A Computationally Improved Optimal Solution for Deadlocked Problems of Flexible Manufacturing Systems Using Theory of Regions

© 2012 Pan, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A Computationally Improved Optimal Solution for Deadlocked Problems of Flexible Manufacturing Systems Using Theory of Regions


Introduction
While competing for a finite number of resources in a flexible manufacturing system (FMS), e.g., robots and machines, each part has a particular operational flow that determines the order in which such resources are needed.However, such competition for shared resources by concurrent job processes can lead to a system deadlock.It occurs when parts are blocked waiting for shared resources held by others that will never be granted.Its related blocking phenomena often incur unnecessary overhead cost, e.g., a long downtime and low utilization rate of some critical and expensive resources, possibly leading to a catastrophic outcome in some highly automated FMS.Therefore, an efficient deadlock control policy must be developed to ensure that deadlocks do not occur.Having received considerable attention in literature, deadlock is normally prevented by using an offline computational mechanism to control the resource requests in order to avert deadlocks.Fanti and Zhou 1 introduce three fundamental methods (i.e.prevention, detection and avoidance) to solve the deadlock problems.Deadlock prevention aims to impose system constraints to prevent a deadlock.Importantly, deadlock prevention algorithms do not require run-time costs since the problems are solved in system design and planning stages.This study belongs to the deadlock prevention field.
Petri nets (PN) 2 have been recognized as one of the most powerful formal methods for modeling FMS.The reason is that they are well suited to represent such FMS characteristics as precedence relations, concurrence, conflict and synchronization.Their analysis methods used for deadlock prevention in FMS include structural analysis and reachability graphs.Deadlock prevention and avoidance schemes have been developed for controlling FMS [3][4][5][6][7][8] by using the former.In particular, deadlock prevention problems are solved using the concept of siphons [3][4][5][6] .Li & Zhou propose an elementary siphon control policy (ESCP) to reduce the redundant siphons to obtain structurally simpler controllers [9][10] .However, they cannot obtain optimal ones.Reachability graph methods are used to obtain the live system behavior [11][12][13][14] .Without confining to a certain class of FMS, they can provide an optimal deadlock controller by adopting the theory of regions 15 .The theory is originally developed for a transition system (TS).A state-based representation with arcs labeled with symbols from an alphabet of events in a TS can be mapped into a PN model.For an elementary TS (ETS) there exists a PN with minimum transition count (one transition for each label) with a reachability graph isomorphic to the original TS.
Uzam 12 follows the theory of regions 15 to define a deadlock-zone (DZ) and deadlock-free zone (DFZ) for preventing deadlocks.Hence, the concept of DZ and DFZ is used to solve ESSPs.An optimal controller can be obtained but suffers from many redundant control places.Ghaffari et al. 13 propose a unique interpretation of the theory of regions and define MF (forbidden marking), MD (dangerous marking), ML (legal marking), and  (the set of marking/transition-separation instances or MTSI).An optimal PN controller synthesis method for FMS is proposed based on both MTSI and the theory of regions.Unfortunately, redundant MTSIs cannot be entirely avoided for large FMS cases.
To reduce redundant control places, Li et al. 16 adopt a combined algorithm based on siphon control and the theory of regions 15 .Its advantage is that the number of separation instances is significantly reduced after some sets of elementary siphons of a system are controlled.However, it fails to determine all sets of MTSIs and its application seems limited to some special nets only.
Uzam and Zhou propose an iterative control policy of liveness enforcement for PNs based on the theory of regions 17 .Less computation is required to obtain a controller.However, as indicated by Li et al 18 , it requires the repeated calculation of reachability graphs.Piroddi et  al. propose a combined selective siphons and critical markings in a reachability graph algorithm to obtain optimal controllers via iterations 19 .They successfully identify the critical uncontrolled siphons and control them to make a deadlock-prone PN live.However, their algorithm also requires the repeated calculation of reachability graphs.Eventually, the controllers are not ordinary (i.e. they contain weighted arcs).This work in this chapter aims to develop a computationally more efficient optimal deadlock control policy by using the theory of regions.It focuses on dead markings in a reachability graph.The concept of a crucial MTSI (CMTSI) is proposed to synthesize optimal controllers.The proposed method can reduce the computational burden of the MTSI method 13 and redundant control places [12][13] .The experimental results indicate that it is the most efficient policy among all known ones [12][13]16 that can design optimal controllers. Secton 2 presents the basic definitions and properties of PNs and the theory of regions.Section 3 describes the proposed policy.Section 4 presents the experimental results.Section 5 gives the comparisons.Conclusions are made in Section 6.

