Virtual Control Policy for Binary Ordered Resources Petri Net Class

Transcription

1 sensors Article Virtual Control Policy Bary Ordered Resources Petri Net Class Carlos A. Rovet, Tomás J. Concepción Elia Esr Cano Computer Systems Engeerg Department, Technological University Panama, , El Dorado, Panama City, Republic Panama; (C.A.R.); (T.J.C.); (E.E.C.); Tel.: (C.A.R.); (T.J.C.); (E.E.C.) Academic Edirs: Vladimir Villarreal Carmelo R. García Received: 20 May 2016; Accepted: 22 July 2016; Publhed: 18 August 2016 Abstract: Prevention avoidance deadlocks sensor networks use wormhole routg algorithm an active research doma. re are diverse control policies will address th problem beg our approach a new method. In th paper we present a virtual control policy new specialized Petri net subclass called Bary Ordered Resources Petri Net (BORPN). Essentially, it an ordary class constructed from various state maches share unitary a complex m, which allows branchg jog processes. reduced structure th new class gives advantages allow analys entire system s behavior, which a prohibitive task large systems because complexity routg algorithms. Keywords: BORPN class; deadlock; Petri nets; resource allocation systems; siphons 1. Introduction concept liveness closely related absence deadlocks systems. A deadlock occurs if one state a system becomes fitely unattaable by an unanswered resource request. vivacity property establhes system should reach all states which it was designed, which why th property helps characterize absence deadlocks. For th reason, liveness a desirable property concurrent systems share simultaneously, because it allows all states system be reached. From pot view resource allocation systems (RAS), goal guarantee all desired states will be reached usg requested by system durg a given time. perspective resource allocation systems will be used model systems through a Petri net model, ree, are used conservatively;, y are not created or destroyed. As it known, a Petri net a mal graphical ol can be used abstraction analys concurrent dcrete events dynamic systems such as sensors networks. In th paper, we will guarantee absence deadlocks system through search liveness property obtaed by analys Petri net model under study. It well known deadlocks occur more ten systems with concurrency, which are best described by Petri nets. Furrmore, possibilities modelg Petri nets are not limited by technology because it a mamatical model with a graphical representation usg a bipartite graph. Normally, way synsize analyze concurrent systems usg Petri nets through subclasses with strengths address specific problems. ree, we will expla new control policy Petri nets called Bary Ordered Resources Petri Net abbreviated as BORPN. Th a subclass previously extg classes such as S 4 PR [1,2] ES 3 PR Petri nets classes [3], which have been used address problems deadlock. It well known a reduced structure allows us improve algorithms analyze Petri net model. It an ever-present desire literature reconcile skills reduce Petri net model, while avoidg extensive calculations. A similar strategy mentioned approaches [4,5] where Boolean calculations were used avoid complex operations. Intuitively, ordary bary decions diagrams Sensors 2016, 16, 1307; doi: /s

2 Sensors 2016, 16, (OBDD) have been used as reduced data structures encodg functions specific domas [6,7]. re are or new applications, such as sensor networks, where re are attacks through wormholes. Durg th action, attackers do not need comprome all sensor nodes. packets are received one location are sent anor location. Later, y can tamper with selectively ward data or messages drupt functions sensor network. To prevent th situation, we can reconfigure routes messages sensor networks, but th can cause deadlock situations. BORPN class a specialized class with a reduced structure faces deadlock problem a wide range dtributed systems, particularly routg algorithms. Its structure reces algorithms durg analys process because it avoids overuse memory large calculatg operations detect structural objects such as siphons. Or novelty methods use vecrs set legal markgs by addg control places as [8 10]. As shown well-known property Commoner, some structural objects such as siphons are closely related properties basic behavior Petri nets, like liveness absence deadlocks, where a part system works but anor remas locked. structural analys Petri net model allows us demonstrate some properties through siphons ensure liveless model, however, it a process uses o much memory; some cases, it even impossible due problem state explosion. Several methods allow reduction exponential number calculations as lear equations or equalities, symmetries, modular design, etc. A new approach presented [11] works with higher-level objects, avoidg wasted memory termediate steps. method based on graph ory through which manipulations maximum are strongly connected subgraphs graph [12]. Th article organized as follows. Section 2 cludes defition BORPN class its basic properties. Section 3 dedicated ensure property liveness exts th Petri net class, cludes an example modelg a routg algorithm applied sensor networks. Section 4 provides method controllg siphons usg prung relation. Fally, conclusions are given Section 5. For a basic defitions properties about Petri nets we recommended [13,14]. 2. Petri Net Class Properties 2.1. BORPN Class Properties In th section we will expla BORPN Petri net class properties [15,16]. Th class a subclass S 4 PR [1,2] ES 3 PR [3] Petri nets, ree, all extg oretical results se networks can be applied th subclass, however contrary reasong not possible. BORPN class has valuable structural mation given by its reduced structure comes from restriction imposed on transitions BORPN class, because it only perms bary operations. Each transition could take or release as a unit, much like behavior a routg algorithm as type wormhole requests releases channels as transport messages like chas bits or flits. BORPN class defed as follows: Defition 1. ( Petri Net Class Ordered Bary Resources). Let s say I N = {1, 2,..., m} a fite set dices. A Petri Net Class Ordered Bary Resources a strongly connected Petri net, self-loops-free N = P, T, C where: (1) P = P 0 P S P S a partition such : (a) P S = i IN P Si, P Si = y P Si P Sj =, all i = j. (b) P 0 = i IN { p0i }. (c) P R = {r 1, r 2,..., r n }, n > 0. (2) T = T a T r a partition such : (a) T a = i IN T ai, T ai =, T a P R, each i, j I N T ai T aj =, all i = j.

