The Impact of Time on the Session Problem Injong Rhee Jennifer L. Welch Department of Computer Science CB 3175 Sitterson Hall University of North Caro

Transcription

1 The Impat of Time on the Session Problem Injong Rhee Jennifer L. Welh Department of Computer Siene CB 3175 Sitterson Hall University of North Carolina at Chapel Hill Chapel Hill, N.C Abstrat The session problem is an abstration of synhronization problems in distributed systems. It has been used as a test-ase to demonstrate the differenes in the time needed to solve problems in various timing models, for both shared memory (SM) systems [2] and messagepassing (MP) systems [4]. In this paper, the session problem ontinues to be used to ompare timing models quantitatively. The session problem is studied in two new timing models, the periodi and the sporadi. Both SM and MP systems are onsidered. In the periodi model, eah proess takes steps at a onstant unknown rate; different proesses an have different rates. In the sporadi model, there exists a lower bound but no upper bound on step time, and message delay is bounded. We show upper and lower bounds on the time omplexity of the session problem for these models. In addition, upper and lower bounds on running time are presented for the semi-synhronous SM model, losing an open problem from [4]. Our results suggest a hierarhy of various timing models in terms of time omplexity for the session problem. 1 Introdution Early work in distributed omputing usually assumed one of two extreme timing models: either 0 the ompletely synhronous model, in whih proesses operate in lokstep rounds of omputation, or the ompletely asynhronous, in whih there are no upper bounds on proess step time or message delay. Sine both of these timing assumptions are often unrealisti, researhers began to investigate the impat on distributed omputing if those timing assumptions are relaxed or tightened to some extent in order to reflet the real time situation. This question has been studied for a variety of problems, inluding Byzantine agreement [7, 8, 1, 13], mutual exlusion [3], leader eletion [5], transation ommit [6], and the session problem [2, 4]. The (s; n)-session problem, first presented in [2], is an abstration of the synhronization needed to solve many distributed omputing problems. Therefore, it is an important tool for understanding the behavior of distributed systems under different timing onstraints. Informally, a session is a minimal-length omputation fragment that involves at least one synhronization" step by every proess in a distinguished set of n proesses. An algorithm that solves the (s; n)-session problem must guarantee that in every omputation there are at least s disjoint sessions and eventually all the n proesses beome idle. We study the problem in two different interproess ommuniation models: shared memory and message passing. In the shared memory model, proesses ommuniate only by means of shared variables. Eah variable is shared by no more than b proesses, where b is a onstant relative to the total number of proesses. In the message passing model, ommuniation is done by exhanging messages aross a network. A proess an broadast a message at a step; the message is guaranteed to be delivered to every proess after some finite time. The relevant timing aspets of a model are the

2 lower bound on proess step time, 1, the upper bound on proess step time, 2, and additionally, for the message passing model, the lower bound on message delay, d 1, and the upper bound on message delay, d 2. The running time of an algorithm for the (s; n)-session problem is the maximum time, over all omputations, until all the n proesses beome idle. If there is an upper bound in real time on 2 and d 2, then it makes sense to measure the running time in terms of real time. If not, then the ommon way to measure the running time is with rounds. A round is a minimal omputation fragment in whih every proess takes at least one step. Arjomandi, Fisher and Lynh [2] studied the (s; n)-session problem in synhronous and asynhronous shared memory models. Synhronous means that 1 = 2, a finite number. Asynhronous means that 1 and 2 are infinite. Their results showed a signifiant time omplexity gap between the synhronous and asynhronous models, namely that s rounds are suffiient for the synhronous ase but (s 1)blog b n rounds are neessary for the asynhronous ase, where n is the size of the distinguished set of proesses. The impliation is that no ommuniation is needed at all in the synhronous ase, but it is needed for every session in the asynhronous ase. (The blog b n fator is essentially the ost of ommuniation when no more than b proesses an aess any shared variable.) Attiya and Mavroniolas [4] addressed the problem in semi-synhronous and asynhronous message passing systems. Semi-synhronous means that 1 > 0, 2 and d 2 are finite, and these onstants are known to the proesses. They modeled the asynhronous system differently than [2]: they let 1 =0 and d 1 = 0, while 2 and d 2 are finite. Their results also indiated an important time separation between semi-synhronous and asynhronous networks, again based on whether or not ommuniation is neessary. We present almost mathing upper and lower bounds for the session problem in the semisynhronous shared memory model. Our bounds are similar to those in [4] for the message passing model when the ost for information propagation in the shared memory model is substituted for the message delay. They indiate that if the time for one ommuniation is less than that for one step multiplied by the ratio of 2 and 1, the model behaves like the asynhronous; otherwise it behaves like the synhronous (inflated by the ratio). Mavroniolas [12] has also independently developed the same lower and upper bounds for the shared memory semi-synhronous model. We introdue two new timing models for the (s; n)-session