Deadlock Analysis and Routing on Free-Choice Multipart Reentrant Flow Lines Using a Matrix-Based Discrete Event Controller *

Transcription

1 D02-REG0071 Deadlock Analysis and Routing on Free-hoice Multipart Reentrant Flow Lines Using a Matrix-Based Discrete Event ontroller * Jose Mireles Jr 1, Member IEEE, and Frank L. Lewis 2, Fellow IEEE. 1 Instituto de Ingeniería y Tecnología de la Universidad Autónoma de iudad Juárez, Ave. del harro 610 Nte. Juárez MÉXIO. 2 Automation & Robotics Research Institute, of The University of Texas at Arlington (UTA) 7300 Jack Newell Blvd. S., Fort Worth, TX , USA. s: jmireles@arri.uta.edu, flewis@controls.uta.edu Abstract. We present an analysis for deadlock avoidance in manufacturing Free-choice Multipart Reentrant Flow-lines (FMRF). In FMRF, shared resources are not dedicated for certain jobs; some jobs have multiple resource choices in reentrant flow lines, i.e. routing decisions to be made. A Discrete Event (DE) supervisor for deadlock avoidance dispatching, which framework uses a rule-based matrix formulation, is used for routing/dispatching FMRF systems. The development of deadlock-free dispatching rules is derived from circular wait (W) analysis for possible blocking situations. We analyze the so-called critical siphons, and certain critical subsystems on FMRF systems to develop deadlock-free dispatching. The DE controller guaranties deadlock avoidance by limiting the work-inprogress in the critical subsystems associated with each W. This is the least-restrictive dispatching policy that avoids deadlock. In this paper, we calculate in matrix form all constructions needed for implementation of DE controllers using online-deadlock-free dispatching rules, for the case of regular FMRF systems. Keywords: Deadlock Avoidance, Petri nets, Discrete Event Systems, Reentrant flow lines, Intelligent control. 1 Introduction The use of shared resources to sequence machine jobs is a major problem in the implementation of Discrete Event Systems (DES) or Flexible Manufacturing Systems [Buzacott et al. 86]. While some resources manipulate or machine single parts in a DES, others manipulate or machine multiple parts for several products in the manufacturing process. It is even harder when DES have multiple choice (or free choice) reentrant flow lines where no jobs have predetermined resources assigned. Then, routing as well as dispatching decisions are needed. If the assignments of resources for specific machine jobs are not correctly sequenced, serious problems in the performance of the DES can be obtained. These problems include blocking and system deadlock [Banaszak et al. 90, Hsieh et al. 94, Kumaran et al. 94, Ezpeleta et al. 95, Jeng et al. 95, Huang et al. 96, Xing et al. 96, Fanti et al. 00, Lewis et al. 98]. Deadlock occurs when blocking develops in a circular wait [Wysk et al. 91, Hyenbo et al. 95] situation, which is a fatal condition * Research supported by ARO Grants DAAD , and DAAD from U.S.A, and by Programa de Mejoramiento del Profesorado (PROMEP) from México. 1

2 that eventually stops all activity in the flow lines involved. Therefore, it is very important that the DE controller properly sequences jobs and assigns resources. Little rigorous work has been done for dispatching in Multipart Reentrant Flow-lines (MRF) with finite buffers. However, we have shown in [Mireles et al. 02] analysis of constructions needed for deadlock-free dispatching on these MRF systems. These constructions are calculated in matrix form for efficient real time control. But, due to the added dispatching routing alternative on Free-hoice MRF (FMRF) systems, the constructions developed for deadlock avoidance on MRF systems are not valid. Also, all references mentioned above are for reentrant flow lines with no routing. In this paper we provide an analysis of constructions needed for FMRF systems, by showing an analogy over MRF systems. We show how to analize and compute in matrix notation the constructions needed for deadlock avoidance in a cretain large class of regular FMRF systems with routing. We extend to systems with routing the notions of circular waits, their critical siphons, and certain critical subsystems. Regular systems do not contain ritical Resources (R) or key resources [Gurel et al. 00]. These R are critical structuredplaced resources that migh lead to Second Level Deadlock (SLD) [Fanti et al. 97] situations in MRF/FMRF systems. Deadlock avoidance is performed on regular FMRF systems by restricting the work-in-progress (WIP) in certain critical subsystems. This is a rigorous notion related to the idea of ONWIP [Spearman et al. 90]. All computations are performed using straightforward matrix algorithms, including computation of these critical subsystems, for any FMRF in a certain general class, regular systems [Gurel et. al. 00]. We demonstrate that this matrix formulation is computationally efficient for control of FMRF systems. 2 Matrix-Based Discrete Event ontroller A Discrete Event ontroller (DE) for manufacturing workcells was described in [Lewis et al. 93a-94b, Pastravanu et al. 94a, Tacconi et al. 97, Mireles et al. 01a-b]. This DE is based on matrices, and it was shown to have important advantages in design, flexibility and computer simulation. The framework of the Discrete Event ontroller is described as follows. Let v be the set of tasks or jobs used in the system, r the set of resources that implement/perform the tasks, u the set of inputs or parts entering the DES. The DE Model State Equation is described as where: x = F v F r F u F u (1) x is the task or state logical vector, is the job sequencing matrix, F v v r u uc 2

