Parallel disrete-event simulation is an attempt to speed-up the simulation proess through the use of multiple proessors. In some sense parallel disret

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1 Exploiting intra-objet dependenies in parallel simulation Franeso Quaglia a;1 Roberto Baldoni a;2 a Dipartimento di Informatia e Sistemistia Universita \La Sapienza" Via Salaria 113, 198 Roma, Italy Abstrat This paper introdues the notion of weak ausality that models the intra-objet parallelism in parallel disrete event simulation. In this setting, a run where events are exeuted at eah objet aording to their timestamp is a orret run. The weak ausality relation allows to dene the largest sub-set of all runs of a simulation that are equivalent to the timestamp-based run. Finally, we desribe an appliation of weak ausality to optimisti synhronization (Time Warp) by introduing a synhronization protool that redues the number of rollbaks and their extent. Key words: Causality, Optimisti Synhronization, Rollbak-Reovery, Disrete Event Simulation, Time Warp 1 Introdution A disrete event simulation onsists of a model of the simulated system and the exeution of a set of events ourring in a virtual time (run). The exeution of an event usually moves the simulation model from one state to another and produes other events to be exeuted at their virtual times. The orretness riterion is ausality whih states that simulation results are orret if virtual time does not derease while exeuting events in a run. 1 quaglia@dis.uniroma1.it 2 baldoni@dis.uniroma1.it This artile appeared in Information Proessing Letters, vol.7, no.3, Preprint submitted to Elsevier Preprint 28 September 1999

2 Parallel disrete-event simulation is an attempt to speed-up the simulation proess through the use of multiple proessors. In some sense parallel disreteevent simulation an be viewed as generated by the following observation: \divide the simulation model into many sub-models as independent as possible of eah other and then exeute events at sub-models as onurrently as possible". More tehnially (2), the simulated system is partitioned into a number of subsystems that are modeled by ommuniating objets (other papers refer suh objets as logial proesses). An objet may shedule an event at virtual time t for another objet by sending to it a message with timestamp t. The advantage over a sequential simulation omes from the fat that previous ausality denition beomes the following weaker loal onstraint: the orretness of results is guaranteed if eah objet exeutes its events in non-dereasing timestamp order 3. Two main synhronization protools have been proposed to guarantee loal ausality. The rst, namely pessimisti synhronization (1), allows an objet to exeute an event with timestamp t after there is the ertainty that no event with timestamp less than t will be sheduled for that objet in the future of the simulation exeution. The seond, known as optimisti synhronization (or Time Warp) (7), lets an objet exeute simulation events whenever they are available and uses a hekpoint-based rollbak as a mean to reover from out of timestamp order exeutions. However, loal ausality does not exploit all potential parallelism inside an objet. For example, let us onsider two events a and b at the same objet, whose timestamps are ts(a) and ts(b) respetively, with ts(a) < ts(b). If the out of timestamp order exeution of a and b produes the same simulation result as if they were exeuted in timestamp order, then a and b are, in a sense, independent due to an intra-objet parallelism. From an operational point of view this independene, whih might allow out of timestamp order runs, has been used in Time Warp with lazy anellation (3), in Time Warp with lazy rollbak (9) and in the study of the super-ritiality of synhronization protools (4; 6; 8). For example, Time Warp with lazy anellation assumes that rolled bak events are orret unless their re-exeution produes a dierent result (so, it allows some out of timestamp order runs). Another example is Time Warp with lazy rollbak, whih works as follows: when an event is reexeuted due to a rollbak, the resulting objet state is ompared to the state 3 The notion of ausality is well known in distributed systems and has been formalized by Lamport in (5) by means of the happened-before relation. That relation models events of a distributed omputation as a partial order, then distributed algorithms based on a logial time (timestamp) have been given to produe a total ordering of events onsistent with the happened-before relation. On the ontrary, in the simulation ontext, the logial time, produed by the ausality, denes an a priori total ordering on events and then parallel disrete event simulation extrats many sub-orders to be exeuted one for eah objet. 2