problem: the periodi and the sporadi. In the periodi model, for eah proess there exists an unknown onstant suh that the proess makes one step at every period of the onstant. In the message-passing variant, d 2 is finite and known. The upper bounds for both the shared memory and message passing models are the time for the slowest proess to take s steps plus the time for one ommuniation. The lower bounds for both are the maximum of the time for the slowest proess to take s steps and approximately the time for one ommuniation. Our results indiate that the periodi model, whih requires one ommuniation, falls in between the synhronous and asynhronous models, whih require no and s 1 ommuniations respetively. In the sporadi model, there exists a nonzero lower bound 1, but no upper bound, on the time between any two onseutive steps of any proess. The sporadi shared memory model is essentially equal to the asynhronous shared memory model and is not onsidered. For the message passing model, the message delay is within [d 1, d 2 ], where d 1 0, d 2 is finite, and both are known. The ombination of the lower bound on step time and upper bound on message delay allows proesses to make inferenes about the omputation, namely, that enough time has elapsed so that a message must have arrived. The lower bound on the persession time is maxfb u 4 1 K; 1 g, where u = d 2 d 1 and K = 2d21 d. 2 u=2 The upper bound on the persession time is minf(b u 1 +3) fl+u; d 2 + flg, where fl is the largest step time by a proess before termination. As the message delay approahes a onstant (i.e., d 1 approahes d 2 ), the per-session time beomes maxf0; 1 g = 1 for the lower bound and minf3fl;d 2 +flg = O(fl) for the upper bound, whih is like the synhronous model. As the message delay flutuates within a bigger interval (i.e., d 1 approahes 0), the per-session time beomes maxfd 2 ; 1 g=d 2 for the lower bound and minf(b d2 1 +3) fl+d 2 ;d 2 + flg = O(d 2 +fl) for the upper bound, whih is like the asynhronous model. These two timing onstraints are inspired by onstraints with the same names ommonly used

3 in many real-time problems, espeially in sheduling of real time tasks for a uniproessor [9, 10,11] where the period of task ourrenes onforms to the onstraints. In pratie, as quoted in [10], periodi timing onstraints are used in appliations suh as avionis and proess ontrol when aurate ontrol requires ontinual sampling and proessing of data. The sporadi timing onstraint is assoiated with event-driven proessing suh as responding to user inputs or non-periodi devie interrupts; these events our repeatedly, but the time interval between onseutive ourrenes varies and an be arbitrarily large. Therefore, the sporadi timing onstraint models proesses that an be bloked for an arbitrarily long (but finite) time waiting for a ertain ondition to be true or a ertain event to our, but that annot make two onseutive steps faster than a ertain lower bound. Table 1 summarizes the bounds. L means lower bound, U means upper. In the periodi model, min and max are the smallest and the largest step times respetively of all proesses. The bounds for the asynhronous shared memory ase are in rounds. The bounds from [4] have been onverted in three aspets for purposes of omparison: (1) That paper onsiders point-to-point networks; thus the results inlude a fator of the network diameter. In our model, d 2 subsumes the diameter fator; we have replaed all ourrenes of the diameter fator with 1. (2) In [4], the onstant 1 is used as the value of 2 ;wehave replaed all appropriate ourrenes of 1 with 2. (3) [4] assumes that all proesses take their synhronized first steps at time 0, resulting in one session at time 0; although we assume that all proesses start at time 0, we don't assume that all take a synhronized step at time 0. We rather assume that all steps (inluding the first step) obey the timing onstraints of a speifi model starting time 0. Our results indiate that the periodi model is more effiient than the semi-synhronous system when max = 2, 2 1 < 2 and n is onstant relative tos. The lower bound for the sporadi system and the upper bound for the periodi system suggest that the periodi system is more effiient than the sporadi system if max is smaller than b u 4 1 K» d 2u 2(d 2 u=2) and n is onstant relative to s.in shared memory, the sporadi system is learly less effiient than the semi-synhronous one, but the relationship between the sporadi and the semisynhronous systems for message passing is rather unlear and understanding it requires further study. The rest of the paper is organized as follows. Setion 2 ontains the definition of system models and Setion 3 desribes how to aomplish ommuniation in the shared memory model. Setion 4 onerns the periodi model, Setion 5 the shared memory semi-synhronous, and Setion 6 the message-passing sporadi. Please note that our lower bound proof tehnique ombines those in [2, 4]. Some proofs are omitted or only skethed due to spae onstraints. 2 Definitions 2.1 Systems The generalized system model definition for shared memory and message passing models is similar to that defined in [2]. There are finite sets P of proesses and X of shared variables. A proess onsists of a set of internal states, inluding an initial state. Eah shared variable has a set of values that it an take on, inluding an initial value. A global state is a tuple of internal states, one for eah proess, and values, one for eah shared variable. The initial global state ontains the initial state for eah proess and the initial value for eah shared variable. A step ß onsists of simultaneous hanges to the state of some proess and the values of some numberofvariables, depending on the urrent state of that proess and urrent values of the variables. More formally, we represent the step ß with a tuple ((s; p; r); (u 1 ;x 1 ;v 1 );:::(u k ;x k ;v k )), where s and r are old and new states of a proess p 2 P ; u i and v i are old and new values of a shared variable x i 2 X for all i. We say that step ß is appliable to a global state if p is in state s and x i has value u i for all i in the global state. A system is speified by desribing P, X, and set ± of possible steps. For all proesses p 2 P and all global states g, there must exist some step involving proess p that is appliable to global state g. A omputation of a system is a sequene of steps ß 1 ;ß 2 ;::: suh that: (1) ß 1 is appliable to the initial global state, (2) eah subsequent step is appliable to the global state resulting from the previous steps, and (3) if the sequene is infinite, then every proess takes an infinite number of steps. That is, there is no proess failure. A timed omputation of

