p₁₂ + 2p₁₃ + p₄ + 2p₆). From the deadlock, we only considered the operation places (p₄ + 2p₆) and omitted other places. The total token number in operation places was 3. To limit the step not to enter p₄ + 2p₆, d the dead marking must be avoided. This means that u₄ + u₆ ≤ 2 and the control token is 2 [12]. $D_{p1}$ cis calculated as follows.

$$D_{p1}=\begin{array}{c}\begin{array}{c}\\ p_4\end{array}\\ p_6\end{array}\begin{array}{c}\begin{array}{cc}{t}_5& \begin{array}{cc}{t}_6& \begin{array}{cc}{t}_8& t_9\end{array}\end{array}\end{array}\\ \left[\begin{array}{cc}1& \begin{array}{cc}-1& \begin{array}{cc}0& 0\end{array}\end{array}\\ 0& \begin{array}{cc}0& \begin{array}{cc}1& -1\end{array}\end{array}\end{array}\right]\end{array}$$

(2)

CP 1 is obtained as D_c1 = −t₅ + t₆ − t₈ + t₉ by using $D_{\mathrm{c}1}=-L{·\mathit{D}}_{\mathit{p}\mathit{1}}$. When (p₂₁, p₂₂) = (2, 1), the dead marking is 2cp₁ + p₁₁ + p₁₂ + 2p₁₅ + 2p₃ + p₇. When u₃ + u₇ ≤ 2 and D_c2 = −t₄ + t₅ − t₉ + t₁₀, $D_{p2}$ is calculated as follows.

$$D_{p2}=\begin{array}{c}\begin{array}{c}\\ p_3\end{array}\\ p_7\end{array}\begin{array}{c}\begin{array}{cc}{t}_4& \begin{array}{cc}{t}_5& \begin{array}{cc}{t}_9& t_{10}\end{array}\end{array}\end{array}\\ \left[\begin{array}{cc}1& \begin{array}{cc}-1& \begin{array}{cc}0& 0\end{array}\end{array}\\ 0& \begin{array}{cc}0& \begin{array}{cc}1& -1\end{array}\end{array}\end{array}\right]\end{array}$$

(3)

When (p₂₁, p₂₂) = (2, 2), the dead marking is p₁₁ + p₁₂ + p₁₄ + 2p₃ + 2p₆. When u₃ + u₆ ≤ 3 and D_c3 = −t₄ + t₅ − t₈ + t₉, $D_{p3}$ is calculated as follows.

$$D_{p3}=\begin{array}{c}\begin{array}{c}\\ p_3\end{array}\\ p_6\end{array}\begin{array}{c}\begin{array}{cc}{t}_4& \begin{array}{cc}{t}_5& \begin{array}{cc}{t}_8& t_9\end{array}\end{array}\end{array}\\ \left[\begin{array}{cc}1& \begin{array}{cc}-1& \begin{array}{cc}0& 0\end{array}\end{array}\\ 0& \begin{array}{cc}0& \begin{array}{cc}1& -1\end{array}\end{array}\end{array}\right]\end{array}$$

(4)

CPs are presented in

Table 1. Three CPs.

(p21, p22)	Dead Marking	Tokens	CP
(1, 2)	p11 + p12 + 2p13 + p4 + 2p6	2	Dc1 = −t5 + t6 − t8 + t9
(2, 1)	2cp1 + p11 + p12 + 2p15 + 2p3 + p7	2	Dc2= −t4 + t5 − t9 + t10
(2, 2)	p11 + p12 + p14 + 2p3 + 2p6	3	Dc3 = −t4 + t5 − t8 + t9

and Figure 5.

Table 2. Comparison of results.

Item	Reference	This Study
CP number	3	3
CP token	7	7
RS	336	336

presents a comparison of the CP number, the number of tokens in each CP, and the reachability states (RSs) with those in Ref. [6].

4. Conclusions

We identified deadlocks through the repeated analysis of small blocks and the application of the PI method. The developed algorithm in this study was experimentally proven to be feasible [18,19,20,21,22]. Optimal results were obtained by employing the theory of regions [23] and dividing the PNS structure into subnets to identify CPs. By utilizing simple variant tokens in idle places, we constructed the subnet PNS, thereby simplifying deadlocks in the reachability graph. This approach enables easy access to CPs and successfully mitigates the state exploration problem [18,19,20,21,22]. In future research, it is necessary to extend the applicability of the algorithm to larger PNSs and obtain optimal results.