J Lundelius and N Lynch. An upper and lower bound for clock synchronization. Information and Control, Vol. 62, Nos. 2/3, September 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....internal clock synchronization algorithm [11] They also proposed a clock synchronization algorithm which provides an optimal clock drift rate. A lower bound of BDCFE7G HFC I2JLK B 29556 on how tight clocks can be synchronized in the absence of failures and clock drift was published in [7], where denotes the maximum number of processes, and C=E:G CFIRJ are the maximum and the minimum communication delays, respectively. An upper and lower bound for the maximum deviation achievable by any synchronization algorithms was derived in [3] The authors show that the temporal ....

....on the value of correct clocks but also on the value of faulty clocks, i.e. the approximation of any clock by any two correct processes is within . This agreement is typically achieved by using a broadcast diffusion approach to disseminate the values of the clocks. Using the bound derived in [7] and considering that two hardware clocks can drift apart by up to during one round, the maximum deviation achievable using this approach is at least BZC=E7G[H8C I2JLK B] 8 159 . In [6] an algorithm is presented which achieves a maximum deviation of BZC=E7G H CFI2JLK neglecting the ....

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J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Control, 62(1):190--204, 1984.

....results in an infinite execution in which each processor takes infinitely many steps, but none of the configurations is legal. Let us make a make a short digression here. Theorem 1 has an interesting corollary for clock synchronization: one of the popular schemes for clock synchronization [LL84] is repeated averaging . Roughly speaking, in the repeated averaging rule each node sets its value to be the average value of its neighbors, while advancing the clock if this average is close enough to its own value. 2 3 4 3 3 4 3 2 1 (a) 1 2 3 4 3 2 4 3 2 1 (b) 3 2 3 4 ....

J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Control, 62(2-3):190--204, 1984.

....modeling decisions and a stimulus for the development of algorithm verification techniques. Similar results should be possible for real time systems. Some examples of complexity results that have already been obtained for real time systems are the many results on clock synchronization, including [8, 11, 17, 20, 32] (see [31] for a survey) In this paper, we embark on a study of complexity results for real time systems. We begin this study by considering timing based variations of certain problems that have previously been studied in asynchronous concurrent systems. In particular, we study a variant of the ....

....composition of the manager and the clock, with the I O automaton hiding operating applied to hide the TICK actions) See Fig. 2. Note that the timed automaton model forces us to model the step time of the manager process explicitly. Other models (e.g. the one used for clock sychronization in [32]) might avoid this level of detail by hypothesizing that the manager s steps are triggered only by input events such as clock ticks or requests. We regard such a model (informally) as a limiting case of our model, as the upper bound on manager step time approaches zero. GRANT(D manager . TICK ....

WELCH, J. L., ^NI) LYNCH, N. (1984), An upper and lower bound for clock synchronization, Inform. and Control 62, Nos. 2/3 (August/September 1984), pp. 190-204.

....time from the synchronization path and achieves precision in the order of 10s using commercial o# the shelf technology. Unknown system properties like non deterministic message delay and synchronization message pattern make it di#cult to apply bounds on the achievable precision as proposed by [8, 7, 10, 12, 11]. Therefore we propose an analysis of clock synchronization algorithms under completely unknown system specifications in terms of message delay and message pattern. Instead of bounds on the achievable precision, we propose two properties that describe good algorithms. Safe synchronization never ....

Jennifer Lundelius and Nancy Lynch. An upper and lower bound for clock synchronization. Information and Control, 62(2/3):190--204, August/September 1984.

....in case a loss of consistency is caused by a violation of the fault hypothesis. The basic algorithms that provide this consistent distributed computing base (clock synchronization and membership) have been analyzed by formal methods and are implemented once and for all in silicon [56] 57] 58] [59], 60] 61] The application does not need to be concerned with the implementation and validation of the complex distributed agreement protocols that are needed to establish consistency in a distributed system. The architecture is replica deterministic, which means that any observed ....

J. Lundelius and N. Lynch. An Upper and Lower Bound for Clock Synchronization. Information and Control, 62(2/3):190--204, 1984.

....for n processes even if all message journeys are confined to a single hop. In [ 15 ] an O(n) message solution relies on the presence of an embedded ring topology, which excludes tree topologies, though these are a natural topology for the data farming applications of concern for us. Other work [ 16 ] though important theoretically, relies on complete graphs. Finally, statistical post processing of the trace record [ 17 ] represents an alternative approach. A method involving interpolation between start and end timing pulses [ 18 ] was initially explored but, though suitable for gauging ....

