Generalized State Classes of Time Petri Nets for Timeliness Analysis

Generalized State Classes of Time Petri Nets for Timeliness AnalysisD IANXIANG X U+*, J IACUN W ANG++ AND R ICHARD A. V OLZ++Department of Computer ScienceTexas A&M UniversityCollege Station, TX 77840, USA++Nortel Networks2201 Lakeside Blvd, MS-99203G50Richardson, TX 75082, USASubmitted as a short paper to IEEE Transactions on Robotics and Automation Special issue on Analysis and Control of Automated Manufacturing Systems via Timed Petri-Net Models.A BSTRACTTime Petri Nets (TPN’s), as an expressive formalism for modeling distributed real-time systems, are usually analyzed through classical state classes or clock-stamped state classes. There are two issues, however, that have not been effectively or efficiently addressed: 1) how to facilitate the timeliness analysis, particularly the end-to-end time delay of task execution, and 2) how to compare state classes for equality in order to reduce state space. In the state class approach, dynamic firing domains consisting of relative firing intervals of individual transitions and timing inequalities for pairs of transitions have to be transformed into a canonical form, which has polynomial complexity. This makes the generation of state class graphs more complicated and the computation of end-to-end delay of task execution inconvenient. In the clock-stamped state class approach, the conditions for determining the fireability of transitions in general TPN’s are loosened indirectly because of the use of global firing domains, which may lead to a bigger reachable set than that obtained by using the state class approach, and only state class trees, rather than graphs, can be generated. To solve these problems, this paper uses dynamic, relative firing intervals for the determination of transition fireability and global firing domains for the timeliness analysis. For general reachability analysis, a state class graph is generated only in terms of relative firing intervals. Whenever a goal is reached through a certain schedule, global firing domains are applied to analyze the end-to-end delay of the schedule. A simple yet typical manufacturing system is used to demonstrate our approach.Keywords: Time Petri nets, state class, reachability analysis, timeliness analysis, real-time systems.* Corresponding author: Dianxiang Xu, xudian@.I. I NTRODUCTIONPetri nets [1, 2] have been used to model a variety of discrete event systems [3, 4] due to their power in modeling asynchronous events, parallelism, contention, and synchronization. It has been of high interest to extend and use Petri nets by adding time features for time dependent systems [5]. Introduction of time into transitions, places or arcs increases the modeling power as well as the complexity of net analysis. Several extended models of Petri nets have been proposed to deal with the timing issues [6], such as timed Petri nets [7, 8], stochastic timed Petri nets [9], and time Petri nets (TPN’s) [10]. Among these models, TPN’s are most widely used for real-time system specification and verification [5, 11-13]. In TPN’s, event synchronization is represented in terms of a set of pre- and post-conditions associated with each individual action of the modeled system, and timing constraints are expressed in terms of minimum and maximum amount of time elapsing between the enabling and the execution of each action. This allows for explicit modeling of time-dependent concurrency and parallelism. It has been shown [5] TPN’s are expressive for most of the temporal constraints while some of these constraints are difficult to represent only in terms of durations, e.g. in Ramchandani’s Timed Petri nets [8].The most fundamental and useful method for analyzing TPN’s, like for many other formal models, is reachability analysis [5]. It permits the automatic translation of behavioral models into a state transition graph made up of a set of states, a set of actions, and a succession relation associating states through actions [11]. The state transition graph makes explicit such