3 is the resource requirements matrix, F r F u F uc is the input matrix, is the conflict resolution matrix, and u c is a conflict resolution vector. This DE equation is performed in the AND/OR algebra. That is, multiplication AND, addition represents logical represents logical OR, and the over-bar means logical negation (used as in [Pastravanu et al. 94].) The job sequencing matrix Fv reflects the states to be launched based on the current finished jobs. The resource requirement matrix F r represents the set of resources needed to fire possible job states this is the matrix used by [Kusiak et al ]. The input matrix F u determines initial states fired from the input parts. The conflict resolution matrix F uc prioritizes states launched from the external u dispatching input, which has to be derived via some decision making algorithm [Panwalkar et al. 77, Graves 81, Mireles et al. 01]. For the case of FMRF systems, Least Slack dispatching policies need to be considered for appropriate routings [Kumar 93,94]. For a complete DE formulation, one must introduce additional matrices, S r and S v, as described next. The state logic obtained from the state equation is used to calculate the jobs to be fired (or task commands), to release resources, and to inform about the final products produced by the system. These three important features are obtained by using the three equations: Start Equation (task commands) v S = S x (2) V Resource Release Equation r S = S x (3) r Product Output Equation y = S x (4) y Fig. 1 shows the DE based on the matrix formulation as used to control job sequences and resource assignment of a workcell. Subscript s on the vectors v and r denotes start. Thus, v and r are outputs from the workcell measured by sensors, while v s and r s are commands to the workcell to begin jobs or set resources as released. 2.1 Matrix Formulation and Petri-Nets There is a very close relationship between the DE just described and Petri Net (PN) tools [Murata 89, Peterson 81, Zhou et al ]. The Incidence Matrix [Peterson 81] of the PN equivalent to the DE controller is obtained by defining the activity completion matrix and the activity start matrix as: Activity ompletion Matrix F = F F F F ] (5) [ u v r uc 3

4 T T T T T Activity Start Matrix S = [ S S S S ] (6) Then, the PN s Incidence Matrix is defined as u v r uc M T T T T [ u u v v r r uc uc T = S F = S F, S F, S F, S F ], (7) where S u and S uc are zero matrices. Rule Based DE ontroller x * ontroller state monitoring logic x = F u F v F r F u u v r uc u * c onflict resolution rules Deadlock Avoidance rules u c Job start logic v = S x S Resource release logic V r = S x S r.. Task complete logic y = S x y Parts in p in Start tasks v s Start resource r s Output y Work ell Parts present u Tasks completed v Resource released r Products pout Plant commands Plant status Fig. 1. Matrix Formulation DES controller If we define the set X containing the elements of x (the state controller vector), and A as the set of v activities containing the vectors v and r, (i.e. A= ), then it has been shown that (A,X,F T, S) is a PN r [Pastravanu et al. 94ab]. This allows one to directly draw the PN of a system given the matrices F and S or vice versa. 4

5 If the marking vector m(t) from a PN is defined as T T T T m(t) = [ u ( t), v( t), r( t), u ( t) ] T. (8) c For a specific time iteration t, then the PN marking transition equation [Peterson 81] is T T m( t + 1) = m( t) + M x = m( t) + [ S F] x( t). (9) 2.2 omplete Dynamical Description for DES A major gap in PN theory has been its inability to provide a complete dynamical description of a DES. The marking transition equation (9) provides a partial description [Peterson 81], but it is not known in the literature how to generate the allowable firing vector x(t). This deficiency is repaired by using the matrixbased DE controller equation (1) together with the PN marking transition equation. The key is to note that the vector x(t) in (9) is identical to the vector x in the DE equation (1) at time t. To put the DE eqs. into a format convenient for simulation, one may write (1) as x = F m or x( t) = F m ( t) = [ F F F F ] [ u v r u ]( t) u v r uc The complete dynamical description of the DES, as described in [Mireles et al. 01], is provided by the PN marking eq. (9), plus the DE equation (10). Note that we are using the Timed Places Petri Net (TPPN) representation of PNs [López-Mellado 95]. A complete development of an implementation and simulations using this DE supervisor/framework is provided in [Mireles et al. 01]. c (10) 3 Matrix Analysis of FMRF Systems with routing: Internal Deadlock onstructions In this section we show analysis of deadlock structures using matrices for reentrant flowlines with routing. These constructions yield computationally efficient algorithms for deadlock-free dispatching of FMRF systems. Least restrictive deadlock-free policies are given in Section 4. In Section 5, we show an example illustrating the formulation of these constructions. onsider the definition of Multiple Reentrant Flow-lines (MRF 1 ) and the assumptions considered in [Gurel et.al 00], which basically define the sort of discrete-part manufacturing systems that can be described by a Petri net. The assumptions are: We assume there are no machine failures. No preemption. A resource cannot be removed from a job until it is complete. Mutual exclusion. A single resource can be used for only one job at a time. 5

6 Hold while waiting. A process holds the resources already allocated to it until it has all resources required to perform a job. After each resource executes one job, it is released immediately. In addition to this assumptions, and since we consider having a Free-hoice Multipart Reentrant Flow-Line, we got the non-restrictive capability that Some jobs have the option of been machined in one resource from a set of resources (routing of jobs), and each resource might be used for different jobs (shared resources.) Job/part routings are deterministic and are provided by a dynamic controller. An example of a class of FMRF is given next. Define a sequence of jobs J k needed to manufacturate a product type k. Define a set of resources R. Then, for a job j i from sequence J k, one resource from a set of resources r k i R might be utilized for machining job j i for product k. Figure 2 shows a diagram of an example of a FMRF system. For this case, the sequence of jobs J k needed to manufacturate k is {j 1, j 2, j 3 }. The set of resources r k 1 available for performing job j 1 is { r 1, r 3, r 4 } R. The terms b i in the figure are the buffers of parts waiting for each job type j i. The FMRF PN representation of the product k of this example is shown in figure 3. Product k J k Sequence of jobs needed for k b 1 j 1 b 2 j 2 b 3 j 3 r k 1 r k 2 r k 3 Selection of resources r 1 r 2 r 3 r 4 r 5 Fig. 2. Example showing a class of a FMRF system. For simplicity in further representations of PNs in the remaining of this paper, we might not draw all resources in the PN. But, we will identify all jobs related for each resource. That is, from previous 6