4 Model Shared Memory Message Passing Syn. L s 2 [2] s 2 U s 2 [2] s 2 Peri- L maxfs max ; blog 2b 1 (2n 1) min g maxfs max ;d 2 g odi U s max + O(log b n) max s max + d 2 Semi- L minfb ;blog b n 2 g (s 1) minfb ;d g (s 1) [4] syn. U minf(b ) 2 ;O(log b n) 2 g (s 1) + 2 minf(b ) 2 ;d g (s 1) + 2 [4] Spor- L See Asyn. SM maxfb u 4 1 K; 1 g (s 1) adi U See Asyn. SM minf(b u 1 +3) fl+u; d 2 + flg (s 1) + fl Asyn. L (s 1) blog b n [2] (s 1) d 2 [4] U (s 1) O(log b n) [2] (s 1) (d )+ 2 [4] Table 1: Bounds for the Session Problem a system is a omputation ß 1 ;ß 2 ;::: together with a mapping T from positive integers to real numbers that assoiates a real time with eah step in the omputation. T must be nondereasing and, if the omputation is infinite, inrease without bound. We will abuse notation and let T (ß i ) indiate the time at whih step ß i ours Shared Memory Model (SMM) We speialize the general system into the shared memory system in whih proesses ommuniate with eah other by means of shared variables. Eah step ß has the property that k =1. That is, it involves only one shared variable. A proess an read and write a shared variable in a single atomi step, and we don't assume any upper bound on the size of the variables. We let b be the maximum number of proesses that aess any single variable, in all the steps of the system. We assume b is onstant relative to the number of proesses Message Passing Model (MPM) We speialize the general system into the message passing system, in whih proesses ommuniate with eah other by exhanging messages. P = R[fNg, where R is the set of regular proesses and N is the network. The network shedules the delivery of messages sent among the regular proesses. X = fnetg [fbuf p : p 2 Rg, where the values taken on by eah variable are sets of messages. net models the state of the network, i.e., the set of messages in transit. buf p holds the set of messages that have been delivered to p by the network but not yet reeived by p. A step of a proess p in R onsists of p reeiving the set M of messages in its buffer buf p, and based solely on those message and its urrent state, hanging its loal state and sending out some message m to all the regular proesses. The result is to set buf p to empty and to add (m; q) tonet, for all q in R. So, the step involves two shared variables, buf p and net. A step of N is to deliver some message of the form (m; q) innet to q. The result is to remove (m; q) from net and add m to buf q. Aordingly, the step also involves two shared variables, net and buf q. This definition of the MPM is an abstrat model of a reliable strongly onneted network with any topology. In a timed omputation, eah message has a delay, defined to be the differene between the time of the step that adds it to net and the time of the step that removes it from net. If the message is never removed, then it has infinite delay. The delay only ounts the time in transit in the network and does not inlude the time that the reipient takes to reeive the message. That is, the time elapsed between the delivery step and the step whih finally removes the message from the buffer is not ounted toward the message delay, even if the message remains in the buffer for a long time before the reipient piks it up from its buffer. 2.2 The Real Time Constraints For eah timing model onsidered, we define the set of admissible timed omputations to onsist of timed omputations whih obey the stated ondition on the step times of all proesses in the SMM

5 (all regular proesses in the MPM) and, additionally for the MPM, the stated ondition on the message delay. Synhronous There exist onstants 2 and d 2 suh that in every timed omputation, for every p in P (p in R for MPM), the time between every pair of onseutive steps of p is 2, and the delay ofevery message is d 2. Thus 2 and d 2 are known" to the proesses and an be used in algorithms. Asynhronous In every timed omputation, every proess takes an infinite number of steps and every message is eventually delivered. Periodi There exists a onstant d 2 suh that in every timed omputation, for every p i in P (p i in R for MPM), there exists onstant i suh that the time between every pair of onseutive steps of p i is i, and the delay of every message is in [0;d 2 ]. Thus the i 's are unknown but d 2 is known. Semi-Synhronous There exist onstants 1 > 0, 2 and d 2 suh that in every timed omputation, for every p in P (p in R for MPM), the time between every pair of onseutive steps of p is in [ 1 ; 2 ] and the delay ofevery message is in [0;d 2 ]. Thus 1, 2 and d 2 are known. Sporadi There exist onstants 1, d 1, and d 2 suh that in every timed omputation, for every p in P (p in R for MPM), the time between every pair of onseutive steps of p is at least 1, and the delay of every message is in [d 1 ;d 2 ]. Thus 1, d 1, and d 2 are known. 