properties as deadlock and reachability [14], and allows for the automatic verification of ordering relationships among task execution times [12]. Two major approaches for the reachability analysis of TPN models are classical state classes [5] and clock-stamped state classes [13]. As will be discussed in detail, however, there are two issues that these approaches have not effectively or efficiently addressed: 1) how to facilitate the timeliness analysis, particularly the end-to-end time delay of task execution, and 2) how to compare state classes for equality in order to reduce state space when generating a state class graph. These issues are important for practical analysis of distributed real-time system models.In the classical state class approach [5], timing inequalities representing the relationships among the firing times of transitions are introduced in dynamic firing domains regarding the fact that relative, dynamic firing intervals of individual transitions alone are inaccurate for timeliness analysis. Accordingly, the timing inequalities for transition pairs, although independent of the determination of transition fireability under a given state class, are also considered for the comparison of state classes (i.e. whether or not two state classes are equal depends on their markings, dynamic firing intervals as well as the inequalities). To do so, dynamic firing domains have to be transformed into a canonical form, which has polynomial complexity [5, 11]. In general, the inequalities make the generation of state class graphsmore complicated. So, this approach is inefficient in deriving the end-to-end delay of a task schedule. In contrast, the clock-stamped state class approach [13] uses the global time mode for state classes, and the end-to-end delay can be easily obtained. Since two schedules reaching the same marking generally have different global time delays, this approach does not compare clock-stamped state classes for equality and thus only generates state class trees, rather than state class graphs. As a result, the reachability analysis is less efficient due to the increased complexity of state space. Another critical problem with the clock-stamped state class approach is that it loosens the conditions of determining the fireability of transitions because of the use of global firing domains. As a consequence, it makes some unreachable markings reachable in some cases. Therefore, this approach is not quite suitable for the timeliness analysis either.To solve the above problems, this paper presents an effective approach, using relative firing intervals for the determination of transition fireability and global firing domains for the timeliness analysis. For reachability analysis of a general TPN model, we generate the state class graph only in terms of dynamic firing intervals of individual transitions without considering the timing inequalities on transition pairs. Whenever a goal is reached through a certain schedule in the state class graph, global firing domains are applied to analyze the end-to-end delay of the schedule. A simple yet typical manufacturing system will be used to demonstrate our approach.The rest of this paper is organized as follows. To make the paper self-contained, section 2 gives a brief introduction to TPN’s, state classes, and clock stamped state classes, and describes the problems with the two approaches. In section 3, generalized state classes are introduced for the reachability and timeliness analysis of TPN’s. Section 4 presents the modeling and analysis of a simple manufacturing system. Section 5 concludes this paper.II. T IME P ETRI NETS, S TATE C LASSES AND C LOCK-S TAMPED S TATE C LASSES In this section, we first define TPN’s that will be used, then briefly review state classes and clock-stamped classes and analyze the issues that they have not efficiently or effectively addressed.A. Time Petri NetsA TPN is a tuple (P, T, B, F, M0, SI) where:(1) P = {p1, p2, …, p m} is a finite nonempty set of places.(2) T = {t1, t2, …, t n}is a finite nonempty set of transitions.