Generating Finite-State Global Behavior of Reconfigurable Automation Systems: DEVS Approach Moon-Ho Hwang Department of Electrical and Computer Engineering Wayne State University Detroit, MI 48202, USA Email: [email protected]

Abstract— This paper introduces a method of achieving the finite-state global behavior of a reconfigurable automation system that consists of sub-components and which can be another component in a higher-level system. The approach introduced here is based on the discrete event system specification (DEVS) that has been regarded as a powerful simulation tool for modular and hierarchical systems, but its global states for whole behavior can be infinite. To achieve the finite state behavior that is needed for quantitative and qualitative analysises of a system, this paper introduces a new class of DEVS, called schedule-controllable DEVS (SCDEVS) that can be seen as a middle class between the ordinary DEVS and schedule-preserved DEVS (SP-DEVS) that was proven that the global behavior of SP-DEVS networks can be abstracted in a finite number of states. The condition for finite-state space of SC-DEVS networks is addressed in the paper. In addition, the comparison between SC-DEVS and SP-DEVS in terms of the number of states and the time consumed in the generation procedure is illustrated by using a cluster tool example.

Fig. 1.

Layout of a Cluster Tool

I. I NTRODUCTION To reduce the lead time in system development, as well as to increase flexibility and adaptability, reconfigurability has been widely accepted in various applications of automation engineering from computer architecture [1] to manufacturing systems [2]. One of the key ideas of reconfigurability is modularity; that is, a technology leading a system consisting of sub-components, called modules and interconnecting with each other so that the system achieve its unified objectives with high performance. Fig. 1 shows a typical cluster tool consisting of process modules, cassette modules and a transport module. In the real cluster tool, there are couplings between its controller and manipulator for flows of sensing signals and actuating commands, as well as material flows as shown in Fig. 2. We can see that the modularity is highly related to the hierarchical modelling functionality because a newly constructed module built-on predefined sub-components should be another component in a higher level [3]. It is not difficult to imagine that the cluster tool shown in Fig. 1 is used in a fabrication lab that can be operated in a company. In order to accommodate the modular and hierarchical feature, Zeigler has extended the discrete event system specification (DEVS) from the non-modular version [4] to the

modular and hierarchical version by introducing the concept of DEVS networks, called coupled DEVS, and showing that DEVS is closed under the coupling operation [5]. Due to its modular and hierarchical feature, DEVS has been widely spread out to various engineering areas such as computer [6], manufacturing [7], traffic [8] and ecological disasters [9]. However, the method of analysis in DEVS formalism has been restricted to simulation because a DEVS network generally has its infinite states space caused by its real-valued time base. Since simulation is a method to trace a possible behavior of a system an infinite number of simulation runs can be required to verify the whole behavior of the DEVS model. Thus, the simulation-based analysis has been criticized for its incompleteness. To overcome the problem of the infinite-state space of DEVS, recently a class of DEVS, called schedule-preserved DEVS (SP-DEVS), has been introduced [10]. By enforcing the strict conditions such as 1) rational-valued time advance function and 2) preserving the internal schedule from an external input, SP-DEVS can achieve the property being closed under the coupling operation with the finite states space.

Fig. 2.

Information & Material Flows of a Cluster Tool Fig. 3.

However, it is a fact that the strict restriction of SP-DEVS can be criticized for loss of modeling flexibility; for example, once its get into a state whose next schedule is in a infinite time, then it cannot return to any state whose next schedule is in a finite time. We call it the ODNR phenomena (once it dies, it never returns) that causes the waiting self-loop that can lead an increased number of states (we will see it in Section IV). This paper introduces a new class of DEVS, called schedulecontrollable DEVS (SC-DEVS), that are conditionally closed under the coupling operation with a finite number of states. SC-DEVS might be regarded as the middle class between ordinary DEVS and SP-DEVS because its properties in terms of modeling power and closure under coupling are in the middle of DEVS and SP-DEVS. The reason why we introduce SC-DEVS is that it has no problem of the ODNR, so the SCDEVS model can be active and passive in terms of its schedule. It leads to the reducction of the number of global states by eliminating waiting self-loop in SP-DEVS. The article is organized as follows. Section II-A introduces SC-DEVS. The SC-DEVS network structure and the condition for closure of coupling operations are addressed in Section III-B. A cluster tool was exemplified in generating finite state behavior of a reconfigurable manufacturing system, and their performance comparisons between SP-DEVS and SC-DEVS are illustrated in Section IV. Finally, the conclusion and further research are presented in Section V. II. S CHEDULE -C ONTROLLABLE DEVS When using SC-DEVS formalism, every leaf node in the hierarchy is described by atomic SC-DEVS. This section gives us the definition of SC-DEVS and the behavior of SC-DEVS by a simple milling machine example. A. Atomic SC-DEVS An atomic SC-DEVS specifies the dynamic behavior. Definition 1 (Atomic SC-DEVS): An atomic model of SCDEVS is a 10-tuple, M =< X, Y, S, τ, δx , ρ, δτ , λ, s0 , SF > where, • X is a finite set of input events. • Y is a finite set of output events.