2.3 The Session Problem We now state the onditions that must be satisfied for a system to solve the (s; n)-session problem (also alled an (s; n)-session algorithm). (1) Eah proess in P (in R for the MPM) has a subset of idle states. The set ± of steps of the system guarantees that one a proess is in an idle state, it always remains in an idle state. (2) There is a distinguished set Y of n variables alled ports; Y is a subset of X in the SMM and the set of buf's in the MPM. There is a unique proess in P (in R for the MPM) orresponding to eah port, whih is alled a port proess. (3) Let p be a port proess whih orresponds to a port y. A port step is any step involving p and y. A session is any minimal sequene of steps ontaining at least one port step for eah port in Y. In every admissible timed omputation, there are at least s disjoint sessions and eventually all port proesses are in idle states. In the timing models with finite upper bounds on step time and message delay, we measure the running time of an algorithm in real time as follows. An algorithm runs in time t if, for every admissible timed omputation, every proess is in an idle state by time t. In the ase of the asynhronous and sporadi models, step time and/or message delay isunbounded (but finite). For these ases, we measure the running time in rounds [14, 2, 4]. A round is a minimal-length omputation fragment in whihevery proess appears at least one. An algorithm runs in r rounds if, in every admissible timed omputation C, the prefix of C before all proesses are idle onsists of at most r disjoint rounds. It is also informative in these models to express the time omplexity of an algorithm in terms of a new parameter fl, the largest step time during the omputation of the algorithm before all the proesses are idle. The values of fl is dependent on a partiular omputation of the algorithm. This type of per-omputation based time omplexity measure is also used in [1]. 3 Communiation in SMM In desribing our algorithms, we use a subroutine alled broadast as a generi operator for ommuniation in both of the ommuniation models. In the MPM, the broadasting of message m by proess p is taken are of by the network. It takes at most d time for a message to be reeived by all proesses in the MPM. However, the ommuniation in the SMM is onstrained by the number of proesses whih an aess a shared variable. Therefore, broadasting in the SMM involves relaying messages from proess to proess by means of shared variables. In a b-bounded shared memory system, we an build a tree networks of proesses and shared variables by making port and port proesses the leaves of the tree. This network an aomplish the neessary ommuniation to propagate a peie of information originaing from a proess to all other proesses in O(log b n) steps. In this paper, when we say broadast in the SMM, it implies all the interation of proesses

6 in the tree network to aomplish the broadasting. Throughout this paper, we only desribe the role of port proesses in an algorithm and assume that broadast enapsulates the interations among port proesses and other proesses whih partiipate in the tree-network ommuniation. In addition, we use the term step" interhangeably with port step"; when neessary, we make the proper distintion. 4 The Periodi Model The periodi model and the synhronous model are similar in that a proess takes steps at regular time intervals, yet they differ from eah other in that there is no bound on the relative speed of proesses in the periodi model. We first present an algorithm A(p) for the (s; n)-session problem in this model and then show that for all periodi algorithms whih solve the (s; n)-session problem, there exists a omputation of A whih takes at least maxfs max ;d 2 g time for the MPM and maxfs; blog b ng max for the SMM. Algorithm A(p): (This algorithm runs in the MBM and the SMM.) Eah port proess aesses its own port s 1 times and at its s 1th step, broadasts the fat. It enters an idle state after it hears that all other proesses have taken s 1 steps and it has taken at least one more port step. Theorem 4.1 A(p) solves the (s; n)-session problem in time s max + d in the MPM and time s max + O(log b n) max in the SMM, where max = maxf i : p i 2 P g. Theorem 4.2 No MP periodi algorithm for the (s; n)-session problem runs in time less than maxfs max ;d 2 g. Theorem 4.3 No SM periodi algorithm for the (s; n)-session problem runs in time less than maxfs max ; blog 2b 1 (2n 1) min g. Proof: Suppose that s max blog 2b 1 (2n 1) min. Sine all proesses must take at least s steps to have s sessions, s max is obviously the lower bound. Suppose that s max < blog 2b 1 (2n 1) min. By way of ontradition we assume that there exists an algorithm A whih solves the (s; n)-session problem in the periodi SMM in time Z stritly less than blog 2b 1 (2n 1) min. We prove that there exists an infinite admissible omputation of A that ontains less than s sessions. Let (ff; T ) be the admissible timed omputation in whih proesses take steps in round robin order and eah proess's ith step ours at time i min Eah onseutive group of steps for p 1 through p jp j is a round. (Round i ours at time i min and onsists of the i th step of eah proess.) Sine all proesses should enter idle states by time Z in ff and all the step time periods are equal to min in (ff; T ), there are at most r = bz= min rounds required until termination in ff. We will perturb (ff; T ) in order to get a new admissible timed omputation (ff 0 ;T 0 ). We will prove that there exists at least one port proess in (ff 0 ;T 0 ) whih enters an idle state before another port proess takes any step, resulting in an admissible omputation that ontains less than s sessions. Fix any port proess p 0 and hange p 0 s step time period to be blog 2b 1 (2n 1) min. Run A with this modified set of proesses to get a new timed admissible omputation (ff 0 ;T 0 ). We define a subround to be a minimal omputation fragment offf 0 that involves all proesses exept p 0. Avariable v is ontaminated in subround k of ff 0 if there exists j» k and proess p 6= p 0 suh that v's value in the global state of ff 0 following p's step on subround j is not equal to v's value in the global state of ff following p's step in round j. We define no variable to be ontaminated in subround 0. A proess p is ontaminated in subround k of ff 0 if p 6= p 0 and there exists j» k suh that in subround j of ff 0, p aesses a variable that is ontaminated in subround j. We define no proesses to be ontaminated in subround 0. Let P (t) be the set of proesses that are ontaminated in subround t, and let V (t) be the set of variables that are not ontaminated in subround t 1 but are ontaminated in subround t. Let P t and V t satisfy the reurrene equations: P 0 = V 0 =0, V t =2 P t 1 + 1, and P t =(b 1) V t + P t 1. Lemma 4.4 jp (t)j»p t and jv (t)j»v t for 0» t» r, where r = bz= min. Proof: By indution on t. The key points are that p 0 ontributes at most one variable to V (t), while eah ontaminated proess ontributes at