(3) B: P × T → N is the backward incidence function.(4) F: T × P → N is the forward incidence function.(5) M0 is the initial marking. (P, T, B, F and M0 together define a Petri net)(6) SI is a mapping of static intervals. ∀t ∈T, SI(t) = [SEFT(t), SLFT(t)], where SEFT(t) andSLFT(t) are non-negative rational numbers, SEFT(t) is the static earliest firing time, and SLFT(t) is the static latest firing time.A state of a TPN is a pair S = (M, I) consisting of a marking M and a firing interval set I. I is a vector of firing times.The number of entries in this vector is given by the number of the transitions enabled by M. A state is reached from the initial state by a given sequence of firing times corresponding to a firing sequence. Since any reachable marking may be reached from the initial marking by different sequence of firing times corresponding to the same firing sequence, the state space is generally infinite.B. State ClassesA state class of a TPN, as an aggregated pseudo-state associated with a firing sequence, represents all states reachable from the initial state by firing all feasible firing values corresponding to the same firing sequence [5]. Let Enabled(M) be the set of transitions enabled under marking M, and τ(t) be the firing time of t relative to the time point when M is reached. A state class is a pair C = (M, D), in which M is the marking of all states in the class, and D is the firing domain defined as a set of linear inequalities. For clarity, we classify the linear inequalities into two categories: the dynamic firing intervals of individual enabled transitions (i.e. a i≤τ(t i) ≤ b i, for any t i∈ Enabled(M) ) and the inequalities of timing constraints on pairs of enabled transitions (i.e. a ij≤τ(t i)- τ(t j) ≤ b ij for any t i, t j∈ Enabled(M) and t i≠t j). Note that, whether or not a transition enabled by M is firable under state class (M, D) depends on the dynamic firing intervals. The inequalities are in essence used for the timeliness analysis and the comparison of state classes when the state class graph is generated. In order to make use of dynamic firing domains, they need to be reduced into a canonical form. It has been proven that one such canonical representation exists uniquely, and an algorithm for this purpose has polynomial complexity [5, 11]. [11] shows timeliness analysis can be conducted by repetitive exploration of the firing intervals and inequalities in the state classes. In comparison, we will show it can be achieved in a more convenient and effective way by using global firing intervals. The generation of state class graph is then made more efficient because there is no need to consider the inequalities in state classes.Without considering the inequalities, we denote a dynamic firing domain D as {D(t)= [EFT(t), LFT(t)]: t∈Enabled(M)}, where D(t) is the dynamic firing interval of t, and EFT(t) and LFT(t) are the dynamic earliest firing time and latest firing time of t, respectively. An enabled transition t is firable under state class C if EFT (t) ≤min{LFT(t i), t i∈Enabled(M)}. Suppose Firable(C) is the set of firable transitions under state class C, and MLFT(C) = min {LFT(t i), t i∈ Firable(C)}is the minimum of latest firing times of all firable transitions in Firable(C). The construction of the state class graph of TPN can be briefly outlined as follows: The initial state class is C0 =(M0, D0), where M0 is the initial marking and D0= {[SEFT(t), SLFT(t)]: t∈Enabled(M0) } (at the initial state, the dynamic firing interval for any enabled transition is equal to its static firing interval); Given a state class C k= (M k, D k), a successor state class C k+1= (M k+1, D k+1) can be created for some t ∈ Firable(C k) by following steps:• The actual firing interval for t under class C k is [EFT k(t), MLFT k(t)]. M k+1 is obstained according to the firing rule of classical Petri nets.