Tight Polynomial Bounds for Steady-State Performance of Marked GraphsJ. Campos�Dpto. de Ingenier��a El�ectrica e Inform�aticaUniversidad de Zaragoza, Spain. G. ChiolayDipartimento di InformaticaUniversit�a di Torino, Italy.J.M. Colom�Dpto. de Ingenier��a El�ectrica e Inform�aticaUniversidad de Zaragoza, Spain. M. Silva�Dpto. de Ingenier��a El�ectrica e Inform�aticaUniversidad de Zaragoza, Spain.AbstractThe problem of computing both upper and lower boundsfor the steady-state performance of timed and stochasticMarked Graphs is studied. In particular, Linear Pro-gramming problems de�ned on the incidence matrix ofthe underlying Petri nets are used to compute tight (i.e.,reachable) bounds for the throughput of transitions forlive and bounded Marked Graphs with time associatedwith transitions. These bounds depend on the initialmarking and the mean values of the delays but not on theprobability distributions (thus including both the deter-ministic and the stochastic cases). Connections betweenresults and techniques typical of qualitative and quanti-tative analysis of Petri models are stressed.1 IntroductionOne of the main problems in the actual use of timed andstochastic Petri net models for the performance evalu-ation of large systems is the explosion of the compu-tational complexity of the analysis algorithms. Exactperformance results are usually obtained from the nu-merical solution of a Markov chain, whose dimension isgiven by the size of the state space of the model. Simpli-�ed methods of computational complexity polynomial onthe size of Petri net description would provide a strikingbreak-through in this �eld, but results of this type areunlikely to be achievable for the computation of exactperformances for general models.In this paper we study the possibility of obtaining (up-per and lower) bounds on the steady-state performanceof Marked Graphs (MG), a well known subclass of Petrinets that allow only concurrency and synchronization butno choice. In particular we study the throughput of tran-sitions, de�ned as the average number of �rings per unittime. From this quantity, applying Little's formula it is�This work has been supported by project PA86-0028 ofthe Spanish Comisi�on Interministerial de Ciencia y Tecnolog��a(CICYT).yThis work was performed while G. Chiola was visiting the ET-SII Zaragoza, with the �nancial support of the Caja de Ahorros dela Inmaculada de Zaragoza.

possible to derive all average performance estimates ofthe model. Under these restrictions we will show resultsthat can be computed in polynomial time on the size ofthe net model, and that depend only on the mean valuesand not on the higher moments of the probability distri-butions of the random variables that describe the timingof the system. Our approach di�ers from that of Mol-loy [Mol85] in that we analyze a net with a given initialmarking instead of computing a limiting behaviour forincreasing number of tokens. In a sense we have takena complementary approach with respect to that of Bru-ell and Ghanta [BG85], that assumed exponentially dis-tributed random delays and bounded the e�ect of syn-chronization. Our independence of the probability dis-tribution can be viewed as a useful generalization of theperformance results, since higher moments of the delaysare usually unknown for real cases, and di�cult to esti-mate and assess. Moreover we show that both upper andlower bounds, computed by means of proper Linear Pro-gramming problems, are tight, in the sense that for anyMG model it is possible to de�ne families of MG modelswith stochastic timings, such that the steady-state per-formances of the timed PN models are arbitrarily closeto either bound.In a Petri net there is an obvious relation between theconcepts of steady-state behaviour and that of repeatable�ring sequences: sequences of transitions that are repeat-able only a �nite number of times cannot contribute tothe steady-state performance of the model. In the partic-ular case of safe MGs, this relation can be made evidentby looking at the occurrence nets that describe their be-haviours [WT85].Figure 1 depicts an example of a live and safe MG.It is easily seen that only sum and \max" operators areneeded to compute the performance: indeed the actualcycle time in this example is the random variable =�1+max(�2; �3)+ �4 (where �i denotes the �ring delay oftransition ti), therefore the average cycle time is � =E[�1] +E[max(�2; �3)] +E[�4] = �1+E[max(�2; �3)] + �4(where �i denotes the average �ring delay of transi-tion ti). The idea is that of computing fast boundsbased only on the knowledge of the �rst moments of1

p 1

p 2

p 3 p 5

p 4

t1

t 2

t 3

t4

Figure 1: Example of a safe MG.PDF: the sum is independent of the probability distri-bution (for linearity); since for non-negative variablesxi � maxi(xi) � Pi xi, E[maxi(xi)] can be boundedby maxi(E[xi]) � E[maxi(xi)] � PiE[xi]. Thereforefor the net in Figure 1 we can write:�1 +max(�2; �3) + �4 � � � �1 + �2 + �3 + �4The paper is organized as follows. Section 2 containsa discussion of the implications that the introduction ofa timing semantics has on the behaviour of a Petri netmodel. In particular the concepts of enabling bound andweak ergodicity are de�ned. Sections 3 and 4 present theupper and lower bounds, respectively. Section 5 containssome concluding remarks and considerations on possibleextensions of the work. In the Appendix the notationand some classical results for Petri nets are collected.2 Stochastic interpretation ofnetsIn the original de�nition, Petri nets did not include thenotion of time, and tried to model only the logical be-haviour of systems by describing the causal relations ex-isting between otherwise unrelated events. This approachshowed its power in the speci�cation and analysis of con-current systems in a non-interleaved way, i.e. in a primi-tive way independent of the concept of time. Neverthelessthe introduction of timing speci�cation is essential if wewant to use this class of models for an evaluation of theperformance of distributed systems.2.1 Timing and �ring processSince Petri nets are bipartite graphs, historically therehave been two ways of introducing the concept of timein them, namely, associating a time interpretation witheither places [Sif78] or transitions [Ram74]. Moreover,in the case of timed transition models, two di�erent �r-ing rules have been de�ned: single phase (atomic) �ring,and three phase (start-�ring with deletion of the inputtokens, delay, and end-�ring with creation of the outputtokens). In summary, using timed place or timed transi-tion models with three phase �ring we can de�ne a policy