7 most two. Also, in the worst ase a proess beomes ontaminated as soon as possible, proesses only beome ontaminated due to the variables that just beome ontaminated, and eah variable ontaminates at most b 1 other proesses. Soving the reurrene equation, we get P t = (2b 1)t 1 : 2 Thus the total number of proesses that are ontaminated in subround r is at most n 1. Sine less than n proesses are ontaminated in subround r, at least one port proess p 6= p 0 is in the same state at the end of subround r in ff 0 as it is at the end of round r in ff an idle state. But p 0 has not taken a step yet. Thus (ff 0 ;T 0 ) is an admissible timed omputation that ontains less than s sessions. Contradition. (Note that log 2b 1 (2n 1) approahes log b n as b and n inrease.) 5 Semi-Synhronous Model In this setion, we address the upper and lower bounds in the semi-synhronous shared memory model. The semi-synhronous algorithm in [4] an be adapted to work in the shared memory semisynhronous model simply by replaing the ommuniation primitives (send and reeive) with the expliit propagation of information through the tree network of shared variables using the broadast subroutine desribed in Setion 2. The proof of the lower bound for the semisynhronous SMM is rather ompliated, beause the propagation of information relies on reading and writing shared variables, and also beause omputations onstruted in the proof must satisfy the real time onstraints. Theorem 5.1 There is no semi-synhronous algorithm whih solves the (s; n)-session problem in the SMM within time stritly less than minfb ; blog b ng 2 (s 1). Proof: Let B = minfb ; blog b ng. If 2» 2 1, then B» 1 and it is obvious that the bound holds sine every proess must take at least s steps to have s sessions. Suppose 2 > 2 1. Assume, by way ofontradition, that there exists a semi-synhronous algorithm, A, whih solves the problem in SMM within time Z stritly less than B 2 (s 1). Then dz=(b 2 )e»(s 1). Let (ff; T ) be the admissible timed omputation in whih proesses take steps in round robin order and eah proess' ith step ours at time i 2. Eah onseutive group of steps for p 1 through p jp j is a round. There are t = dz= 2 e rounds required until termination in ff. Let ff = fifl, where fi ontains the first t rounds of ff. Following the proof of Theorem 1 in [2], we will show that there is a reordering fi 0 of fi that results in the same global state as fi but that ontains at most s 1 sessions. Thus ff 0 = fi 0 fl is a omputation with at most s 1 sessions. We then will show how to time the events in ff 0 to produe an admissible timed omputation (ff 0 ;T 0 ) with at most s 1 sessions, a ontradition. We onstrut a partial order» fi on the steps in fi, representing dependeny. Let ff» fi fi for every pair of steps ff and fi in fi, and say fi is dependent on ff, ifff=fior if ff preedes fi in ff and ff and fi either involve the same proess or involve the same variable. Close» fi under transitivity. The following laim is not diffiult to prove. Claim 5.2» fi is a partial order, and every total order of steps of fi onsistent with» fi isaomputation whih leaves the system in the same global state as fi does. Let fi = fi 0 ;fi 1 ;:::;fi m where m = dz=(b 2 )e. Eah fi k (exept possibly the last one) onsists of B rounds. Let y 0 be an arbitrary port in Y. For all k, 1» k» s 1, we show that there exists a port y k and two sequenes of steps ffi k and ψ k, suh that the following properties hold. (p y i is the orresponding port proess to y i,1»i»s 1.) (i) ffi k ψ k is a total ordering of the steps in fi k, onsistent with» fi. (ii)ffi k does not ontain any step by proess p y k 1 whih aesses y k 1. (iii)ψ k does not ontain any step by proess p y k whih aesses y k. Then define fi 0 to be ffi 1 ψ 1 ffi 2 ψ 2 :::ffi m ψ m.

8 For all k, 1»k»m,y k,ffi k, and ψ k are defined indutively. If there is some port variable that is not aessed by any step in fi k, then let y k be that port, ffi k the empty sequene, and ψ k = fi k. Otherwise (every port variable is aessed in fi k ), let fi k be the first step in fi k that aesses y k 1. As a onsequene of a very general result proved in [1], there exists a port variable y k suh that: (iv) it is not the ase that fi k» fi ff k, where ff k is the last step in fi k that aesses y k. We now assign times (the mapping T 0 )toevery step in fi k and then let fik 0 be any total ordering of the steps in fi k onsistent with the times. We then define ffi k and ψ k. ffl For eah proess p 2 P, if there are some steps of p in fi k whih ff k is dependent on, we let ß be the step that ourred last among them. We retime ß and all the steps of p that happened earlier than ß suh that the first step of p in fi k ours at 2 1 B(k 1) + 1, the next step ours 1 time later, and so on. ffl For eah proess p, if there are some steps of p in fi k whih are dependent onfi k,we let ß be the step that ourred first among them. We retime ß and all the steps of p that ourred later than ß suh that the last step of p in fi k ours at 2 1 Bk, the