Proceedings of the 34th Conference on Decision & Control New Orleans, LA - December 1995

.TAI511:40

Asymptotic Throughput of Continuous Timed Petri Nets GUY COHENtt

STEPHANE GAUBERT~*

JEAN-PIERRE

QUADRAT$-

t Centre Automatique et Syst&mes, Ecole des Mines de Paris, 35 rue Saint-Honor6, 77305 Fontainebleau Cedex, France. e-mail: cohen@cas.

ensmp. fr

$ INRIA. Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France. * e-mail: {stephane .Gaubert, Jean-Pierre . Quadrat }@inria. fr

of the horizon,and for the correspondingPetriNets(that wecdl undiscounted),thereexistsan asymptotic (hroughput (mean number of tiringsof a giventransitionper time unit). UndiscountedPetri Nets are chmacterizedby the followingsimplestructurtiproperty:therewe as manyinput as output,arcsat each place. Then, the routingpolicy is uniquelydefined(onehasto routethetokensequallytowmdsdownstreamarcs). WeshowthatundiscountedPetri Netsadmit P-invariants(linearcombinationof markings invariantby firing of transitions). They atso admit Tinvariants(sequencesof fings preservingthe marking) whichmebestinterpretedas an input-outputhomogeneity property:if we distinguishbetweeninputtransitions(representinge.g.the availabilityof raw materitis)andoutput transitions(representingfinishedparts)and if we addone unitof eachinputmatend. thenoneobtainsonemoreunit of each finishedpart. Finflly, we introducethe class of Peti Netswithpotential, obtined fromundiscountedPetri Netsviarescdings (changesof units).Thisclassis suited to the modelingof productionsystemsin whichpartsare producedaccordingto differentratios. A more completepresentationin a systemtheoretical spiritwillbe found in [8], to whichthe readeris referred for omittedproofs. Here,we focuson the maincomputationalconsequenceofthisapproach,thatis.theasymptotic throughputformula.

Abstract We setup a connectionbetweenContinuousTimedPeti Nets (the fluid version of usual Timed Petri Nets) md Markovdecisionprocesses.Wecharacterizethe subclass of ContinuousTimedPetri Nets correspondingto undiscountedaveragecoststructure.Thissuhlass satisfiesconservationlaws and shows a linear growth: one obtains as mere applicationof existingresultsfor DyraunicProgrammingtheexistenceof anmymptoticthroughput.This rate cm be computedusing Howard-typealgorithms.or by an extensionof the wellknowncycle timeformulafor timed event graphs. We present an illustratingexmple and brieflysketchthe relationwiththe discretecase. Keywords— Peti Nets, Dynamic Programming, Markov Decision Processes. Discrete Event Systems, Maxplus algebra. I. Introduction The fact that a subclassof DiscreteEventSystemscan be representedby Iin= equationsin the (min.+)or in the (max.+) semiringis now atmost classical [7], [2]. The (rein,+)linearitytilows the presenceof synchronization and saturationfeaturesbutprohibitsthemodelingof many interestingphenomenasuch as “birth” md “death” processes(multiplicationof tokens)and concurrency.More recently[3], [8], it was rdized that undersomeassumptions on the routing policies, these additiond features couldberepresentedby moregeneralrecurrences,involving both conventionallinear systemsand (rein,+) linear systems.Fromthecontroltheoreticalpointof view,these are polynomial systemsover the (rein,+)tigebra, that is, the exact (rein,+)counterpartof conventionalpolynomial discretetimesystems.This approachwas outlinedin [8], wherein p,articulmthe(rein,+)andogueofVolterraexpansion wasgiven. Another(simpler)pointof viewis basedon Markovdecisionprocesses.As shownin [8],the “poiynomid”Petri Net equationscan be interpretedas thedynamicprogrammingequationsofa canonicalMarkovdecisionprocessassociatedwiththenet,equippedwithan additivediscounted cost. Moreexpliciteiy,thecounterfunction(numberof firings) of transitionq at time t is equal to the value functionat stateqfortheassociatedMarkovdecision process in horizont. Of particularinterestis thecaseof undiscounted costs: thenthe vatuefunctiongrowslinearlyas a function

0-7803-2685-7/95 .