3 Sensors 2016, 16, (b) T r = i IN T ri, T ri =, T a P R, each i, j I N T ri T rj =, all i = j. (3) For all r I N, subnet N i generated by P Si { p 0i } Tai T ri a strongly connected state mache, such each cycle contas p 0i duces a mimal T-Semiflow. (4) For each r P R re a mimal P Semiflow, Y r {0, 1} P, such {r} = y r P R, y r [r] = 1, P 0 y r =, y P S y r =. (5) P S = r Pr ( y r \ {r}). whole model Petri net BORPN compres various strongly connected networks represented by N i where i N +. A loop-free Petri net exts if only if t T t t = beg state mache (SM). Where each SM corresponds a subnet. se SMs ger build a sgle Petri net BORPN class. deadlock problems are related unachievable states produced by diverse processes, a simultaneous manner, reta ask, thus generatg a loop does not allow any evolution processes. As shown Defition 1, Paragraph 1, P places are partitioned three groups representg: (a) process places P S ; (b) idle places P 0i representg pendg messages; (c) P R. Resource places represent availability, due RAS perspective, cannot be created or destroyed by processes. structure BORPN class requires each cycle contas idle places P 0i. If a process starts, it acquires a mark from idle place when process ends, mark idle place should return. In or words, durg evolution process, various can be used, although y must be released when process completed. Liveless property sought ensure completion processes thus have system free deadlocks. Each process requires use at least one resource, however it must be acquired or released as a unit. For above reason, Y r component a Boolean vecr due peculiar behavior th network. Transitions a BORPN have a particular behavior followg our approach RAS perspective. We consider process behavior resembles a pipe, which can be partitioned process units. first processg unit acquirg last unit release m. Th approach restricts behavior transitions allows us model particular systems more accurately, unlike traditional approaches. ree, we are able model a process represents transportation objects (messages, items, etc.) via networks or warehouse dtribution centers. Defition 1, Paragraph 2 related transitions are partitioned two djot sets. Transitions T a T r mean acquire release, respectively. ree, { t i, t j } T t i P R = 1 t j P R = 1 where i = j. However, th restriction does not impede bifurcations, one useful charactertic represent complex systems where deadlock problems are. In [13] where it proved a state mache strongly connected t = t = 1 live, which why it duces an variant property conservation marks places. For a BORPN, th property establhed Defition 1, Section 3, where all i I N, subnet N i generated by N i = P 0i P Si, T ai T ri, C i where i N + a strongly connected state mache, such each cycle contas a place p 0i. Each cycle contag idle place closes a circuit duces a T-Semiflow on path. Fally, Defition 1, Paragraphs 4 5 are related variant structural properties idle places respectively. Thus, any r P R re a mimum P-Semiflow where Y r {0, 1} P. Process places joed with resource r are known as carrier places H. se places charge availability while representg a process state, as shown by Defition 2. Defition 2. Consider N a BORPN P R set resource locations. set carrier places H r support mimal P-Semiflow H r = Y r \ {r} resource where r P R. A BORPN a state mache strongly connected with, ree all transitions have a sgle pot entry/exit process would have a sgle place put/output resource. Thus, transitions may be characterized as enabled or dabled, by markg place resource as shown Figure 1 more mally explaed Defitions 3 4.