....led to ordering errors. The maximum error for varying round trip times can be calculated from equation 9 using the measured minimum round trip time, 6:0 Theta 10 Gamma5 s, a clock drift of 10 Gamma5 s and a clock resolution of 10 Gamma6 s. With a measured net bandwidth of approximately 16 MHz on the Paramid and with the diameter of the network only three, the maximum error for our set up is well within a 0.5 ms boundary. After four timings, the refresh period was set once and for all. It would probably be too disruptive to re sample at a later time and would add to the complexity ....

J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Control, 62(2-3):190--204, 1984.

....use local clocks which do not indicate the same time. However, aside from relatively considerations, it usually holds that there is some bounded proportion between elapsed 1 local time spans [Vit84] Techniques such as message passing [Lan78] can be used to keep local clocks almost synchronized [LL84]. In this paper we do assume that local clocks su#er a small bounded drift [KTZ88] Since network delays relate to the actual transaction origination times but the receiving site associates transaction arrivals to their (logical) timestamps, one has to relate logical and global time in order to ....

Lundelins J., Lynch N., "An upper and lower bound for clock synchronization ",Information and Control 62(2-3) 190-204, 1984

....tolerate message losses. In this paper, we present a clock synchronization protocol that enhances the IEEE 802.11 standard by achieving high precision even in the presence of message losses. 2 2 Clock synchronization In [Lam 78] the necessity of clock synchronization in general is motivated, LuL 84] gives a lower bound on the precision that can be achieved (with deterministic algorithms) and proposes an algorithm achieving that precision. In [Cr 89] a probabilistic algorithm is introduced that achieves a higher precision (at the price of being non deterministic) Examples of fault tolerant ....

J. Lundelius, and N. Lynch. An upper and lower bound for clock synchronization. Inf. Control 62, 1984, pp. 190-204.

....delay is expected to vary in the range [d min ; d max ] with d var = d max Gamma d min being referred to as the message delay variation and d var =2 being referred to as the reading error. Naturally, d var affects the worst case clock synchronization tightness ( Delta int ) Lundelius and Lynch [4] proved that in a system with n clocks and message delay variation d var the clock synchronization tightness cannot be better than d var (1 Gamma 1=n) Such a lower bound holds under the strong assumptions that all clocks run at a perfect rate and that there are no failures in the system. Under ....

....implementation, and they revealed that the message delay can vary by several hundreds of milliseconds. A theoretical lower bound for the worst case clock synchronization tightness ( Delta int ) that can be achieved in an environment with message delay variation d var is d var (1 Gamma 1=n) [4]. According to this lower bound, one should expect the Delta int of our implementation to be at least as large as the message delay variation (d var ) in the UNIX environment. Our experiments show that one of the algorithms that was tested, namely SWA, can beat this lower bound with very high ....

J. Lundelius and N. Lynch, "An Upper and Lower Bound for Clock Synchronization," Information and Control, No. 62, 1984, pp. 190-204.

....to vary in the range [d min ; d max ] with d var = d max Gamma d min being referred to as the message delay variation and d var =2 being referred to as the reading error. Naturally, d var affects the internal clock synchronization tightness in the system ( Delta int ) Lundelius and Lynch [5] proved that in a system with n clocks and message delay variation d var the clock synchronization tightness cannot be better than d var (1 Gamma 1=n) Such a lower bound holds under the strong assumptions that all clocks run at a perfect rate and that there are no failures in the system. Under ....

....algorithms and they revealed that the message delay can vary by several hundreds of milliseconds. A theoretical lower bound for the worstcase clock synchronization tightness ( Delta int ) which can be achieved in an environment with message delay variation d var is d var (1 Gamma 1=n) [5]. According to this lower bound, one should expect the Delta int of our implementation to be at least as large as the message delay variation (d var ) in the UNIX environment. Our experiments show that one of the algorithms that was tested, namely SWA, can beat this lower bound with very high ....

J. Lundelius and N. Lynch, "An Upper and Lower Bound for Clock Synchronization," Information and Control, No. 62, 1984, pp. 190-204.

....delay d max in the system. Message delay is expected to vary in the range [d min ; d max ] with d var = d max Gamma d min being referred to as the message delay variation. Naturally, d var affects the internal clock synchronization tightness in the system ( Delta int ) Lundelius and Lynch [10] proved that in a system with n clocks and message delay variation d var the clock synchronization tightness cannot be better than d var (1 Gamma 1=n) 4 Use convergence function to compute a clock correction term from the clock deviation vector. Apply clock correction term to local clock. ....