• For each inherited transition t j (t j≠t) that is enabled under both M k and M k+1, EFT k+1(t j) = max(0, EFT k(t j)-MLFT(C k)), and LFT k+1(t j) = LFT k(t j)-EFT k(t)• For each newly enabled transition t j under M k+1 (not enabled under M k), EFT k+1(t j) = SEFT(t j) and LFT k+1(t j) = SLFT k(t j).Note that, dynamic firing intervals of individual transitions alone are inaccurate for the timeliness analysis. For example, suppose the initial marking for the TPN in Fig. 1 is (1, 0, 1, 0. 0)T. According to the semantics of TPN’s, the correct time span for reaching marking (0, 1, 0, 1, 0)T by firing t2 and t1 is [3, 5]. Using the state class method, the relative firing intervals for t1 after t2 fires at sometime during [2, 4] is [max{0, 3-4}, 5-2], i.e. [0, 3]. Obviously, the intervals [2, 4] and [0, 3] have no clear relationship with the correct time span [3, 5].C. Clock-Stamped State ClassesA clock-stamped class [13] of a TPN consists of three parts: 1) a marking M, 2) an "global" firing domain corresponding to firing intervals (relative to the beginning of execution) of all firable transitions in the state class, and 3) a clock stamp that corresponds to the moment when the state class is reached with the clock value relative to the beginning of execution.Fig 1. A TPN ExampleThe clock stamped state class method has two critical issues. First, the fireability of transitions under a clock stamped state class cannot be correctly determined in terms of the global firing intervals. For the above example, when marking (0, 1, 0, 1, 0)T is reached, the time stamp for the state class is [3, 5], andthe firing domains for newly enabled transitions t3 and t4 are [1, 3] + [3, 5] = [4, 8] and [4, 5] + [3, 5] = [7, 10], respectively. According to the firing rules of clock stamped state classes, both t3 and t4 are firable because both EFT(t3) = 4 and EFT(t4) = 7 are less than min{LFT(t3), LFT(t4)} = 8. This is incorrect regarding the semantics of TPN’s due to the fact that t4is never firable (t3must fire before t4, which disables t4). In essence, clock stamped state classes are not suitable for the reachability analysis of general TPN’s. Second, since time stamps in a firing schedule are increasing monotonically, the clock stamped state class method can only generate state class trees, rather than state class graphs. This makes state space even more complex.III. T IMELINESS AND R EACHABILITY A NALYSISIn this section, we generalize the concept of state classes by integrating classical state classes with global time mode for the timeliness analysis of TPN’s, such that:• relative firing domains are used to determine the fireability of transitions, and• absolute firing domains are used to calculate the time span of a firing schedule.For general reachability analysis, the state class graph of a TPN is generated only in terms of dynamic firing intervals. The timeliness of a firing schedule is then conducted separately using global firing intervals.A. Generalized State ClassesA generalized state class (GSC) is a 4-tuple C = (M, D, AD, ST) where(1) M is a marking.(2) D is a relative firing domain, i.e., a set of constraints on the values of the time to fire fortransitions enabled by current marking M. For an enabled transition t i, D(t i) represents its relative firing interval. Let EFT(t i) be the left bound of D(t i) (the relative earliest firing time) and LFT(t i) be the right bound of D(t i) (the relative latest firing time).(3) AD is an global (absolute) firing domain, i.e., a set of constraints on the values of the time to firefor transitions enabled by current marking M. For an enabled transition t i, AD(t i) represents its absolute firing interval. Let AEFT(t i) be the left bound of AD(t i) (the absolute earliest firing time) and ALFT(t i) be the right bound of AD(t i) (the absolute latest firing time).