3 produed by the previous exeution of that event; if those states are the same, rolled bak events whose exeution is dependent on the produed state are not re-exeuted. However, the lak of a denition for intra-objet parallelism is the ause of (possibly) unneessary overhead in both previous protools. Speially, Time Warp with lazy anellation, requires rolled bak events to be re-exeuted (this is neessary to hek their orretness), and Time Warp with lazy rollbak requires state omparison. Both overheads ould be removed if rolled bak events are known to be independent of the event ausing the rollbak due to an intra-objet parallelism. Basing on observations of previous paragraphs, in this paper we introdue the relation of WEak Causality (WEC) that models the intra-objet parallelism. WEC states that two events must be exeuted at an objet aording to their timestamp if their out of order exeution would produe a dierent result. This relation is based on onits reated by events that operate on the same data as well as on the timestamps of events. WEC allows to dene a set of simulation runs that move the simulation to the same nal state of a pure timestampbased simulation run (these runs are said to be equivalent to a timestampbased run). Then an optimisti synhronization protool in whih rollbaks our only upon violations of WEC is presented. This protool exhibits a potential redution of the number of rollbaks and of their extent ompared to previous solutions. The remainder of the paper is organized as follows. Setion 2 provides the model of an objet. Setion 3 introdues the WEC relation. In the same setion, a theorem is proved whih states that any simulation run onsistent with WEC is equivalent to the timestamp-based run. A remark on the set of runs originated by WEC onludes Setion 3. The appliation of WEC to optimisti synhronization is desribed in Setion 4. 2 Model of an objet A parallel disrete event simulation onsists of a set of objets, eah objet exeutes a sequene of events (e 1 ; : : : ; e k ). Eah objet has a state onsisting of a set of variables fv 1 ; : : : ; v`g aessed by read and write operations. Eah variable v k has an initial value, namely init(v k ), whih is assigned by an initial titious write operation. The exeution of an event e moves the objet's state from to (with 6= ) and may generate new events. Eah event exeution is deterministi i.e., given a state, the exeution of e will always produe and will always generate the same new events. When an event e is generated by objet O, it is sheduled for a given virtual time (timestamp), 3

4 denoted ts(e) (we assume without loss of generality that to eah event in an objet orresponds a distint timestamp), then a message with the ontent of e and ts(e) is sent by O to an objet O (not neessarily distint from O) that will exeute it 4. Timestamps will then totally order the set of events at an objet: Denition 2.1 Let e and e be two events exeuted at an objet, e preedes e, denoted e < t e, if ts(e) < ts(e ). Eah event exeution reads and/or writes variables forming the objet's state atomially. Hene, eah event an be modeled as a set of read and write operations on elements in the objet's state. We dene the read set (resp. write set) of an event e, denoted R(e) (resp. W (e)), the set of the state variables read (resp. modied) while exeuting e. RW (e) represents the set obtained by the union of R(e) and W (e). We assume that if a state variable v is both read and written by the event e, then the read operation takes plae before the write one. An exeution of a sequene of events at an objet is alled run 5. A run moves an objet from its initial state to a nal state. A timestamp-based run represents an exeution in whih events are proessed at an objet aording to a non-dereasing order of their timestamps, it then represents the orret run with respet to the ausality riterion. The nal state of a timestampbased run is denoted ts. A run is equivalent to the timestamp-based run i it ontains the same set of events of the timestamp-based run and moves the state of the objet from to the nal state ts. The set of runs that are equivalent to the timestamp-based run is alled orret runs set denoted R. 3 The weak ausality relation In this setion we dene the WEak Causality (WEC) relation between events ourring at an objet. By using suh a relation we show that events that are not WEC related an be exeuted in any order as they will produe a run in the orret runs set. We present WEC through an introdutory example, then we provide a formal denition. Finally, we prove a theorem stating that any run onsistent with WEC is equivalent to the timestamp-based run. 4 Note that seeds for the random number generation either are elements of the objet's state or are part of the ontent of an event. 5 As we are interested in modeling the intra-objet parallelism, in the reminder of the paper we onsider a run as an exeution at an objet. 4

5 3.1 An introdutory example and the WEC denition Let us suppose to have a timestamp-based run at an objet with six events e 1, e 2, e 3, e 4, e 5 and e 6 and that the variables v 1, v 2, v 3 and v 4 form the state of the objet. Moreover to eah event are assoiated the following read/write sets: R(e 1 ) = fv 3 g; W (e 1 ) = fv 1 ; v 2 g R(e 2 ) = fv 2 g; W (e 2 ) = fv 3 g W (e 3 ) = fv 3 g R(e 4 ) = fv 1 g; W (e 4 ) = fv 4 g R(e 5 ) = fv 2 g; W (e 5 ) = fv 4 g R(e 6 ) = fv 3 ; v 4 g Note that in the timestamp-based run, event e 2 reads a value of the variable v 2 written by e 1 while e 3 writes v 3 previously written by e 2. Also e 4 reads v 1 that was written in the timestamp-based run by e 1 and so on. This means that e 4 ould be proessed just after the exeution of e 1 as it does not onit on state variables with e 2 and e 3. Following these arguments, we get four runs whih are able to move the simulation to the same nal state of the timestamp-based run: (a) e 1 ; e 2 ; e 4 ; e 3 ; e 5 ; e 6 (b) e 1 ; e 2 ; e 4 ; e 5 ; e 3 ; e 6 () e 1 ; e 4 ; e 2 ; e 5 ; e 3 ; e 6 (d) e 1 ; e 4 ; e 2 ; e 3 ; e 5 ; e 6 Previous observations suggest to formalize the denition of onit between events as follows: Denition 3.1 Let e and e be two events belonging to a run r. e onits with e, denoted e < C e, if the following prediate P is true: P (W (e) \ RW (e ) 6= ;) _ (R(e) \ W (e ) 6= ;) By using the notion of onit between events, let us introdue the WEC relation that models the intra-objet parallelism: 5