A Milling Machine

S is a non-empty and finite states set. τ : S → Q+,∞ is the maximum sojourning time of a 0 state where Q+,∞ denotes a set of non-negative rational 0 numbers with infinity. • δx : S × X → S is the external transition function. • ρ : S × X → {0, 1} is the reschedule indicating function. • δτ : S → S is the internal transition function. ² • λ : S → Y is the internal output function where Y ² means Y ∪{²} and ² is the non-event such that ² 6∈ X ∪Y . 1 • s0 ∈ S is the initial state. • SF ⊆ S is a set of acceptable states. ¥ Example 1 (Milling Machine): Let’s start to take a look at a very simple model for atomic SC-DEVS. Fig. 3(a) shows a simple machining center consisting of an input buffer and a machining table. The input buffer and the machining table are modelled by atomic SC-DEVS, respectively shown in Fig. (b) and Fig. (c). In state transition graphs of Fig. (b) and (c) a node s, τ (s) means the state s with τ (s), for example, EE, 1 is s = EE and τ (EE) = 1. And the state transition δx (s, ?x) = ?x s0 is illustrated as s −→ s0 if ρ(s, x) = 1, while we use a ?x dashed line s 99K s0 when ρ(s, x) = 0. The internal transition δτ (s) = s0 with its output event λ(s) =!y is illustrated by !y s −→ s0 . And the initial state is indicated by an arrow, while an element of acceptable states is denoted by doubled lines; for example, the state EE is both the initial and an acceptable state. These graphical notations are used throughout this paper. The mathematical presentation of the input buffer is as follows: MBuf f er =< X, Y, S, τ, δx , ρ, δτ , λ, s0 , SF > where X = {?xi, ?ii}; Y = {!o}; S = {EE, OE, EO, OO}; τ (EE) = 1, τ (OE) = 1, τ (EO) = ∞, τ (OO) = ∞; δx (EE, ?mi) = OE, δx (EO, ?ii) = EE, δx (EO, ?mi) = OO, δx (OO, ?ii) = OE; δτ (EE) = EE, δτ (OE) = EO, ρ(EE, ?mi) = 0, ρ(EO, ?ii) = ρ(EO, ?mi) = ρ(EO, ?ii) = 1; λ(EE) = ², λ(OE) =!0; s0 = EE; SF = {EE}. ¤ • •

B. Behavior of Atomic SC-DEVS Definition 2 (Behavior of Atomic SC-DEVS): 1 This paper assumes only one initial state, but it is easily extended to a set of initial states, as in [10]

III. SC-DEVS N ETWORK This section introduces another formalism that provides the functionality of hierarchical model construction, called coupled SC-DEVS or SC-DEVS network. A. Definition of SC-DEVS network Definition 3 (Structure of SC-DEVS Network): A SC-DEVS network is a 7-tuple, N =< X, Y, C, Zxx , Zyx , Zyy , select >

Fig. 4.

Trajectories of Input Buffer

Given M =< X, Y, S, τ, δx , ρ, δτ , λ, s0 , SF >, the total states set Q considers the remaining time r at s ∈ S such that Q = {(s, r)|s ∈ S, 0 ≤ r ≤ τ (s)}

(1)

where • X(Y ) is the finite set of input (output) events. • C = {Mi |Mi =< Xi , Yi , Si , τi , δxi , ρi , δτ i , λi , s0i , SF i >} is the finite set of sub-component SC-DEVSs, which can be atomic SC-DEVS or coupled SC-DEVS models. It is assumed that all of the leaf nodes are atomic SC-DEVS models and hierarchical depth to all leaf notes are finite. S • Zxx ⊆ X × Xi is the external input coupling Mi ∈C