(4) ST is the time stamp of the GSC class, which is a (global) time interval.Before we define the transition firing rules, let us explain what we want the absolute firing domain AD and the time stamp ST to be: (1) For an enabled transition t i, AD(t i) gives the global firing time interval of t i, where by “global” we mean the values are counted relative to the beginning of the net’s execution from the initial GSC class C0,defined as C0= (M0, D0, AD0, ST0), where M0 is the initialmarking, D 0 and AD 0 contain all static firing time intervals of the transitions enabled at M 0, and ST 0 = [0, 0]. (2) ST indicates the global time delay interval in which the net runs from the initial GSC class C 0 to current GSC class C. Theorem 1 in next subsection will show this property.Now we consider the firing rules that guide the timeliness analysis of a TPN. An enabled transition t j is said to be firable at GSC-class C k if EFT k (t j ) ≤ min{LFT k (t i ), t i ∈ En(C k )}, where En(C k ) is the set of all enabled transitions at C k . Let Fr(C k ) be the set of firable transitions at GSC class C k , andMLFT(C k ) = min{LFT k (t i ), t i ∈ Fr(C k )},MALFT(C k ) = min{ALFT k (t i ), t i ∈ Fr(C k )},where MLFT(C k ) / MALFT(C k ) define the minimum of relative/absolute latest firing times of all firable transitions in Fr(C k ). We divide the firable transitions in Fr(C k ) into two groups: (1) inherited firable transitions that were firable before C k is reached, and (2) newly firable transitions that begin firable at C k . The firing of transition t f ∈ Fr(C k ) changes GSC class C k to C k+1. Let C k = (M k , D k , AD k , ST k ) and C k+1 = (M k+1, D k+1, AD k+1,ST k+1). We define firing rules as follows:Step 1. Calculate )(~f k t D , the feasible relative firing interval for firing transition t f . It is obtained by shifting right bound of D (t f ) to MLFT (C k ) while keeping its left bound unchanged, that is, )]( ),([ )(~k f k f k C MLFTt EFT t D = LetST k+1 =[AEFT k (t f ), MALFT(C k )]Step 2. Calculate firing intervals of inherited firable transitions in GSC class C k+1.2.1 Let )(1f k k t B M M −′=′+ and collect (inherited) firable transitions at M ′k+1.2.2 Let D k+1 = D k , AD k+1 = AD k and delete from D k+1 and AD k+1 all entries whose correspondingtransitions are disabled by M ′k+1.2.3 For each inherited firable transition t j (t j ≠ t f ) at M ′k+1, letEFT k+1(t j ) = max{0, EFT k (t j )-MLFT(C k )}LFT k+1(t j ) = LFT k (t j )-EFT k (t f )AEFT k+1(t j ) = max{AEFT k (t j ), AEFT k (t f )}.ALFT k+1(t j ) = ALFT k (t j )Step 3. Calculate the firing intervals of newly enabled transitions after firing t f .3.1 Let )(11f k k t F M M +′=++ and collect newly enabled transitions: they are enabled at M k +1 butnot firable at virtual marking M ′k +1.3.2 Add into D k +1, AD k+1 entries for corresponding newly enabled transitions at M k +1: if t j (t j ≠ t f )is a newly enabled transition at M k +1, thenD k+1(t j) = SI(t j).AD k+1(t j) = SI(t j) + ST k+1 .3.3 If t f is still enabled at M k+1 after firing, letD k+1(t f) = SI(t f).AD k+1(t f) = SI(t f) + ST k+1.Note that by Step 3.3, a transition, which is still enabled after its firing, is always treated as a newly enabled one. This simplifies the treatment of states in which a transition has sufficient tokens in its input places to permit multiple firings. The treatment of this situation, usually referred to as multiple enabledness [5], requires multiple firing intervals be associated with a single transition and involves a number of semantic subtleties that are not relevant to the objective of this paper.B. Timeliness AnalysisWe have described the transition firing rules that guide the evolution of GSC classes. Now we show how GSC classes are suitable for timeliness analysis. Theorem 1 illustrates what the absolute firing domain AD and the time stamp ST in a GSC class exactly stand for. Corollary 1 indicates what can be gained from the generation of the reachability graph. To facilitate our description, we denote firing schedule t0 t1…. t n-1 that transforms C0 to C n by (C0, t0)(C1, t1)…(C i, t i)…(C n-1, t n-1)C n.Theorem1. Let C i = (M i, D i, AD i ST i ) be a reachable GSC from C0 = (M0, D0, AD0,,ST0) through firing schedule t0 t1…. t n-1. Then(1) ST i is the global time (interval) when GSC class C i is reached;(2) If t j∈Fr(C i), then AD i(t j) is the global firing time interval of t j.Proof:(1) From the preconditions, we know there must be a firing schedule starting with C0 and ending with C