for con ict resolution independent of the time speci�ca-tion but we cannot model preemption; on the other hand,using timed transition models with single phase �ringwe can model preemption but we cannot de�ne con ictresolution policies indepent of the timing speci�cation(unless introducing the concept of immediate transitions[AMBC84] that adds one degree of freedom, but is stillpart of the timing speci�cation).In any case, from the analysis of the di�erent choices,it follows that the only e�ect of these di�erent timing in-terpretations on the performance evaluation of a modelis due to the di�erent implications that the choices haveon the resolution of con icts. Since in the context of thiswork we are considering a class of nets in which there areno con icts, we do not have to fully choose one partic-ular interpretation: we only say that we consider timedtransition MGs.2.2 Single versus multiple server seman-tics: enabling boundAnother possible source of confusion in the de�nition ofthe timed interpretation of a PN model is the concept of\degree of enabling" of a transition (or re-entrance). Inthe case of timing associated with places, it seems quitenatural to de�ne an unavailability time which is inde-pendent of the total number of tokens already presentin the place, an this can be interpreted as an \in�niteserver" policy from the point of view of queueing theory.In the case of time associated with transitions, it is lessobvious a-priori whether a transition enabled k times ina marking should work at conditional throughput 1 ork times that it would work in the case it was enabledonly once. In the case of Stochastic PNs with exponen-tially distributed �ring times associated with transitions,the usual implicit hypothesis is to have \single server"semantics (see, e.g., [FN85a], [Mol82]), and the case of\multiple server" is handled as a case of �ring rate de-pendent on the marking; unfortunately this trick cannotwork in the case of more general probability distributions,and in particular cannot be used in the case of determin-istic timings. This is the reason why people working withdeterministic timed transition PN prefer an in�nite serversemantics (see, e.g., [Zub85], [RP84], [HV87]). Of coursean in�nite server transition can always be constrainedto a \k{server" behaviour by just reducing its enablingbound to k, as we will see later.Therefore the in�nite server semantics appears to bethe most general one, and for this reason it will beadopted in this work. However this generality of the in�-nite server assumption will be payed in terms of complex-ity of the algorithms for the computation of performancebounds. Indeed the performance of a model with in�niteserver semantics does depend on the maximum degree ofenabling of the transitions. For this reason we introducehere a concept of degree of enabling of a transition: theenabling bound E(t). It allows us to generalize the clas-sical concept of enabling of a transition (a generalization2

of the concept of liveness of a transition is presented in[CCS89]).DEFINITION 2.1 Enabling bound. Let hN ;M0i be amarked PN, 8t 2 TE(t) = max ks:t: 9M 2 R(N ;M0) : M � kPRE[t]From the above de�nition it appears clear how it ispossible to obtain a k{server transition from an in�niteserver one: adding one place that is both input and out-put (with multiplicity 1) for that transition and markingit with k tokens.The de�nition above refers to a behavioural prop-erty that depends on the reachability graph of a PN.Since we are looking for computational techniques atthe structural level, we can also introduce the structuralcounterpart of the concept. Structural net theory hasbeen developed from two complementary points of view:graph theory [Bes86] and mathematical programming (ormore speci�cally linear programming and linear algebra)[SC87]. Let us introduce our structural de�nition fromthe mathematical programming point of view; essentiallyin this case the reachability condition is substituted bythe weaker (linear) constraint that markings satisfy thenet state equation: M =M0 + C~�, with M;~� � 0.DEFINITION 2.2 Structural enabling bound: 8t 2 TSE(t) = max ks:t: 9~� � 0;M =M0 + C~� � kPRE[t]Note that, the de�nition of structural enabling boundreduces to the formulation of a Linear Programmingproblem [Mur83] using matrix C (the incidence matrix ofthe PN). For MGs the behavioural concepts always col-lapse into the structural ones. In fact, according to Theo-rem A.1, for any MGM 2 R(M0) i�M =M0+C~�^~� �0. This allows an e�cient computation of enabling boundbased on the Linear Programming problem that charac-terizes the structural enabling bound.In case of non-strongly connected MGs it is possibleto obtain SE(t) = 1 for some transition t; this createsno harm: because of the assumption of an in�nite serversemantics, it only implies that the timing of that transi-tion does not a�ect the steady-state performance of themodel.2.3 Ergodicity, measurability, boundsIn order to be able to speak about steady-state perfor-mance we have to assume that some kind of \averagebehaviour" can be estimated on the long run of the sys-tem we are studying. The usual assumption in this caseis that the system models must be (strongly) ergodic (seede�nitions of ergodicity in [FN85b]). This assumption isvery strong and di�cult to verify in general; moreover,it creates problems when we want to include the deter-ministic case as a special case of a stochastic model (see

[CCS89]). Thus we introduce the concept of weak ergod-icity that allows the estimation of long run performancesalso in the case of deterministic models.DEFINITION 2.3 The marking process of a stochasticmarked net is weakly ergodic (or measurable in long run)i� the following limit exists:M def= lim�!1 1� Z �0 M(s)ds < ~1The �ring process of a stochastic marked net is weaklyergodic (or measurable in long run) i� the following limitexists: ~�� def= lim�!1 ~�(�)� < ~1According to the above de�nitions and the propertieslisted in the Appendix, strong connectivity and bounded-ness have equivalent meaning for live MGs. From Prop-erties A.2, A.3, the following result follows:COROLLARY 2.1 Strongly connected live MGs haveweakly ergodic marking process.3 Upper bounds for stochasticstrongly connected MGsIn this and the next Sections, performance bounds forstrongly connected (and thus structurally bounded) MGsare presented. Strong connectivity of a graph is a wellknown problem of polynomial time complexity.3.1 Upper bound for the steady statethroughputLet us take into account just the �rst moments of theprobability density functions associated with transitions.In the following, let �i be the mean value of the randomvariable associated with the �ring of transition ti, and Dthe diagonal matrix with elements �i, i = 1; :::;m.The limit expected �ring count vector per time unit is~�� = lim�!1 ~�(�)� (1)and the mean time between two consecutive �rings of aselected transition, ti, �i = 1~��i (2)Then the components of Pre �D � ~�� � �i (where ~�� hasbeen normalized for having the ith component equal 1)represent the product of the number of tokens reservedfor �ring the transitions and the mean length of time thatthese tokens reside in each place between two consecutive�rings of ti.LetM be the limit vector of the average number of to-kens in all places (i.e.M = lim�!1 1� R �0 M(s)ds). Then,3

