Sufficient Conditions for a Flexible Manufacturing System to be Deadlocked

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1 Paper 0, INT 0 Suffiient Conditions for a Flexile Manufaturing System to e Deadloked Paul E Deering, PhD Department of Engineering Tehnology and Management Ohio University deering@ohioedu Astrat In reent years, researhers have een interested in sheduling algorithms to avoid deadlok in Flexile Manufaturing Systems (FMS) FMS are disrete event systems haraterized y the availaility of resoures to produe a set of produts Raw parts, whih elong to various produt types, enter the system at disrete times and are proessed onurrently while sharing a limited numer of resoures In suh systems, a situation may our in whih parts eome permanently lok This is alled deadlok This paper presents the suffiient onditions for deadlok to exist in a FMS; it models a FMS using digraphs to alulate slak, knot, order and spae; it identifies three types of iruits that are fundamental in determining if a FMS is in deadlok Introdution Moving the wrong part in a manufaturing system ould plae the live (deadlok-free) system into a deadloked state or dead state The only reourse would e to manually resolve the deadlok and reset the FMS to a live state Clearly, avoiding deadlok altogether would lead to inreased prodution and dereased laor osts To prevent manual deadlok resolution a Deadlok Avoidane Algorithm (DAA) was developed in [] The DAA did not allow the system to enter any dead states and proved suffiient onditions for the system to e live The DAA introdued the idea of spae If spae > 0 of all losed paths in the manufaturing then deadlok would e avoided The only prolem was that some live states were deteted dead states See figure The DAA in [] only proved suffiient onditions for a system to e live Proeedings of The 0 IAJC-ASEE International Conferene ISBN

2 Figure : Live States deteted as Dead This paper will prove suffiient ondition for the manufaturing system to e dead and is the partial results of referene [] This paper is organized as follows: the first setion disusses previous researh on deadlok in a FMS; the next setion defines a mathematial model of a manufaturing systems; iruit parameter slak, knot order and spae is then defined; the next setion introdues three types of iruit uses to proves suffiient onditions for a manufaturing systems to e dead Related Researh Many researhers use Petri nets [-9] as a formalism to desrie deadlok in a manufaturing system Banaszak and Krogh [] proposed a deadlok avoidane algorithm (DAA), whih developed a restrition poliy ased on prodution route information to guarantee that no irular wait situations would our Their DAA is suffiient for avoiding deadloks ut is not an optimal solution Viswanadham, Narahari, and Johnson [6] developed a deadlok avoidane algorithm that employed a look-ahead poliy This algorithm did not detet all deadloked states, and the authors suggested using a reovery mehanism in ase of system deadlok Zhou and DiCesare [7] and Zhou [8] generalized the sequential mutual exlusions (SME) and parallel mutual exlusions (PME) onepts and derived the suffiient onditions for a Petri net (PN) ontaining suh strutures to e ounded, live, and reversile In general, PN solutions are suitale for manufaturing systems that ontain few resoures ut eome very ompliated for larger systems Another formalism to desrie the manufaturing system is to use graphs [0-0] In this approah, the verties represent resoures and the edges represent part flows etween resoures Wysk, Joshi, and Yang [6] were the first to develop a speialized direted graphial struture alled a wait relation graph (WRG) to model a manufaturing system In referene [6], they developed a string manipulation proedure that yields a set of ontrol ations to detet and reover from primary deadlok Cho, Kumaran, and Wysk [0] used system status graphs to develop the onept of simple and non-simple ounded iruits with empty and non-empty shared resoures to detet part flow deadlok and impending part flow deadlok This method introdued the onept of a ounded iruit to detet deadlok The method deteted deadlok ased on harateristis of this ounded iruit The methods in referenes [6] and [0] ould only handle single apaity resoures Fanti, Maione, and Turhiano [] used a graph alled working proedure digraph and developed a simple graph-theoreti method for deadlok detetion and reovery in systems with multiple apaity resoures This algorithm did not prevent deadlok from ourring either, ut it suggested a suitale reovery strategy Judd and Faiz [] expanded on the original formulation proposed y referene [6] and the first to define slak, order, and spae to avoid deadlok This method provided suffiient onditions for deadlok y satisfying a set of linear inequalities Lipset, Proeedings of The 0 IAJC-ASEE International Conferene ISBN

