DEADLOCK AVOIDANCE IN BATCH PROCESSES. M. Tittus K. Åkesson

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1 DEADLOCK AVOIDANCE IN BATCH PROCESSES M. Tittus K. Åkesson Univesity College Boås, Sweden, Chalmes Univesity of Technology, Gothenbug, Sweden, Abstact: A Peti net based appoach fo the modeling of batch plants as well as poducts is pesented. Allocating non-shaable esouces fo poduct ecipes can lead to deadlocks. In this pape we focus on the detection of potential deadlock situations using a gaph stuctue deived fom synchonization issues that aise when moving a batch between esouces. It is shown that diffeent gaph pattens define potential deadlocks. These pattens even suggest how to implement a Peti net supeviso to avoid these deadlocks. Keywods: Peti nets, gaph theoetic models, supevisoy contol, modelling. INTRODUCTION Batch pocesses take an impotant place in pocess industies. A batch pocess involves a sequence of opeations that ae caied out on a discete quantity of mateial within a piece of opeating equipment. A ecipe specifies the sequence of opeations to be executed and all possible ways the plant can by utilized in ode to poduce the desied poduct. A discete supeviso can be used to coodinate the simultaneous execution of a numbe of ecipes within the same plant. Its task is to ensue that all ecipes can be teminated coectly, i.e. it pevents actions that will lead the system into a deadlock. This is done by allowing/disallowing the ecipes access to the plant s diffeent esouces. Thus, the task of the supeviso is to pevent esouce allocations that can lead to deadlock. Note that the poposed supeviso should only pevent allocations if they can lead to a deadlock. If thee ae seveal possible paths though the plant it is not the task of the supeviso to choose one of them. Instead, this is done by a schedule, e.g., an optimization algoithm, o a human opeato, and is outside the scope of this pape. Deadlock avoidance techniques can, if desied, esult in minimally estictive solutions, i.e., nothing moe than necessay is pevented. The main dawback with deadlock avoidance is that it is computationally expensive. In fact, if a minimally estictive solution is equied, deadlock avoidance belongs to the class of NP-had poblems and is thus computationally intactable fo aleady elatively small poblems. We use Peti nets as ou modeling tools fo ecipes as well as esouces. The task of avoiding deadlocks can then be divided into two pats: the identification of potential deadlock situations and the fomulation of estictions (placement of monitos to avoid these deadlocks. Holloway et al. (997 give an oveview ove the wok that has been done to implement contol mechanisms based on aleady existing mutual exclusion constaints. Thee ae, howeve, fewe esults available that focus on the identification of these constaints. In this pape we divide the plant into two sub-systems: pocessos (units and the tanspoting system esponsible fo mateial tansfes. It is shown that it is enough to solve the deadlock poblem in these two sub-systems sepaately, thus educing the oveall computational complexity consideably. In Tittus and Lennatson (998a and Åkesson and Tittus (998 diffeent techniques have been pesented to solve the deadlock poblem in the tanspoting system. Hee, we concentate on the identification of deadlocks that can occu when booking pocessos. Using coloed diected gaphs to descibe the booking poblem we identify gaph pattens that define potential deadlocks fo a esticted class of poblems. Futhemoe we show that these gaph pattens even detemine the placement of so-called monitos to avoid deadlocks. It is woth mentioning that the poposed methodology to detect and esolve deadlocks is not esticted to batch pocesses but to evey kind of pocesses booking nonshaable esouces in the way specified in the pape. We stat by biefly eviewing the plant and ecipe models peviously intoduced in Tittus and Åkesson (997, 999. They focus on esouce allocation and synchonization issues. Next, potential deadlock situations ae pointed out and chaacteized. Afte defining coloed diected gaphs, we descibe deadlock pattens and how to place Peti net (PN monitos. We conclude the pape with some outlook on futue wok. 2. MODELS We model both esouces and ecipes by means of labeled, odinay PNs because of thei explicit suppot of paallelism and modula specifications. These PN ae defined by the 7-tuple PN = fp T P e P ost m i g. P and T ae finite, non-empty sets of places p i and tansitions t i, espectively. Pe and Post ae subsets of P T and denote out-acs fom and in-acs jp to places, espectively and m i : P! Z j 0 defines