4 Sensors 2016, 16, Defition 3. Consider a BORPN, with set places set process places. A transaction enabled on markg or (dabled on markg) summarized o ( ) iff M markg p ( ) (, ) o ( ( ) (, )). Sensors 2016, 16, Figure 1. Markg enabled dabled places. Defition 4. Consider a BORPN, beg set places set process Defition 3. Consider N a BORPN, with P R set places P S set process places. places. Transition enabled on markg or (dabled on markg) summarized or ( ) A transaction iff t T enabled on markg or (dabled on markg) summarized mpe or (mpe) iff M markg p as ( ) (, ) or ( ( ) (, )). p t P S M markg p M (p) PRE (p, t) or (M (p) < PRE (p, t)). For transitions release,, markgs are enough m so y can be Defition 4. Consider N a BORPN, beg P R set places P S set process fired, but set acquire,, re should be markgs so places. Transition t T enabled on markg or (dabled on markg) summarized mre or (mre) iff y can be fired. transition from left side Figure 1 enabled process markg p t P S M markg p as M (r) PRE (r, t) or (M (r) < PRE (r, t)). enabled markg, which opposite transition from right side dabled For transitions process markg release, process T dabled resource markg resource. A path a r, markgs mpe are enough m so y can be T-Semiflow as, where each fired, but set acquire, T = 2 does not satfy one four conditions are a, re should be markgs mre mpe so y can sufficient be fired. necessary t extence a deadlock [14]. Th type route does not have 1 transition from left side Figure 1 enabled process markg enabled retention wait markg, condition which because opposite it only requests transition a resource from complete right side entire dabled process. ree, process markg each process place dabled belongs resource th markg type resource. T-Semiflow A path a T-Semiflow as follows: N as X, H where each X = 2 does where not satfy one. Due four conditions structure are sufficient class, necessary some places will extence never be cluded a deadlock a deadlocked [14]. Th type situation route does are not known have as retention places-withoutdeadlock. wait condition because it only requests a resource complete entire process. ree, each p i place belongs th type T-Semiflow as follows: p P S H r p r = p r = where r Defition P 5. Consider a BORPN, with set places set process R. Due structure BORPN class, some places will never be cluded a deadlocked places. A place called deadlock-free-place iff situation are known as places-without-deadlock.. Defition places 5. Consider meet requirements N a BORPN, with Defition P 5 will be fired when y have markg mpe, R set places P S set process places. A so place y p never belong a state deadlock, however, y can m a structural part a siphon. i P S called deadlock-free-place iff p i P R = BORPN places Class meet requirements Defition 5 will be fired when y have markg mpe, so y Th never class belong defed a state face deadlock, however, problems y can concurrent m a structural systems part [16] such a siphon. as sensor networks use wormhole routg algorithms followg our RAS viewpot processes BORPN Class an ordary Petri nets class where P-Semiflow a resource a bary vecr, which why Th re class a directed defedpath face between deadlock problems transitions concurrent take systems resource [16] such as sensor transitions networks release useit. wormhole routg algorithms followg our RAS viewpot processes. BORPN an ordary Petri nets class where P-Semiflow a resource a bary vecr, which why re adefition directed path 6. (A between directed path). transitions A directed path take a sequence resource places transitions release,,, it. such {,,, } from where, 1 Defition {, } N. 6. (A directed path). A directed path a sequence places transitions p 1 t 1, p 2 t 2,..., p k t k such {t 1, t 2,..., t k } from p 1 p k where t i p i P S t i P i+1 P S, 1 i k {i, k} N +. When th directed path related a class it known as a resource zone, as shown Defition When th 7. directed Th charactertic path related very important a BORPN class order it known avoid as extensive a resource structural zone, asanalys shown Defition Petri net 7. model, Th charactertic hence reducg very important amount states order avoid system extensive will structural be analyzed. analys Petri net model, hence reducg amount states system will be analyzed.

5 Sensors 2016, 16, Defition 7. (Zone a resource). Consider N i = P i, T i, C i, i N + N a BORPN P R set. zone a resource set places holders r P R tercepts net N i as Z r i,j = H r N i, where i N + N j N + P. subdex i represents BORPN class, if re more than one zone, j subdex will crease. When dex j omitted, we assume only one zone th resource. In lear processes re only one path between capture release resource, however non-lear processes re will be more than a catch or release resource. Corollary 1 states extg structure a zone lear processes. Corollary 1. (Zone a resource lear processes). Consider N a BORPN with only leal processes, where P R set places. Consider Z r i,j = {p 1... p k }, such k N + P zone a resource r net N i j = 1. place p x Y r, x = 1... k, where (p x ) Y r = P x+1. So s such p 1 Z s i,1 p k Z s i,2. In a strongly connected state mache re could be non-lear processes, so th network, zone a resource should be generalized consider different acquitions releases. Due RAS approach process, places first part process are set S A where A means acquirg. places are mataed latter part process belong set S R, where R means releasg. Corollary 2 describes structure a zone non-lear processes. Corollary 2. (Zone a resource non-lear processes). Consider N a BORPN with non-leal processes, where P R set places {r, s} P R. Consider Zi,j r = (S A S R ), where S A r, S R r. In th way, p i S A, p j S R such (p i ) Y r = p j, i = j. So, p x Z s i,1 S A p k Z s i,2 S R, x = k. zong would produce an overlap on zones. When re an overlap between different resource zones, it will be known as teams. concept team comes from pot view where a process acquires/releases various strict order. y are all workg ger progress process as a team. Moreover, teams are a characterization th order will be used describe new BORPN class. Defition 8 summarizes th concept team a N i. Defition 8. Consider N a BORPN, beg P R set. A team a set places where { r i, r j } PR, satfyg (1) P X Y ri Y rj N i = P X P R = P X P R ; (2) p i Y ri N i ( PX Y rj ) = ; (3) p j Y rj N i (P X Y ri ) =. set P X P S N i exts because tersection or overlappg between two Petri net model, however it should be a unique set. second condition prevents extence more than one set P X through put output. Fally, prevent subsets between becomg volved, a particular set locations must ext. re must be a place does not belong P-Semiflow or resource volved. If previous conditions are met, re will be an overlappg among all volved, th would be called a team. A Petri net where all belong a team a BORPN class, additionally, it necessary all places a process belong any P-Semiflow team. Fally, each transition where resource team acquired, re only one way, strongly connected state mache, reach each transition where resource released. Defition 9. (Properties a resource BORPN class). A Petri net a BORPN class if all belong a team p i P S such p i Y r =, r P R.