3 Deering, and Judd [4] extended referene [] to preisely quantify neessary and suffiient onditions for deadlok to exist In this researh, they redefined the order of a knot, defined a speial state alled an evaluation state, and defined the onept of order redution The approah was to put the system into an evaluation state and then ompute the order Deering [] and [] improved referene [4] y further refining the order of a knot and evaluation state, as well as eliminating the need for order redution Zhang, Judd, and Deering [7] developed a deadlok avoidane algorithm (DAA) ased on referenes [4] and [], whih avoided deadlok and was exeuted in polynomial time Zhang, Judd, and Deering [9] expanded upon referenes [7] and [] to quantify the suffiient onditions for a system state to e live and derived the liveness neessary and suffiient onditions for an evaluation state Zhang and Judd [0] extended referene [9] to allow hoie in proess flow or flexile part routing Modeling a Manufaturing System An FMS onsists of a set, R, of finite resoures, suh as roots, uffers, and mahines, whih produe a finite set, P, of produts Eah resoure r R has a apaity of ap(r) units that an perform the required operations The apaity funtion an e extended to a set of resoures, that is: ap( R) = ap( r), for any R R r R () For eah produt p P, the proess plan plan( p) = r r Kr defines the sequene of resoures that are required to produe p Resoure r m is the terminal resoure for produt p It is assumed that all proess plans are fixed, finite, and sequential A part is an instane of a produt that flows through the system At any given time, a manufaturing system is working on a set Q of parts The funtion lass( q ) returns the produt p to whih part q elongs m A manufaturing system an e represented y a WRG, G = ( V, A) Eah vertex represents a resoure; that is, V=R A direted ar is drawn from vertex r to vertex r, if r immediately follows r in at least one proess plan Eah ar will e laeled with the part(s) that will flow through it A sugraph G = ( R, A ) G of an WRG onsists of a suset of the resoures and ars of G, so that all the ars in A onnet resoures in R The union (intersetio, denoted y G G ( G G ), of two sugraphs is the union (intersetio of the omponent resoure and ar sets A path P = R p, A ) is a sugraph ( p whose resoures and ars an e ordered in the list r ar aka n rn where eah ar in, the list onnets the resoures on either side When speifying a path, writing the ars is redundant Therefore, only the resoures will e enumerated when a path is defined A simple path is a path with no repeated elements in the ordered list A losed path is a path Proeedings of The 0 IAJC-ASEE International Conferene ISBN

4 with the same first and last element A simple iruit is a losed path with no repeated elements in the ordered list, exept the first and last elements The funtion n( q ) returns a positive integer that represents the position in plan[lass( q )] of the operation that is urrently proessing q When a new part q is added to the system, then n ( q ) = As the part is moved from resoure to resoure aording to its plan, n( q ) is inremented until it reahes the end of its plan and exits the system The state n of a manufaturing system is a vetor ontaining the urrent n(q) for all q Q A state n of a manufaturing system is live if a sequene of part movements exist that will empty the system A state n of a manufaturing system is dead, or deadloked, if it is not live Given a manufaturing system G = ( R, A), let a A and r R Then, the funtion tail( a ) returns the resoure at the tail of the given ar; the funtion head(a ) returns the resoure at the head of the ar A unit of the resoure r = tail( a) is said to e ommitted to ar a if it is proessing a part q whose next resoure in its proess plan is head( a ) It is important to note that the numer of resoure units ommitted to the outgoing ars of r an e less than the numer of usy units This happens when some of the usy units are eing used for terminal operations A resoure unit is free if it is not ommitted to an ar; y this definition, a usy unit that is not ommitted is still termed free A resoure is free if any of its units are free A resoure is empty if it ontains no parts The ommitment funtion om( a, returns the numer of resoure units that are ommitted to ar a when the system is in state n The ommitment funtion is extended to a set of ars as follows: om( A, = om( a,, for any A A a A A part is enaled if either the next resoure in its proess plan ontains at least one resoure unit that is not usy, or the part is in the last step of its proess plan Suppose that the system is in state n 0 ; there exists an ar a suh that resoure r = head( a) is free and the part in the resoure r = tail( a) is ommitted to a Then, when r finishes its operation, this part an e moved to resoure r This proess is alled propagation The th symol n k is used to denote the state of the system after the k propagation A part q in WRG G an e shifted to resoure r if it an e propagated to r without propagating any other part in G A part q in WRG G is said to have a free exit if it an shift its terminal resoure r m in G Slak, Knot, Order, and Spae This setion will summarize the major onepts and results from [,, 4, 6] This setion defines the onept of slak, knot, order, and spae The slak is the numer of free resoure units availale for parts to flow on a sugraph () Proeedings of The 0 IAJC-ASEE International Conferene ISBN