The total state transition function δ : Q × X ∪ Y ∪ {²} → Q is defined as follows. For (s, r) ∈ Q, e ∈ X ∪ Y ∪ {²}, δ((s, r), e) = (s0 , r0 ) = q where  (δx (s, e), τ (δx (s, e))) if e ∈ X ∧ ρ(s, e) = 1    (δ (s, e), r) if e ∈ X ∧ ρ(s, e) = 0 x q= (2)  (δτ (s), τ (δτ (s))) if r = 0 ∧ e = λ(s)    (s, r) otherwise Notice that the remaining time at the new state generated δx can be controlled by ρ in SC-DEVS. This is the reason why the DEVS is named as schedule-controllable. By using the total state function, we can define the state trajectory function δˆ : Q × (T, X ∪ Y ∪ {²}) → Q such that for q ∈ Q, t ∈ R+, 0 and e ∈ X ∪ Y ∪ {²}, ( δ(q, e) for ω = (t, e) ˆ δ(q, ω) = (3) 0 ˆ δ(δ(q, ω ), e) for ω = ω 0 (t, e) Example 2 (Behavior of Input Buffer): Let’s take a look at the behavior of the input buffer shown in Fig. 3(b). We assume that q = (EE, τ (EE)) = (EE, 1) and this model starts at time t = 0. As time passes, its total state has been changing from (EE, 1) to (EE, 0). At time t1 if there is no external input; its total state is changed from (EE, 0) to (EE, 1) ˆ because δ((EE, 1), ²[0,1] ) = (EE, 1) where ²[0,1] means there is no event within time [0, 1]. In this example, an input event ?mi occurs at time t2 = 1.5, which makes the state changed into (OE, 0.5). Since δx (OE) = EO and λ(OE) =!o, the state trajectory from ˆ (EE, 1) with ω[0,t3 ] = (t2 , ?mi)(t3 , !o) is δ((EE, 1), ω[0,t3 ] = ˆ ˆ δ(t ˆ 2 , ?mi), (t3 , !o)) = δ((EE, 1), (t2 , ?mi)(t3 , !o)) = δ( ˆ δ((OE, 0.5), (t3 , !o)) = δ((OE, 0), !o) = δτ (OE) = EO. In the same manner, we can trace the state trajectory with a given timed sequence of input and output events. ¤

•

relation. Zyx ⊆

S Mi ∈C

Yi ×

S

Xi is the internal coupling

Mi ∈C

•

relation. S Yi → Y is the external output coupling Zyy =

•

function. select : 2C − ∅ → C is a tie-breaking function

Mi ∈C

¥ Example 3 (Milling Machine): The structure shown in Fig. 3(a) is described by structure of SC-DEVS as follows. X = {i}; Y = {o}; C = {MB , MM } where MB and MM are instances of MBuf f er and MM illingT able , respectively; Zxx = {(?m, ?miB )} where ?miB indicates ?mi of model B; Zyx = {(!oB , !iM ), (!oM , ?iiB )}; Zyy (!oM ) =!o; select({B, M }) = M. ¤ B. Behavior of SC-DEVS Network Since atomic SC-DEVS has been defined in Section II-A, we would explain the behavior of SC-DEVS networks using atomic SC-DEVS-like form as follows. Definition 4 (Behavior of SC-DEVS Network): Given a SC-DEVS network, N =< X, Y, C, Zxx , Zyx , Zyy , select >, the global behavior of N called a coupled SC-DEVS model is MN =< X, Y, S, τ, δx , ρ, δτ , λ, s0 , SF > where • X(Y ) is the set of input events (output events) defined in N • S = {(. . . , (si , ri ), . . .)|(si , ri ) ∈ Qi , Mi ∈ C}. +,∞ • τ : S → Q0 is defined for s ∈ S, τ (s) = τ ((. . . , (si , ri ), . . .)) = M inimum ri Mi ∈C