6 Sensors 2016, 16, Some features a BORPN class: 1. itial fal states are a collapsed state, called idle place. 2. options between paths are permitted, but iterations or loops are not. 3. cannot be created nor destroyed. 4. are shared between paths. 5. Resource places have a mark dicatg availability. 6. A state could use several. 7. order which are allocated must be same as y are released. 8. Transitions acquire or release but never both events at same time. behavior several systems can be described terms states systems, sce se states ir changes have a physical meang. Based on th, we can say an itial mark represents lack activity system allows start processes. BORPN class conservative with due P-Semiflow, so all reachable markgs will represent possible states system from an acceptable itial markg. marks places P 0i represent maximum amount processes waitg same Petri net or state mache. marks places P R model availability, ree a mark enough represent it. process place P S lacks itial markg because th markg represents lack activity system. Defition 10. Consider N = P 0 P S P R, T, C a BORPN net. An itial markg m 0 acceptable N iff: (1) i I N, m 0 [ p0i ] > 0. (2) p P S, m 0 [p] = 0. (3) r P R, m 0 [r] = 1. Usually, order implement policy deadlock prevention it necessary consider itial markg Petri net model. In our avoidance control policy deadlock, Petri net model does not get modified, so itial markg remas unchanged. In order apply our policy deadlock prevention it necessary add virtual make Petri net model free deadlocks, which will keep same itial markg as previous. All same, if new places processes are added, y rema empty itial markg. Our control policy deadlock does not add new processes Petri net model, so no idle place will be added. 3. Liveness Analys BORPN Class In th section, liveless property characterized by siphons. A siphon a set places, when y become empty, rema ever alike. For, all output transitions places a vacuum siphon will be dabled ever because at least one pot entry (belongg siphon) will be empty ever. Empty siphons can be represented as circular waits, due fact a siphon exts complex overlappg loops, where each cycle represents a set empty. In [15] siphons break by addg virtual until a Petri net model free deadlocks obtaed. From anor perspective, siphons are prevented from losg marks usg logical functions guarantee liveness property Petri net model. BORPN class has diverse properties related siphon structure its volved, which why concept th type structurally safe net could be troduced, due bary markg processes places places. structure BORPN class ensures all reachable states have Boolean states. Defition 11. A Petri net system N called a Structurally Safe Net class iff each place p P R P S, exts a P-Semiflow y N + P R P S such p y y.m 0 1.

7 Sensors 2016, 16, followg results affirm all vecrs markg R(N, m 0 ) \ {P 0i } i N + N, except those idle places, belong set {0, 1}. idle place satfies needed conditions be an implicit place because all put transitions also have anor pot entry. Due th feature markg idle places could be generated from markg or places. For se nets, reasong possible through Boolean calculation where manipulation dialg could be made usg ol ordary bary decion diagrams (OBDDs). One strategy reduce number elements have be treated simultaneously, which produces a well-defed network, however th dcussion beyond scope th article. Lemma 1. Consider N, m 0, N = P 0 P S P R, T, C, a BORPN Petri net. Consider m a dead mark, such m RS (N, m 0 ) τ T set dead transitions belongs m. set τ accomplhes τ > 1. Pro. We tested th result by contradiction. Let us suppose τ = 1 re a transition t τ dead a markg m RS (N, m 0 ). As t a dead mark implies t S A S A r as required by Corollary 2, th t we have mre mpe states. From mpe we can shoot t transitions p S p m [p] = 0 m 0 could be reached. But as τ = 1 sce system well defed (as establhed Defition 1) any mimal T-Semiflow contag t could be fireable from m 0 which a contradiction with t beg dead m. Th contradicts hypos τ = 1 we conclude τ > Liveness orem Liveness property states implementation program (process) eventually reaches a desirable state. Th property its structural characterization BORPN class a very important characterization supports followg oretical results. orem 1 summarizes th result. orem 1. net N livg iff re not an empty siphon D, where D P R 2 exits mpe, p D mre, r D. Pro. We proved th result by contradiction. ) If D P R 2 n r 1, r 2 D where r 1 = p r 2 = p ; p, p N i as r belongs a zone a resource Zi,j r = H r N i, i N + N by Defition 7, sce mpe exts we can verify p p = r 1 p p = r 2 where p, p N j (exts such p p because re an arc from r t p ; r 2 t t p by construction) but as y r [r] = 1 by Defition 1, mre, r D so D empty net N not livg. ) If mpe, p D; mre, r D D P R 2 n exts r, we can fire transitions r all active satfy condition p = r but as p belongs holders r, by Defition 2, re more than a resource siphon D, th declares re anor resource r satfies p H r net N. As y r [r] = 1 n we will reach a mre, mpe D P R 2 we can conclude an empty siphon Modelg a Basic Routg Algorithm Th subsection provides an example modeled through Petri nets transport a message uses a wormhole routg algorithm. Figure 2a shows model transport a message usg a wormhole routg algorithm (physical or virtual objects) composed three nodes or stations two duplex channels nomated C A C B. It can be deduced if nodes or end stations (1 3) each want send objects simultaneously, a deadlock may occur central node. It should be mentioned we assume node, or station 2, unable send or receive messages, but able ward messages nodes/remag stations.