6 Denition 3.2 Let e and e be two events belonging to a run r. e is weakly ausally related to e, denoted e W?! EC e, if: (a) (e < t e ) ^ (e < C e ); or (b) there exists an event ^e exeuted at the same objet suh that: (e W?! EC ^e) ^ (^e W?! EC e ). Figure 1 depits the WEC relations of the previous example. Hene events exeuted at an objet an be modeled as a partial order be = (E; W?! EC ), where E represents the set of all events exeuted at the objet. Two events e and e suh that :(e W?! EC e W EC ) and :(e?! e) are said to be onurrent and are denoted ejje. Conurrent events an be exeuted in any order. e 2 e 3 e 1 e 4 e 5 e 6 Fig. 1. The partial order on the sequene of events (e 1 ; : : : ; e 6 ) 3.2 Equivalene between linear extensions of be and the timestamp-based run We prove that any linear extension of be represents a run equivalent to the timestamp-based one (i.e., a run in the orret runs set). Prior entering the proof, we introdue some simple notations. For eah state variable v 2 R(e), we denote as V ts e (v) the value of v read by the event e in the timestamp-based run (ts), and we denote as V e (v) the value of v read by the event e in a run whih is a linear extension of be distint from ts. Theorem 3.1 Let e 2 E be a simulation event and be a linear extension of be with 6= ts. For any state variable v 2 R(e) we have V e (v) = V ts e (v). Proof (by indution). Let us onsider the sequene of events orresponding to a linear extension with 6= ts. Let us denote as e i (with i 1) the i-th event of the sequene (i.e., the i-th event of the linear extension ). The theorem is proved by indution on i. Base Step. i = 1. In this ase e i is the rst event of the linear extension. For eah variable v 2 R(e i ) then e i reads the initial value init(v), that is V ts (v) = init(v). Suppose by ontradition that V (v) 6= V (v), in this ase ei ei ei we have that in ts there must exist an event e suh that e < t e i and e writes onto v a value distint from init(v). In this ase the write set of e and the read set of e i interset (i.e., e < C e i ), so we have by Denition 3.2 that e W?! EC e i meaning that e i annot be the rst event of any linear extension of be. So the ase assumption is ontradited and the laim follows. 6

7 Indution Step. We suppose the result is true for all the events e i with i > 1 and show that is holds for the event e i+1. Suppose by ontradition that there exists a variable v 2 R(e i+1 ) suh that V ts (v) 6= V (v). We have three ei+1 ei+1 ases: (a) there does not exist any event e 2 E suh that e is distint from e i+1 and v 2 W (e) (i.e., no event distint from e i+1 writes v). In this ase the event e i+1 reads from v the initial value init(v) either in or in ts. Thus the assumption is ontradited and the laim follows; (b) all the events e 2 E distint from e i+1 and writing v are suh that e i+1 < t e (i.e., all the events whih are distint from e i+1 and write the variable v have timestamp larger than ts(e i+1 )). In this ase, for all suh events e then e i+1 W EC?! e meaning that in the linear extension annot exist any event e j with j < i + 1 (i.e., e j preedes e i+1 in ) whih writes v. As for ase (a), the event e i+1 reads from v the initial value init(v) either in or in ts. Thus the assumption is ontradited and the laim follows; () there exists at least one event e 2 E suh that v 2 W (e) and e < t e i+1 (i.e., there exists at least one event e whih writes v and has timestamp less than ts(e i+1 )). In this ase e W?! EC e i+1 meaning that e must neessarily preede e i+1 in the linear extension (i.e., e = e j with j < i + 1). Let e m be the event in E with the highest timestamp and satisfying the ondition of ase (). Then e m (with m < i + 1) is the latest event whih writes v and preedes e i+1 both in ts and in. We have supposed by ontradition that V ts (v) 6= V (v). In ei+1 ei+1 this ase during the exeution of, e m writes on v a value distint from the one that would have be written in ts. As the exeution of e m is deterministi, then the only way for this to our is that e m reads in values dierent from the ones that would have been read in ts. This ontradits the indution step assumption that all the i events (inluding e m ) that preede e i+1 in atually read the same values that would have been read in ts. Therefore the laim follows. Corollary 3.2 Any linear extension of be is equivalent to ts. Proof. By Theorem 3.1 we have that in any linear extension of be all the events in E read the same values that would have been read in the timestampbased run. As the exeution of eah event is deterministi, then eah event will produe the same updates and generates the same new events of ts (those updates on state variables will lead to the same nal state of ts, namely ts ), therefore, by denition, the linear extension is equivalent to ts. 3.3 The set of orret runs originated by WEC Let us now show that the set of runs R W EC onsisting of the linear extensions of be = (E; W?! EC ) is the largest set ontaining orret runs with respet to the 7