step before that ours 1 time earlier, and so on. ffl All other steps in fi k are assigned the same time as they are under T (the original timing). Let ffi k be all the steps that happened up to time(ff k ) inluding ff k, and let ψ k be the remainder. Lemma 5.3 fi 0 is onsistent with» fi. Proof: For any k, 1» k» s 1, pik any two steps, ß and ß 0 in fi k suh that ß» fi ß 0. Thus T (ß)» T (ß 0 ). (Reall that T is the original timing.) We only need to prove that T 0 (ß)» T 0 (ß 0 ), where T 0 is the new timing. Eah ofßand ß 0 belongs to either ffi k or ψ k and is either retimed or not. If ß is retimed in ffi k and ß 0 is not retimed in ffi k, then T 0 (ß)» T 0 (ß 0 ) sine T 0 (ß)» T (ß) and T 0 (ß 0 ) T (ß 0 ). All other ases an be proved similarly. Lemma 5.4 (fi 0 ;T 0 ) is admissible. Proof: We need to prove that all steps in (fi 0 ;T 0 ) satisfy the real time onstraint imposed on the semi-synhronous model. By the onstrution, no two onseutive steps by a proess in the system are loser than 1 in (fi 0 ;T 0 ); therefore, the lower bound on step time is preserved. We now show that the maximum time between any two onseutive steps of a proess is 2. Let ß and ß 0 be two onseutive steps of some proess p. First assume that ß and ß 0 both our in fi k for some k, ß is the i-th step of p in fi k and ß 0 is the i + 1-st. If ß and ß 0 are both in ffi k or are both in ψ k, then either there is no hange in their times or they are retimed to be 1 apart. Now suppose ß is in ffi k and ß 0 is in ψ k. By onstrution, T 0 (ß 0 ) T 0 (ß) = B = minfb ; log b ng 1 + 1» : Sine 1 < 2 =2, this differene is less than 2. Now suppose ß ours in fi k 1 and ß 0 ours in fi k. In the worst ase, ß is retimed to our at x 2 =2 and ß 0 is retimed to our at x + 2 =2, where x is the time at the end of fi k 1. (This is true by the definition of B.) So the time between ß and ß 0 is at most 2. Lemma 5.5 fi 0 ontains less than s sessions. Lemma 5.5 is true beause of the way ψ k 1 and ffi k are defined. The theorem now follows. 6 The Sporadi Model In the MPM, a lower bound 1 on step time and lower and upper bounds [d 1 ;d 2 ] on message delay are imposed. The orretness of our sporadi algorithm A(sp) depends on the following observation: If a proess p i reeives a message m from a proess p j at time t, then the message must have been sent no later than t d 1, beause it takes at least d 1 time for a message to be delivered. All the messages reeived by p i after t + d 2 d 1 must have been sent

9 after m was, beause it takes at most d 2 time for a message to be delivered. Using the above fat, eah proess broadasts a message at every step arrying its knowledge on the number of sesssions happened by the time that the step ours. After reeiving a message m whih says there are at least k 1 sessions in the system, a proess waits for d 2 d 1 time. After that, the proess waits to reeive at least a message from every proess. it is lear that there are at least k sessions in the system by the time beause every message reeived after t + d 2 d 1, where t is the time that m is sent, must have been sent after the time there are at least k 1 sessions. We first proeed by presenting the algorithm A(sp). A message is denoted m(i; V ), where i is the identifier of a sending proess p i and V is an integer in [0;s 1]. We let Λ be a don't are value for either of the fields and u = d 2 d 1. A(sp) for proess p i : B := b u 1 +1; ount := session := 0; msg buf := temp buf := ;; while( session < s 1) read buf i and let the set of messages obtained be M; msg buf := msg buf [ M; if for all j 2 [n], m(j; session) isinmsg buf then /* ondition 1 */ ount := 0; session := session + 1; elsif (ount > B) then temp buf := temp buf [ M; if for all j 2 [n], at least one m(j; Λ) is in temp buf then /* ondition 2 */ ount := 0; session := session + 1; temp buf := ;; end if; end if; broadast m(i; session); ount := ount + 1; end while; Enter an idle state. Theorem 6.1 A(sp) solves the (s; n)-session problem within time minfb u 1 +1fl+(u+2fl);d 2 +flg(s 2) + d 2 +2fl: Proof: Consider an arbitrary admissible timed omputation C of A(sp). The following lemma (proof omitted) an be used to prove the theorem. Lemma 6.2 There exists at least one step in C in whih a proess sets its session to k, 0» k» s 1. Let p i k be the first proess whih sets session i k to k 0. To inrement session, a proess must reeive a message(s) whih notifies the proess that there is at least one session after the last update to session. Let M k be the set of messages reeived by p i k that auses p i k to set session i k to k. (We define M 0 to be the empty set.) In more detail: If ondition 1 was true, M k is the set of message m(j; session i k 1) for all integers j 2 [n] in msg buf. If ondition 2 was true, M k is the set of messages in temp buf at the time. Assuming that m k is the message whih issent last among M k,we prove the following lemma. Lemma 6.3 Let ß be the step whih sends m k. There are at least k sessions by the time ß ours in C. Proof: We proeed by indution on k. For the basis, when k =0,itisalways true that there are at least 0 sessions in C. Indutively when k>0, assuming the lemma is true for k 1, we show that when ß ours, there are at least k sessions. Let fi be the step that sent m k 1 and ff be the step in whih p i k 1 sets session i k 1 to k 1. For p i k to update its session, one of onditions 1 and 2 in the algorithm must hold. First, assume that ondition 1 is true. Aording to the algorithm, a proess broadasts a message with a new session value k 1 after it sets its session to the new value k 1, before whih time there were k 1 sessions in C beause the indution hypothesis ditates that there were k 1 sessions in C when m k 1 was sent. Beause ff is the first step to set session i k 1 to k 1, all messages in M k,must have been sent when or after ff ours. Beause all proesses take at least one step to send messages in M k after there were at least k 1 sessions, there must be at least k sessions in C by the time that message m k is sent. Seond, assume that ondition 2 is true. Sine p i k is the first proess whih sets session i k to k,