i, that is, (C0, t0)(C1, t1)…(C i-1, t i-1)C i. The proof of the theorem is carried out by induction on i. For the basis case (i = 0), C0 is the initial class. Obviously we have: 1) ∀ t j ∈Fr(C0),. AD0(t j)is exactly the static firing time interval of t j, which is also the global firing time. And 2) ST0 = [0, 0], which is the arriving time of C0. Therefore, the theorem holds for i = 0.Now assume that the assertion holds for i ≤ k. Consider i = k + 1. It follows from the equation ST k+1 = [AEFT k(t k), MALFT(C k)]where AEFT k(t k) is the absolute earliest firing time of t k, and MALFT(C k) is the minimum latest firing time of all firable transitions in Fr(C k), hence the actual latest global firing time of t k. Therefore, ST k+1 is the global firing time interval of t k. Because firing a transition takes no time, ST k+1 is also the global arriving time of state class C k+1, which is reached by firing t k+1. So (1) holds.(2) Suppose that a transition t j is firable at C k+1. There are three different cases of t j.Case 1. t j is a newly enabled transition at M k+1and t j≠t k.AD k+1(t j) = SI(t j) + ST k+1 = [SEFT(t j) + AEFT k(t k), SLFT(t j) + MALFT(C k)],where, AEFT k(t k) is the earliest (global) arriving time of state class C k+1, SEFT(t j) the static (relative) earliest firing time when t j is enabled at C k+1, so SEFT(t j) + AEFT k(t k) is the earliest global firing time of transition t j; MALFT(C k) is the latest (global) arriving time of state class C k+1, SLFT(t j) the static (relative) latest firing time when t j is enabled at C k+1, so SLFT(t j) + MALFT(C k) is the latest global firing time of transition t j. Therefore, AD k+1(t j) is the global firing time interval of t j.Case 2. t j = t k.Because we ignore multiple-enabledness, so t k is viewed as a new enabled transition at M k+1. Thus the conclusion drawn in Case 1 also applies to this case.Case 3. t j is an inherited transition.In this case, it follows from Step 2 thatAD k+1(t j) = [max(AEFT k(t j), AEFT k(t k)), ALFT k(t j)].According to the assumption made for i ≤ k, [AEFT k(t k)), ALFT k(t j)] is the global firing time interval of transition t j at state class C k. The latest global firing time of t j at C k+1 should be the same as it is at C k; however, the earliest global firing time of t j at C k+1 must take the larger value of AEFT k(t j)and AEFT k(t k), because t j is supposed to fire after C k+1 is reached. So, AD k+1(t j) is the global firing time interval of transition t j at state class C k+1.Thus the theorem holds. t According to theorem 1, ST i gives the exact global time interval when GSC class C i is visited. Corollary 1. Let C i = (M i, D i, AD i, ST i) and C j = (M j, D j, AD j, ST j) be two reachable GSC classes of a TPN where C j is reachable from C i. If ∀ t j ∈Fr(C i), t j is a newly enabled transition, then the time span that the TPN runs from C i to C j is ST j −ST i.Proof:Because all firable transitions at C i are newly enabled, so the future behavior of the TPN starting from C i is reached is independent of the history before C i is reached. Suppose that if the TPN starts running from class C i at time 0, it will reach class C i at global time interval [x, y]. Then we know that if the TPN starts running from class C i at time a, it will reach class C i at global time interval [x+a, y+a]. Futhermore, if the TPN starts running from class C i at time interval ST i= [a, b], it will reach class C i at global time interval ST i = [min{x+a, x+b}, max{y+a, y+b}] = [x+a, y+b]. Because the time span that the TPN runs from C i to C j is independent of the starting time, so it follows from Theorem 1 that the time span is [x, y], or ST j −ST i. t Corollary 1 is very useful for timeliness analysis. As mentioned in [12], the key issue of timeliness analysis is to verify whether a marking can be reached with timing constraints. Corollary 1 shows that theconcept of GSC classes helps establish quantitative timing relationship between any two reachable classes in a firing schedule.C. Reachability Analysis through State Class GraphsThe major purpose of generalized state classes is meant for the timeliness analysis of firing schedules. General reachability analysis is conducted in two separate steps. 