Petri nets 2
A Petri net (PN) is a 5-tuple N = (P, T, F, W, M0) where P is a finite set of places; T is a finite set of transitions, with P  T ≠  and P  T =  ; F  (P×T)  (T×P) is the set of all directed arcs, W: (P×T)  (T×P) → is the weight function where = { 0, 1, 2, …}, and M0: P → is the initial marking.A PN is said to be ordinary, denoted as (P, T, F), if f  F, W(f) = 1.
[N] + (p, t) = W(p, t) is the input function that means the multiplicity of a directed arc from p to t if (p, t)F.
[N] -(p, t) = W(t, p) is the output function that means the multiplicity of a directed arc from t to p if (t, p)F.The set of input (resp., output) transitions of a place p is denoted by •p (resp., p•).Similarly, the set of input (resp., output) places of a transition t is denoted by •t (resp., t•).A PN structure (P, T, F, W) is denoted by N. A PN with a given initial marking is denoted by (N, M0).
A PN is said to be pure if no place is both input and output places of the same transition.The so-called incidence matrix [N] of a pure Petri nets is defined as spurious ones (with respect to the state equation) 20 .They may also be the solutions of the state equation but not reachable markings.In this work, ones just focus on the reachable markings.
A transition t is said to be live if for any M R(N, M0), there exists a sequence of transitions whose firing leads to M' that enables t.A PN is said to be live if all the transitions are live.A PN contains a deadlock if there is a marking M  R(N, M0) at which no transition is enabled.Such a marking is called a dead marking.Deadlock situations are as a result of inappropriate resource allocation policies or exhaustive use of some or all resources.Liveness of a PN means that for each marking M  R(N, M0) reachable from M0, it is finally possible to fire t, t  T through some firing sequence.(N, M0) is said to be reversible, if  M  R(N, M0), M0  R(N, M).Thus, in a reversible net it is always possible to go back to initial marking (state) M0.A marking M is said to be a home state, if for each marking M  R(N, M0), M is reachable from M. Reversibility is a special case of the home state property, i.e. if the home state M = M0, then the net is reversible.

Theory of regions and synthesis problem 13
The theory of regions is proposed for the synthesis of pure nets given a finite TS 15 , which can be adopted to synthesize the liveness-enforcing net supervisor (LENS) for a plant model [12][13] .For convenience, our method follows the interpretation of the theory of regions in 13 .
First of all, let T be a set of transitions and G be a finite directed graph whose arcs are labeled by transitions in T. Assume that there exists a node v in G such that there exists a path from it to any node.The objective of the theory of regions is to find a pure PN (N, M0), having T as its set of transitions and characterized by its incidence matrix [N](p, t) and its initial marking M0, such that its reachability graph is G and the marking of node v is M0.In the following, M denotes both a reachable marking and its corresponding node in G.
Consider any marking M in net (N, M0).Because (N, M0) is pure, M can be fully characterized by its corresponding incidence vector where M   is the firing vector of path M  .For any transition t that is enabled at M, i.e., t is the label of an outgoing Consider now any oriented cycle  of a reachability graph.Applying the state equation to a node in  and summing them up give the following cycle equation: where  is an oriented cycle of G, ( ) t   is a firing vector corresponding to , and C is the set of oriented cycles of G.
According to the definition of G, there exists an oriented path M from M0 to M. Applying (1)   along the path leads to There are several paths from M0 to M. Under the cycle equations, the product [N](p,) M   is the same for all these paths.As a result, M  can be arbitrarily chosen.The reachability of any marking The above equation is called the reachability condition.Notably, (3) is necessary but not sufficient.Hence, spurious markings are beyond this paper.
It is clear that the cycle equations and reachability conditions hold for any place p.For each pair (M, t) such that M is a reachable marking of G and t is a transition not enabled at M, t should be prevented from happening by some place p.Since the net is pure, t is prevented from happening at M by a place p iff The above equation ( 4) is called the event separation condition of (M, t).The set of all possible pairs (M, t) where M is a reachable marking and t is not enabled at M is called the set of event separation instances or marking/transitions-separation instances (MTSI) 13 .Symbol  is used to represent the set of MTSI in this paper.To solve the control problem,  is identified.The corresponding control places can then be found to prevent the transitions of the controlled system from firing in order to keep all legal markings only.