5 Definition : The slak of any sugraph G = ( R, A ) G is given y slak( G, = ap( R ) om( A, n ) () A losed path in a WRG G is in primary deadlok in state n if slak(, = 0 Definition : Let e any two losed paths in a WRG of a manufaturing system If onsists of exatly one resoure with a apaity of one, then this resoure is alled a knot with respet to Definition : Let e two losed paths in a WRG G Path is onneted to if 0 and a part urrently exists in the system that must propagate from to without leaving Definition 4: Given two losed paths, then are ross-onneted if is onneted to is onneted to Definition 5: Let the losed path in state n onsist of two losed paths,, suh that = = k, where k is a knot The order of knot k with respet to the losed path in state n is defined as:, if and are ross onneted order(,, ) = k n 0, otherwise (4) The order of any simple iruit is zero Definition 6: Let e a losed path in a WRG G in state n that ontains m knots Then, the order of is given y: m order(, = order( k,, Definition 7: Let e a losed path in a WRG G of a manufaturing system in state n The free spae on a losed path is the differene etween the slak and the order: i= i (5) spae(, = slak(, order(, C, G (6) where C G is the set of all losed paths in G The following theorem proves that if all losed paths of a WRG G have spae greater than zero, G is live Proeedings of The 0 IAJC-ASEE International Conferene ISBN

6 Theorem : Let C G e the set of all losed paths in a non-empty WRG G in state n If, spae(, > 0, C G (7) then G is live Proof: See referene [] Suffiient Conditions for a System to e Dead The previous setion proved suffiient onditions for a manufaturing system to e live; that is, if the spae of all losed paths in a manufaturing system is greater than zero, then the system is live This setion will prove suffiient onditions for a manufaturing system to e dead Unfortunately, this annot e proven in the general ase, sine there is insuffiient information in the WRG to determine these onditions However, when the system is in a speial system state alled an evaluation state, it an e shown that a manufaturing system is dead if one of the losed paths equals zero The following example will demonstrate this more learly Example : Let the WRG G in Figure e in state n Suppose that the proess plans for parts a, appear as presented in Tale Assume that the state of the system is n = [ n( a), n( ), n( )] = [,,] Tale depits the order and spae omputations for this system Tale Proess plans for example Part Proess Plan a r rr 4 r 4r r r r r r, r r a, a, a Figure Manufaturing system for example r r 4 a Tale Order and spae omputations for example Proeedings of The 0 IAJC-ASEE International Conferene ISBN

7 Ciruit Order Spae Sine the spae of the union etween is zero, the method previously presented in referene [] annot onlude whether the system is live or dead This revised method will show that the order in whih parts flow through order-one knots is required to desrie suffiient onditions for a dead system For example, knowing that parts a and must pass through r efore any other parts an leave and that the spae of the union etween is zero, will allow researhers to know that the system in Figure to e dead This setion will ontain three parts: the first setion shows neessary and suffiient onditions whih render asi losed path as dead; the next two setions show suffiient onditions for deadlok of hained omplex losed paths Basi Closed Paths Definition 8: A asi losed path is a losed path in a WRG G in state n suh that order (, = 0 Theorem : Given a asi losed path = (R,A) in state n If spae(, = 0 then is dead Proof: See referene [] Theorem allows us to onlude that spae greater than zero of a asi losed path is neessary and suffiient for the system to e live The next setion addresses a partiular losed path that ontains order-one knots Chained Closed Paths This setion defines a hained losed path and introdues a speial state alled an evaluation state A series of definitions, some lemmas and a theorem will prove that if a hained losed path is in an evaluation state and its spae is equal to zero, then the hained losed path is dead Definition 9: A hained losed path is a losed path ontaining one or more order-one knots with respet to, suh that an e deomposed into a set of asi losed paths whih interset at only the order-one knots The following is a simple example of a hained losed path: Proeedings of The 0 IAJC-ASEE International Conferene ISBN