2 the initial making of the PN. is a set of events and : T! associates an event to each tansition. Futhemoe, let p and p denote the tansition afte and befoe place p, espectively. Analoguously, t and t denotes the place afte and befoe tansition t, espectively. 2. Resouce Models In this pape we distinguish between two geneic classes of plant esouces, namely pocessos and tanspot devices. Pocessos ae typically tanks, eactos and othe containe like units, fully equipped with contol modules and othe devices to manipulate a batch. Tanspot devices, on the othe hand, have as thei main task to open and close connections between pocessos, causing and peventing mateial flow. Typical examples ae valves and pumps. We exemplify pocessos with tanks and tanspoting devices with on/off valves, and use safe PN (Muata (989 as ou modeling tool. Since we use the models fo supeviso synthesis, and we only conside the booking issue, the poposed models eflect the booking status of the esouces. As mentioned in the model epesentation section, an event is associated with evey tansition. Recipes and esouces shae some events, and the pinciple is that ecipes initiate events and esouces follow them. The initiation of an event in the ecipe is only possible if the esouce tansition labeled with the same event can be fied simultaneously. Pocessos When a ecipe has booked a pocesso P i, it has full and exclusive access to P i and the associated equipment. Figue (a shows the geneic PN model of a pocesso. Initially the place UNBOOKED contains one token, indicating that this place is active. When a ecipe wants to book pocesso P i, it ties to initiate the booking event b in P i :s PN, denoted b(p i. If place UNBOOKED contains a token, the token is moved to place BOOKED, and othe ecipes ae pevented fom booking the pocesso. When the ecipe is finished with pocesso P i it initiates the unbooking event u, denoted u(p i, and othe ecipes can book the pocesso. Note that the models do not indicate whethe the pocesso cuently is executing an opeation o not. bl ( i ul ( i bt ( unbooked (a bt ( ut ( 2 (b ut ( 2 i up ( i (c booked opeation {( b P, b( L } up ( i k u( L opeation Fig.. (a Geneic model fo a pocesso P i. (b Line model L i. (c Segment of a booking ecipe moving mateial fom P i to P k via L. Tanspot devices When moving mateial fom one pocesso to the next, a numbe of valves have to coopeate to open and close cetain connections. Also, sometimes it is enough to block some tanspot devices (i.e., make sue the tanspote stays closed duing mateial tansfe in ode to secue a line fo a mateial tanspot. In this case we do not need to book the tanspote, but only pevent othe ecipes fom booking it. Thus, besides the natual states UNBOOKED and BOOKED, two exta states in which the valve is kept closed ae needed. Two block states ae needed since a valve with two outlets can be blocked by at most two ecipes, each ecipe blocking one of the valve s outlets. A valve with moe than two outlets then has a coespondingly lage numbe of places indicating blocking. In accodance with ou intoductoy discussion a set of vitual line esouces, called connection lines o just lines, is ceated. Fo each possible connection between any two pocessos a line esouce can automatically be ceated fom infomation about the plant s topology. A PN model of a line with thee tanspot devices is given in Fig. (b. The line model ensues full flexibility when booking tanspot devices. 