J. Lundelius and N. Lynch, "An Upper and Lower Bound for Clock Synchronization," Information and Control, No. 62, 1984, pp. 190-204.

....clock synchronization, which is beyond the scope of this thesis. 1. 2 Previous Work Different variants of the clock synchronization problem have been the target of a vast amount of research from both practical viewpoint (e.g. 26, 6, 24, 28, 1, 15] and theoretical viewpoint (e.g. [16, 19, 7, 13, 33, 3], surveys [31, 30] and references therein) the exact definition of the problem depends both on the intended use of the clocks and on the specific underlying system. The large number of variants is justified by the wide spectrum of applications. One of the popular variants studied theoretically ....

....variants is justified by the wide spectrum of applications. One of the popular variants studied theoretically is internal synchronization in the case where all clocks in the system are assumed to run exactly at the rate of real time (we call such clocks drift free hereafter) Lundelius and Lynch [19] consider the case in which there is a communication link between each pair of processors, and message latency bounds are identical for all links in the system. For this case, they present a synchronization algorithm 1 In this thesis, numbers range over R [ f1; Gamma1g unless explicitly ....

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J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Computation, 62(2-3):190--204, 1984. 144

....is a small bound Sigma on the amount by which a correct clock is changed at each resynchronization. 1 The value of f depends on the failure mode assumed. For instance, to tolerate f processor Byzantine failures without authentification, f must satisfy f n 3 , with n the number of processors [LL84]. Irisa A Taxonomy of Clock Synchronization Algorithms 9 Another way to rule out trivial solutions is to require that logical clocks are permanently within a narrow envelope of real time (property 3) Property 3 (Accuracy) For any correct processor p i 2 P, and for any real time t, there exists ....

J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Control, 62(2/3):190--204, August 1984.

....disseminating GPS time to all nodes in a distributed system is still a challenge. As more and more distributed real time systems are now being built atop COTStype LANs, the most desirable solution would be using the data network for time distribution, as done e.g. in NTP [Mil91] However, LWL84] revealed that the worstcase synchronization tightness achieved by any clock synchronization scheme depends on the worst case uncertainty ( maximum variability, jitter) in the end to end transmission delay. For typical LANs, lies in the ms range, which makes it impossible to use a simple ....

Jennifer Lundelius-Welch and Nancy A. Lynch. An upper and lower bound for clock synchronization. Information and Control, 62:190--204, 1984.

....of two categories, deterministic or probabilistic. A deterministic method assumes the existence of a upper bound on communication delays. As long as all message delays experienced are less than the upper bound a deterministic method can always successfully read a remote clock. It has been shown in [5] that 10 if a deterministic method assumes a bounded message transmission delay that the smallest interval n clocks can be synchronized to is (max Gamma min) 1 Gamma 1=n) where max is the upper bound, and min is the lower bound on message delays. However if message delays are considered ....

N. Lynch J. Lundelious. An upper and lower bound for clock synchronization. Information and Control, 62:190--204, 1984.

....results in an infinite execution in which each processor takes infinitely many steps, but none of the configurations is legal. Let us make a make a short digression here. Theorem 1 has an interesting corollary for clock synchronization: one of the popular schemes for clock synchronization [LL84] is repeated averaging . Roughly speaking, in the repeated averaging rule each node sets its value to be the average value of its neighbors, while advancing the clock if this average is close enough to its own value. Corollary 1 Repeated averaging does not stabilize. Proof Sketch: The scenario ....

J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Control, 62(2-3):190--204, 1984.

....nodes. In this case the delay between node transmissions is minimal and may be tolerated. However, as the number of nodes in the network grows larger, LCA will impose greater delays between node transmissions in the TDMA communication scheme and may be unacceptable. Additionally, it has been shown [11] that as communications increase the amount of skew in a synchronous timer also increases, thereby degrading the performance of the overall system or introducing additional delay and overhead. The Max Min heuristic [12] was developed to extend the notion of 1 hop clusters and generalizes cluster ....

Jennifer Lundelius and Nancy Lynch. An Upper and Lower Bound for Clock Synchronization. Information and Control, Vol. 62 1984.