Controller synthesis method
In this section, an efficient controller synthesis method is developed based on the theory of regions.Please note that all transitions of the PN models are regarded as controllable ones.

Supervisory control problem
It is assumed that a deadlock-prone PN model contains at least a dead marking in its reachability graph at which no transition is enabled.Its reachability graph contains dead and live zones.Consequently, this study attempts to propose a method to prevent the controlled systems from entering a dead zone/marking.
A dead marking cannot enable any transition and thus cannot go to any other markings.We can formally define the dead marking MD as follows.
Definition 1: The set of dead markings MD = {M R(N, M0)| at M, no transition is enabled}.
Definition 2: A zone consisting of all dead markings is called a dead zone, denoted by ZD.
Once a marking enters a dead zone, the system is dead.If there is no dead zone in a reachability graph, the system is called a live one.
The goal of the work is to control a deadlock-prone system such that it is live.All markings of a reachability graph can be divided into three groups: legal markings (ML), quasi-dead markings (MQ), and dead markings (MD).

Definition 3:
The set of quasi-dead markings MQ = {MR(N, M0)| M must eventually evolve to a dead one regardless of transition firing sequences}.
Definition 4: A zone consisting of all quasi-dead markings is called a quasi-dead zone, denoted by ZQ.
Definition 5: A zone consisting of all quasi-dead and dead markings, i.e., ZI = ZD  ZQ, is called an illegal zone.
Markings except quasi-dead and dead markings are legal ones.Once a legal marking is enforced into the illegal zone, the net will eventually become deadlock.
Definition 6: A zone consisting of all legal markings is called a legal zone, i.e., ZL = R(N, M0) -ZI.
Ramadge and Wonham show that a system has the maximally permissive behavior if the system behavior equals ZL 21 .In other words, one must remove all the markings in illegal zone (i.e.quasi-dead and dead markings) from R(N, M0) if one wants to obtain the maximally permissive behavior.Ghaffari et al. propose the MTSI method to achieve their deadlock prevention based on the theory of regions 13 .However, the set of all MTSIs from the reachability graph must be identified.As a result, we can conclude that their method is computationally inefficient.A more efficient method is thus needed as described next.

Crucial MTSI (CMTSI)
Two types of CMTSIs are defined as follows.A Type II CMTSI contains a legal marking that cannot reach a dead marking with one single transition's firing as shown in Figure 2. Given a dead marking in M"D, the shortest transition firing sequence needs to be found.The main reason is based on the fact that, for a dead marking, the length of the firing sequence from the initial marking to CMTSI is the longest path than those from the initial marking to MTSIs.Hence, the solutions of MTSIs will be totally covered by the solution of CMTSI.For example, as shown in Figure 3, σ * is the shorter path since |σ * | < |σ| (i.e.|σ * | =1 and |σ| = 3).
Remark 1: A dead marking is always with its corresponding CMTSI.As a result, the corresponding CMTSI is of either Type I or II.Type I may be viewed as a special case of Type II CMTSI by defining  * = 0 (no need to enter ZQ but directly to ZD).Type I CMTSI will be processed first in our proposed method.In the following, Theorems 1-3 will help readers to understand how to choose CMTSIs, which are with the same firing sequence of legal markings, from Types I and II.
Theorem 1: If a dead marking M  M'D is associated with two different CMTSIs, only one CMTSI needs to be controlled.
Proof: Assume that a dead marking M is with both CMTSIs {Mi, tm} and {Mj, tn} as shown in Figure 4.According to the state equation, Mi According to (4), realizing either CMTSI, e.g., {Mi, tm},    Proof: Assume that a dead marking M is associated with both Type II CMTSIs {Mp, tr} and {Mq, ts}.Mp and Mq reach two different quasi-dead markings Mp and Mq via tr and ts's firing, respectively as shown in Figure 6.
According to the state equation, Mp    Control places are then found after CMTSIs.They are used to keep all markings of the controlled system within the legal zone.Here, Figure 8(a) taken from existing literatures 22 is used to demonstrate how to identify two types of CMTSIs from its reachability graph (i.e. Figure 8(b)).Assume that all transitions of PN models are immediately in this case.Therefore, one can easy identify there are two dead markings M'D (i.e.M2) and M"D (i.e.M15) and four quasi-dead markings (i.e.M11, M12, M13 and M14).Additionally, the markings M0, M1, M3-M10 are the legal markings.Based on the mentioned above, there are two sets of CTMSIs in the reachability graph system due to the two dead markings in the system.As a result, one can infer that {M0, t2} belongs to type I CMTSI and {M7, t1} belongs to type II.In this Petri net system model, there are only one type I CMTSI and only one type II.