10 session i k must have taken on k 1 aording to the proof of Lemma 6.2. Let t be the time when fi ours at whih time m k 1 was sent and t 0 be the time that p i k sets session i k to k 1. The message m k 1 must arrive atbuf i k 1 at time between [t + d 1 ;t+d 2 ] beause of the bounds on message delay. Thus, t 0 t d 1. Sine ount in the algorithm is reset whenever session is updated, when ount i k is equal to B most reently before when p i k sets session i k to k, say, at time t 00, at least time B 1 > u must have elapsed sine t 0. So, t 00 > t 0 + u t + d. Therefore, all messages reeived at t 00 or later must be sent after time t, at whih time there were k 1 sessions by the assumption. Sine at least one message is sent by eah proess after time t, there must be at least one additional step by all proesses between time t and the time ß ours. Therefore, there must be at least k sessions by the time ß ours. To analyze the time omplexity of the algorithm A(sp), we use the atual maximum step time fl sine in our sporadi model the upper bound on the step time is not available. We define for eah k, 2» k» s 1, T k = maxft : p i sets session i to k at time t in C for all p i 2 Rg. Lemma 6.4 For eah k, 2» k» s 1, T k+1» T k + minfb u 1 +1)fl+(u+2fl);d 2 +flg. Proof: Aording to the algorithm, a proess broadasts a message at every step. Thus, if proess p i reeives a message from proess p j at time t, it will reeive at least one more message from p j by time» t + u +2fl. Let p i k be the last proess to set session i k to k and p i k+1 be the last proess to set session i k to k +1. Wenow look at eah of the possible ases whih may ause session i k+1 to be updated to k +1: If ondition 2 is true when session i k+1 is updated to k +1, p i k+1 has made at least B = b u +1 steps sine the last update to session i k+1. Beause a proess must wait, sine then, at most u +2fl time to reeive another set of messages sent from all proesses, at most (b u +1)fl+(u+2fl) time has elapsed. If ondition 1 is true when session i k+1 is updated to k +1, let t be the time at whih p i k broadasts m(i k ;k); note that by definition, t = T k. Message m(i; k) must be reeived by p i k+1 by time t + d 2 + fl. Sine both onditions take at most minfb u + 1fl +(u+2fl);d 2 +flgtime to be true sine the last update to session, the lemma follows. From Lemmas 6.2 and 6.3, it follows that there are at least s 1 sessions at the time that m s 1 is sent. All proesses will eventually set their session's to s 1 ( sine session an't be bigger than s 1). Eah proess sets session to s 1 beause it reeives a ertain message. Therefore, there is at least one additional step by all proesses after there have been s 1 sessions in C. Thus, there are at least s sessions in C. By the algorithm, initially it takes at most d 2 +2fl to reeive at least one message from all proesses in order to aomplish the first session. Using Lemma 6.4, it is lear now that it takes at most minfb u 1 +1fl+(u+2fl);d 2 +flg(s 2)+d 2 +2fl. (This equals minf(b u 1 +3) fl+u; d 2 +flg(s 1)+fl if d 1 < b u 1 +1 fl). We now prove the lower bound. Theorem 6.5 No sporadi algorithm solves the (s; n)-session problem in the MPM within time < maxfb u 4 1 K; 1 g(s 1) where K = 2d21 (d 2 u. 2 ) Proof: The general struture of this proof follows that of Theorem 5.1. Let B = b u 4 1. When B K» 1, the lower bound holds beause a proess must exeute at least s steps to ahieve s sessions. Suppose that B K > 1. Assume, by way of ontradition, that there exists a sporadi algorithm, A, whih solves the (s; n)-session problem in the MPM within time Z stritly less than B K (s 1). Then dz=(b K)e»(s 1). We show that there exists an admissible timed omputation of A whih does not inlude s sessions. Let (ff; T ) be the admissible timed omputation in whih regular proesses take steps in round robin order and eah proess' ith step ours at time i K, and all message delays are exatly d 2. Eah onseutive group of steps for p 1 through p n is a round.

11 Therefore, there are r = dz=ke rounds required until termination in ff. Let ff = fifl, where fi ontains the first r rounds of ff. We will show that there is a reordering fi 0 of fi that ontains at most s 1 sessions. Thus ff 0 = fi 0 fl is a sequene with at most s 1 sessions. In order to get an admisssible omputation fi 0, we will assign new times (T 0 )toevery step in fi and let fi 0 be any total ordering of the steps in fi onsistent with the times, and then we will prove that (fi 0 ;T 0 ) is an admissible timed omputation whih results in the same global state as fi. A ontradition. Then we will show how to reorder fi to produe admissible timed omputation (ff 0, T 0 ) that results in the same global state as ff. Let fi = fi 1 :::fi m where m = dz=(b K)e. Eah fi k,1»k»m(exept possible the last one) onsists of B rounds, and for some sequene i 0 ;i 1 :::i m of integers in [1;n], eah omputation fragment fi k onsists of ffi k ψ k suh that: (i) ffi k does not ontain any step by proess p i k 1. (ii) ψ k does not ontain any step by proess p i k. Then define fi 0 to be ffi 1 ψ 1 ffi 2 ψ 2 :::ffi m ψ m. As in Lemma 5.5, we prove that fi 0 has at most s 1 session. Lemma 6.6 fi 0 has at most s 1 sessions. Sine all proesses in fl are in idle states, ff 0 has at most s 1 sessions. We need to show how to reorder every step in fi to get an admissible timed omputation (ff 0 ;T 0 ) whih preserves properties (i) and (ii), and results in the same global state as ff. Let us first assign times (a new mapping T 00 ) to every step, ß, infiinluding all the steps of the network N suh that T 00 (ß) = T (ß) 21 K. That is, every proess exept N takes a step at every 2 1 and eah round ours at every 2 1. Sine the delivery steps of N are also remapped, the message delay is redued to d 2 21 K = d 2 u. 2 C 0 = (fi; T 00 ) is an admissible timed omputation beause fi is a omputation, and the step times and message delays obey the sporadi time onstraint. From C 0, we onstrut (fi 0 ;T 0 ), an admissible omputation whih results in the same global state as fi 00 (and fi). We map T 00 to T 0 in order to get i k, i k 1, ffi k and ψ k for all k,1»k»m. For all k, hoose i k arbitrarily, as long as i k 6= i k 1. For all 0» j» m, let t j = B 2 1 j. t j is equal to the ending time of fi j in C Let ß be all steps of p i k and all the steps of N that deliver messages to p i k in fi k. Retime ß suh that T 0 (ß) =t k 1 +(T 00 (ß) t k 1 )=2. 