•

δx : S × X → S is defined as δx ((. . . , (si , ri ), . . .), x) = (. . . , (s0i , ri0 ), . . .) where (s0i , ri0 ) =

  if Cond1 (δxi (si , xi ), ri ) where (δxi (si , xi ), τi (δxi (si , xi ))) if Cond2   (si , ri ) otherwise Cond1 = (x, xi ) ∈ Zxx ∧ ρi (si , x1 ) = 0 and Cond2 = (x, xi ) ∈ Zxx ∧ ρi (si , xi ) = 1. • ρ : S × X → {0, 1} is defined such that for s = (. . . , (si , r( i ), . . .) ∈ S and x ∈ X 0 if ρi (si , xi ) = 0∀(x, xi ) ∈ Zxx ρ(s, x) = 1 otherwise Let the set of imminents, IM M (s) = {Mi |Mi ∈ C ∧ ri = τ (s)}. Imminents are the set of components that have minimum remaining time qi .r and Mi∗ = select(IM M (s)). • δτ : S → S is defined as δτ ((. . . , (si , ri ), . . .)) = (. . . , (s0i , ri0 ), . . .) where (s0i , ri0 ) =  0 0 (si∗ , τi∗ (si∗ )) if Mi = Mi∗  (δxi (si , xi ), ri ) if (yi∗ , xi ) ∈ Zyx   (si , ri ) otherwise where s0i∗ = δτ i∗ (si∗ ) and yi∗ = λi∗ (si∗ ). ² • λ : S → Y is defined as λ((. . . , (si , ri ), . . .)) = Zyy (λi∗ (si∗ )). • s0 = (. . . , (si , τi (si )), . . .). • SF = {(. . . , (si , ri ), . . .)|∀Mi ∈ C, (si , ri ) ∈ QF i }. ¥ Because its states are associated with components’ remaining times that are real numbers, the number of states, |S| in the behavior of a SC-DEVS network is infinite. For achieving the finite state global behavior of a SC-DEVS network, we will apply the time abstraction method which has been introduced in [10]. C. Relative-Schedule Abstraction Definition 5: Suppose that N =< X, Y, C, Zxx , Zyx , Zyy , select > is a SC-DEVS network and MN =< X, Y, S, τ, δx , ρ, δτ , λ, S0 , SF > is its global behavior. Let (si , ri ) ∈ Qi , Mi ∈ C. Then s = (. . . , (si , ri ), . . .), s0 = (. . . , (s0i , ri0 ), . . .) ∈ S, s is equivalent to s0 in terms of their relative-schedules, denoted by s ∼ =t s0 , if ∀Mi ∈ C, 0 0 0 si = si and ri − r = ri − r where r = M inimum ri and r0 = M inimum ri0 . Mi ∈C

Mi ∈C

m m m MN =< X, Y, S m , τ m , δxm , ρm , δτm , λm τ , S0 , SF > is said t to be ∼ = relative-schedule abstraction of MN if S m = {s ∈ S|∀s0 ∈ S s.t. s ∼ =t s0 , τ (s) ≥ m 0 m τ (s )}; τ = τ |S m →Q+,∞ ; λτ = λ|S m →Y ² ; δxm = 0 δx |S m ×X→S m ; ρm |S m ×X→{0,1} ; δτm = δτ |S m →S m ; S0m = S0 ∩ S m ; SFm = SF ∩ S m ; ¥ Example 4 (Behavioral Model of Machining Center): Let’s consider the behavioral model that is the relativeschedule abstraction of the machining center constructed in Example 3. Even though there are an infinite number of states in the model, we can get a set of states and their state transition as shown in Fig. 5 when gathering the state generated by discrete events. In Fig. 5, the first total state is for the input buffer and the later one is for the milling table. For example, (((EO, ∞), (M, 3)), 3) indicates the total states of the input

Fig. 5.