8 Sensors 2016, 16, Figure 2b shows a Petri net with two state maches, where mache SM1 models flow object from node or station 1 node or station 3. mache SM2 models flow reverse direction, which means, objects from node or station 3 node or station 1. From our RAS Sensors perspective, 2016, 16, 1307 are channels system y are represented by places 8 14 we call C A C B have a mark dicatg wher y are available or not. Th Petri net allocation belongsystems BORPN acquire class, which release suitable modelg same aorder. wide As range mentioned resource orem allocation 1, systems liveness acquire se types release networks related same order. extence As mentioned siphons orem are 1, not liveness marked sufficiently se types m. networks network related Figure extence 2b has a siphons D med are not by marked followg sufficientlyplaces m. ={ network,,, Figure,, 2b }, has where a siphon y Dare med not marked by followg with markg places D = = {p 2, p 3, p 5., punder 6, C A, Cth B }, where markg, y areoutput not marked transitions with places markg m siphon = p 1 + p 4. Under th are dead markg, siphons output D unmarked. transitions Obviously, places re siphon a deadlock t 2 t 6 are dead system siphons does not D unmarked. guarantee Obviously, liveness re property, a deadlock as Petri net system shows does not deadlock. guarantee liveness property, as Petri net shows deadlock. Figure Figure Modelg Modelg bary bary ordered ordered petri petri net net (BORPN) (BORPN) class class transport transport a message message usg usg wormhole wormhole routg routg algorithm. algorithm. (a) (a) Representation Representation System; System; (b) Petri (b) net Petri model net model system. system. 4. Explanation Method Controllg Siphons 4. Explanation Method Controllg Siphons diverse extg methods hlg deadlocks through literature can be classified based diverse m extg which methods y work hlg two big strategies: deadlocks those through strategies literature modify can be classified structure based processes m which those y do work not modify two big strategies: structure those strategies model. capacity modify add structure or not new virtual processes those a charactertic do not modify used th structure policy makes model. a clear capacity dtction add which or not strategy new virtual beg used. a selection charactertic used strategy th depends policy makes kd a problem clear dtction be solved; which th strategy because beg some used. applications selection or areas it strategy could be depends impossible kd use both problem strategies. be To solved; expla th a deadlock because prevention some applications policy, we will or areas use a it BORPN could be Petri impossible net model, its use respective both strategies. structural To expla analys, a deadlock prung prevention graph policy, we used will use a obtag BORPN Petri mimal net model, siphons its respective th net structural [17]. analys, prung graph used obtag mimal siphons th net [17] Prung Relation 4.1. Prung Relation A specialized algorithm compute mimal siphons begng from set unique mimal A specialized siphons one algorithm resource Dcompute R P R, as set mimal up Defition siphons 1.4. begng ree, from order set compute unique all mimal mimal siphons siphons, one we consider resource 2 DR P PR, as set up Defition 1.4. ree, order compute R 1 subsets, each one contag a different non-empty subset all. mimal siphons, An easywe wayconsider construct 2 P R each 1 subsets, one se each one 2 P contag a different non-empty subset R 1 cidate siphons by union. mimal An siphons easy way D 1 correspondg construct each one cluded se 2 P R 1 cidate siphons beby constructed. union mimal resultsiphons union D 1 correspondg a siphon, which because cluded each one opers siphon be a constructed. siphon, but, general, result it not mimal. union a Th siphon, non-mimality which because can are each from one process opers places, p, a siphon, become but, non-essential, general, it i.e., not re mimal. are anor Th non-mimality put placescan are transitions from process p places, allow p, become remove nonessential, only possibility, i.e., re are th anor case, put places se new places transitions be resource p places, allow different remove those p. make only p. ppossibility, essential. se th resource case, places se appear new places union be resource operation. places, or different source those non-mimality make p essential. se resource places appear union operation. or source non-mimality are when none places a siphon Dr D 1 become non-essential. In th case, full mimal siphon Dr contaed siphon resultg from union operation, ree it will be not mimal. Moreover, th case, re not a mimal siphon contag tended set because, at least, one m, r, cannot belong because Dr contaed. From previous dcussion