provided that the previous limit exists, M ��i is the vec-tor of products of the mean number of tokens and thelength of one cycle and we have:M � �i � Pre �D � ~�� � �i (3)From this inequality, the minimum cycle time associatedwith transition ti, �mini , can be derived. We take intoaccount that �mini must be such that inequality (3) holdsand for some place pj the equality is reached:�mini = Pre(j) �D � ~�� � �iM(j) (4)Since the vector M is unknown, (4) cannot be solved.Making the product with a P-semi ow Y for any reach-able marking M :Y T �M0 = Y T �M = Y T �M (5)Now, from (3) and (5):Y T �M0 � �i � Y T � Pre �D � ~�� � �i (6)And the minimum cycle time in steady state is:�mini = maxY 2fP-semi owg Y T � Pre �D � ~�� � �iY T �M0 (7)Of course, an upper bound for the throughput of ti is1�mini .Let us formulate the previous lower bound for the cy-cle time in terms of a particular class of optimizationproblems that we introduce now.DEFINITION 3.1 A fractional programming problemis an optimization problem of the form [Mur83]:max f(x) = cT � x+ �dT � x+ �subject to A � x = b; x � 0A 2 Rv�u; b 2 Rvc; d 2 Ru; �; � 2 RA fractional programming problem is said to be homo-geneous if coe�cients � and � are equal to zero.THEOREM 3.1 For any net, the minimum cycle timeassociated with transition ti can be computed by the fol-lowing homogeneous fractional programming problem:�mini = max Y T � Pre �D � ~�� � �iY T �M0s.t. Y T � C = 0 (FPP1)Y � 0; ~1T � Y > 0PROOF: Just notice that the optimum can be alwaysreached with an elementary P-semi ow. If Y � gives us

the optimal solution, Z�, and it is not an elementaryP-semi ow, then Y � = kXi=1(aiYi) (8)with Yi elementary P-semi ows and ai > 0;Pki=1 ai = 1(i.e. Z� =Pki=1(aiY Ti �Pre �D �~�� ��i)). Then ai = 1k 8i,all Y Ti �Pre�D �~�� ��i must be equal (if not, the maximumof them would give a larger value than Z�) and all Yi areoptimal solutions of the problem.Q.E.D.Because for MGs ~�� � �i = ~1, writing � = D � ~1 thefollowing can be directly stated:COROLLARY 3.1 The minimum cycle time forstrongly connected MGs can be obtained by solving thefollowing fractional programming problem:�min = max Y T � Pre � �Y T �M0s.t. Y T � C = 0 (FPP2)Y � 0; ~1T � Y > 0THEOREM 3.2 The minimum cycle time for livestrongly connected MGs can be obtained by solving thefollowing linear programming problem:�min = max Y T � Pre � �s.t. Y T � C = 0; Y T �M0 = 1 (LPP1)Y � 0PROOF: The problem (FPP2) can be rewritten into alinear problem just taking into account that if we write:�min = max Y T � Pre � �qs.t. Y T � C = 0; q = Y T �M0Y � 0; ~1T � Y > 0Then, because Y T �M0 > 0 (guaranteed for live MGs),we can change Yq by Y and obtain the problem (LPP1) inwhich~1T �Y > 0 is redundant (Y T �M0 = 1 =) ~1T �Y > 0)and can be removed.Q.E.D.It is well known that the simplex method for the solu-tion of linear programming problems gives good resultsin practice, even if it has exponential theoretical worstcase complexity. In any case an algorithm of polyno-mial theoretical worst case complexity can be found in[Kar84].Theorem 3.2 shows that the problem of �nding an up-per bound for the steady-state throughput in a stronglyconnected stochastic MG can be solved looking at thecycle times associated with each P-semi ow (cycles forMGs) of the net, considered in isolation. These cycletimes can be computed making the summation of the4

average �ring times of all the transitions involved in theP-semi ow, and dividing by the number of tokens presentin it.The above bound is the same that has been obtainedfor strongly connected deterministic MGs by other au-thors (see for example [Ram74], [RH80], [Mur85]), buthere it is considered in a practical LPP form. For thesenets, the reachability of this bound has been shown([Ram74], [RH80]). Since deterministic timing is justa particular case of stochastic timing, the reachabilityof the bound is assured for our purposes. Even more,the next result shows that the previous bound cannotbe improved only on the base of the knowledge of thecoe�cients of variation for the transition �ring times.THEOREM 3.3 For strongly connected MGs with ar-bitrary values of mean and variance for transition �ringtimes, the bound for the cycle time obtained from (FPP2)cannot be improved.PROOF: Let �2i the arbitrary variance associated withtransition ti. We know from [Ram74] that for determin-istic timing the bound is reached. Let �i be the average�ring time associated with transition ti. Then there ex-ists a sequence of families of m distributions with means�i and variances �2i , i = 1; :::;m, for which the cycle timetends to the one obtained from (FPP2).Consider the family (for varying values of the param-eter 0 � � < 1):X�i;�i(�) = � �i� with probability 1� �i�i(�+ 1���i ) with probability �iwhere �i = �2i (1� �)2�2i (1� �)2 + �2iNow, taking � closer to 1 for the previous family of ran-dom variables, the cycle time tends to the bound given by(FPP2). This is because only \max" and sum operatorsare needed to compute the cycle time and the previousfamily of random variables behaves closer to determinis-tic variables when � tends to 1, i.e.lim�!1E[max(X�i;�i(�); X�j ;�j (�))] = max(�i; �j)and, of course, 8 0 � � < 1E[X�i;�i(�) +X�j ;�j (�)] = �i + �jQ.E.D.A polynomial computation of the minimal cycle timefor deterministic timed strongly connected MGs was pro-posed in [Mag84], solving the following linear program-ming problem:�min = min s.t. �C � z + M0 � Post � � (LPP2) � 0; z � 0

To investigate the relationship between (LPP1) and(LPP2) let us consider the dual problem of (LPP2):�min = max Y T � Post � �s.t. Y T � C � 0; Y T �M0 � 1 (DPP2)Y � 0Since MGs are consistent nets, the restriction Y T �C �0 of (DPP2) can be substituted by Y T � C = 0 (i.e. therestriction of (LPP1)). For all Y such that Y T � C = 0,Y T � Post = Y T � Pre. Now, for live nets, 8Y 2 Nn;Y 6= 0 such that Y T �C = 0 then Y T �M0 � 1. Thus therestriction Y T �M0 � 1 of (DPP2) can be substituted byY T �M0 = 1 for live nets (i.e. the restriction of (LPP1)).Then for live strongly connected MGs, the generalproblem (FPP1) takes the linear form (LPP1) which isequivalent to (LPP2) formulated in [Mag84] for deter-ministic systems. For non-live nets the problem (FPP1)has unbounded optimal solution (see Theorem 3.4). Thiscan be easily understood since non live nets have a nullthroughput (in�nite minimal cycle time).3.2 Interpretation and derived resultsLinear programming problems give an easy way to deriveresults and interpret them. Just looking at the objec-tive function of the problem (FPP2) the following mono-tonicity property is obtained: the optimum value for theminimum cycle time decreases if � decreases or if M0 in-creases.PROPERTY 3.1 Let N be a strongly connected MGand � the mean times vector.i) For a �xed �, if M 00 �M0 (i.e. increasing the numberof initial resources) then the throughput upper boundof hN ;M 00; �i is larger than or equal to the one ofhN ;M0; �i (i.e. �min0 � �min).ii) For a �xed M0, if �0 � � (i.e. for faster resources)then the throughput upper bound of hN ;M 00; �0i islarger than or equal to the one of hN ;M0; �i (i.e.�min0 � �min).The next property is strongly related to the reversibil-ity of live MGs.LEMMA 3.1 For live strongly connected MGs, thebound obtained with the problem (LPP1) does not changefor any reachable marking.PROOF: Let us consider the minimum cycle time for amarkingM =M0+C �~� in terms of a linear programmingproblem: �min = max Y T � Pre � �s.t. Y T � C = 0; Y � 0M =M0 + C � ~�Y T �M = 1M � 0; ~� � 05