1) Generating state class graph without considering the time stamps and global firing domains. Two state classes are viewed as identical if and only if they have the same firable transitions and corresponding relative firing intervals. This is similar to the traditional state class graph, except the timing inequalities on transition pairs. Since there are no timing inequalities on transition pairs, the functional reachability analysis is generally more efficient. 2) If a goal marking is reached in the state class graph, analyzing the time delay of the schedule that reaches the goal. An example illustrating this will be given in next section.IV. A N E XAMPLETo illustrate our approach, this section describes the modeling and analysis of a simple yet typical manufacturing system, which is composed of 5 machines and 1 assembler, as shown in Figure 2. The system receives two types of parts (A and B) as inputs. A-parts go to machine 1 for processing, whereas B-parts go to machine 2 for processing. After being processed by machine 2, B-parts go to either machine 3, or machines 4 and 5 for processing. After a pair of A-part and B-part gets processed, they are sent for assembly, where a final product is produced. Assume that the processing of machine 1 on an A-part takes 2 to 9 time units, the processing of machine 2, machine 3, machine 4, and machine 5 on a B-part takes 1 to 3, 3 to 7, 1 to 2, and 2 to 5 time units, respectively. The assembler takes 2 to 4 time units.Fig 2. A manufacturing system.The TPN model of this system is shown in Fig. 3. Table 1 describes its places and transitions.Fig 3. The TPN model for the example manufacturing systemTable 1. Legends of the TPN in Fig. 3.Place Descriptionp1 A-part bufferp2 B-part bufferp3 A-part ready for assemblyp4 B-part processed by machine 2p5 B-part being processed by machine 5p6 B-part ready for assemblyp7 Buffer for final productsTransition Description Time intervalt1 Machine 1 works on A-part [2, 9]t2 Machine 2 works on B-part [1, 3]t3 Machine 3 works on B-part [3, 7]t4 Machine 4 works on B-part [1, 2]t5 Machine 5 works on B-part [2, 5]t6 Assembler works on a pair of A- and B- parts [2, 4]Let us first analyze the timeliness of schedule t2t4t5t1t6. The initial GSC class is C0 = (M0, D0, AD0, ST0) whereST0= [0, 0].M0= (1 1 0 0 0 0 0)T,D0= {D0(t1): [2, 9] , D0(t2): [1, 3]}.AD0= {AD0(t1): [2, 9] , AD0(t2): [1, 3] }.Thus,MLFT(C0) = min{ LFT0(t1), LFT0(t2) } = {9, 3} = 3.MALFT(C0) = min{ ALFT0(t1), ALFT0(t2) } = {9, 3} =3Both t1 and t2 are firable. Firing t2 will reach GSC class C1 = (M1, D1, AD1, ST1). Following Step 1 in section III-A, the dynamic relative firing interval for t2 is [1,3], and ST1 = [AEFT0(t1), MALFT(C0)] = [1, 3]. Following Step 2, we have:M′1 = M0−B(t1)= (1 0 0 0 0 0 0)T.D1(t1) = [0, 8].AD1(t1) = [2, 9].By Step 3, M1 = M′1 + F(t1)= (1 0 0 1 0 0 0) T, under which there are two newly enabled transitions t3 and t4.D1(t3) = [3, 7].D1(t4) = [1, 2].AD1(t3) = [3, 7]+ ST1= [4,10].AD1(t4) = [1, 2]+ST1 = [2, 5].Under GSC class C1, t1 and t4 are firable. But t3 is not firable because EFT1(t3)=3 > min{LFT1(t1), LFT1(t3), LFT1(t4)} = 2. Note that, AEFT1(t3)=4 < min{ALFT1(t1), ALFT1(t3), ALFT1(t4)} = 5 . The global firing intervals cannot be used to determine the fireability of t3 (in the clock-stamped state class method, t3 would be firable). If t4 fires, its relative firing interval is [1, 2]. Similarly, we can get a new GSC class, say C2, where:M2 = (1 0 0 0 1 0 0)T.ST2 = [2, 5].D2(t1) = [0, 7].AD2(t1) = [2, 9].D2(t5) = [2, 5].AD2(t5) = [2, 5]+ST2 = [4, 10].Under C2, both t1 and t5 are firable. The relative firing interval of t5 is [2, 5]. Firing t5 reaches GSC class C3 such that:M3 = (1 0 0 0 0 1 0)T.ST3 = [AEFT2(t5), min{ALFT2(t1), ALFT2(t5)}] = [4, 9]D3(t1) = [0, 5].AD3(t1) = [max{ELFT2(t1), ELFT2(t5)}, ALFT2(t1)] = [4, 9].Under C3, only t1 is enabled and firable. Firing t1 leads to GSC class C4, where:。