8 model introdued in Setion 2. To this purpose, we rst show that any other relation WEC', weaker than WEC, dened on events allows runs that are not equivalent to the timestamp-based one (so any relation weaker that WEC may generate runs whih are not orret). Then we show that any relation WEC' whih is not weaker than WEC annot originate a set of orret runs R W EC whih is larger than R W EC. To obtain a relation WEC' weaker than WEC we relax the notion of onit between events. Hene, let us assume a onit be determined by a prediate P suh that P ) P. By means of a ase analysis, we show that for eah prediate P satisfying previous relation, there exists a run allowed by WEC' whih is not equivalent to the timestamp-based run. We obtain the following ases: P F ALSE In this ase WEC' allows any out of timestamp order exeution. Let onsider a run with two events, say e 1, e 2 suh that e 1 < t e 2, and let v be the only variable forming the objet's state. Events are suh that: W (e 1 ) = fvg, W (e 2 ) = fvg. Trivially, if events are exeuted out of timestamp order, then the nal state of the objet is dierent from ts. P W (e) \ RW (e ) 6= ; Let us onsider a run with two events, e 1 and e 2 suh that e 1 < t e 2, and let v 1 and v 2 be variables forming the objet's state. Suppose events are suh that: R(e 1 ) = fv 1 g, W (e 1 ) = fv 2 g, W (e 2 ) = fv 1 g. Under W?! EC, e 1 jje 2, hene both runs (e 1 ; e 2 ) and (e 2 ; e 1 ) are allowed. However, the run (e 2 ; e 1 ) moves the state of the objet to a value dierent from ts sine e 1 reads the state variable v 1 updated by e 2 and writes v 2. P R(e) \ W (e ) 6= ; Let onsider a run with two events, e 1 and e 2 suh that e 1 < t e 2, and let v 1 and v 2 be variables forming the objet's state. Suppose events are suh that: W (e 1 ) = fv 1 ; v 2 g, W (e 2 ) = fv 1 g. Under W?! EC, we have e 1 jje 2 hene both runs (e 1 ; e 2 ) and (e 2 ; e 1 ) are allowed. However, the run (e 2 ; e 1 ) moves the state of the objet to a value dierent from ts sine e 1 re-updates v 1 previously written by e 2. Let us now investigate on the size of the set of runs R W EC allowed by a relation WEC' whih is not weaker than WEC. This relation is originated by a prediate P suh that P 6) P (reall that in this ase WEC' allows only orret runs). The following ase analysis shows that R W EC R W EC : P, P In this ase the two onit prediates are equivalent, so we have that for any pair of events < e; e > belonging to E, if ejje under WEC' then ejje under 8