Since Y T �M = Y T �M0 this problem is equivalent to:�min = max Y T � Pre � �s.t. Y T � C = 0; Y � 0M =M0 + C � ~�Y T �M0 = 1M � 0; ~� � 0Since the restrictionsM =M0+C �~�; M � 0 and ~� � 0do not a�ect the solution, they can be removed withoutchanging the optimum of this problem with respect tothe one of (LPP1).Q.E.D.The next is a result on the complexity of the veri�-cation of liveness for MGs. It has been recently pointedout in [ES89] by using quite di�erent arguments and tech-niques. Here liveness is characterized by the �niteness ofthe cycle time.THEOREM 3.4 Liveness of a strongly connected MGcan be decided in polynomial time.PROOF: We know that for strongly connected MGs, live-ness and deadlock-freeness coincide. Then for decidingliveness of a strongly connected MG it is enough to studythe �niteness of the optimal value of (LPP1).For strongly connected MGs, the optimal value of(LPP1) is a lower bound for the cycle time. If this opti-mal value is in�nite the cycle time is unbounded so thenet is non live. If the optimal value of (LPP1) is �nite,since it is reachable for some (deterministic [Ram74] aswell as stochastic) timing (cfr. Theorem 3.3), the netmust be live.Q.E.D.COROLLARY 3.2 The problem (LPP1) has un-bounded solution i� 9Y � 0; Y 6= 0 such that Y T �M0 = 0and Y T � C = 0This result has the following topological interpretation:the problem (LPP1) has unbounded solution i� there ex-ists an unmarked circuit in the strongly connected MG.4 Lower bounds for strongly con-nected MGs4.1 Basic result for 1{live MGA trivial lower bound in steady-state performance fora live PN with a unique repetitive �ring count vector[CCS89] is of course given by the sum of the �ring timesof all the transitions weighted by the �ring count vectoritself. Since the net is live all transition must be �reable,and the sum of all �ring times multiplied by the numberof occurrences of each transition in the (unique) averagecycle of the model corresponds to any complete sequen-tialization of all the activities represented in the model.

This lower bound is always reached in an MG consist-ing of a single loop of transitions and containing a singletoken in one of the places, independently of the highermoments of the PDFs (this observation can be triviallycon�rmed by the computation of the upper bound, whichin this case gives the same value).To improve this trivial lower bound let us �rst considerthe case of 1{enabled MGs (i.e. strongly connected MGin which E(t) = 1 for all transitions t). Of course liveand safe MGs are guaranteed to be 1{enabled, but theresult that we are going to present apply to more generalcases. If we specify only the mean values of the transition�ring times and not the higher moments, we may always�nd stochastic models whose steady-state throughput isarbitrarily close to the trivial lower bound, independentlyof the topology of the MG (only provided that it is 1{enabled). Let us give a formal proof of this (somewhatcounter-intuitive) result.LEMMA 4.1 8� � 0 there exists a family of randomvariables xi�(�) with expected value E[xi�(�)] = � 80 <� � 1, 8i � 0 and with coe�cient of variation rangingfrom 0 to 1 for decreasing values of 0 < � � 1, and �xedvalues of i > 0. This family is de�ned as:xi�(�) = � 0 with probability 1� �i��i with probability �iPROOF: E[xi�(�)] = � is straightforward to compute.E[(xi�(�))2] = �2�i implies that the coe�cient of variationis 0 for � = 1, and that it tends to 1 as � ! 0 providedthat i > 0.THEOREM 4.1 For any live and safe MG with a spec-i�cation of the mean �ring times �j for each tj 2 T itis possible to assign PDFs to the transition �ring timessuch that the average cycle time is � = Pj �j � O(�)80 < � � 1, independently of the topology of the net (andthus independently of the potential maximum degree ofparallelism intrinsic in the MG). (We use here the no-tation O(f(x)) to indicate any function g(x) such thatlimx!0 g(x)f(x) � k 2 IR.)PROOF: by construction, we will show that the asso-ciation of the family of random variables xj�1�j (�) witheach transition tj 2 T yields exactly the cycle time� claimed by the theorem. To give the proof we willconsider a sequence of models ordered by the index oftransitions, in which the q-th model of the sequence hastransitions t1; t2; : : : tq timed with the random variablesxj�1�j (�), and all other transitions immediate (�ring inzero time); the jT j-th model in the sequence representsan example of reachability of the lower bound, indepen-dent of the net topology. Now we will prove by inductionthat the q-th model in the sequence has a cycle time�q =Pqj=1 �j �O(�)Base: q = 1 : trivial since the repetitive cycle thatconstitute the steady{state behaviour of the MG contains6