9 Sensors 2016, 16, are when none places a siphon D r D 1 become non-essential. In th case, full mimal siphon D r contaed siphon resultg from union operation, ree it will be not mimal. Moreover, th case, re not a mimal siphon contag tended set because, at least, one m, r, cannot belong because D r contaed. From previous dcussion about non-mimality result union a subset siphons D 1, th section we defe mal ols results allowg sieve 2 P R 1 cidate siphons, order reta only those givg re mimal ones. first ol so called prung relation defed on set D 1. We will say siphon D r D 1 prunes siphon D x D 1, r = x, if only if T rx = D r D x = 0 ( two siphons share some transition) U r = r T rx ( T rx D x P S ) = Ø (re ext common transitions with an put process place belongg D x also resource r puts se transitions). cidate elements be pruned from siphon D x by siphon D r are each one pairs (t, t P S ), where t U r. cidate place be removed from one se pairs t P S. Th place can be removed, because it becomes non-essential D r D x, only if ( t P S ) U r, i.e., all its output transitions belong previously defed set U r. In order represent th prung relation we will defe a graph named Prung Graph (PG). Defition 12. ([11]). Let N be a BORPN net P R set resource places. Prung Graph (PG) N a graph G = (V, E) where, (1) V = P R ; (2) E V x V all r, x P R, r = x, (r, x) E iff U r = r T rx ( T rx D x P S ) = Ø with T rx = D r D x. Be aware BORPN S 4 PR, hence all extg oretical results S 4 PR can be applied th subclass. Given a prung graph G = (V, E), we defe prung subgraph G G duced on set vertices V V, as graph G = (V, E (V V )). Associated a prung graph or subgraph we defe followg three labellg functions. Defition 13. ([11]). Let N be a BORPN net P R set resource places. Prung Graph (PG) net N. 1. Resource labellg function. S: P R S 1, where all r P R, S(r) = D r S Arc labellg function. L: E 2 T x P s, where (r, x) E, L(r, x) = {(t, p) t r T rx ( T rx S(x) P S ) ; T rx = S(r) S(x) ; {p} = t P S }. 3. Prung labellg function. K G : V 2 P s, where all r V, K G (r) P S computed by algorithm [11] Virtual Resources Our methodology ences liveness property by addg places as virtual terms a Petri net model. se kd are implemented over extg physical as routg restrictions avoid fulfillg four classical necessary conditions extence deadlocks. ma idea systematically add virtual until we obta a model fits with orem 1. Prung Graph Petri net model Figure 3 shown Figure 4. Th graph was obtaed usg algorithm presented [11]. obtaed prung relation functions are as follows: L(R, S) = {t 9, p 7 }; L(R, T) = {t 9, p 7 }; L(S, T) = {t 8, p 6 }; L(T, S) = {t 3, p 2 }; L(T, R) = {t 3, p 2 }; L(S, R) = {t 2, p 1 }. Employg algorithm presented [11] over Prung Graph (G) Figure 3 were computed three Strongly Connected Subgraphs called G 1, G 2 G 3. subgraph contas followg vertexes: G 1 = {R, S}, G 2 = {R, T} G 3 = {S, T}. se subgraphs represent three mimal siphons called D 1, D 2, D 3, respectively. siphons conta followgs places: D 1 = {R, S, p 2, p 3, p 4, p 8, p 9, p 10 }, D 2 = {R, T, p 3, p 4, p 5, p 8, p 9, p 10 } D 3 = {S, T, p 3, p 4, p 5, p 7, p 8, p 9 }.

10 Sensors Sensors 2016, 2016, 16, 16, Figure A BORPN Petri Net. Figure Figure Prung Prung Graph Graph Figure Figure Employg Influence Subnet algorithm presented [11] [11] over over Prung Graph Graph (G) (G) Figure 3 were were computed three three Strongly Connected Subgraphs called called G1, G2 G2 G3. G3. subgraph contas Sce siphons belongg net, we are able fd fluence subnet which followg vertexes: G1 G1 = {R, {R, S}, S}, G2 G2 = {R, {R, T} T} G3 G3 = {S, {S, T}. T}. se se subgraphs represent three three mimal characterized by put prung graph becomes not-markg-resource-enabled siphons called called D1, D2, D2, D3, D3, respectively. siphons conta followgs places: (mre). maximal strongly connected subgraphs G G Prung Graph (G) summarize D1 D1 = {R, {R, S, S, p2, p2, p3, p3, p4, p4, p8, p8, p9, p9, p10}, p10}, D2 D2 = {R, {R, T, T, p3, p3, p4, p4, p5, p5, p8, p8, p9, p9, p10} p10} D3 D3 = {S, {S, T, T, p3, p3, p4, p4, p5, p5, p7, p7, p8, p8, p9}. p9}. mimal siphons BORPN Petri net model. relation among siphons implicated a Prung Graph (PG) contas duces a subnet. fluence area a resource r given by Influence Subnet its holders H places as show Defition 2. se places are holdg capacity or availability Sce Sce while siphons representg belongg a process state. net, net, we we By Lemma are are able able 1, fd fd siphons fluence a BORPN subnet will which need have characterized more than by by one put put a siphon. In th prung way, each graph prung becomes relation not-markg-resourceenabled clude ( ors ). )., maximal which strongly are connected characterized subgraphs as Implicated G' G' Resources G (I). Prung We cangraph malize (G) (G) summarize defition Implicated mimal siphons Resources (I) are as BORPN follows: Petri Petri net net model. relation among siphons implicated a Prung Graph Graph (PG) (PG) contas duces a subnet. fluence area area a Defition resource r 14. among two could given given Let N by by be a its its BORPN holders net H places G = (V, as as E) show show Prung Defition Graph se se net places N. Let are are D be holdg mimal siphon capacity N or or availability duce a strongly connected while while subgraph representg G with D a R process P R. For state. state. each By By arc Lemma E r,x 1, 1, G' r, siphons x D R exts a BORPN an implicated will will need need have have (I) such more more than than I one one N r,x resource D R a I = siphon. (( Y In In r th th Y way, way, x ) each each P prung S ) D R relation. among two two could could clude ors, which are are characterized as as Implicated Resources (I). (I). We We can can fluence malize subnet defition obtaed by Implicated terrelation Resources among (I) (I) are are as as follows: implicated E G, (r, x) E possible because label arcs conta mation about places transitions Defition D. Let Let By Defition be be a BORPN 12, net net arcs G = (V, (V, subgraphs E) E) Prung are labeled Graph Graph by a net net set.. common Let Let D be be transitions mimal Гsiphon T among implicated duce a strongly connected plus a set subgraph places. G' G' with with se DR places DR PR. PR. For For are each each private arc arc E'r,x process E'r,x G' G' places r, r, x DR DR exts exts an an implicated (I) (I) such such I N r,x r,x DR DR I = (( (( ) PS) DR. DR.