Global Behavior of the Machining Center

buffer and the milling table are (EO, ∞) and (M, 3), so the remaining time of that state is 3. ¤ Even though we can achieve the relative-schedule abstraction of the SC-DEVS network of Example 1 as shown in Fig. 5, generally speaking, we can fail to create the atomic SCDEVS model with finite states. But fortunately, a condition guaranteeing finite states by the relative-schedule abstraction is identified in the next section. It will give us a clue for enforcing finite state behavior in a given SC-DEVS. D. Condition for Behavior having Finite-States Suppose that N =< X, Y, C, Zxx , Zyx , Zyy , select > whose sub-components are atomic SC-DEVS models and s = (. . . , (si , ri ), . . .) ∈ S m , x ∈ X. The states are to be categorized according to δx ((. . . , (si , ri ), . . .), x) = (. . . , (s0i , ri0 ), . . .) as follows. 1) R(s, x) = {Mi ∈ C|ri 6= ri0 } is a set of rescheduled components. a) RF (s, x) = {Mi ∈ R(s, x)| ri0 < ∞} is a set of components rescheduled by finite time. b) RI(s, x) = {Mi ∈ R(s, x)| ri0 = ∞} is a set of the components rescheduled by infinite time. 2) C(s, x) = C −R(s, x) is a set of continued components. a) CF (s, x) = {Mi ∈ C(s, x)|ri0 < ∞} is a set of continued components with a finite-time schedule. b) CI(s, x) = {Mi ∈ C(s, x)|ri0 = ∞} is a set of continued components with an infinite-time schedule. Definition 6 (Partially Rescheduled SC-DEVS Network): Suppose that an external event, x ∈ X, is put into SC-DEVS N . Then, by x, s ∈ S m is said to be 1) rescheduled if RF (s, x) 6= ∅ ∧ CF (s, x) = ∅, 2) partially rescheduled if RF (s, x) 6= ∅ ∧ CF (s, x) 6= ∅ and 3) continued otherwise. The N is said to be partially rescheduled if there exists s0 ∈ S m such that s0 is partially rescheduled by x ∈ X. ¥ m Theorem 1: |S m | of MN is finite if N is not partially rescheduled. Proof: Let’s consider s = (. . . , (si , ri ), . . .) ∈ S m where (si , ri ) ∈ Qi and assume that for each Mi in C, Mi is an atomic SC-DEVS model. By definition of atomic SC-DEVS, the number of possible si value is finite because |Si | is finite. We need to check the number of total state (si , ri ) in S m . Let’s consider the number of remaining time ri0 at a s0i ∈ Si when an internal state transition occurs such that δτ (s) =

s0 = (. . . , (s0i , ri0 ), . . .). Let g S ∈ Q+ 0 be the greatest common divisor such that for all si ∈ Mi ∈C Si . Then every internal state transition happens at a certain time g ∗ n where nsi ∈ N+ 0 (non-negative natural numbers). That means, after time abstract minimization, the number of ri at a si ∈ Si generated by internal state transitions is finite. Let’s consider the number of ri0 in s0i ∈ S m when an external state transition occurs. The state transition considering instance sequence of external transition from s by x can be rewritten by using IR(s, x) and IC(s, x) as follows. δx ((. . . , (si , ri ), . . .), x) = (. . . , (s0i , ri0 ), . . .) where ( (s0i , τi (s0i )) if Mi ∈ R(s, x) 0 0 (si , ri ) = (s0i , ri ) if Mi ∈ C(s, x) Suppose ∀s ∈ S, ∀x ∈ X, s is not partially rescheduled; that is, s is either rescheduled or continued. By Definition 6, s is rescheduled by x, if RF (s, x) 6= ∅ ∧ CF (s, x) = ∅ . For this rescheduled case, δx (s, x) = (. . . , (s0i , ri0 ), . . .) where ( (s0i , τi (s0i )) if Mi ∈ RF (s, x) (4) (s0i , ri0 ) = (s0i , ∞) otherwise Thus r0 = M inimum(ri ) = M inimum(τi (s0i )). And for Mi ∈C

Mi ∈RF (s,x)

τi (s0i ) is fixed so r0 is independent from the occurrence time t of x and unique s0 is generated from s and x. By Definition 6, s is continued by x, if RF (s, x) = ∅ and CF (s, x) 6= ∅. Thus δx = (. . . , (s0i , ri0 ), . . .) where ( (s0i , ri ) for Mi ∈ CF (s, x) 0 0 (5) (si , ri ) = (s0i , ∞) otherwise Here, since ri0 < ∞ for Mi ∈ CF (s, x), r0 = M inimum(ri ) = M inimum(ri ). Even though r0 is depenMi ∈C

Mi ∈CF (s,x)

dent of the happening time of x, however, ri0 −r0 is constant in the continued case. Thus by the relative-schedule abstraction, only one s0 can be achieved by (s, x). In addition, by the assumption of finite depth of coupling and atomic leaf nodes, even if Mi ∈ C is a coupled SC-DEVS model, then |S m | is finite. Example 5 (Finiteness of Global Behavior Model): In order to identify whether the system in Fig. 3(a) has its finite states or not, we need to check if the network is partially rescheduled or not. Obviously, there is an input event ?i and it has coupled with ?mi of B. Notice that only two states ((EE, r1 ), (E, ∞)) and ((EE, r2 ), (E, ∞)) are responding to the external event ?i such that δx ((((EE, r1 ), (E, ∞)), r1 ), ?i) = (((OE, r1 ), (E, ∞)), r1 ) and δx ((((EO, ∞), (M, r2 )), r2 ), ?i) = (((OO, ∞), (M, r2 )), r2 ) and R(((EE, r1 ), (E, ∞)) , ?i) = ∅ and R(((EO, ∞), (M, r2 )), ?i) = ∅. That means, these responding states to the external event ?i are continued so this network is not partially rescheduled. Therefore, its global behavior can be achieved by the relative-schedule abstraction as shown in Fig. 5. ¤

Fig. 6.