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11, Continuous Timed Petri Nets under Stationary Routing Policies

Fig. 1. A TimedPetriNet.

The definitionof ContinuousTimed Petri Nets is syntaxicdly very similarto that of conventiontiTimedPetri Nets:oneh’asto spwify thetopologyof thegraph,the initi~ marking,and[hedurations.The maindifferenceliesin 2029

the functioning and the interpretationof the system,since fluidsinsteadof tokenscirculatein the net. Definition11.1(CTPN). A Continuous Timed Petri Net withMultipliers(CTPN)is a valuedbipartitegraphgiven by a 5-tupleA’ = (T, Q, JM,m, ~) with the following characteristics. 1. T is a finite set whose elementsare ctild places.

Placesshouldbe seen as reservoirs,with inputand output pipes,in whicha liquidflowsaccordingto a dynwics describedlateron. 2. Q isa finiteset whoseelementsarectiled transitions. Transitionsmixtheflowscomingfromtheplacesimmediatelyupstre’amin givenproportionsmd instantaneously, and pourtheresultingliquidin downstreamplacesdso in given proportions. 3. M E (Rt)Px ‘UQ ‘F.

The multiplier MPq (resp. .MgP)givesthe numberof edgesfromtransitiong to place p (resp. from place p to trmsition q). Weallownoninteger numberof edges.The zerovahtefor M codestheabsence of edge. Wesay that vertex(placeor transition)r is upstream vertexs if JM,,# O.EquivNently,s is downstream r. Wedenoteby routthesetof verticesdownstrem vertex r andby rl”thesetof upstreamvertices.Multipliersdetermine the mixingand dispatchingproportionsas follows: transitionq ties MqP molecules of fluidfrom each upstrem placep, and producesMPI~molecules3of fluidin each downstra place p’. The mixingprocessat transition q continuesas longx all the upstreamplacesare non empty. When a place is upstreamseveral ~sitions. we assumethatthereis a routingmechanismfixingwhichproportionsoftheflowshouldbe sentto theconcurrentdownstrem transitions.Keepingthe discreteterminology,we will still cdljring of transitionq theconsumptionof M~P moleculesin each upstrem placep and theproductionof MPI~moleculesin each downstreamplace,but now,trartsitionfiringsare countedwithred numbers. 4. m c (R+)p represents the initial marking: mP gives the amount of fluid initially avtilable in place p.

5. r E (R+)p (holdingtimes): TPgives the sojourn timein placep. i.e. the minimaltime from the entry of a moleculein placep to its avaibilityfor the firingof downstreamtransitions.Thisdelaymay be causedfor exmple by a prep’wationtime requiredfor heatingor homogenizingthe fluid.

Definition11.2(Routing Policy). A (stationary. origin independent) routing policy is a map p : Q x T + P,+, such that VP C ~+ 2qCPti Pf/P = 1. Pqp determinesthe proportionof fluidrouted to the downstreamtransitionq by placep. The inititi stockof fluidmP is routedwiththe sameproportions.

We next give the dynamic equationssatisfiedby the TimedPetri Net. Counter functions are &ssociatedwith nodes of thegraph: 1, 2P(t) denotesthecumulatedquantityof fluidswhich hasenteredplacepup to timet, includingtheinitialstock; 2. Zq(t) denotes the cumulatednumber of firingsof trmsition q up to time t. Allthesecumulafed quantitiesareofcoursenondecreasing functions. Wefocushereon the au[onornousregime,thatis wemsame thatthe countertrajectoriesZr (t), r,s G Q u T are frozenfor t < 0 at a given initialcondition.and we let thesystemevolvefreely for t ~ O.A moregenertiinputoutputapproach(theset of transitionsbeingpartionnedin inputistate/output transitions)is detailedin [8],[9]. Weintroducethe notation

PropositionD.3. The counter variables of a CTPN satisfy thefollowing equations6 (la) (lb) qcp’” Remark /1.4. From

(l). we deduce the transition-to-

transitionequation: Z~(t) = P