8 Example : Consider the manufaturing system in Figure Assume that all a part types flow to the right from to, and that all part types flow to the left from to In this state, resoures r to r are order-one knots The manufaturing system an e deomposed into three simple iruits,, Let = In this example, is a hained losed path, sine an e deomposed into asi losed paths so that eah iruit intersets eah other at only the order-one knots (ie = r and = ) r a a a a r r r r4 Figure A hained losed path a The following example will help the reader oneptualize the need for an evaluation state: Example Suppose that a part exists in all the resoures shown in Figure 4 exept r 4 Eah part is ommitted to the outgoing ar of its resoure Assume that all part a types must flow to iruit efore ompletion and parts d and d must flow to iruit efore ompletion Call this state n This state may, or may not, e dead, depending on the ultimate destination of part in the resoure r 7 Case Suppose part must move to resoure r 4 and then to r 5 and exit the system Clearly, in this ase, state n is a live state Case Suppose part must flow to r 4 ommit to iruit Then state n is a dead state Proeedings of The 0 IAJC-ASEE International Conferene ISBN

9 ase flow to to exit r8 r7 a r a d r6 r a r4 d r5 ase exit Figure 4 Manufaturing system for example To distinguish and to evaluate these two ases, the dynamis of the part rossing through the knot should e analyzed more losely Notie that in oth ases, all part a s must ross knot r 4 efore any other part on an leave But the part rossing dynamis are different on iruit in the two ases Notie that in ase, part an leave iruit efore part d must ross knot k In other words, a resoure may eome free on efore part d must ross the knot In this state, we onlude that state n is not in an evaluation state The method in the referene [] annot determine if deadlok exists y omputing the spae in state n Notie that in Case, part must ross knot r 4 efore any other part an leave In this situation, no part an esape efore the rossing must our The state of the system in ase is onsidered to e an evaluation state These ideas motivated the following definitions Definition 0: Let e two losed paths in a WRG G suh that = k where k is an order-one knot If a part q on propagates to k ommits to an ar on, then q is said to ross knot k Definition : A asi losed path in a WRG G is always in an evaluation state Definition : An empty hained losed path in state n is in an evaluation state Definition : Let a non-empty hained losed path e in state n A hained losed path an e divided into two losed paths,, at any order-one knot k suh that = k Then hained losed path is in an evaluation state if all order-one knots are empty, and Proeedings of The 0 IAJC-ASEE International Conferene ISBN

10 for eah order-one knot k, two parts, q and q exist, suh that a part q must ross from to efore any other part an leave, and is in an evaluation state after the move; and part q must ross from to efore any other part an leave, and is in an evaluation state after the move The system in Example is in an evaluation state Resoures r and r are order-one knots For order-one knot r, part a must ross from to efore any other part an leave, and part must ross to efore any other part an leave For order-one knot r, part a must ross from to efore any other part an leave and part must ross to efore any other part an leave After moving either part a or part, oth are in evaluation states The next lemma will show how the parts are ommitted when a hained losed path is in an evaluation state Lemma : Given a hained losed path = ( R, A) in a WRG G that is in an evaluation state n, spae (, = 0 if, and only if, all order-one knots are empty in and all other resoures in are filled ommitted to resoures on Proof: See referene [] The next two lemmas are preliminary results that are required to prove the final theorem of this setion Lemma : Given a hained losed path that is in an evaluation state n, if spae(, = 0 then a part q exists suh that when it is moved, it will fill an order-one knot ommit an outgoing ar of that knot on Proof: See referene [] Lemma : Given a non-empty hained losed path that is in an evaluation state n 0, if spae(, n0 ) = 0 then propagating any part will reate a hained losed path suh that and spae(, n ) = 0 Proof: See referene [] Definition 4: If any sugraph in a WRG G is dead, then G is dead Proeedings of The 0 IAJC-ASEE International Conferene ISBN