Controller SC-DEVS Model for Cluster Tool

IV. G LOBAL B EHAVIOR G ENERATION OF THE C LUSTER T OOL A. Model Description This section uses a typical reconfigurable manufacturing system for generating global behavior example. Let’s consider the cluster tool whose layout and information/material-flow couplings are as shown in Fig. 1 and Fig. 2, respectively. In this target system, we would like to operate a cassette having three slots for accommodating a batch of wafers when a cassette arrives at a cassette module (CM). All wafers are assumed to have identical processes that are sequentially performed from the first process to the 3rd, whose processing times are 20, 50, and 30 seconds, respectively. Each processing module (PM) has a single capacity at one. After processing the 3rd PM, a wafer is sent to the slot of the cassette from which the wafer came out. The transportation model (TM) provide the transportation functionality from a source model to a destination module for each waiting wafer. The rotation speed of TM is 36 degrees a second (for the 360-degreed rotation, it takes 10 seconds) and the stretching & shrinking times of the arm are both 2 seconds. 1) Controller: A SC-DEVS model for the cluster-tool controller, called CTCtrl, is modelled as shown in Fig. 6. Its interface and state variables, internal state transitions, and external state transitions are summarized in Tables I, II and III, respectively. Here, ncm, nsl, and npm stand for the number of cassette modules (=2), slots (=3), and process modules (=3), respectively. In CTCtrl, the maximum sojourning time τ of all states is equal to 1. The output function λ is defined as λ(s) = !OTM:(TM.s,TM.d) if phase = CmdTransfer; λ(s) = !OPM[index]:tp if phase = CmdProcess; λ(s) =!OCM[index]:1 if phase = CndCM-Out; And ρ returns 0 for all states and input events 0. That is, the controller CTCtrl is a SP-DEVS model that can be also a SC-DEVS. 2) Plant Models: The models in the plant are constructed using SP-DEVS and SC-DEVS. Fig. 7(a), (b) and (c) are SP-DEVS-based models: the cassette module, the transport module and the process module, respectively. The sensing delay of arriving and the actuating delay of any works are assumed as ts at CM, TM and PM.

Fig. 7.

Atomic DEVS Models for a Cluster Tool

TABLE I I NTERFACES AND S TATE VARIABLES OF C ONTROLLER CTC TRL port name ?ICM[ncm] ?IPM[npm] ?ITM !OCM[ncm] !OPM[npm] !OTM

name phase CM[ncm][nsl] PM[npm] W[ncm][nsl] TM tp index

Interface port value bool;/*0: Empty (E), 1: Ocuppied (O)*/ integer;/* 0:E, 1:O, 2: Finished */ bool; /*1: picked, 0: placed */ bool; /*1: send out a cassette */ float; /* tp : processing time */ struct { integer hs , vs , hd , vd ; }; output /* indexes of source & destination */ State Variables type {Wait, CmdTranfer, CmdProcess, CmdCM-Out }; {OccupiedByNew (ON), Sending (S), Empty (E), Reserved (R), OccupiedByFinished (OF) }; {ON, S, E, R, OF, Processing (P) }; integer;/* current location index of wafer[i][j]*/ struct { integer hs , vs , hd , vd ; }; float; /* processing time of PM index*/ integer;/* commend target index of CM or PM*/ type input input input output output

B. Experiments 1) Effect of scanning/response speed: In this experiment, we tested ts with 0.5, 0.25 and 0.1 unit times. Fig. 7(d), (e), and (f) illustrates SC-DEVS models for CM, TM, and PM. The basic algorithm of generating global behavior SCDEVS networks is similar to that of SP-DEVS [11]. We would like to evaluate the performance of generating global behavior depending on ts in terms of the number of states (|S|) and state transitions (|E|) and its generating time required. Table IV summarizes the results of generating global behavior in SPDEVS models and a SC-DEVS model. In the SP-DEVS case,