2.2 Recipe Models Poduct models o ecipes can be specified on diffeent levels of abstaction. One can distinguish between the plant-independent geneal ecipe, which descibes the opeations to be applied without efeence to any specific equipment to be used, and the contol ecipe, which, besides the opeations, also shows the aw mateials path though a specific plant. Recipes on an even lowe abstaction level descibe the detailed contol that has to be applied. In this pape we will concentate on models fo the plant-dependent ecipe. These models then can be used togethe with the peviously intoduced plant models to synthesize a discete supeviso that coodinates the concuent execution of ecipes. Each ecipe is modeled by an individual PN. In Tittus and Åkesson (997, 999 geneic PN building blocks have been intoduced fo the following basic functions: ( opeation: diffeent phases applied to the whole batch o a pat of the batch, (2 move: moving the batch fom one unit to anothe, and, as special cases of (2: (3 join: meging of two pats of a batch fom two souce units into a thid taget unit and (4 split: sepaating a batch fom one souce unit into two taget units. Fo the pesent wok it is enough to conside the booking model fo a simple mateial tansfe between two pocessos. Figue (c shows a segment of a booking ecipe that stats with an opeation in pocesso P i (denoted by a PN place. The mateial tansfe is initiated by simultaneously booking the taget pocesso, P k, and the connecting line object, L. This booking stategy makes it possible to sepaate the deadlock analysis when booking pocessos fom the deadlock analysis in the tanspoting system. Simultaneous booking is fomalized by using the connected event fb(p i b(l g. The tansition labeled with the event connection can only be fied if all the connected

3 events can be fied at the same time. Afte the mateial tansfe has been accomplished, the line and, afte execution of optional post-pocessing, the souce pocesso ae eleased. The batch is now contained in pocesso P k whee the next opeation can take place. Thus, the booking ecipe is put togethe of these building blocks, which in tun contain events booking and unbooking pocessos and lines. 3. DEADLOCK SITUATIONS Thee ae fou necessay conditions fo deadlock to occu: ( mutual exclusion, i.e., a esouce can only be used by one pocess at a time, (2 hold and wait, i.e., pocesses must exist which ae holding esouces while waiting fo othes, (3 no pe-emption, i.e., a esouce can only be eleased voluntaily by a pocess, and (4 cicula wait, i.e., a cicula chain of pocesses must exist such that each pocess holds esouces which ae being equested by the next pocess in the chain. In ou applications, the below mentioned pocesses coespond to ecipes booking and unbooking esouces. In this section we will show that deadlock is elevant when booking pocessos and connections between them. Let us fist conside the booking of pocessos. If we deem pe-emption an unacceptable altenative then the above conditions suely apply to the booking of pocessos when moving mateial. A simple example will illustate the point. Assume two ecipes R and R 2 that need to move mateial fom pocessos P to P 2 and fom P 2 to P, espectively. If R has booked P and R 2 has booked P 2 with both ecipes now waiting fo the othe to elease the esouce, then the system has deadlocked. The poblem can also aise when booking two o moe lines that have two o moe tanspoting devices in common. If each line books one valve and then waits fo the othe valve to be eleased the system locks. The following lemma educes the complexity of the oveall system s deadlock analysis significantly and is the eason fo the division of the system into pocessos and tanspoting system. Lemma. Given the booking models intoduced in the pevious section, all possible deadlock situations can be detected consideing pocesso booking and bookings in the tanspoting system sepaately. PROOF. We have to show that all occuences of deadlocks obtained by consideing all esouces ae a subset of the deadlocks obtained by consideing each sub-system sepaately. Accoding to ou ecipe models, each ecipe has to own one o moe souce pocessos befoe attempting to book the taget pocesso and the coesponding line. We