only one (single-server) deterministic transition with av-erage �ring time �1 = �1.Induction step: q > 1 : taking the limit � ! 0, thenewly timed transition tq will �re most of the times withtime zero, thus normally not disturbing the behaviourof the other timed transition, and not contributing tothe computation of the cycle time, that will be just�q�1 = Pq�1j=1 �j � O(�) (as in the case of model q � 1)with probability 1� �q�1. On the other hand, the newlytimed transition has a (very small) probability �q�1 ofdelaying its �ring of a time �q�q�1 , which is at least orderof 1� bigger than any other �ring time in the cycle, sothat in this case all other transitions will wait for the�ring of tq after having completed their possible current�rings in a time which is O(�) lower than the �ring timeof tq itself (i.e., �q�q�1 = �q�1O(�) ). Therefore we obtain that�q = (1��q�1)�q�1+�q�1( �q�q�1�O(�)) =Pqj=1 �j�O(�).Q.E.D.Until now we have shown that the trivial sum of theaverage �ring times of all transitions in the net consti-tutes a tight (reachable) lower bound for the performanceof a live and safe MG (or more generally of a 1{enabledstrongly connected MG, but otherwise independently ofthe topology) in which only the mean values and nei-ther the PDFs nor the higher moments are speci�ed forthe transition �ring times. Let us now extend this resultto the more general case of k{enabled strongly connectedMGs, and see whether we can derive some reachable lowerbound.An intuitive idea could be to try to derive a lowerbound for MG containing transitions with enablingbound k � 1 (remember that for MGs E(t) = SE(t))by taking the algorithm used for the computation of theupper bound in the case of non-safe MG, and substitutein it the \max" operator with the sum of the �ring timesof all transitions involved. After some manipulation toavoid counting more than once the contribution of thesame transition, one can arrive at the formulation of thefollowing value for the maximum cycle time.THEOREM 4.2 For any live and bounded MG with aspeci�cation of the mean �ring times �j for each tj 2 Tit is not possible to assign PDFs to the transition �ringtimes such that the average cycle time is greater than�max =Xj �jE(tj)independently of the topology of the net (and thus inde-pendently of the potential maximum degree of parallelismintrinsic in the MG).PROOF: we give in the following a proof of this resultsby constructing some auxiliary MG models. These auxil-iary models are obtained by adding structural constraintson the �ring of the transitions with respect to the orig-inal one, in such a way that the performance may onlyremain the same or decrease, and than verify that theirmaximum cycle time is � � �max.

4.1.1 Construction to demonstrate the TheoremLEMMA 4.2 Any strongly connected MG can be con-straint to contain a main cycle including all transitions,without changing their enabling bound. This main cyclecontains a number of tokens equal to the maximum of theenabling bounds among all transitions. In addition thereare other minor cycles that preserve the enabling boundsfor transitions with bound lower than the maximum. Theidea behind this constrain is to introduce a structural se-quentialization between all transitions, thus potentiallyreducing the degree of parallelism between the activitiesmodelled by the transitions. In other words from the par-tial order given by the initial MG structure we try to de-rive a total order without changing the enabling bound.PROOF: To construct an MG of the desired form we canapply the following iterative procedure that interleavestwo non-disjoint cycles into a single one. Since the MGis strongly connected each node belongs to at least onecycle; moreover, since the original MG is �nite and eachcycle cannot contain the same node more than once, thiscycle interleaving procedure must terminate after a �nitenumber of iterations. To reduce the number of cycles, im-plicit places created after each iteration can be removed.The iteration step is the following:1. take two arbitrary non-disjoint cycles (unless theMG already contains a main cycle including allnodes, there always exists such a pair of cycles be-cause the MG is strongly connected);2. combine them in a single cycle in such a way thatthe partial order among transitions given by the twooriginal cycles is substituted by a compatible butotherwise arbitrary total order. This combinationcan be obtained by adding new places that are con-nected as input for a transition of one cycle and out-put for a transition of the other cycle that we decidemust follow in the sequence determined by the newcycle we are creating;3. mark the new places added in such a way that thenew cycle contains the same number of tokens as themaximum of the number of tokens in the two originalcycles.Consider as an example of the application of this iterativestep the net depicted in Figure 2a. This net containsonly two cycles, namely t1,t2,t4, and t1,t3,t4; we canthen add either the cycle t1,t2,t3,t4 or t1,t3,t2,t4; Figure2b depicts the resulting net in case we choose to addthe second cycle. In this case only place p6 (from t3 tot2) needs to be added to obtain the longer cycle, andit should be marked with one token, so that the newcycle comprising places p1,p3,p6,p4 contains two tokens,as the original cycle p1,p2,p4 (while the other originalcycle p1,p3,p5 contained only one).The above procedure is applied iteratively until alltransitions are constrained into a single main cycle. In7

p 1

p2 p3

p5p4

t 1

t 2t 3

t 4

p1

p 2 p 3

p 5p4

t 1

t 2 t 3

t 4

p 6

p1

p3

p 5p4

t 1

t2 t 3

t 4

p 6

a) Original net with t and t concurrent.

23

b) Transformed net: t and t sequentialized and p made implicit.

2

23

c) Elimination of implicit places. Main loop: p , p , p , p . Minor cycle: p , p , p .

11

33

45

6Figure 2: Example of structural sequentialization.our example, we need not iterate the procedure since wealready have obtained a cycle containing all transitions ofthe MG. At this point we can identify and eliminate theimplicit places that have been created during the cyclesinterleaving procedure. We obtain then an MG composedby one main cycle containing NM = maxt2T E(t) tokensthat connects all transitions, and a certain number of mi-nor cycles containing less tokens than NM that maintainthe enabling bound of the other transitions. In our ex-ample we can easily see that place p2 become implicit inFigure 2b, so that it can be eliminated, �nally leadingourselves to the MG depicted in Figure 2c.Q.E.D.It is evident that the MG transformed by applying theabove Lemma has a cycle time which is greater than orequal to the cycle time of the original one, since someadditional constraints have been added to the enablingof transitions: thus the cycle time of the transformed MGis a lower bound for the performance of the original one.Now if NM = 1 in the above lemma, we re-�nd the lowerbound of Theorem 4.1. In the case of NM > 1 we canshow that the cycle time of the transformed net cannotexceed �max as follows.PROOF for Theorem 4.2: without loss of generality,assume that transitions in the net resulting from the ap-plication of Lemma 4.2 are partitioned in two classesS2 and S1, with enabling bounds K2 = NM > 1 andK1 < NM , respectively (the proof is easily extended tothe case of more than two classes). Construct a newmodel containing only K1 tokens in the main cycle; atthis point all transitions behave as K1{servers, so thatthe cycle time is given by the sum of the �ring timesof all transitions, divided by the total number of cus-tomers in the main loop K1; moreover the delay timefor the transitions belonging to class S1 is simply given