TABLE II I NTERNAL S TATE T RANSITIONS OF C ONTROLLER CTC TRL C#

Pre-Condition

C1-1

∃j, k s.t. CM[j][k]=ON∧PM[0]=E

C1-2

¬ C1-1 ∧∃i, j s.t. j=i+1∧ PM[i]=OF∧PM[j] = E

C1-3

¬ C1-2 ∧∃j, k s.t. PM[npm-1]=OF∧ W[j][k]=npm-1

C2

¬ C1-3 ∧∃i s.t. PM[i]=ON

C3

¬ C2 ∧∃i, ∀j, s.t. CM[i][j] = OF

Post-Condition phase := CmdTranfer; CM[j][k]=S; PM[i]=R; W[j][k]=ncm; TM:=(j,k,ncm,0); phase := CmdTranfer PM[i]=S; PM[j]=R; W[l][m]:=j for W[l][m]=i TM:=(ncm+i,0,ncm+j,0); phase := CmdTranfer PM[npm-1]=S; W[j][k]:=j TM:=(ncm+npm-1,0,j,k); phase := CmdProcess; PM[i]:=P; index :=ncm+i; tp := tp (PM[i]); phase := CmdCM-Out; CM[i]=S; index:=i;

we varied ts as 0.5, 0.24, and 0.1 (so 1/ts is 2, 4, and 10). As 1/ts increases, the number of states and transitions are almost linearly increased, as well as generating time. Actually, the response time of sensing and actuating becomes almost 1/1000 seconds in the real as the sensor and actuator technologies have been improved. But if we apply 1/1000 seconds in SPDEVS models, their global states space explodes, even though it is bounded in a finite number. SC-DEVS models shown in Fig. 7(d),(e), and (f) can be regarded as SP-DEVS in the case of limts →0 . The waiting self-loop transition δτ (s) = s with τ (s) → 0 can be eliminated by changing τ (s) = ∞ in the SC-DEVS case so that all performance indices for generating global behavior are the best in the cluster tool experiment, as summarized in Table IV.

TABLE III E XTERNAL S TATE T RANSITIONS OF C ONTROLLER CTC TRL Input Event !ICM[i]:1

Pre-Condition CM[i]=E TM.hs
!ITM:1 !ITM:1 !ITM:0 !ITM:0 !IPM[i]:1 !IPM[i]:2 !IPM[i]:0 !ICM[i]:0

Post-Condition CM[i]:=O; W[i][k]=i for k=0 to nsl-1 CM[i][k]=OF for k=0 to nsl-1 CM[TM.hs ][TM.vs ]:=E; PM[TM.hs −ncm]:=E; CM[TM.hd ][TM.vd ]:=OF; PM[TM.hd −ncm]:=ON; PM[i]:=ON; PM[i]:=OF; PM[i]:=E; CM[i]:=E;

TABLE IV E FFECT OF G ENERATING G LOBAL B EHAVIOR BY ts 1/ts 2 4 10 -

|S|

|E|

SP-DEVS Cases 73,022 74,884 117,247 120,133 281,365 288,601 SC-DEVS Case 9,662 10,382

Time (sec.) 76.0 130.0 418.0 10.0

2) Effect of Reconfiguration: When we want to improve the performance of a system, we can usually add one component to the module that is considered as the bottleneck. We create an experiment with the effect of adding a new PM that performs the 2nd process so that now we have parallel processes in the 2nd process. We use the ts is 0.5 in SP-DEVS case. And we can modify the controller as 1) npm = npm + 1 and the C1-2 case as shown in Table V. The reconfiguration effect of the additional parallel PM is summarized in Table VI in terms of the number of states |S| and transitions |E| and their generation times. As we can see in Table VI, |S| and |E| increase 251% (216%) and 250% (215%), respectively and their generation time becomes slower by 270% (250%) in SP-DEVS cases of ts = 2 (ts = 4). Meanwhile, |S| and |E| increase 176% and 175%, respectively and their generation time is slower by 180% in SC-DEVS case.