Procedure of deadlock prevention policy
Next, quasi-dead, dead, and legal markings are identified.Based on [12][13] , the maximally permissive behavior means all of legal markings (ML) and the number of reachability condition equations equals |ML|.Additionally, all CMTSIs can be obtained such that the legal markings do not proceed into the illegal zone.The proposed deadlock prevention algorithm is constructed as Figure 9.
Theorem 5: The proposed deadlock prevention policy is more efficient than the method proposed by Ghaffari et al. 13 Proof: The theory of regions is used to prevent the system deadlocks by both our deadlock prevention policy and the conventional one.All MTSIs can be controlled by the two control policies.Since (  )  , the use of CMTSI can more efficiently handle the synthesis problem than that of MTSI 13 .
[ N](pc, t1) After listing all reachability conditions, all legal markings can be determined.In detail, M0-M6, M8, and M13-M19 are legal.Hence, the following reachability conditions are obtained.
M 0(pc)  0 (9) Furthermore, two optimal control places CP1 and CP2 can be obtained when ( 5) and ( 7)-( 23) are solved.Their detailed information is: By the same way, using ( 6) and ( 7)-( 23), one can find two optimal control places, Cp3 and Cp4.M0 Notably, CP1 and CP4 are the same.Therefore, a redundant control place (CP4) can be removed.As a result, the system net can be controlled with the three control places CP1, CP2 and CP3.The optimally controlled system net (N1H, M0) is obtained as shown in Table 1.
It is worthy to emphasize that the three control places are obtained by using two CMTSIs and 36 equations under our control policy.However, six MTSIs/ESSPs and 108 equations have to be solved in two existing literatures 12,24 .
Additional Control Places M0(Cpi) Example II: This example is taken from 23 and is used in 12,16,24 .Here, the PN model of the system, denoted as (N2, M0), is shown in Figure 12.  2. Notice that {(M56, t9)} in M77 is a redundant one.Tables 3-4 show the event separation condition equations based on 18 CMTSIs.Here, the procedure of our method is introduced as follows.Due to the space limitation, we use only one example (i.e.dead marking M9) to illustrate how to prevent legal markings from leading to dead one M9 by using a CMTSI.To do so, 'C1 = {(M65, t1)} can be located due to M9.The event separation condition equation can then be identified as follows.

MD Event Separation Condition Equations
Next, three different cycle equations are: Finally, 205 reachability condition equations can be listed.They represent the sequence of all legal markings from the initial one.Moreover, control place CP1 can be computed, i.e., M0 (CP1) = 2, CP1 = {t6, t13}, and CP1 = {t1, t11}.Similarly, other control places are obtained as shown in Tables 5-6.Finally, the controlled net is obtained by adding the six control places as shown in Table 7.It is live and maximally permissive with 205 reachable markings.However, 59 ESSPs and nine control places are required in 12 .It hints that 59 sets of inequalities are needed in 12 , while only 18 sets of inequalities suffice using our algorithm.Hence, our policy is more efficient than that in 12 .
To compare the efficiency of the deadlock prevention methods, the proposed one is examined in the system net (N2L1, M0).One can realize that M57 is the only deadlock marking in R(N2L1, M0).Based on our method, only the dead marking M57 needed to be controlled.
Here, only one  = {(M44, t9)} is needed.Obviously, the involved necessary equations are much less than those of the conventional one.We can obtain the same controlled net as that in 16 .Hence, the proposed concept of CMTSIs can be used in their approach 16 to improve its computational efficiency significantly as well.

Comparison with existing methods
One can attempt to make a comparison with the previous methods 12,16,24 in terms of efficiency.The first one proposed by Uzam For Example II, eight MTSIs are required to obtain the six control places under Algorithm L. Hence, one can infer that its performance is better than that of Algorithm U.Only one set of CMTSI is needed to obtain the same control result by Algorithm P. As a result, one can conclude that our proposed policy is more efficient than the other two methods.
To examine and compare the efficiency of the proposed method with those in 16,24  In Table 12, the number of MTSIs calculated by Algorithm L can be controlled, but Algorithm P is more efficient in these cases.For instance, when M (p15) = M (p18) = M (p19) = 5, || L = 192, meaning that one still has to solve 192 MTSIs when Algorithm L is used.However, || P =108.Only 108 MTSIs need to be solved by using Algorithm P. Importantly, the computational cost can be reduced by using our proposed method when it is compared with those in 12,16 .In conclusion, Algorithm P is more efficient in large reachability graph cases than those in [12][13] .