show that all cyclic waits involving both lines and pocessos ae coveed by consideing lines and pocesso booking sepaately. Thee ae two ways fo two ecipes to fom a cyclic wait involving pocessos and lines. The fist case is that both ecipes have to equie access to the same line. Since each line is defined by the pocessos it connects, this can only be the case if the pocessos fom a cyclic wait. Thus this case is coveed by deadlock analysis fo pocessos. The othe case is that deadlock between two lines can cause deadlock in the oveall booking system. Assume two ecipes, R and R 2. R has booked pocesso P and line L!2 and waits fo pocesso P 2, while R 2 has booked pocessos P 2 and P 3 and waits fo line L 2!3. Assume futhemoe that L!2 and L 2!3 cannot be booked simultaneously. This kind of situations ae uled out by the fact that the booking models equie simultaneous booking of both taget pocesso and line. The extension of the easoning to moe ecipes is staight fowad DEADLOCK AVOIDANCE Usually one distinguishes between two steps in the implementation of deadlock avoidance policies. Fist, deadlock situations need to be identified and fomulated as geneal mutual exclusion constaints (GMEC with espect to the booking of esouces. The second pat consists of the implementation of suitable mechanisms in ode to enfoce the MEC without esticting the booking behavio moe than absolutely necessay. The above mentioned GMECs can easily be implemented by monitos. A monito is a PN place whose initial making in a geneal setting epesents the numbe of available esouces. Its outgoing and incoming tansitions epesent, espectively, the acquisition and elease of these esouces (see Fig. 4. In Holloway et al. (997 can be found an oveview ove solutions to the GMEC poblem fo PNs. Avoidance policies that can be computed in polynomial time have been pesented fo the pice, howeve, that these policies in geneal give a somewhat estictive solution, i.e., the esulting supeviso is geneally not maximally pemissive. We focus now on the identification of deadlock situations aising when booking pocessos. A detailed discussion of deadlock avoidance in the tanspoting system can be found in Tittus and Lennatson (998b and Åkesson and Tittus (998. Afte some peliminaies on coloed, di-gaphs, we show how to tansfom the deadlock poblem into a gaph stuctue and identify gaph pattens that coespond to deadlocks. 4. Booking of Pocessos We will hee take a close look at pocesso booking and the infomation essential fo deadlock analysis. Assume a ecipe that consists of a staight sequence of pocessos that ae booked and unbooked. The PN in Fig. 2(a shows the sequence of booking and unbooking events that can occu unde the assumption that the whole batch is moved each time. The souce pocesso is eleased by the ecipe as soon as the batch tanspot has been completed. Even though some postpocessing might be equied befoe eleasing the pocesso, the pocesso is no longe needed by the ecipe and will soone o late be eleased. Since time does not play any ole fo ou deadlock analysis the unbooking events can be omitted and the infomation necessay fo the analysis is fomulated as a labeled PN as shown in (Fig. 2(b

4 u( P b( P3 u( P2 b( P 3 (a (b A special case of the elementay cycle is the elementay coloed cycle defined as follows: Definition 4. An elementay coloed cycle (ECC is an elementay cycle whee no two edges have the same colo. Fig. 2. Event eduction in a booking sequence fo deadlock-analysis puposes Fo analysis puposes we define the pocesso-waitelationship. A pocesso P is said to wait fo a i pocesso P 2 in a ecipe i, denoted P! P 2,if the availability of P 2 is an immediate equiement fo the elease of P in ecipe i. The thus defined binay pocesso-wait elation is non-symmetic and non-tansitive. Fo a given system with a numbe of ecipes, all the pocesso-wait elations can be captued in a coloed digaph, which we will denote the pocesso-wait elationship gaph (PWRG. 4.2 The Pocesso-Wait-Relationship Gaph Definition 2. A coloed digaph is given by the quatuple G C = fv E C g. V = fv ::: v n g denotes the set of