7 example, the jobs r k 1 are shown in PN figure 3 as r k 1a, r k 1b, and r k 1c. These jobs are jobs perfomed by r 1, r 3, and r 4, respectively. Then, if we rename the jobs names by its machine resources and the number of appearances of jobs from such resources, instead of the type of jobs, we can simplify the PN diagram. For the example, since r 3 is shared for jobs r k 1b, and r k 3a, we can rename such jobs by simply r 3a and r 3b. Also, we will remove the buffer resources b i, and highlight just its jobs by b i. So that figure 4 is a simplified figure from the PN from figure 3, with the understanding that it is only a drawing simplification from the previously defined FMRF system. r 3 r 1 r 2 r k 1a k in r k 1b r k 2a r k 2b r k 1c b 1 b 2 b 3 r 5 r k 3a r k 3b k out r 4 Figure 3. Petri Net system structure example of a FMRF. r 1a k in b 1 r 3a b 2 b 3 r 4a r 2a r 4b r 3b r 5a k out Figure 4. Simplified Petri Net system structure example of a FMRF. For the class of FMRF systems satisfying the assumptions and definitions just given, deadlock can occur only if there is a circular wait relation among the resources [Jeng-Dicesare 95, Gurel et al. 00, Fanti et al. 97]. ircular wait relations are ubiquitous in reentrant flowlines and in themselves do not present a problem. However, if a circular wait relation develops into circular blocking, then one has deadlock. But, as long as dispatching is carefully performed, the existence of circular wait relations presents no problem. However, we must examine the FMRF structure for such circular wait relationships, and identify other structures, as well as the presence of regular system [Gurel et al. 00] structures, or Second Level Deadlock 7

8 structures [Fanti et al. 97] to apply the respective deadlock avoidance policy. The definition of a regular system and Second Level Deadlock structures are given later. In the next section we analyze the FMRF system for the search of circular waits and their internal structures needed for deadlock free dispatching. 3.1 ircular Waits: Simple ircular Waits and their Unions. In this section, we present a matrix procedure to identify all circular waits (W) in MRF/FMRF systems. W structures are need for deadlock analysis and are special wait relationships among resources described as follows. Given a set of resources R, for any two resources r i, r j R, r i is said to wait for r j, denoted r i r j, if the availability of r i is an immediate requirement to release r j, or equivalently, if there exists at least one transition t r j r i. ircular waits among resources are a set of resources r a, r b, r w, which wait relationships among them are r a r b r w, and r w r a. The simple ircular Waits (sw), are primitive Ws which do not contain other Ws. To identify such sw, one has to construct first a resources wait relation digraph [Harary 72]. onsider a digraph D=(R,A), where R is the set of nodes and A={a ij } is the set of edges, with a ij drawn if r i r j (in other words, each a ij represents all transitions in r j r i ). The digraph of resources is easily obtained from the matrix formulation of the system, by getting W r = (S r F r ) T. (11) Each one in the w ij elements from W r, represents that the digraph has an arc from resource i to resource j (by bypassing a transition r j r i.) The algorithm we used to identify all sw in FMRF systems uses a string algebra approach used by [Hyenbo et al. 95]. We use a binary string-like algebra (MATLAB code available at an extension of the algorithm presented in [Mireles et al. 02], which was limited to nine resources in the digraph. The new binary algorithm, which is limited to 126 resources, returns a matrix called w, and its number of rows is the number of sw contained in the DES, and its columns correspond to the resources present in the system (defined and ordered as in the rows of S r ). In this matrix, w, each entry of one in position (i,j) means that each resource j is included in the i th sw. However, due to the complexity of the Free-hoice extension of the MRF systems, and due to the diversity of loop paths that a set of resources contained in a sw might have (see section 3.2, theorem 1), we needed to identify not only the resources that compose each sw, but also the transitions that link them. This will give us specific information needed to locate siphons and certain critical subsystems needed for 8