Conclusion
The proposed policy can be implemented for FMSs based on the theory of regions and Petri nets, where the dead markings are identified in its reachability graph.The underlying notion of the prior work is that many inequalities (i.e.MTSIs) must be solved to prevent legal markings from entering the illegal zone in the original PN model.One must generate all MTSIs in a reachability graph and require high computation.This work proposes and uses CMTSI to overcome the computational difficulty.7][28][29] The proposed method can reduce the number of inequalities and thus the computational cost very significantly since CMTSIs are much less than MTSIs in large models.Consequently, it is optimal with much better computational efficiency than those existing optimal policies [12][13]16 . Morebenchmark studies will be desired to establish such computational advantages of the proposed one over the prior ones.It should be noted that the problem is still NP-hard the same as other optimal policies due to the need to generate the reachability graph of a Petri net.The future research is thus much needed to overcome the computational inefficiency of all these methods.
or p is marked with at least W(p, t) tokens, as denoted by M[t>.A transition may fire if it is enabled.In an ordinary net, it is enabled iff p  •t, M(p)  1. Firing t at M gives a new marking Msuch that  p  P, M(p) = M(p) -W(p, t) + W(t, p).It is denoted as M[t>M.M indicates the number of tokens in each place, which means the current state of the modeled system.When Mn can be reached from M0 by firing a sequence of transitions σ, this process is denoted by M [σ > Mn and satisfies the state equation Mn = M + [N]   .Here,   is a vector of non-negative integers, called a firing vector, and ( ) t   indicates the algebraic sum of all occurrences of t in .The set of all reachable markings for a PN given M0 is denoted by R(N, M0).Additionally, a definition of linearized reachability set (using the state equation) is defined as (N, R M0)={M: M = M0 + [N](•   )}.This definition is suitable for the incorporation of the state equation into a set of linear constraints.The markings in (N, M0) -R(N, M0) are called R

Definition 7 :
Type I CMTSI:  = {(M, t)|M ML, t  T, and  M  MD, M ML, and t T such that M [t > M and M [t > M}.Denote the set of all the dead markings related to  as M'D, i.e., MD = {M  MD |  (M, t)   such that M [t  M}.They are called type I deadlocks.Definition 7 explains a legal marking that can evolve into a dead or legal zone as shown in Figure 1 through a single transition's firing.For those dead markings that are not type I deadlocks, we need to introduce Type II CMTSI and deadlocks.Definition 8: A zone consisting of all type I deadlocks (M'D) is called type I dead zone, denoted by Z.Definition 9: k is defined as a transition firing sequence starting in a quasi-dead marking (MQ) and ending in a deadlock marking in MD where i = |σk| is the number of transitions in σk, called its length.Denote a firing sequence with the shortest length (i.e., smallest i) from any quasi-dead marking to M' as σ * (M') given M'  MD -M'D.Definition 10: Type II CMTSI :  = {(M, t)|M  ML, t  T, and  M  MQ, M  ML, M  MD, t T, and a firing sequence σ = σ * (M ) from M to M such that M [t > M , M [t > M, and M [σ > M}.The set of dead markings associated with Type II CMTSI is denoted as M"D, called type II deadlocks.MD = {M  MD |  (M, t)  , M  MQ and a firing sequence  from M to M such that M[tM and σ = σ * (M )}.

Figure 1 .
Figure 1.A structure of Type I CMTSI.

Remark 2 :
which in turn implies M0 + [N]N](•tn)  -1 and vice versa.Hence, only one CMTSI needs to be controlled.Based on Theorem 1, if a dead marking M  M'D is associated with more than two CMTSIs, only one of them needs to be controlled.

Figure 2 .
Figure 2. A structure of Type II CMTSI.

Figure 3 .
Figure 3.The shorter path σ * in Type II CMTSI given a dead marking.