2. Let ff be all steps of p i k 1 and all the steps of N that deliver messages to p i k 1 in fi k. Retime ff suh that T 0 (ff) =t k (t k T 00 (ff))=2. 3. All other steps in fi k are assigned the same time as they are under T 00. Fix up the states of the network in fi so that the state of the network is onsistent with all the send and reeive steps of regular proesses in fi. Let ffi k be the prefix of fik 0 up to the last step of p ik, and let ψ k be the remainder. Let fik 0 be any total ordering onsistent with T 0. We now prove the following lemma: Lemma 6.7 (fi 0 ;T 00 ) is an admissible timed omputation whih results in the same global state as fi. Proof: The time period of fi k in C 0,1»k»m, is equal to to B2 1 sine fi k in C 0 onsists of B rounds (exept the last one) and the step time is 1. Sine no step is retimed outside the time boundary of (fi k ;T 0 ), the time period of (fi k ;T 00 ) is also equal to (fik 0 ;T00 ). In (fi; T 00 ), the message delay of all messages is bigger than B2 1 by the definition of B, so that the messages sent infi k are never reeived in fi k. In (fi 0 ;T 0 ), the messages sent infik 0 are never reeived in fik 0 too beause no step is retimed outside the the time boundary of (fi k ;T 00 ). By the onstrution, in fi 0, the delivery steps of all messages are retimed with the steps that reeive the messages, so that every message is delivered always before reeived. Every step in fi 0 reeives the same set of messages as the orresponding step in fi does. Sine states of proesses are updated based only on the urrent state and the set of message

12 reeived, fi 0 is a omputation whih leads to the same global state as fi. Now, to prove that (fi 0 ;T 0 ) is admissible, we need to show that (fi 0 ;T 0 ) obeys the sporadi time onstraints. First, it is lear that every omputation step time in fi 0 is bigger than the minimum step time 1 by the onstrution. Seond, we need to prove the delay ofany message sentinfi 0 is within [d 2 u; d 2 ]. For any message m sent in fi 0, let ß i be the step of a proess in R whih sends m and ß j be the step of N whih delivers m. We need to prove that T 0 (ß j ) T 0 (ß i )is in [d 2 u; d 2 ]. Without loss of generality, assuming ß i is in fi 0 k, T 0 (ß j ) T 0 (ß i ) = T 00 (ß j ) T 00 (ß i )+ [T 0 (ß j ) T 00 (ß j )] [T 0 (ß i ) T 00 (ß i )]: It an be proved that for any step ß, u=4» T 0 (ß) T 00 (ß)» u=4: T 00 (ß j ) T 00 (ß i ) = B 2 1» d 2 u=2 from the onstrution of C 0. Therefore, it is lear that T 0 (ß j ) T 0 (ß i ) is always within [d 2 u; d 2 ]infi 0. Sine there exists an admissible timed omputation (fi 0 ;T 0 )ofawhih has at most s 1 sessions by Lemma 6.7 and Lemma 6.6, this ontradits the assumed existene of algorithm A. Therefore, Theorem 6.5 now follows. Aknowledgments: We thank Kevin Jeffay for helpful omments. This work was partially supported by an NSF PYI Award CCR , an IBM Faulty Development Award, and NSF grant CCR Referenes [1] H. Attiya, C. Dwork, N. Lynh and L. Stokmeyer, Bounds on the Time to Reah Agreement in the Presene of Timing Unertainty," Pro. 23rd ACM Symp. on Theory of Computing, 1991, pp [2] E. Arjomandi, M. Fisher and N. A. Lynh, Effiieny of Synhronous versus Asynhronous Distributed Systems," Journal of the ACM, Vol. 30, No. 3, 1983, pp [3] H. Attiya and N. A. Lynh, Time Bounds for Real-Time Proess Control in the Presene of Timing Unertainty," Pro. 10th Real-Time Systems Symp., De. 1989, pp [4] H. Attiya and M. Mavroniolas, Effiieny of Semi-Synhronous versus Asynhronous Networks," Pro. 28th Allerton Conf. on Communiations, Control and Computing, Ot. 1990, pp [5] B. A. Coan and G. Thomas, Agreeing on a Leader in Real-Time," Pro. 11th Real-Time Systems Symp., De [6] B. A. Coan and J. L. Welh, Transation Commit in a Realisti Timing Model," Distributed Computing, Vol. 4, 1990, pp [7] D. Dolev, C. Dwork and L. Stokmeyer, On the Minimal Synhronism Needed for Distributed Consensus," Journal of the ACM, Vol. 34, No. 1, Jan. 1987, pp [8] C. Dwork, N. Lynh and L. Stokmeyer, Consensus in the Presene of Partial Synhrony," SIAM Journal of Computing, Vol. 12, No. 13, Nov. 1983, pp [9] K. Jeffay, The Real-Time Produer/Consumer Paradigm: A Paradigm for the Effiient, Preditable Real-Time System," University of North Carolina at Chapel Hill, Department of Computer Siene, submitted for publiation. April [10] K. Jeffay, D. F. Stanat and C. U. Martel, On Optimal, Non-Preemptive Sheduling of Periodi and Sporadi Tasks," Pro. 12th IEEE Real-Time Systems Symp., De. 1991, pp [11] C. L. Liu and J. W. Layland, Sheduling Algorithms for Multiprogramming in a Hard Real- Time Environment," Journal of the ACM, Vol. 20, No. 1, Jan. 1973, pp [12] M. Mavroniolas, Effiieny of Semi-Synhronous versus Asynhronous Systems: Atomi Shared Memory," TR-03-92, Aiken Computation Lab., Harvard University, Jan [13] S. Ponzio, Consensus in the Presene of Timing Unertainty: Omission and Byzantine Failures," Pro. ACM Symp. on Priniples of Distributed Computing, Ot. 1991, pp

13 [14] G. Peterson and M. Fisher, Eonomial Solutions for the Critial Setion Problem in a Distributed System," Pro. 9th ACM Symp. on Theory of Computing, 1977, pp