9 constructions of our deadlock policy for FMRF systems (see section 3.2, theorem 3). For instance, and related to connectivity between places and transitions, if we define (by duality of W r ) W t = (F r S r ) T, (12) We will get a digraph of transitions, W t. That is, W t is a digraph having arcs from transition t i to transition t j (by bypassing a resource t j t i.) Then, one can identify loops among transitions by using string algebra. But, even if we calculate two outcomes from the string algebra, transition loops and resource loops, we might not be able to identify which set of transition loops correspond to which set of resource loops (due to the behavior of the algorithm.) That is why, we need the general digraph matrix 0r Sr W =, (13) Fr 0t where 0 r is a zero-matrix having nxn elements, 0 t is a zero-matrix having mxm elements, n be the number of resources or rows (column) of S r (F r ), and m be the number of transitions or rows (columns) of F r (S r ). This digraph W can be easily draw from the corresponding PN of the system, but having only the arcs interconnecting transitions and resources, and erasing all job places. For instance, for our PN example, this drawing is obtain by erasing figure 4 from figure 3. Then, if we use this digraph matrix W with our binary algorithm to identify loops, we will get both results obtained from W r and W t, say circular wait of resources wr and circular wait of transitions wt, by getting w = [ wr* wt* ], (14) where all rows of wa are rearranged in different sequence in wa* (for a be r or t ). A special remark follows. Each i th row from wr* contains resources from the i th sw, and accordingly corresponds to the i th row of wt*, loop of transitions that closes the i th sw loop. In other words, from each row i of w, i th wt* (i th wr* ) has the set of arcs that closes a loop in the digraph W r (W t ), composed by the resources (transitions) from the i th loop wr* (i th loop wt*.) Note that the number of ones on each row of wt* is the same as in wr*. The dimensions of w are cx(n+m), for c be the total number of sw, n be the number of resources, and n be the number of transitions. So, wr* is a cxn matrix, and wt* is a cxm matrix. Unfortunately, to be able to analyze FMRF systems (and it also happens for MRF systems) and its possible deadlock structures, we need to identify all Ws, not only simple circular waits. These Ws are examined in our deadlock policy. The entire set of Ws are the sw plus the circular waits composed of unions of non-disjoint sw (unions through shared resources among sw.) In [Mireles et al. 02] we show 9

10 code that calculates all Ws from the sets of all sw; it uses a Gurel s algorithm (from [Lewis et al. 98]) in matrix form for efficiency of computations. Using this code, we obtain two resulting matrices, out and G. out provides the set of resources which compose every W (in rows), that is, an entry of one on every (i,j) position means that resource j is included in the i th W. G provides the set of composed Ws (rows) from unions of sw (columns), that is, an entry of one on every (i,j) position means that j th sw is included in the i th composed W. Then, to calculate the set of loop resources and loop transitions included on each W, we use the following constructions W r = G T wr*, and (15) W t = G T wt*. (16) W r and W t correspond to each other. That is, every i th row from W r contains the W of resources (ordered in its columns) correspond to the W of transitions (columns) from the i th row from W t. Since deadlock situations are highly associated with circular waits of resources, we will refer in subsequent sections of the paper to W as the loops W r, unless specifically referred to W t loops. 3.2 Deadlock Analysis for Routing Systems: ritical Siphons and ritical Subsystems. In this section, we apply PN and matrix-based notions to calculate specific PN-place sets associated with each W. The determination of these sets is required so that we can identify possible circular blocking (B) [Ezpeleta et al. 95, Xing et al. 96, Lewis et al. 98, Gurel et al. 00] phenomena or deadlock situations. After computing the sets, we will provide computationally efficient matrix-based algorithms for a least restrictive deadlock-free dispatching policy in section 4. These PN-place sets are highly tied to siphons associated with each W. A siphon set has a behavioral property that if it is token-free under some marking, then it will remain token-free under each successor marking. Such property may leads to B, i.e. deadlock. A set of places S is a siphon if and only if for all places p i S one has p i U j {p i } for some {p j } S. Three important sets associated with the W are the siphon-job sets, J s (), the critical siphons, S c (), and critical subsystems, J o (). The critical siphon of a W is the smallest siphon containing the W. Note that if the critical siphon ever becomes empty, the W can never again receive any tokens. This is, the W has become a circular blocking. The siphon-job set, J s (), is the set of jobs which, when added to the set of resources contained in W, yields the critical siphon. The critical siphons of that W are the conjunction of sets J s () and. The critical subsystems of the W, are the job sets J() from that not contained in the siphon-job set J s () of. That is J o ()= J()\ J s (). The job sets of W are defined by J() = r J(r), for J(r)=r J, where J is the set of all jobs. 10

11 We now provide computational tools to determine the siphon-job sets J s (), the critical siphons, S c (), and critical subsystems, J o (), for every W. To determine such sets, we need to calculate the set of adding transitions T + = \ Wt, clearing transitions T = N + \ Wt, and neutral transitions T =. T T + T is the set of transitions that, when fired, add tokens to the W. On the contrary, T is the set of transitions which, when fired, subtract tokens from. T is the set of transitions external to Wt that, N when fired, do not add or subtract transitions from, just move tokens in. and are the set of input and output transitions from. These sets of transitions are important in keeping real-time track of the tokens inside every W, and hence in determining the status of tokens inside the critical siphon. In order to implement efficient real-time control of the DES, we need to compute these sets in matrix form. Then, we need to determine for each W, quantities and in matrix form, which are needed for calculation of the adding and clearing transitions. In matrix form, quantities and are denoted as d and d respectively, computed as d = out S r, and (17) d = out F r T. (18) Now, we are able to calculate the adding transitions T + = \ W t in matrix form T p = d ( d W t ), (19) the clearing transitions T = \ Wt in matrix form T m = d ( d W t ), (20) N And the neutral transitions T = + T T in matrix form T n = T m T p (21) where operation A B represents an element-by-element logical AND operation between matrices A and B. + N For each circular wait, these matrix forms contain the set of transition vectors T, and T arranged in the rows of matrices Tp, T m, and T n respectively. That is, an entry of one on every (i,j) position in matrix T p (T m, T n ), means that j th transition is a + N T ( T, ) transition belonging to that i th (row set from the) composed W. T T 11