11 Theorem : Given a non-empty hained losed path that is in an evaluation state n 0, if spae(, n0 ) = 0, then is dead Proof: See referene [] Complex Closed Paths Closed paths that are not asi losed paths or hained losed paths are lassified as omplex losed paths This setion will introdue omplex losed paths It will also e shown, if a omplex losed is in an evaluation state and it ontains a path with spae equal to zero, then this is suffiient for determining if the system is dead We will first define a omplex losed path and its various omponents, then follow these definitions with an example Definition 5: A omplex losed path is a losed path that ontains one or more order one knots that is not a hained losed path Definition 6: A omplex path an e deomposed into two paths, one eing a hained losed path and the other is alled the auxiliary losed path The intersetion of the auxiliary losed path intersets and the hained losed path must ontain one or more order one knots of the hained path Definition 7: A ypass path is the portion of the auxiliary path that does not interset the hained losed path Definition 8: The first ar on the ypass path is a ypass ar Consider the following example Example 4: Suppose that the system in Figure 5 has the following parts and proess plans as depited in Tale Tale Proess plans for example 4 Part Proess Plan a r r rr 4 r 4r r6 r d r r5 r6 r Assume that the system is in state n = n( a ), n( a ), n( ), n( d ), n( d )] [,,,, ] [ = Proeedings of The 0 IAJC-ASEE International Conferene ISBN

12 r 5,d a r d r 6 d a a d d r d a a r a r 4 a Figure 5 Complex losed path for example 4 The system onsists of three simple losed paths: = r rr6 r, = r rr 4r, and = r r r5 r6 r r The order( r,, = Clearly, the manufaturing system in Figure 5 is not a asi losed path The system annot e a hained losed path either sine ( ) is not a knot Aording to Definition 5, the system in Figure 5 is a omplex losed path The omplex losed path an e deomposed into a hained losed path, (ie,, and an auxiliary losed (ie, ) Closed path is an auxiliary losed path sine the intersetion of and the hained losed path ontain the order-one knot r The simple path r r5 r6 is a ypass path that joins together Ar a on resoure r is a ypass ar sine it leaves r along the auxiliary losed path We next define the evaluation state for a omplex losed path Definition 9: A omplex losed path in a WRG G is in an evaluation state if its ypass ars are not ommitted Definition 0: An empty sugraph that is a omplex losed path in a WRG G in state n is in an evaluation state Definition : A WRG G is in an evaluation state if all losed paths in G are in an evaluation state The system in Example 4 is in an evaluation state This is eause part a in resoure r is not ommitted to the ypass ar a The hained losed path is in as evaluation state per Definition The spae of the hained losed path is zero Clearly, the system is dead The next theorem proves this onept in general Theorem : Given a omplex losed path p that is in an evaluation state n 0, if any hained losed path * p has spae ( *, n0 ) = 0, then p is dead Proeedings of The 0 IAJC-ASEE International Conferene ISBN