by D1 = Ptj2S1 �j . Now if we increase the number oftokens in the main loop from K1 to K2 the delay timeof S1 cannot increase, so that the contribution of S1 tothe cycle time cannot exceed D1 for each of the �rst K1tokens. Under the hypothesis that the throughput ofthe system is given by the inverse of �max (i.e., assum-ing X = 1�max ), the average number of tokens of themain loop computed using Little's formula cannot ex-ceed N1 = XD1, therefore the average number of tokensavailable to �re transitions in S2 cannot be lower thanN2 = K2 �N1 = K2 K2�K1K1 Ptj2S1 �j +Ptj2S2 �jPtj2S2 �j + K2K1 Ptj2S1 �jOn the other hand, we need onlyN2 = XD2 = K2 D2Ptj2S2 �j + K2K1 Ptj2S1 �jtokens to sustain throughput X in subnet S2, so that weare assuming a delay in S2D2 � K2 �K1K1 Xtj2S1 �j + Xtj2S2 �jNow we claim that this is the actual maximum delaybecause the �rst K1 tokens can proceed at the maxi-mum speed in the whole net, thus experiencing only de-layPtj2S2 �j in subnet S2, while the remaining K2�K1tokens can also queue up for travelling through S1, thusexperiencing an additional delay of 1K1 Ptj2S1 �j each.Q.E.D.4.2 Reachability of the lower boundThe lower bound in performance given by the computa-tion of �max as de�ned in Theorem 4.2 can be shown to8

be reachable for any MG topology and for some assigne-ment of PDF to the �ring delay of transitions, exploitingthe reachability of the trivial bound shown in Theorem4.1 for 1{enabled MGs.THEOREM 4.3 For any strongly connected MG witha speci�cation of the mean �ring times �j for each tj 2 T ,and for all 0 < � � 1, it is possible to assign PDFs to thetransition �ring times such that the average cycle timeis: �max =Xj �jE(tj) �O(�)independently of the topology of the net (and thus inde-pendently of the potential maximum degree of parallelismintrinsic in the MG).PROOF: by construction, in a very similar way than inthe case of Theorem 4.1. The only technical di�erence isthat now, without any loss of generality, we assume �rstof all to enumerate transitions in non-increasing orderof enabling bound, i.e., rename the transitions in sucha way that 8ti; tj 2 T , i > j =) E(ti) � E(tj).Then, as in the case of Theorem 4.1, we will show thatthe association of the family of random variables xj�1�j (�)with each transition tj 2 T yields exactly the cycle time�max claimed by the theorem. To give the proof we willconsider a sequence of models ordered by the index oftransitions, in which the q{th model of the sequence hastransitions t1; t2; : : : tq timed with the random variablesxj�1�j (�), and all other transitions immediate (�ring inzero time); the jT j-th model in the sequence representsthe resulting model that is expected to provide the ex-ample of reachability of the lower bound. Now we willprove by induction that the q-th model in the sequencehas a cycle time �q = qXj=1 �jE(tj) �O(�)Base: q = 1 : trivial since the repetitive cycle that consti-tute the steady{state behaviour of the MG contains onlyone (E(t1){server) deterministic transition with average�ring time �1 = �1=E(t1).Induction step: q > 1 : taking the limit � ! 0, eachserver of the newly timed transition tq will �re most ofthe times with time zero, thus normally not disturbingthe behaviour of the other timed transition, and not con-tributing to the computation of the cycle time, that willbe just �q =Pq�1j=1 �jE(tj) � O(�) (as in the case of modelq � 1) with probability 1 � �q�1. On the other hand,each of the servers of the newly timed transition has a(very small) probability �q�1 of delaying its �ring of atime �q�q�1 , which is at least order of 1� bigger than anyother �ring time in the cycle. Now if E(tq) = 1, thenthe proof is completed, since also 8j > q E(tj) = 1 byhypothesis, and we reduce to the induction step of theproof of Theorem 4.1. Instead if E(tq) > 1 then we canconsider E(tq) consecutive �rings of tq , and compute the

average �ring time as the total time to �re E(tq) timesthe transition, divided by E(tq). Now if we consider mconsecutive �rings of instances of transition tq we obtainan average delay:0Xj=m�1 (1� �q�1)j�(q�1)(m�j) (m� j)�q�(q�1) = �q(1 +O(�))Therefore the average cycle time of the q{th model willbe�q = (1�O(�q�1))�q�1+ �qE(tq) (1+O(�)) = qXj=1 �jE(tj)�O(�):Q.E.D.4.3 A polynomial algorithm to computethe lower boundFirst of all we recall that in the case of live MGs theenabling bound equals the structural enabling bound foreach transition; thus we present a characterization of theproblem of the determination of the structural enablingbound in terms of a Linear Programming problem, whichis known to be solvable in polynomial time.For any transition t 2 T , the computation of the struc-tural enabling bound SE(t) can be formulated in termsof the following LPP:SE(t) = max �s:t: M =M0 + C~�M � �PRE[t]M � 0; ~� � 0by de�nition. Then we can observe that the vector M isredundant in the system of linear inequalities, so that wecan remove it, obtaining:SE(t) = max �s:t: M0 + C~� � �PRE[t]M0 + C~� � 0; � � 0Alternatively, we can switch to the dual LPP:SE(t) = min Y TM0s:t: Y TC � 0Y TPRE[t] = 1Y T � 0Now we can recall that strongly connected MGs areconsistent nets with a single minimal T-semi ow whichis the vector~1, so that the constraint ~� � 0 can be relaxedin the primal problem. The e�ect on the dual problem ofthis relaxation is the transformation of the �rst constraintinto Y TC = 0. In other words, the dual problem for thecomputation of SE(t) can be rewritten as follows:SE(t) = min Y TM0s:t: Y TC = 0Y TPRE[t] = 1Y T � 09

This LPP is less complex to solve with the simplex algo-rithm than the original dual problem because it involvesthe introduction of fewer slack variables.Because of the minimization requirement, the optimumof the objective function is always achieved with elemen-tary P-semi ows Y . In case of MGs, these elementaryP-semi ows can only be elementary cycles, so that wecan give the following interpretation of the dual LPP innet terms: the enabling bound for a transition t of astrongly connected MG is given by the minimum num-ber of tokens contained in any cycle of places containingtransition t. In a non-strongly connected MG there canbe no such cycle, so that this number can be in�nite.As �nal remarks we can state the following:a) Liveness for a strongly connected MG can be abyproduct of a more general (polynomial complex-ity) computation:hN ;M0i is a live MG () 8t 2 T SE(t) > 0:b) If the MG is known to be live for M0, and 9t 2 Tsuch that SE(t) = 1, then 8t0 2 T belonging to thesame cycle denoted by Y in the corresponding LPP,SE(t0) = 1.5 ConclusionsIn this paper we have addressed the problem of comput-ing upper and lower bounds for the throughput of sys-tems modelled by means of strongly connected stochasticMGs. Both bounds can be computed by means of properLinear Programming problems on the incidence matrixof the net, whose solution is known to be of worst casetheoretical polynomial complexity. As a by-product, wecan characterize the liveness of a MG in terms of non-nullthroughput for all its transitions, so that we obtained analternative proof of a recently obtained result on the poly-nomiality of the liveness problem for MGs [ES89]. Thisshows an example of possible interleaving between quali-tative and quantitative analysis for timed and stochasticPetri nets.The upper bound on throughput for MGs was �rst pro-posed by Ramchandani in 1974, and then re-discoveredand/or re-interpreted by many others, in the frameworkof the study of the exact performance of timed Petri netswith deterministic timing. The contributions given bythis paper in this sense are three: an alternative refor-mulation in terms of Linear Programming problems; theproof that this case represents an upper bound in perfor-mance independently of the probability distribution alsoin the framework of stochastic Petri nets; the proof thatthe upper bound is reachable not only by deterministicbut also by stochastic models, with arbitrary values ofcoe�cient of variations.The lower bounds on throughput presented in this pa-per as well as the concept of enabling bound for tran-sitions are new results. The lower bound in through-put consisting in the inverse of the sum of the �ring