nodes, E = fe ::: e m g the set of edges and C = f ::: p g the set of colos which is identical to the set of ecipes. The patial function : (V E C! V defines the stuctue of the gaph including the coloing of the edges. The gaph s nodes v i 2 V coespond to the system s pocessos P i 2P, the one-to-one mapping (v i = P i establishing this coespondance. The stuctue and colo of the digaph is given by (v i e =v j ( (v i! (v j. Each tansition in the pocesso booking ecipes is mapped to a unique edge in the gaph. We denote this coespondance by the one-toone mapping : E! T, with T denoting the set of tansitions in ecipe. Futhemoe, we define a wait-path between two pocessos, P i and P k, denoted by P i,! P k, if thee exists a set of pocessos such that P m n i! P j! l :::! P k. The same opeato (,! also defines the coesponding wait-path in the digaph, i.e., v i,! v k. The case whee each edge in a wait path has the same colo, say, we denote as a uni-coloed wait-path. Fo that we wite P i,! Pk with denoting the path s colo. Fo the deadlock analysis discussed late a numbe of special pattens in the PWRG ae of impotance. Definition 3. An elementay cycle L e is a sequence of distinct nodes hv v 2 ::: v n i and edges he e 2 ::: e n i with v i 2 V and e i 2 E such that thee exists an C fo each i = ::: n ; such that (v i e i =v i+, and (v n e n =v. An example of an elementay cycle is shown in Fig. 3(a. Accoding to Def. 3, each node only appeas once in the sequence and the colo of the connecting nodes is abitay. Anothe special case of an elementay coloed cycle is a so-called atomic counteflow. Definition 5. An atomic counteflow is an elementay coloed cycle of length two, i.e., consisting of two nodes. Atomic counteflows can be connected to fom counteflows as follows. Definition 6. A counteflow of the PWRG G C is a sequence of nodes hv v 2 ::: v n i, v i 2 V, such that (v! (v 2! :::! (vn and (v n! 2 (v n;! 2 :::! 2 (v with 2 C and 6=. Futhemoe, thee does not exist a node v i 2 V such that eithe (v i! (v and (v! 2 (v i, o (v i! 2 (v n and (v n! (v i. We then call v and v n the counteflow s tuning nodes. An example of a counteflow is given in Fig. 3(b. Note that each counteflow has two colos and that two edges of diffeent colo oiginate and teminate in each node except the tuning nodes. The tuning nodes have only one oiginating and one teminating edge. In the coesponding plant a counteflow coesponds to two batches going though the same sequence of pocessos but in opposite diections. (a V V 2 V 3 V n... (c V n (b Fig. 3. Examples of (a an elementay cycle, (b a counteflow, and (c a coloed counteflow stuctue. Counteflows can be combined to fom (coloed counteflow stuctues. Befoe we define these moe fomally we need to give a definition of tuning nodes fo counteflow stuctues. Definition 7. A tuning node v t in a counteflow stuctue is a node that is connected to one othe node v via exactly one in- and one out-ac of diffeent colos. i That is, thee exists one node v such that v t! v and v! j v t, i 6= j and no othe node is connected to v t. Definition 8. A (coloed counteflow stuctue is a set of counteflows that ae connected at tuning nodes epesenting mutual esouces such that the esulting gaph satisfies the following equiements:

5 A subset V ST of the counteflows tuning nodes ae also tuning nodes in the counteflow stuctue. We denote these as stuctue tuning nodes. (8v i 2 V ST : 9v k 2 V ST such that (v i,! (v k fo an abitay and i 6= k. Each coloed path oiginates and teminates in a stuctue tuning node, v t and v t2, espectively. Assume we extact all stuctue tuning nodes and connect each pai of nodes, say v ti and v tj, fo which thee exists a colo such that (v ti,! (v tj with a coloed ac of colo. Then, the esulting gaph has to be exactly one elementay coloed cycle. The second and thid condition guaantee that the numbe of colos in the counteflow stuctue is equal to jv ST j and a consequence of the last condition is that no two counteflows in the stuctue can have the same combination of colos. Figue 3(c shows an example of a counteflow stuctue. We point out that each node can belong to a numbe of pattens, while each atomic counteflow only can belong to one counteflow. Each counteflow, in tun, can be contained in an abitay numbe of counteflow stuctues. Also note, that nonatomic counteflows ae not special cases of elementay cycles. 