11 Sensors 2016, 16, belong each one siphons. function L labels each arc (r, x) E r, x D R with set pairs (t, p) representg common transition cidate place p be pruned by mimal siphon D r mimal siphon D x when we construct D r D x, ree, label preserves order prung will be used reference, an unequivocal way, each one arc. With th mation we are able fd set Implicated Process Places (IPP) Influence Subnet (IS) N r, x i N i N N, r, x D R. next defition characterizes, a structural way, Implicated Processes places. Defition 15. Let N be a BORPN net G = (V, E) Prung Graph N. Let D be mimal siphon N duce a strongly connected subgraph G G. D R P R are implicated places (I) D S D process places belongs siphon D. For each arc E r, x G r, x D R exts an implicated process places Π N r, x D S such : Π = p {r, x} H p. Explorg set places Π we can obta IS subnet. Π places are common places several are implicated a subgraph G which share from what was acquired net until y will be released. Sce from th pot view subgraph shows th overlap on zones characterize th class net as was stated previous defitions. ree, each Π places give us mation about transitions were used net. followg defition 16 explas IS subnet. Defition 16. Let N be a BORPN net Π set places belong IS subnet which used an implicated resource I a subgraph G'. Let Г be a set shared places transitions IS N r, x i = <, Γ> with i N N, p iff p (H r H x ) P S Π r, x I p (Zi r Zi x) N r, x i, Γ =. places transitions found IS subnet will be used by control policy applied th kd net Controllg Siphons Usg Virtual Resources To control siphons we use mation comg from prung graph G fluence subnet (IS). Each siphon represented prung graph by a strongly connected subgraph G where implicated state set cidate will become virtual. question : which are more appropriate implicated are necessary change net virtual? To select set most appropriate be changed, we use mation computed fluence subnet N r, x i = <, Г>. In th net we can fd set process places transitions use set implicated a directed path each strongly connected state mache net. n with siphons contaed each subgraph computed prung graph by iterations we remove all siphons breakg prung relations among. se processes are falized when y are not strongly connected a subgraph prung graph, ree y are not siphons. breakg process occurs usg virtual replace real. ma idea make an attempt agast prung relation amongst implicated a siphon. From a structural pot view means: add virtual will be acquired or released by transitions belong implicated process places IS. Due structure an BORPN it breaks some dependencies among with virtual. Algorithm 1 controls all strongly connected subgraphs were received from prung graph BORPN class. By iterations, algorithm utilizes each subgraph remove all siphons from net addg virtual. Of mation fluence subnet, IS, common transitions implicated places siphons are known. followg results concern termation Algorithm 1.