Theorem 2 :
If a dead marking M  MD, is associated with two CMTSIs whose markings can reach a same quasi-dead marking M' via their respective single transition's firing, only one CMTSI needs to be controlled.Proof: Assume that a dead marking M is associated with Type II CMTSIs {Mp, tr} and {Mq, ts}.Mp and Mq reaches a quasi-dead markings M' via tr and ts's firing, respectively as shown in Figure 5.According to the state equation, Mp + [N](•tr) + [N](• *   ) = Mq + [N](•ts) + [N](• *   ) M. Arranging the above equation, one can realize that Mp + [N](•tr) = Mq + [N](•ts) .According to (4), realizing either CMTSI, e.g., {Mp, tr}, leads to M0 + [N] N](•tr)  -1, which in turn implies M0 + [N] N](•ts)  -1 and vice versa.Hence, only one CMTSI needs to be controlled.Theorem 3: A dead marking M  MD, is associated with two CMTSIs whose markings can reach two different quasi-dead markings Mp and Mq via two different single transitions' firing.Both need to be controlled if [N](• * r   )  [N](• * s   ).
Mp is not equal to Mq.And also according to the definition of the event separation condition equation, the first set of CMTSI {Mp, tr} leads to the first event separation condition equation is M0 + [N](• 0 p M M    ) + [N](•tr)  -1; and the second set of CMTSI {Mq, ts} leads to the another event separation condition equation is M0 + [N](• 0 q M M    ) + [N](•ts)  -1.Hence, one can infer that Mp and Mq are two different quasi-dead markings if [N](• * r   )  [N](• * s   ).It hints the two event separation condition equations are different.As a result, both CMTSIs need to be controlled.

Figure 4 .
Figure 4.A type I deadlocks associated with two CMTSIs.

Figure 5 .Figure 6 .Remark 3 :
Figure 5. Two CMTSIs connected to the same quasi-dead marking.Definition 12: A legal marking M  ML can be led to a quasi-dead marking Mq via a single transition firing.Mq must eventually evolve to a dead one Md (i.e.Md  MD) after a sequence n = t1t2…tn fires.Denote the set of all the markings on the path from Mq to Md as Mq-d.

Figure 7 .
Figure 7. Two CMTSIs connected to two quasi-dead markings Mp and Mq with r * = s * =  * (MD).

Table 2 .
The Dead Markings and Their Relative CMTSIs.

Table 3 .
The Dead Markings and Relative Event Separation Condition Equations of Type I CMTSI.

Table 4 .
The Dead Markings and Relative Event Separation Condition Equations of Type II CMTSI.

Table 5 .
Control Places from TYPE I CMTSI.

Table 6 .
Control Places from TYPE II CMTSI.

Table 10 .
16,24alled Algorithm U, is totally based on the theory of regions.It solves six ESSPs in Example I. Then three control places are added on the net such that the controlled net is live and reversible.As for Example II, it solves 59 MTSIs.Nine control places are obtained.However, the proposed deadlock prevention policy called Algorithm P solves only two and 18 CMTSIs in Examples I and II, respectively.The other one is proposed by Li et al.16,24called Algorithm L in which only the theory of regions is used in Example I. Notice that both the controlled results of Algorithms L and U are the same in Example I. InExample II, using Algorithm L, eight MTSIs are solved and six control places are computed.However, under the two-stage control policy, only one set of MTSI is needed by using our new policy to obtain the controlled result that is as the same as Algorithm L in Example II.Note that both the definitions of ESSP and MTSI are the same.Hence, ESSP and CMTSI can be regarded as MTSI for the comparison purpose.The detailed comparison results are given in Table10. Hoever, only 18 MTSIs among 59 MTSIs are needed by using Algorithm P. Comparison of the Controlled Systems.

Table 11 .
, indicate the number of reachable markings (states), legal markings, and dead markings under Algorithms U and L, respectively.Additionally, MTSIs of Algorithms U, L, and P are symbolized by || U , || L and || P , respectively.The last column is ra = || P / || U in Table11, and rb = | P / || L in Table12.Notably, Algorithm G 13 can be regarded as Algorithm U in Table11since the number of MTSIs and ESSPs are the same.In table 11, here, Nsep represents the number of MTSIs, and the Nsep/U, Nsep/L and Nsep/P represent the number of MTSIs of Algorithms U, L, and P, respectively.Obviously, the number of || U in the plant model grows quickly from cases 1 to 8.For instance, when M (p15) = M (p18) = M (p19) = 5, || U = 4311, meaning that one must solve 4311 MTSIs when Algorithm U is used.However, since || P = 228, only 228 equations (MTSIs) need to be solved under Algorithm P. As a result, Algorithm P is more efficient than Algorithm U in a large system.Parametersin the Plant and Partially Controlled Models with Varying Markings: U vs. P.

Table 12 .
Parametersin the Plant and Partially Controlled Models with Varying Markings: L vs. P.