4.3 Deadlock Avoidance when Booking Pocessos With these stuctues defined we can now move on to explain how these pattens coespond to deadlock situations and how to avoid deadlock. We assume hee that each ecipe is only executed once. This does, howeve, not diminish the geneality of ou esults since multiple batches of the same ecipe can be modeled by cloning the ecipe. The fist esult concens the ole of elementay coloed cycles fo deadlock analysis. The poof follows fom the definition of deadlock. Lemma 9. Each elementay colo cycle in a PWRG that is not pat of a counteflow defines a geneal mutual exclusion constaint, which, if not enfoced, can lead to deadlock. Note that a counteflow consists of a numbe of ECC of length two. Applying Lemma 9 to ECC that constitute a counteflow can lead to esticted deadlock, that is, the monitos themselves lock the system. Lemma 0. The GMEC defined by each elementay colo cycle of length n and with a set of edges E ECC = fe i g, is enfoced in a minimally estictive way by placing one monito place with n ; initial tokens. The monito has out-acs to the tansitions (e i and in-acs fom the tansitions (e i. PROOF. The only way to beak the cycle is to deny one of the ecipes access to the pocesso in the cycle. This is done by placing a monito with n ; tokens and out-acs to the booking events of the n pocessos in the cycle. That way, not all pocessos in the cycle can be booked at the same time and the cyclic wait is boken. As soon as any of the involved ecipes eleases its pocesso, a token is etuned to the monito. Any less estictive policy would allow all ecipes access to the coesponding pocessos in the cycle and would thus lead to deadlock. Hence, the policy is minimally estictive. 2 Figue 4 shows the pocesso booking models of thee ecipes, the coesponding PWRG, and the monito based on the elementay colo cycle. b( P 3 b( P3 Fig. 4. Pocesso booking model with monito and coesponding PWRG Fo the est of the pape we will now concentate on a sub-class of ecipes, whee each pocesso occus at most once in each ecipe. In industial applications it is not uncommon that a ecipe uses a pocesso only once. The above lemmas, howeve, apply even in the geneal case. Assumption. No pocesso occus moe than once in a ecipe. With this assumption in place we can fomulate some lemmas that link counteflows and counteflow stuctues to the deadlock poblem. Lemma. Each counteflow in the PWRG fomulates a GMEC fo the coesponding batch ecipes. Only one ecipe at a time can be allowed to have access to the pocessos in the counteflow. PROOF. Assume an abitay long counteflow with n nodes epesenting pocessos P though P n with P,! P n and P n,! P. We shall pove the lemma by contadiction. Assume that both ecipes gain access to the pocessos in the counteflow, i.e., has booked P and has gained access to P n.any continuation of ecipe execution will soone o late lead to making that coesponds to an elementay colo cycle consisting of two nodes, and thus deadlock the system. This poves the lemma. 2 Counteflows coespond to the case whee two ecipes un though the same sequence of pocessos but in opposite diections. As the lemma indicates, access to the whole sequence of pocessos has to be esticted as fomulated in the following coollay. The monito token is eleased at the ealiest possible time. Lemma 2. Assume a counteflow W containing a set of edges E W = fe i g and two tuning nodes v t v t2. Futhemoe, let e t t, e f t, et t2, and e f t2 denote the edges going to and fom v t and v t2, espectively. The GMEC defined by W is enfoced in a minimally