12 In terms of these constructions, matrix form sets are described next, indicating one on every entry (i,j) for places that belong to that set existing in every i th W. The job sets described earlier for each W, J(), in matrix form are described by T J = d F v = d S v. (22) The siphon-job sets for MRF systems are defined for each i th W i as J s ( i ):= J( i ) + T. In matrix notation, for MRF systems, this set is found by J smrf = T p F v. (23) Also, for the case of MRF systems, a shortcut matrix formula for the calculation of the siphon-job sets in matrix form, without calculating T p is provided in [Mireles et al. 02], and it is J s = J F ). (24) ( d v However, for FMRF systems, the formula (23) is not valid for some cases. The following two theorems show the reason and cases where (23) is not a valid siphon-set for FMRF systems. Theorem 1 (Existence of one transition-paths in presence of resource buffer in a loop) For a FMRF system having a simple W, sw, loop which contains at least one resource b i (as defined in section 3) and its job b i has at least two precedent transitions t pre = b i, or at least two posterior transitions tpos = b i, then, only one transition from tpre and one from t pos, say t pre and t pos respectively, are part of the corresponding W t loop. Proof: Suppose j a and j b are the jobs types a and b, posterior to and preceding buffers from job b i, assume product name is k. Then, sets of resources r k a and r k b might be used to manufacture jobs j a and j b. So, since the wait relationship of resources for every W containing a buffer resource b i must be b i r b r c r a and r a b i, only one resource from each set r k a and r k b has to be contained in the W r loop, say resources r a and r b, for r a be a resource from the set r k a and r b from the set r k b (might be the same resource if there exist a sw loop b i r a and r a b i, r a =r b. To show in theorem 4). Note that r b = tpre and that b = tpos, are the b i r a i only two transitions included in the W t loop. See figure 5.! Theorem 2 (Different definition and construction of siphon-job sets between MRF and FMRF) In a FMRF system, the set of jobs J s = J ( F ) is different from the set J smrf = T p F v, only if at least one d v resource b i is contained in a W resource loop, and its job b i has at least two posterior transitions t pos = b i. 12

13 Proof: From the definition of set of jobs J s, d only includes only one of the transitions from t pos, since it was proved in theorem 1 that loop path W r passes by only one transition from the set t pos, say this transition is t pos. With at least this transition, bi is excluded from the set Js. However, from the definition of J smrf, the remaining excluded transition(s) from the set t pos, say t exc (or t exc for the case of t pos having only two transitions) are T p transitions, since any t exc b i. This means that b i is wrongly included in the set J smrf, and correctly excluded from J s. See figure 6.! W loop b i r a loop b i : t pos j b b i t pre r b t pre t pos Figure 5. Help figure for theorem 1. j a : t pre t pos j b : b i j a t pre t exc t exc : t pos Figure 6. Help figure for theorem 2. Therefore, (24) must be used for calculating siphon-job sets in FMRF systems, and avoid wrong calculation of the constructions needed for our deadlock avoidance policy, such constructions are described next. The critical subsystems is a set of jobs used for deadlock avoidance policies (section 4) in W structures. These jobs are the p-invariant [Lewis et al. 98] covering job sets, not belonging to the critical siphons job set from a specific W. This set is defined for MRF systems as J o ( i ) = J( i )\ J s ( i ), for all Ws, i. In matrix form, this set is obtained by J omrf = J F ). (25) ( d v However, for the complex routing FMRF systems, this equation is not valid. The general definition J o ( i ) must be particularized for FMRF systems. The proper formulation of sets J o for FMRF systems is J o ( i ) = Wt J( i ), for all corresponding loops W r, i (loop i identified by row i th of W r.) The matrix form of J o ( i ) is obtained from row i th of J o = W t F v. (26) Notice that for FMRF systems, W t {J( i )\ J s ( i )}. The following theorem explains the reason why J o and J omrf are different between MRF and FMRF systems. 13

14 Theorem 3 (ritical subsystems sets in MRF and FMRF systems, importance of W t ) In a FMRF system, the set of jobs J omrf = J ( d Fv) is different from the set Jo = W t F v, only if there exists at least one loop W r containing one resource b i, which job b i has at least two precedent transitions t pre = k least two resources from the set r i-1 can perform jobs type j i-1, preceding job b i. Proof: Define the two resources from r k i-1, as r a and r b, for r a, r b { ( b i b i, i.e. at }. Then, from theorem 1, only one of the transitions bi should be path of the loop W t, say this resource is r a. This means that r b is not contained in the W r set. This results that the critical subsystems J o = W t F v will contain only the job ) ra bi, since ( r a b i) (W t ) and J omrf = J ( contain wrongly both jobs r d Fv) ax and r by from resources r a and r b since all bi ( d ) and { r ax r a, r by r b } b i.! W loop r a b i r k i-1 r ax b i t pos r by r b t pre Figure 7. Help figure for theorem 3. That is why, for FMRF systems, i.e. systems having routing dispatching problems, it is so important to calculate constructions W t. Otherwise, if the standard definition set from MRF systems is used, one will get wrong sets of critical subsystems. Other important construction previously calculated is the union of simple circular waits, the W set. The next theorem defines and highlights one of important the considerations of such unions, other important considerations will be described in the deadlock avoidance section. Theorem 4 (Unions of Ws, importance of W t ) In FMRF systems, critical subsystems from a W might contain several jobs from a single set r k i-1, only if the W is a union of sw which share at least one resource buffer type b i (section 3) having at least two posterior transitions b i. 14