V. C ONCLUSION AND F URTHER R ESEARCH To verify a reconfigurable automation system, this paper introduced SC-DEVS that is a middle class of ordinary DEVS and SP-DEVS and SC-DEVS can be closed under the coupling operation with finite states. A cluster tool was exemplified by TABLE V M ODIFIED PART OF I NTERNAL S TATE T RANSITION OF CTC TRL C#

C1-2

Pre-Condition ¬ C1-1 ∧ PM[0]=OF∧PM[1]=E; PM[0]=OF∧PM[2]=E; PM[1]=OF∧PM[3]=E; PM[2]=OF∧PM[3]=E;

i:=0,j:=1 ∨ i:=0,j:=2 ∨ i:=1,j:=3 ∨ i:=2,j:=3

Post-Condition phase := CmdTranfer PM[i]=S; PM[j]=R; W[l][m]:=j for W[l][m]=i TM:=(ncm+i,0,ncm+j,0);

TABLE VI E FFECT OF G ENERATING G LOBAL B EHAVIOR BY R ECONFIGURATION 1/ts 2 4 -

|S|

|E|

SP-DEVS Cases 183,520 (251%) 186,948 (250%) 253,482 (216%) 258,112 (215%) SC-DEVS Case 17,001 (176%) 18,178 (175%)

Time (sec.) 205.0 (270%) 325.0 (250%) 18.0 (180%)

generating finte state behavior of a reconfigurable manufacturing system. In all experiments, SC-DEVS showed better performance than SP-DEVS with frequently-scanning models. By using SC-DEVS, we have also studied the qualitative verification such as safetyness and fairness checking as well as quantitative verification such as optimistic and pessimistic process time evaluation, called Min/Max processing time evaluation [12]. This finite state approach is expected to extend its scalability via the symbolic state approach that employs the binary decision diagram [13]. ACKNOWLEDGMENT This work was partially supported by the Post-doctoral Fellowship Program of Korea Science & Engineering Foundation (KOSEF). R EFERENCES [1] S. C. Goldstein, H. Schmit, M. Budiu, S. Cadambi, M. Moe, and R. Taylor, “PipeRench: A Reconfigurable Architecture and Compiler,” IEEE Computer, vol. 33, no. 4, pp. 70–77, 2000. [2] Y. Koren, U. Heisel, F. Jovane, T. Moriwaki, G. Pritchow, H. V. Brussel, and A. Ulsoy, “Reconfigurable Manufacturing Systems,” CIRP Annals, vol. 48, no. 2, pp. 527–540, 1999, keynote paper. [3] D. W. R.G. Sargent, J.H. Mize and B. Zeigler, “Hierarchical Modelling for Discrete Event Simulation (Panel),” in Proceedings of the 25th Winter Simulation Conference. Los Angeles, CA: ACM Press, 1993. [4] B. Zeigler, Theory of Modelling and Simulation, 1st ed. New York: Wiley Interscience, 1976. [5] ——, Multifacetted Modeling and Discrete Event Simulation, 1st ed. London,Orlando: Academic Press, 1984. [6] A. Concepcion and B. Zeigler, “DEVS Formalism: A Framework for Hierarchical Model Development,” IEEE Transactions on Software Engineering, vol. 14, no. 2, pp. 228–241, Feb 1988. [7] B. Choi, B. Park, and J. Park, “A formal model conversion approach to developing a DEVS-based factory simulator,” Simulation, vol. 79, no. 8, pp. 440–461, Feb 2003. [8] J. Lee, Y. Lim, and S. Chi, “Hierarchical Modeling and Simulation Environment for Intelligent Transportation Systems,” Simulation, vol. 80, no. 2, pp. 61–76, Feb 2004. [9] L. Ntaimo, B. Zeigler, M. Vasconcelos, and B. Khargharia, “Forest Fire Spread and Suppression in DEVS,” Simulation, vol. 80, no. 10, pp. 479– 500, Oct 2004. [10] M. Hwang and S. Cho, “Timed Analysis of Schedule Preserved DEVS,” in 2004 Summer Computer Simulation Conference, A. Bruzzone and E. Williams, Eds. San Jose, CA: SCS, 2004, pp. 173–178. [11] M. Hwang, “Generating Behavior Model of Coupled SP-DEVS,” in Proceedings of 2005 DEVS Integrative M & S Symposium. San Diego, CA: SCS, April 2005, pp. 90–97. [12] ——, “Qunatitative Verification of Reconfigurable Automation Systems:DEVS Approach,” in Technical Report:2005-01, http://kalman.eng.wayne.edu/mhhwang, 2005. [13] J. Burch, E. Clarke, K. McMillan, D. Dill, and L. Hwang, “Symboling Model Checking: 1020 States and Beyond,” Information and Computation, vol. 98, no. 2, pp. 143–170, 1992.