13 Proof: See referene [] Conlusion Three types of losed paths were identified to prove suffiient onditions for a manufaturing system to e dead A speial state alled an evaluation state was introdued It was showed that if a asi, hained or omplex losed path that is in an evaluation state with spae=0 then the system is dead Unfortunately, determining if a losed path is in an evaluation state is a prolem The prolem is there is insuffiient information in the WRG to determine if a system is in an evaluation state This is a topi of future researh Referenes [] Deering, E P (008), A Simple Deadlok Avoidane Algorithm in Flexile Manufaturing Systems, International Journal of Modern Engineering, vol 9, no, pp 9-6 [] Deering, E P (000), Neessary and Suffiient Conditions for Deadlok in Manufaturing Systems, PhD Dissertation, Ohio University [] Banaszak, Z and B Krogh (990), Deadlok Avoidane in Flexile Manufaturing Systems with Conurrently Competing Proess Flows IEEE Trans on Rootis and Auto, vol 6, no 6, pp 74 7 [4] Barkaoui, K and IB Adallah (995), A Deadlok Method for a Class of FMS, Proeedings of the 995 IEEE Int Conf On Systems, Man and Cyernetis, pp [5] Hsieh, F and S Chang (994), Dispathing-driven Deadlok Avoidane Controller Synthesis for Flexile Manufaturing Systems, IEEE Trans Rootis and Auto, vol 0, no, pp [6] Viswanadham, N, Y Narahari, and T Johnson (990) Deadlok Prevention and Deadlok Avoidane in Flexile Manufaturing Systems Using Petri Net Models, IEEE Trans on Rootis and Auto, vol 6, no 6, pp 7 7 [7] Zhou, M and F DiCesare (99), Parallel and Sequential Mutual Exlusion for Petri Net Modeling of Manufaturing Systems with Shared Resoures, IEEE Trans on Rootis and Auto, vol 7, no 4, pp [8] Zhou, M (996), Generalizing Parallel and Sequential Mutual Exlusions for Petri Net Synthesis of Manufaturing Systems, IEEE Symposium on Emerging Tehnologies and Fatory Automation, vol, pp [9] Ezpeleta, J, J Colom, and J Martinez (995), A Petri Net Based Deadlok Prevention Poliy for Flexile Manufaturing Systems, IEEE Trans on Rootis and Automation, vol, no, pp 7 84 [0] Cho, H, TK Kumaran, and R Wysk (995), Graph-Theoreti Deadlok Detetion and Resolution for Flexile Manufaturing Systems, IEEE Trans on Rootis and Auto, vol, no, pp [] Fanti, M, G Maione, and B Turhiano (996), Deadlok Detetion and Reovery in Flexile Prodution Systems with Multiple Capaity Resoures, Industrial Proeedings of The 0 IAJC-ASEE International Conferene ISBN

14 Appliations in Power Systems Computer Siene and Teleommuniations Proeedings of the Mediterranean Eletrotehnial Conferene, vol, pp 7 4 [] Judd, R P and T Faiz (995), Deadlok Detetion and Avoidane for a Class of Manufaturing Systems, Proeedings of the 995 Amerian Control Conferene, pp [] Judd, R P, P Deering, and R Lipset (997), Deadlok Detetion in Simulation of Manufaturing Systems, Proeedings of the 997 Summer Computer Simulation Conferene, pp 7 [4] Lipset, R, P Deering, and R P Judd (997), Neessary and Suffiient Conditions for Deadlok in Manufaturing Systems, Proeedings of the 997 Amerian Control Conferene, vol, pp 0 06 [5] Lipset, R, P Deering, and R P Judd (998), A Stak-Based Algorithm for Deadlok Avoidane in Flexile Manufaturing Systems, Proeedings of the 998 Amerian Control Conferene [6] Wysk R, N Yang, and S Joshi (99), Detetion of Deadloks in Flexile Manufaturing Systems, IEEE Transations Rootis and Automation, vol 7, no 6, pp [7] Wenle, Z, R P Judd, and P Deering (00), Evaluating Order of Ciruits for Deadlok Avoidane in a Flexile Manufaturing System, Proeedings of the 00 Amerian Control Conferene, pp [8] Fanti, MP, B Maione, S Masolo, and B Turhiano (995), Control Polies Coniliating Deadlok Avoidane and Flexiility in FMS Resoure Alloation, IEEE Symposium on Emerging Tehnologies and Fatory Automation, vol, pp 4 5 [9] Wenle, Z, R P Judd, and P Deering (004), Neessary and Suffiient Conditions for Deadloks In Flexile Manufaturing Systems Based On A Digraph Model, Asian Journal of Controls, vol 6, no, pp 7 8 [0] Wenle, Z and R P Judd (007), Evaluating Order Of Ciruits For Deadlok Avoidane in a Flexile Manufaturing System, Asian Journal of Controls, vol 9, no, pp 0 Biography PAUL DEERING is urrently an Assistant Professor in the Engineering Tehnology and Management Department in the Russ College of Engineering and Tehnology at Ohio University He has worked in the area of Information Tehnology for more than 0 years and has taught many engineering omputer siene ourses for the Russ College Proeedings of The 0 IAJC-ASEE International Conferene ISBN