....logical systems to deal with di#erent time scales have been proposed in [4, 7, 16] However they cannot su#ciently model time based on local clocks which may drift in distributed systems. Also, some researchers have explored methods for agreement on processes time by using clock synchronization [10, 13] and for maintaining local time based on causality by using time stamps [9, 12] However the former methods cannot cope with systems where communication delay is unpredictable. The latter methods lose real time duration between events. The goal of this paper is to develop a formalism for reasoning ....

Lundelius, J., and Lynch, N., An Upper and Lower Bound for Clock Synchronization, Information and Control, Vol.62, p190-204, 1984.

....the clock drifts ae p , which can be balanced by the resynchronization period PS . In our example the latter is mainly responsible for the maximum of 33 s. Furthermore, we can observe that precision (t) remains bounded regardless of the common increase decrease of accuracies ff p (t) According to [6], nothing can be done to beat the precision arising from the transmission delay uncertainties, except by shrinking them with sophisticated timestamping techniques. Our solution [11] for that subproblem inserts a timestamp on thefly into the memory holding a packet, hence sojourn times in waiting ....

J. Lundelius-Welch and N. Lynch. An upper and lower bound for clock synchronization. Information and Control, 62(2/3):190--204, 1984.

....by physical clocks on local processors instead of any global clock. However, such physical clocks are not perfect. Their measurement rates are di#erent and may drift. Indeed, many clock synchronization techniques for compensating for clock drift have already been explored, for example see [40, 43, 45, 48]. However, these techniques are not available in all distributed systems. Even when such techniques are supported, if the required precision of time information is finer than the intervals of clock synchronizations, di#erences and uncertainties among clocks may lead cooperations among distributed ....

.... 0 and #, # 1 and # 2 are given constants in R 0 . 3) Monotone Clock: # is called a monotone clock if for any t 1 ,t 2 #T such that t 1 t 2 : max#(t 1 ) min#(t 2 ) # # Note that the clock given in (2) corresponds a clock adjusted by synchronization mechanisms, for example see [40, 43, 45]. Therefore, the clock can satisfy the condition of synchronized clocks given in [43] when choosing # 1 # 2 to be less than # max : #i, j # I, #t #T : # i (t) # j (t) ##max CHAPTER 4. LOCALITY IN TIME 58 where # i and # j are clock mappings for i th and j th clocks, and # max means ....

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J. Lundelius and N. Lynch. An Upper and Lower Bound for Clock Synchronization. Information and Control, 62:190--204, 1984.

....nodes. In this case the delay between node transmissions is minimal and may be tolerated. However, as the number of nodes in the network grows larger, LCA will impose greater delays between node transmissions in the TDMA communication scheme and may be unacceptable. Additionally, it has been shown [15] that as communications increase the amount of skew in a synchronous timer also increases, thereby degrading the performance of the overall system or introducing additional delay and overhead. Other solutions base the election of clusterheads on degree of connectivity [11] not node id. Each node ....

Jennifer Lundelius and Nancy Lynch. An Upper and Lower Bound for Clock Synchronization. Information and Control, Vol. 62 1984.

....clocks C i , C j it is guaranteed: Precision: External clock synchronization aims at maintaining virtual clocks within some maximum deviation from a time reference external to the system, i. e for each correct clock C i it is guaranteed: Accuracy: Internal clock synchronization algorithms [43,26,30] guarantee precision in case of known bounds on transmission delays of the network. Otherwise, internal clock synchronization is best effort [9,46] and precision d cannot be a priori determined for all t. As accuracy a always implies precision 2a, externally synchronized clocks are also internally ....

J. Lundelius and N. Lynch. An Upper and Lower Bound for Clock Synchronization. Information and Control, Vol. 62, No. 2-3, 1984.

....calculus of time bounds. Dolev et al. 8] use similar techniques to obtain synchronization algorithms for relativistic systems. Much effort has been devoted to studying internal synchronization, where the goal is to synchronize clocks within a system in which real time is not available (see, e.g. [11, 12, 6, 10, 24, 1], surveys [22, 21] and references therein) The approach of comparing the synchronization bounds to the best possible bound for the given execution was first presented by Attiya et al. in [1] where they studied internal synchronization. The work in [1] extended the work of Halpern et al. 10] ....

....as a game against nature, which means that it is assumed that the execution should be taken as if it is generated by an adversary whose aim is to provide as little information as possible within the system specification. The work in [10] in turn, extended the work of Lundelius and Lynch [12], which analyzed internal synchronization in fully connected systems where all link specifications are identical. Much work is also devoted to the issue of fault tolerant clock synchronization (e.g. 21, 13, 7] which falls outside the scope of this paper. As for practical work, two prominent ....