12 Sensors 2016, 16, Algorithm 1. Controllg siphons Input. N, G Local. IS Output. N, G 1. Beg 2. Compute strongly connected subgraphs G G [11] 3. While exts a strongly connected subgraph G G Do 4. Select a strongly connected subgraph G G 5. Compute set I Π from G 6. Compute IS 7. If p N r, x i p ɛ G ; r, x ɛ I I 2 such p = r p = x n 8. Add a virtual r v x v such r v = p r v = p; x v = p x v = p 9. Remove connection r x from transitions, ( r = p r = p) = Ø; ( x = p x = p) = Ø 10. End If 11. Compute G its strongly connected subgraphs G G 12. End While 13. End Lemma 2. Algorithm 1 applied a BORPN class N Prung Graph G, termates controls all strongly connected subgraph G G. Pro. algorithm termates because each iteration loop startg from le 3, we control strongly connected subgraph G G addg virtual removg implicated from fluence subnet. Moreover, all new virtual added fluence subnet was computed le 5 have a new set vertices (virtual ) different those correspondg graph G previously computed. Also, some arcs E G will be replaced G by new arcs connect th virtual. ree, G becomes not strongly connected subgraph. Complexity Algorithm 1 based computed G every time we add a virtual resource. ree, worst-case time complexity th algorithm corresponds one computed [12]. Applyg Defitions BORPN net Figure 3 prung graph Figure 4 we obta sets implicated, implicated places fluence subnet: I 1 = {R, S, T}, Π 1 = {p 2, p 6, p 7 }, 1 = {p 2 }, Г 1 = {t 2, t 3 }, I 2 = {R, S, T}, Π 2 = {p 1, p 2, p 7 }, 2 = {p 7 }, Г 2 = {t 8, t 9 }, I 3 = {R, S}, Π 3 = {p 1, p 7 }, 3 = Ø, Г 3 = Ø, I 4 = {S, T}, Π 4 = {p 2, p 6 }, 4 = Ø, Г 4 = Ø, I 5 = {R, T}, Π 5 = {p 2, p 7 }, 5 = Ø, Г 5 = Ø. We obta virtual places by applyg Algorithm 1 fluence subnet previously computed. new BORPN class with virtual shown Figure 5. In Figure 6 prung graph without a strongly connected graph depicted. Algorithm 1 obtas a modified BORPN class cludg virtual as new virtual channels routg algorithm. Due th new virtual routg algorithms must be changed. ma idea multiplexg until obta new virtual. cycles represented by strongly connected subgraphs which are represented as siphons BORPN class are avoided, hence deadlock also avoided. With se new we obta a prung graph without strongly connected subgraphs, ree new net live. followg lemma states th condition. Lemma 3. new BORPN net N live iff re not a strongly connected subgraph G G. Pro. Algorithm 1 ends when re not a strongly connected subgraph G, each strongly connected subgraph G G correspond a siphon D. By orem 1, if re not a siphon D N n new BORPN net N live.

13 Sensors 2016, 16, Sensors 2016, 16, 1307 Sensors 2016, 16, Figure Figure Modified Modified BORPN BORPN net net with with virtual virtual.. Figure Figure Prung Prung graph graph without without strongly strongly connected connected graph graph Figure Figure Algorithm obtas modified BORPN class cludg virtual as new virtual channels 5. Conclusions Algorithm 1 obtas a modified BORPN class cludg virtual as new virtual channels routg routg algorithm. algorithm. Due Due th th new new virtual virtual routg routg algorithms algorithms must must be be changed. changed. ma Inidea th paper multiplexg we have presented a new control until policy obta new specialized virtual Petri. net subclass called ma idea multiplexg until obta new virtual. cycles cycles Bary represented Ordered by Resources strongly Petri connected Net (BORPN), subgraphs which oriented are represented address deadlock as siphons sues BORPN large represented by strongly connected subgraphs which are represented as siphons BORPN systems class are avoided, allocatehence deadlock a sgle manner also avoided. release With se new we same obta order. class are avoided, hence deadlock also avoided. With se new we obta a Th prung behavior graph without typical strongly wormhole connected routg algorithms subgraphs, used ree sensor new networks. live. BORPN followg class prung graph without strongly connected subgraphs, ree new net live. followg able lemma characterize states th condition. property liveless through structural objects called a siphon. reduced lemma states th condition. structure th new class gives advantages allow analys entire system s behavior, which Lemma Lemma a prohibitive new new taskborpn large net net systems live live dueiff re re complexity not not a strongly strongly a routg connected connected algorithm. subgraph subgraph se.. siphons were computed usg prung graph [11]. With mation given siphons we located sets Pro. Pro. implicated Algorithm Algorithm 1 ends ends when when implicated re re places not not a strongly strongly produce. connected connected Basedsubgraph subgraph on knowledge G, G, each each strongly zone connected connected, subgraph subgraph usg G G G previous correspond correspond sets, we a siphon siphon compute D. D. By By orem fluence1, 1, subnet if if re re which not not a siphon siphon newd virtual places n n arenew located, BORPN BORPN thusnet removg live. live. siphons makg net live. Acknowledgments: Conclusions Conclusions Th work has been supported by Technological University Panama. Author Contributions: Carlos A. Rovet: Ma idea. Elia Esr Cano: Formal methods. Tomás J. Concepción: In In th th paper paper we we have have presented presented a new new control control policy policy specialized specialized Petri Petri net net subclass subclass called called Analyzg Editg. Bary Bary Ordered Ordered Resources Resources Petri Petri Net Net (BORPN), (BORPN), which which oriented oriented address address deadlock deadlock sues sues large large Conflicts Interest: authors declare no conflict terest. systems systems allocate allocate a sgle sgle manner manner release release se se same same order. order. Th Th behavior behavior typical typical wormhole wormhole routg routg algorithms algorithms used used sensor sensor networks. networks. BORPN BORPN class class able able characterize characterize property property liveless liveless through through structural structural objects objects called called a siphon. siphon. reduced reduced structure structure th th new new class class gives gives advantages advantages allow allow analys analys entire entire system s system s behavior, behavior, which which a prohibitive prohibitive task task large large systems systems due due complexity complexity a routg routg algorithm. algorithm. se se siphons siphons were were computed computed usg usg prung prung graph graph [11]. [11]. With With mation mation given given siphons siphons we we located located

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