15 Proof. onsider the case buffer b i has at least two posterior transitions b i, this means that at least two resources type r k i-1 exist. Define these two resources, as r a and r b, such that r a, r b {( ) }. Then, from theorem 1, only one of the transitions bi should be path of each simple W. This means that both resources cannot be contained in a each simple W. Assume two simple Ws, W 1 and W 2, such that resource r a is contained in W 1, r b is contained in W 2, and that { ra, b i r b } b i, i.e. transition { ra b i } W t1, and { rb b i } W t2, where W t1 (W t2 ) is the corresponding W t matrix from W 1 (W 2 ). Then, from the theorem assumption that b i is contained in the union of both Ws W 1, and W 2, this means that {r a, r b, b i } W ru, for W ru be the union of simple Ws W 1 and W 2. This results that the correponding W t matrix from W ru must contain union of both W t1 and W t2, from the corresponding rows of (16) for W 1 and W 2. So that the general expression of critical subsystems J o = W t F v, is particularized for this set J o (W ru )= [W t1 W t2 ]F v, which will contain two jobs { r ax r k i-1. W loop r a b i ra, r by rb}! r k i-1 r ax b i t pos : r by r b t pre Figure 8. Help figure for theorem 4. 4 Deadlock Avoidance In terms of the constructions just given, we now present a minimally restrictive resource dispatching policy that guarantees absence of deadlock for Free-hoice Multi-part Reentrant Flow-lines (FMRF). For efficient implementation in real time of DE controllers, we use matrices for all computations. In this paper, we are considering the case where the system is regular, that is, it cannot contain ritical Resources (R) (so-called structured bottleneck resources or key resources [Xing 96, Gurel et al. 00] existing in Second Level Deadlock (SLD) structures [Fanti et al. 97, 00].) A mathematical regularity test for MRF systems is described in [Mireles et al. 02]. A forthcoming paper will show the analysis of constructions for regularity test for FMRF systems. For the case the MRF/FMRF system is not regular, we can still use this matrix formulation, but with a different dispatching policy designed for irregular systems containing SLD structures, we will consider this casein a forthcoming paper. 15

16 4.1 Dispatching Policy In this section we consider dispatching for regular systems. A matrix test for system-nonregularity is given in the next section. In [Lewis et al. 98] was given a minimally restrictive dispatching policy for regular systems that avoids deadlock for the class of MRF systems which has the first assumptions considered on section 3 for FMRF systems, excluding the additional assumptions. That is, in MRF systems all sets type r k i defined for FMRF systems contain only one resource r i. So that it is not necessary the presence of routing resources b i in MRF systems. In FMRF systems b i are not only resource buffers, but, robotic systems that handle parts from several machines to other several machines, i.e. these act like routers in communication systems. A brief note on deadlock, is that, for the class of MRF and FMRF systems, deadlock is equivalent to a circular blocking (B). There is a B if and only if there is an empty circular wait (W) of resources. This always happens iff the corresponding critical siphon of that W is empty. By construction, in Lewis policy-structure terms, this is equivalent to all jobs of the W being in the associated ritical Subsystem (S). Similarly, in terms of PN, there is a deadlock iff all tokens of the W are in the ritical Subsystem set. Therefore, our first policy, key to deadlock avoidance, is to ensure that the WIP in the S is limited to one job less than the total number of initial tokens in the W (i.e. the total number of resources available in the W). Due to the necessity and sufficiency of all the conditions just outlined, this MAXWIP policy [Lewis et al. 98] is the least restrictive policy that guarantees absence of deadlock. It is very easy to implement. Preliminary off-line computations using matrices are used to compute the S. A supervisor is assigned to each circular wait, and is responsible for dynamic dispatching by counting the jobs in the corresponding S and ensuring that they do not violate the following condition, for each W i, m(j o ( i )) < m o ( i ). (27) That is, the number of enabled places contained in the S for each i must not reach the total number of resources contained in that i. However, for FMRF systems there exists a special case where all critical subsystems are allowed to be full, i.e W become empty without getting into a B, or deadlock. Theorem 5 explains this special case. Theorem 5 (Acceptance of empty circular waits, special case on regular FMRF systems) For a FMRF system having a simple W containing at least one resource b i, which job b i has at least two posterior transitions t pos = b i, then, even if the W is empty, and if any of those transitions tpos is an adding 16

17 transition t p + T, such that t N p T =, this W will never be in deadlock (never be in a blocking situation.) Proof: From definition, a W r is empty if the corresponding set critical subsystems, J o = W t F v, is full. Also, from definition, all transitions W t have posterior and precedent resources from the corresponding set W r, i.e. W t R {W r } and Wt R {W r }. That is, if W r is empty, all transitions W t are not enable. Then, if we determine a transition { W t } able to free a token from a critical subsystem job place to a resource from the W r set, then, deadlock is avoided. From theorem 1, only one t pos from the set t pos = b is included in Wt. This suggests that the other posterior transition t p must be able to free the token from the critical subsystem, and add it to set W r. Since t p bi \ Wt, from definition, t p is an adding transition { T + = \ Wt}. t p fulfills the first necessity condition we have { W t }. Assume that t p is not a clearing transition, that is, t p i \ Wt. Then, t p T N =. This means that tp can only add tokens to W r and not subtract. Now, the last part to demonstrate, is that t p clears tokens from the critical subsystems set, and is not dead. Note that b i {J o = W t F v } { t p }, so, it is contained in the critical subsystem set, and t p is able to substract tokens from b i. Also, notice that a resource r b { r k i \ W r } exists, since t p exists. Since r b is not contained in a circular wait relationship, eventually, it will enable t p, and free the blocking situation. This occurs in regular systems only. For irregular systems, b i belongs to a critical subsystem set from other W, and the clearance of its token depends on conditions of this other W.! b i W loop r a : t pos r b j a b i t pre t p t pre t pos Figure 9. Help figure for theorem 5. 17