times of all transitions divided by their respective en-abling bounds reduces to the trivial sequentialization ofall transitions in the case of safe nets, but has been shownto be reachable with some probability distribution whenthe coe�cient of variation increases. The concept of en-abling bound generalizes the usual one of enabling for atransition, and provides another example of possible in-terleaving between qualitative and quantitative analysisfor timed and stochastic Petri nets.This work can be extended in two directions: by con-sidering classes of Petri nets behaviourally \similar" toMGs, as done in the companion paper [CCS89], or byremoving some behavioural restriction. Work is still inprogress for the case of unbounded MGs, for which theextension of the results presented in this paper is nottrivial. In particular, some \trivial" extensions suggestedby many authors that studied the case of deterministicbounded MGs (like, e.g., [Mag84]) appear not to work inthe case of unbounded MGs.References[AMBC84] M. Ajmone Marsan, G. Balbo, and G. Conte.A class of generalized stochastic Petri nets forthe performance analysis of multiprocessorsystems. ACM Transactions on ComputerSystems, 2(1), May 1984.[Bes86] E. Best. Structure theory of Petri nets: thefree-choice hiatus. In Advanced Course onPetri Nets, L.N.C.S. Springer-Verlag, Berlin,September 1986.[BG85] S.C. Bruell and S. Ghanta. Throughputbounds for generalized stochastic petri netmodels. In Proc. Int. Workshop on TimedPetri Nets, IEEE-CS Press, Torino, Italy,July 1985.[Bra83] G. W. Brams. R�eseaux de Petri: Th�eorie etPratique. T.1. th�eorie et analyse. Masson,Paris, 1983.[BV84] E. Best and K. Voss. Free-choice systemshave home states. Acta Informatica, (21):89{100, 1984.[CCS89] J. Campos, G. Chiola, and M. Silva. Proper-ties and steady-state performance bounds forPetri nets with unique repetitive �ring countvector. In Proc. of 3rd Intern. WorkshopPNPM'89, IEEE-CS Press, Kyoto, Japan,December 1989.[CHEP71] F. Commoner, A. W. Holt, S. Even, and A.Pnueli. Marked directed graphs. Journal ofComputer and System Science, (5):511{523,1971.10

[ES89] J. Esparza and M. Silva. Liveness of boundedfree-choice nets is decidable in polynomialtime. Report, Dpto. Ingenier��a El�ectrica e In-form�atica, Universidad de Zaragoza, Febru-ary 1989.[FN85a] G. Florin and S. Natkin. Les r�eseaux de Petristochastiques. Technique et Science Informa-tiques, 4(1):143{160, 1985.[FN85b] G. Florin and S. Natkin. Les r�eseaux de Petristochastiques. Thesis de Doctorat d'Etat,Universit�e P. et M. Curie, Paris, 1985.[HV87] M. A. Holliday and M. Vernon. A general-ized timed Petri net model for performanceanalysis. IEEE Transactions on Software En-gineering, 13(12):1297{1310, December 1987.[Kar84] N. Karmarkar. A new polynomial time algo-rithm for linear programming. Combinator-ica, (4):373{395, 1984.[Mag84] J. Magott. Performance evaluation of concur-rent systems using Petri nets. InformationProcessing Letters, (18):7{13, 1984.[Mol82] M. K. Molloy. Performance analysis usingstochastic Petri nets. IEEE Transaction onComputers, 31(9):913{917, September 1982.[Mol85] M.K. Molloy. Fast bounds for stochasticpetri nets. In Proc. Int. Workshop on TimedPetri Nets, IEEE-CS Press, Torino, Italy,July 1985.[Mur77] T. Murata. Circuit theoretic analysis andsynthesis of marked graphs. IEEE Transac-tions on Circuits and Systems, 24(7), July1977.[Mur83] K. G. Murty. Linear Programming. JohnWiley & Sons, 1983.[Mur85] T. Murata. Use of resource-time productconcept to derive a performance measure oftimed Petri nets. In Proc. 1985 MidwestSymposium Circuits and Systems, Louisville,U.S.A. August 1985.[Ram74] C. Ramchandani. Analysis of AsynchronousConcurrent Systems by Petri Nets. PhD the-sis, M.I.T. Cambridge, Mass. February 1974.[RH80] C. V. Ramamoorthy and G. S. Ho. Per-formance evaluation of asynchronous concur-rent systems using Petri nets. IEEE Transac-tions on Software Engineering, 6(5):440{449,September 1980.[RP84] R. R. Razouk and C. V. Phelps. Performanceanalysis using timed Petri nets. In Proc. In-ternational Conference on Parallel Process-ing, pages 126{129, August 1984.