6 estictive way by placing a monito containing one token. The monito has out-acs to tansitions (e f t and (e f t2, and in-acs fom tansitions (et t and (e t t2. PROOF. Placement of the monito accoding to the lemma enfoces the GMEC and follows fom the poof of Lemma. A token can be etuned to the monito when the last pocesso in the path coesponding to a tuning node is booked, e.g., booking pocesso P n using the notation fom the poof of Lemma. Even though the monito no longe esticts the othe ecipe fom enteing the counteflow, so does the fact that P n cannot be booked by befoe it is eleased by. And with that has left the counteflow. 2 Unfotunately, we ae to date not able to fomulate the next two statements about the ole of counteflow stuctues fo deadlock although extensive simulation as lemmas since the fomal poof has not been completed yet. Conjectue 3. Each coloed counteflow stuctue in the PWRG fomulates a GMEC fo the coesponding batch ecipes. The numbe of ecipes that ae allowed access to the esouces of the counteflow stuctue is the numbe of tuning nodes minus one. Conjectue 4. Assume the counteflow stuctue WS containing a set of edges E WS = fe i g and a set of tuning nodes V T = fv ti g, i = ::: n. As befoe, let e t ti and ef denote the edges going to and fom ti v ti, espectively. The GMEC defined by WS is enfoced by placing a monito containing n ; tokens. The monito has out-acs to all tansitions (e f ti and inacs fom all tansitions (e t ti. Note that the monito fo the counteflow stuctue does not do away with the need to place a monitos fo each individual counteflow in the stuctue accoding to Lemma Example Figue 5(a shows excepts fom fou pocessobooking ecipes and pat of thei PWRG. Each ac in the PWRG is labeled with its coesponding colo. As can be seen, we can identify one elementay colo cycle, thee counteflows and one coloed counteflow stuctue. Note that we only conside deadlocks caused by bookings of the pocessos. Deadlocks oiginating fom bookings in the tanspoting system ae teated sepaately. Placing one monito fo each deadlock patten accoding to the above lemmas, esults in a deadlock-fee system as can easily be veified by compaing the eachability tees of both the oiginal and the deadlock-fee system. 5. CONCLUSION We have in this pape teated ecipes that only consist of one single batch steam. No join and split tansfes 4 v 3 v 2 b( P3 4 5 b( P3 b( P3 4 5 v v 2 5 v v 2 v v 4 v 4 v 2 Fig. 5. Fou pocesso-booking ecipes, the coesponding PWRG with (v i =P i and the identified deadlock stuctues. have been consideed. Though the extension of the gaph pattens to joins and splits seems to be athe staightfowad, this fact needs to be poved. So fa, thee diffeent pattens in the wait-elationship gaph have been identified to descibe potential deadlock situations. It emains to investigate if thee ae othe gaph stuctues that indicate deadlock in a system with the specifications teated in this pape. Anothe issue to conside is computational complexity of the deadlock solutions. Lewis et al. (998 have shown that monitos that solve deadlock situations elated to elementay colo cycles can be computed in polynomial time. Note, howeve, that the potential numbe of such deadlock situations is exponential in the numbe of esouces. Refeences K. Åkesson and M. Tittus. Modula contol fo avoiding deadlock in batch pocesses. In Poc of WBF 98, 998. L.E. Holloway, B.H. Kogh, and A. Giua. A suvey of Peti net methods fo contolled discete event systems. J. DEDS, 7(2:5 90, 997. F.L. Lewis, A. Guel, S. Bogdan, A. Doganalp, and O. Pastavanu. Analysis of deadlock and cicula waits using a matix model fo discete event manufactuing systems. Automatica, 34(9:083 00, 998. T. Muata. Peti nets: popeties, analysis and applications. Poc. of the IEEE, 77(4:54 580, 989. M. Tittus and K. Åkesson. Discete event models in batch contol. In Poc. 2nd MATHMOD, pages , Vienna, Austia, 997. M. Tittus and K. Åkesson. Peti net models in batch contol. To appea in Mathematical Modelling of Systems., 999. M. Tittus and B. Lennatson. Hieachical supevisoy contol fo batch pocesses. Technical Repot I- 998:2, Contol Eng. Lab., Chalmes Univ. of Technology, Götebog, Sweden, 998a. To be published in Tansactions on Contol Systems Technology. M. Tittus and B. Lennatson. Hieachical supevisoy contol fo batch pocesses. In Poc. ACC 98, Philadelphia, USA, 998b.