18 For implementation purposes, for very DE iteration, a dispatching/decision policy must be selected. Among all possible dispatching policies [Panwalkar et al. 77], one possible policy is the Least Slack (LS) policy, which tries to optimize throughtput and minimize the variance of lateness of jobs waiting to be processed [Kumar 93-94]. However, deadlock situations might appear in certain Ws while trying to optimize throughtput. Therefore, we have to combine LS with our deadlock avoidance test (27). Thus, before we intent to dispatch the LS resolution, we must examine the marking outcome with our deadlock policy. If this resulting outcome does not satisfy (27), then the algorithm denies (pre-filters in real time) firing the transition identified to deadlock the system. This is accomplished by adding an F uc matrix, a conflict resolution matrix, to our matrix based structure (see [Lewis et al. 98] for an example showing the use of F uc ). Later, althought throughput might be comprimised, LS is run again for the next possible allowable firing sequence without deadlock. Therefore, using optimal LS dispatching while permitted, we will try to satisfy in most of the current status of the cell the case m(j o (i)) = m o (i)-1, for efficiently improve throughput. 5 onclusion An analysis for deadlock avoidance in Free-choice Multipart Reentrant Flow-lines (FMRF) is presented. Definition of a general class of FMRF systems was provided. In FMRF systems shared resources are not dedicated for certain jobs; some jobs have multiple resource choices in reentrant flow lines, i.e. routing problems have to be solved. The development of deadlock-free dispatching rules was derived from circular wait analysis for possible blocking situations. Analysis of constructions related to circular wait loops is shown, including transition loops, resource loops, critical siphons, and certain critical subsystems. This analysis was shown as an alalogy of Multipart Reentrant Flow-lines with no Free-choice routing dispatching problems. The main supervisor is a Discrete Event (DE) controller which uses a Matrix-Based framework. The Discrete Event ontroller (DE) guaranties deadlock avoidance by limiting the work-in-progress in the critical subsystems associated with each W. This is the least-restrictive dispatching policy that avoids deadlock. We calculated in matrix form all constructions needed for implementation of DEs using onlinedeadlock-free dispatching rules, for the case of FMRF systems being regular. Future research will cover implementations of FMRF systems with routing dispatching problems, regularity tests for FMRF systems and analysis of critical resources for the case of irregular systems having Second Level Deadlock structures. 18

19 References [1] Banaszak Z. A. and B. H. Krogh. Deadlock Avoidance in Flexible Manufacturing Systems with oncurrently ompeting Process Flows. IEEE Trans. Robotics and Automation, RA-6, pp (1990). [2] Buzacott J.A. and D.D. Yao. Flexible Manufacturing Systems: A Review of Analytical Models. Management Sci. 7, pp (1986). [3] Ezpeleta S. D., J. M. olom and J. Martinez. A Petri Net Based Deadlock Prevention Policy for Flexible Manufacturing Systems. IEEE Trans. Robotics and Automation, RA-11, pp (1995). [4] Fanti M.P., B. Maione, S. Mascolo, and B. Turchiano. Event-Based Feedback ontrol for Deadlock Avoidance in Flexible Production Systems. IEEE Transactions on Robotics and Automation, Vol. 13, No. 3, June [5] Fanti M.P., B. Maione, S. Mascolo, and B. Turchiano. omparing Digraph and Petri Net Approaches to Deadlock Avoidance in FMS. IEEE Transactions on Systems, Man, and ybernetics-part B: ybernetics, Vol. 30, No. 5, October [6] Graves S... A Review of Production Scheduling. Operations Research, vol. 29, no. 4 (1981). [7] Gurel A., S. Bogdan, and F.L. Lewis. Matrix Approach to Deadlock-Free Dispatching in Multi-lass Finite Buffer Flowlines. IEEE Transactions on Automatic ontrol. Vol. 45, no. 11, Nov. 2000, pp [8] Harary F. Graph Theory. Addison-Wesley Pub. o., Massachusetts [9] Hsieh F.-S. and S.-. hang. Dispatching-Driven Deadlock avoidance controller Synthesis for Flexible Manufacturing Systems. IEEE Trans. Robotics and Automation, RA-11, pp (1994). [10] Huang Hsiang-Hsi, F.L. Lewis, D. Tacconi. Deadlock Analysis Using a New Matrix-Based ontroller for Reentrant Flow Line Design. Proceedings of the 1996 Industrial Electronics, ontrol, and Instrumentation. IEEE IEON 22nd International onference on. Vol. 1. pp vol.1 (1996). [11] Hyuenbo., T. K. Kumaran, and R. A. Wysk. Graph-Theoretic Deadlock Detection and Resolution for Flexible Manufacturing Systems. IEEE Transactions on Robotics and Automation, vol. 11, no. 3, pp (1995). [12] Jeng M.D. and F. Diesare. Synthesis Using Resource ontrol Nets for Modeling Shared-Resource Systems. IEEE Trans. Robotics and Automation, RA-11, pp (1995). [13] Kumar, P.R. Re-entrant lines. Queueing Systems: Theory and Applications. Switzerland. vol. 13, pp (1993). [14] Kumar, P.R. Scheduling Semiconductor Manufacturing Plants. IEEE ontrol Systems Magazine, Vol. 14, Issue 6, pp Dec [15] Kumaran, T. K., W. hang, N. ho and R. A. Wysk. A Structured Approach to Deadlock detection, avoidance, and solution in Flexible Manufacturing Systems. Int. Journal Prod. Res., vol. 32, pp (1994). [16] Kusiak, A. and J. Ahn. A Resource-onstrained Job Shop Scheduling Problem with General Precedence onstraints. Working paper, no , Iowa (1991). [17] Kusiak A. and J. Ahn. Intelligent Scheduling of Automated Machining Systems. omputer Integrated Manufacturing Systems, vol.5, no.1, Feb. 1992, pp UK (1992). [18] Lewis, F.L., H.-H. Huang and S. Jagannathan. A systems approach to discrete event controller design for manufacturing systems control. Proceedings of the 1993 American ontrol onference (IEEE at. No.93H3225-0). American Autom. ontrol ouncil. pp vol.2. Evanston, IL, USA (1993 a). [19] Lewis F.L., O.. Pastravanu and H.-H. Huang. ontroller Design and onflict Resolution for Discrete Event Manufacturing Systems. Proceedings of the 32nd IEEE onference on Decision and ontrol (at. No.93H3307-6). IEEE. Part vol.4, pp vol.4. New York, NY, USA (1993 b). [20] Lewis F.L. and H.-H. Huang. ontrol System Design for Flexible Manufacturing Systems, in (A. Raouf and M. Ben Daya) Flexible Manufacturing Systems: Recent Developments, Elsevier (1994 a). [21] Lewis F.L., H.-H.Huang, O.. Pastravanu, A. Gürel. A Matrix Formulation for Design and Analysis of Discrete Event Manufacturing Systems with Shared Resources IEEE International onference on Systems, Man, and ybernetics. Humans, Information and Technology (at. No.94H3571-5). IEEE. Part vol. 2, 1994, pp New York, NY, USA (1994 b). 19