[SC87] M. Silva and J. M. Colom. On the compu-tation of structural synchronic invariants inP/T nets. In G. Rozenberg, editor, Advancesin Petri Nets 1988, pages 386{417, L.N.C.S.Springer-Verlag, Berlin, 1987.[Sif78] J. Sifakis. Use of Petri nets for performanceevaluation. Acta Cybernetica, 4(2):185{202,1978.[WT85] R. P. Wiley and R. R. Tenney. Performanceevaluation of decision-free stochastic timedPetri nets. In Proc. 24th Conf. Decisionand Control, Ft. Lauderdale, FL. December1985.[Zub85] W. M. Zuberek. Performance evaluation us-ing timed Petri nets. In Proc. InternationalWorkshop on Timed Petri Nets, pages 272{278, IEEE-CS Press, Torino,Italy, July 1985.A Petri net de�nitions and nota-tionA.1 Net structureA place/transition net N is a 4-tuple N =hP; T; Pre; Posti, where- P is the set of places (jP j =n),- T is the set of transitions (P \ T = ;) (jT j =m),- Pre (Post) is the pre- (post-) incidence function repre-senting the input (output) arcs Pre : P � T ! IN(Post : P � T ! IN)The pre- and post-set of a transition t 2 T are de-�ned respectively as �t = fpjPre(p; t) > 0g and t� =fpjPost(p; t) > 0g. The pre- and post-set of a place p 2 Pare de�ned respectively as �p = ftjPost(p; t) > 0g andp� = ftjPre(p; t) > 0g.The incidence matrix of the net C = [cij ] (1 � i �n,1 � j �m) is de�ned by cij = Post(pi; tj) � Pre(pi; tj).Similarly the pre- and post-incidence matrices are de-�ned as PRE = [aij ] and POST = [bij ], where aij =Pre(pi; tj) and bij = Post(pi; tj).A.2 Token gameA function M : P ! IN is called a marking. A markingM can be represented in vector form, with the ith com-ponent associated with the ith element of P . A markedP/T net hN ;M0i is a P/T net N with an initial markingM0.A transition t 2 T is enabled in marking M i� 8p 2 PM(p) � Pre(p; t). A transition tj enabled in M can �reyielding a new marking M 0 de�ned by M 0(p) = M(p)�Pre(p; tj)+Post(p; tj) (or in vector form,M 0[i] =M [i]+C[i; j]). The notation M [tiM 0 denotes that transition t11

is enabled in M and that M 0 is reached from M by �ringt in it.A �nite sequence of transitions � = t1t2 : : : tn is a �nite�ring sequence of hN ;M0i i� there exist a sequence ofmarkings such that M0[t1iM1[t2iM2 : : : [tniMn. In thiscase, marking Mn is said to be reachable from M0 by�ring �, and this is denoted by M0[�iMn. Similarly, anin�nite �ring sequence � = t1t2 : : : is de�ned for hN ;M0ii� there exist an in�nite sequence of markings such that8i 2 IN Mi�1[tiiMi.The notation M [�i denotes a �rable sequence � frommarking M . The function ~� : T ! IN is the �ringcount vector of the �rable sequence �, i.e. ~�[t] representsthe number of occurrences of t 2 T in �. If M0[�iM ,then we can write in vector form M = M0 + C~�, whichis referred to as the linear state equation of the net. Amarking M 0 is said to be potentially reachable i� 9~� � 0such that M 0 =M0 + C~�.A.3 Basic propertiesThe reachability set R(N ;M0) is the set of all mark-ings reachable from the initial marking. Denoting byPR(N ;M0) the set of all potentially reachable markingswe have the following relation: R(N ;M0) � PR(N ;M0).L(N ;M0) is the set of all �ring sequences and theirsu�xes in hN ;M0i: L(N ;M0) = f�jM [�i and M 2R(N ;M0)g.A place p 2 P is said to be k{bounded i� 8M 2R(N ;M0) M(p) � k. A marked net hN ;M0i is saidto be (marking) K{bounded i� each of its places is K{bounded. A net N is structurally bounded i� 8M0 themarked nets hN ;M0i are K{bounded for some K 2 IN .A transition t 2 T is live in hN ;M0i i� 8M 2R(N ;M0) 9M 0 2 R(N ;M) such that M 0 enables t. Themarked net hN ;M0i is live i� all its transitions are live(i.e. liveness of the net guarantees the possibility of an in-�nite activity of all transitions). A net N is structurallylive i� 9M0 such that the marked net hN ;M0i is live. Themarked net hN ;M0i is deadlock-free i� 8M 2 R(N ;M0)9t 2 T such that M enables t.A repetitive component is a function (vector) X : T !IN such that X 6= 0 and C �X � 0. A consistent repetitivecomponent (or T-semi ow ) is a repetitive component Xsuch that C � X = 0. A conservative component (or P-semi ow ) is a function (vector) Y : P ! IN such thatY 6= 0 and Y T � C = 0. The support of (T- and P-)semi ows are de�ned by jjX jj = ft 2 T jX(t) > 0g andjjY jj = fp 2 P jY (p) > 0g. A (T- or P-) semi ow I isminimal support i� there exist no other semi ow I 0 suchthat jjI 0jj � jjI jj. A (T- or P-) semi ow is canonical i�the greatest common divisor of its components is 1. A(T- or P-) semi ow is elementary i� it is canonical andminimal support.A net N is repetitive if there exist a repetitive com-ponent X � ~1. A net N is consistent if there exist aT-semi ow X � ~1. A net N is conservative if there exista P-semi ow X � ~1.

A.4 Additional propertiesAn implicit place is one which never restricts the �ringof its output transitions. Let N be any net and Np bethe net resulting from adding a place p to N . If M0 isan initial marking of N , M0 [m0(p) denotes the initialmarking of Np. The place p is implicit in the marked nethNp;M0 [m0(p)i i� L(Np;M0 [m0(p)) = L(N ;M0).A livelock is a maximal subset of strongly connectedstates that have no connections outside the subset itself.DEFINITION A.1 [BV84], [Bra83] A marked Petrinet has the directedness (or con uence) property i� forall pair of reachable markings, M0[�1iM1 andM0[�2iM2,there exist two sequences �01 and �02 such that M1[�01iMand M2[�02iM .DEFINITION A.2 M 2 R(N ;M0) is a home state i�8Mi 2 R(N ;M0) :M 2 R(N ;Mi).DEFINITION A.3 A marked net is reversible i� itsinitial marking is a home state.A.5 Marked GraphsDEFINITION A.4 [CHEP71] MGs are ordinary Petrinets (pre and post incidence functions taking values inf0; 1g) such that j�pj = jp�j = 1;8p 2 P .PROPERTY A.1 MGs are structurally persistentnets.THEOREM A.1 [Mur77] Let hN ;M0i be a live (pos-sibly unbounded) MG. The two following statements areequivalent:i) M 2 R(N ;M0), i.e. M is reachable from M0.ii) Bf �M = Bf �M0, with Bf the fundamental circuitmatrix of the graph, and M � 0.According to the above theorem M 2 R(N ;M0) ,M0 2 R(N ;M). In other words:PROPERTY A.2 Live MGs are reversible.PROPERTY A.3 Let N be a MG.i) N is structurally bounded (i.e. hN ;M0i is bounded8M0) i� it is strongly connected.ii) Let hN ;M0i be live. Then hN ;M0i is bounded i� Nis structurally bounded.12