9 WEC, the onverse is also true. This implies that the set of linear extensions of be = (E; W?! EC ) is idential to the set of linear extensions of be = (E; W?! EC ). P ( P In this ase we have that for any pair of events < e; e > belonging to a set of events E, if ejje under WEC' then ejje under WEC, but the onverse is not neessarily true. Therefore the set of linear extensions of be = (E; W?! EC ) may ontain at least one element whih is not inluded in the set of linear extensions of be = (E; W?! EC ). 4 An appliation of weak ausality to optimisti synhronization In optimisti synhronization (Time Warp) objets exeute events whenever they are available under the optimisti assumption that the exeution order does not violate ausality. Eah time a ausality violation is deteted (i.e., an event is being exeuted whose timestamp is lower than that of some previously exeuted events), then the objet is rolled bak to its state immediately prior the violation and the exeution resumes (7). In this setion we show how WEC an be used to implement a partiular kind of Time Warp synhronization; suh a synhronization will be referred to as WEC-based Time Warp. The advantage of the latter over the original protool lies on the possibility to not roll bak the objet state eah time an out of timestamp order exeution is deteted. In partiular, all the out of timestamp order runs that respet the partial order dened by WEC are allowed by the protool. As formalized in Setion 3, events at an objet are modeled by the partial order be = (E; W?! EC ) and they an be exeuted aording to any linear extension of be. Upon the exeution of an event e, we an identify the set E E x(e) (with E x (e) E) as the set of exeuted events 6. Furthermore we denote as E RL (e) the subset of E E x(e) onsisting of all the events e suh that e < t e. We onsider E RL (e) totally ordered by timestamp. Eah time an objet shedules for exeution an event e suh that E RL (e) = ;, it exeutes e as in lassial Time Warp synhronization. On the ontrary, eah time an event e is sheduled suh that E RL (e) 6= ;, the objet behaves as follows: it traverses E RL (e), starting from the minimum, until it nds an event e r, if any, suh that e < C e r. If e r does not exists then e is exeuted. Otherwise, all events in E RL (e) with timestamp larger than ts(e r ) are undone (i.e., they orrespond to events not exeuted yet), and the state of the objet is rolled bak to its value immediately prior e r 's exeution. Then e is exeuted. 6 Note that events exeuted and then rolled bak are not part of E E x(e). 9

10 In both ases, we obtain a run (with e as last exeuted event) whih is a prex of a linear extension of be. It is lear that the rollbak extent (number of events undone due to rollbak) is, on the average, smaller than the one of the original Time Warp protool. In the best ase it is zero (and no state restoration is exeuted at all), in the worst ase it is equal to the ardinality of E RL (e). 4.1 Operational issues From an operational point of view, the ondition e < C e r annot always be traked before exeuting e as sometimes R(e) and W (e) are determined during e's exeution. In order to overome this problem, we an dene the read and write sets of an event e before e's exeution, denoted RB(e) and W B(e) respetively, as the maximum set of state variables that e an potentially read and write during its exeution (i.e., RB(e) R(e) and W B(e) W (e)). Hene, the rollbak point an be determined replaing RB(e) and W B(e) in the denition of onit between events. The usage of RB and W B overestimates the number of weak ausality dependenies between events but leads to a safe solution. Obviously, WEC-based Time Warp an lead to performane improvement whenever the enoding of Read/Write sets of events and their omparison is less spae-time expensive 7. Finally we remark that, even if there exists an event e r in E RL (e) suh that e < C e r, it might be possible that after the re-exeution of the rolled bak events we get the same result of the previous exeution. This ould happen, for example, if e does not really updates the objet's state. In suh a ase, lazy anellation and lazy rollbak an be embedded in the previous WEC-based rollbak sheme as they are an orthogonal approah to avoid unneessary undoing of events. 5 Summary In this paper we have provided a formal model for the parallelism of events in simulation objets. This parallelism allows to enlarge the set of orret runs of a simulation program. i.e., runs moving the objet to the same state of a timestamp-based run. To model this parallelism, we have introdued a weak ausality relation whih is based on events' onit as well as on events' timestamps. From an operational point of view, we have shown how weak 7 This may happen, for example, either when events an be lassied into types and eah type orresponds to a given Read/Write set on state variables, or when Read/Write sets have small ardinality. 1

11 ausality an be well suited to optimisti (Time Warp) synhronization to avoid unneessary rollbaks in an orthogonal way with respet to both lazy anellation and lazy rollbak. Referenes [1] K.M. Chandy, J. Misra, Distributed Simulation: a Case Study in Design and Veriation of Distributed Programs, IEEE Transations on Software Engineering, SE5 (5) (1979) [2] R.M. Fujimoto, Parallel Disrete Event Simulation, Communiations of ACM, 33 (1) (199) [3] A. Gafni, Spae Management and Canellation Mehanisms for Time Warp, Teh. Rep. TR , University of Southern California, Los Angeles (Ca,USA). [4] M. Gunter, Understanding Superritial Speedup, Proeedings of 8th Workshop on Parallel and Distributed Simulation, 1994, pp [5] L. Lamport, Time, Clok and the Ordering of Events in a Distributed System, Communiations of the ACM, 21 (7) (1978) [6] D. Jeerson, P. Reiher, Superritial Speedup, in: Pro. 24th Annual Simulation Symposium, pp [7] D. Jeerson, Virtual Time, ACM Trans. on Programming Languages and Systems, 7 (3) (1985) [8] S. Srinivasan, P.F. Reynolds, Super-Critiality Revisited, in: Pro. 9th Workshop on Parallel and Distributed Simulation, 1995, pp [9] D. West, Optimizing Time Warp: Lazy Rollbak and Lazy Reevaluation, Master's Thesis, University of Calgary, January