20 [22] Lewis F.L., Gurel A, Bogdan S, Docanalp A, Pastravanu O. Analysis of Deadlock and ircular Waits using a Matrix Model for Flexible Manufacturing Systems. Automatica, vol.34, no.9, Sept. 1998, pp Publisher: Elsevier, UK (1998). [23] López-Mellado E. Simulation of Timed Petri Net Models. Systems, Man and ybernetics, Intelligent Systems for the 21st entury., IEEE International onference on, Volume: 3, pp (1995). [24] Mireles, J. and F.L. Lewis, On the Development and Implementation of a Matrix-Based Discrete Event ontroller. MED01, Proceedings of the 9 th Mediterranean onference on ontrol and Automation. Published on D, reference MED June Dubrovnik, roatia (2001a). [25] Mireles, J. and F.L. Lewis. Intelligent Material Handling: Development and Implementation of a Matrix-Based Discrete Event ontroller. IEEE Transactions on Industrial Electronics. Vol. 48, No. 6, December 01 (2001b). [26] Mireles, J. and F.L. Lewis, A. Gurel. Deadlock Avoidance for Manufacturing Multipart Reentrant Flow Lines Using a Matrix-Based Discrete Event ontroller. Int. Journal of Production Research. To appear (2002). [27] Murata, T. (1989). Petri Nets: Properties, Analysis and Applications. Proceedings of the IEEE, vol.77, no.4, April 1989, pp USA. [28] Panwalkar S.S. and W. Iskander. A Survey Of Scheduling Rules, Operations Research, vol. 26, pp (1977). [29] Pastravanu O.., A. Gürel, F. L. Lewis and H.-H. Huang. Rule-Based ontroller Design Algorithm For Discrete Event Manufacturing Systems. Proceedings of the 1994 American ontrol onference (at. No.94H3390-2). IEEE. Part vol.1, pp vol.1. New York, NY, USA (1994 a). [30] Pastravanu, O.., A. Gürel and F. L. Lewis. Petri Net Based Deadlock Analysis in Flowshops with Kanban-Type ontrollers. 10th ISPE/IFA International onference on AD/AM, Robotics and Factories of the Future ARs & FOF `94. Information Technology for Modern Manufacturing. onference Proceedings. ORI Publications. pp Kanata, Ont., anada (1994 b). [31] Peterson J.L. Petri Net Theory and the Modeling of Systems. Prentice-Hall. 1981, Englewood liffs, NJ, USA (1981). [32] Spearman M.L., D.L Woodruff, and W. Hopp ONWIP: a pull alternative to kanban. Int. J. Prod. Res., Vol. 28, No. 5, 1990, pp (1990). [33] Spearman, M.L., D.L. Woodruff, and W.J. Hopp, ONWIP: a Pull Alternative to Kanban. Int. J. Production Research, vol. 28, no. 5, pp (1990). [34] Tacconi D.A. and Lewis F.L. A New Matrix Model for Discrete Event Systems: Application to Simulation. IEEE ontrol Systems (1997). [35] Wysk, R. A., N. S. Yang and S. Joshi. Detection of Deadlocks in Flexible Manufacturing ells. IEEE Trans. Robotics Automet., RA-7, (1991). [36] Xing K. Y., B.S. Hu, and H.X. hen. Deadlock Avoidance Policy for Petri-Net Modeling of Flexible Manufacturing Systems with Shared Resources. Automatic ontrol, IEEE Transactions on, vol. 41 issue 2, Pp Feb [37] Zhou, M., F. Diesare and A. A. Desrochers. A Hybrid Methodology for Synthesis of Petri Net Models for Manufacturing Systems. IEEE Trans. Robotics and Automation, vol. 8, no. 3, pp (1992). [38] Zhou M., F. Diesare. Petri Nets Synthesis for Discrete Event ontrol of Manufacturing Systems, Kluwer Academic Publishers (1993). 20