J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Computation, 62(2-3):190--204, 1984.

....systems to deal with different time scales have been proposed in [4, 7, 16] However they cannot sufficiently model time based on local clocks which may drift in distributed systems. Also, some researchers have explored methods for agreement on processes time by using clock synchronization [10, 13] and for maintaining local time based on causality by using time stamps [9, 12] However the former methods cannot cope with systems where communication delay is unpredictable. The latter methods lose real time duration between events. The goal of this paper is to develop a formalism for reasoning ....

Lundelius, J., and Lynch, N., An Upper and Lower Bound for Clock Synchronization, Information and Control, Vol.62, p190-204, 1984.

....on message transmission, but require that their probability distribution is known. Deterministic protocols require the existence of an upper bound max to message transmission delay. We briefly review some papers concerning the topic of clock synchronization, beginning with deterministic ones. In [7] the problem of synchronizing n clocks in a fully connected network is considered. Even if the clocks run at the same rate of real time and there are no failures, it is impossible to synchronize clocks closer than (1 Gamma 1 n ) where = max Gamma min. In [2] clock synchronization is ....

J.Lundelius, N.Lynch "An upper and a lower bound for clock synchronization", Information and Control, n. 62, 1984, pp 190-204.

....the problem of synchronizing the clocks of processors in a failure free distributed system when the hardware clocks do not drift but there is uncertainty in the message delays. Our goal is to develop closed form expressions for how closely the clocks can be synchronized. Lundelius and Lynch [5] showed that the local clocks of n processors in a fully connected network with the same uncertainty u on each link cannot be synchronized any more closely than u Delta (1 Gamma 1 n ) They provide a simple algorithm that achieves this bound. Halpern et al. 3] subsequently extended this ....

....bounds for k ary m cube networks, first for networks without wrap around and then for networks with wrap around. We conclude in Section 5 with some open problems. 2 Definitions and Prior Results We first describe our formal model and problem definition (following [2] which is based on those in [5] and [3] Then we summarize previous results upon which our results rely, namely the conversion of any communication network to an equivalent clique and the technique of shifting executions. 2.1 Model We consider a set of n processors p 0 through p n Gamma1 , connected via point to point links. ....

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J. Lundelius and N. A. Lynch, "An Upper and Lower Bound for Clock Synchronization," Information and Control, vol. 62, pp. 190-204, 1984.

.... or GPS (Global Positioning System) A number of deterministic clock synchronization algorithms has been published [4, 8, 10, 14] Most of them are structured around periodic rounds of broadcast communication and address fault tolerance aspects; for a survey see [13] An important result [11] fixes the upper bound ffi = max Gamma min) 1 Gamma 1=N) on Universit e Catholique de Louvain y Universit a degli Studi di Pisa the synchronization precision for deterministic algorithms executing in a distributed system with N nodes. Here min and max are the minimum and maximum network ....

J. Lundelius and N.A. Lynch. An upper and lower bound for clock synchronization. Information Control, 62:190--204, 1984.

....by physical clocks on local processors instead of any global clock. However, such physical clocks are not perfect. Their measurement rates are different and may drift. Indeed, many clock synchronization techniques for compensating for clock drift have already been explored, for example see [40, 43, 45, 48]. However, these techniques are not available in all distributed systems. Even when such techniques are supported, if the required precision of time information is finer than the intervals of clock synchronizations, differences and uncertainties among clocks may lead cooperations among distributed ....

....f 0 g and ffi, ffl 1 and ffl 2 are given constants in R 0 . 3) Monotone Clock: 8 is called a monotone clock if for any t 1 ; t 2 2 T such that t 1 t 2 : max8(t 1 ) min8(t 2 ) u t Note that the clock given in (2) corresponds a clock adjusted by synchronization mechanisms, for example see [40, 43, 45]. Therefore, the clock can satisfy the condition of synchronized clocks given in [43] when choosing ffl 1 ffl 2 to be less than ffl max : 8i; j 2 I; 8t 2 T : j8 i (t) 0 8 j (t)j ffl max CHAPTER 4. LOCALITY IN TIME 58 where 8 i and 8 j are clock mappings for i th and j th clocks, and ffl max ....

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J. Lundelius and N. Lynch. An Upper and Lower Bound for Clock Synchronization. Information and Control, 62:190--204, 1984.

....order to enforce time constraints. Clock synchronization in distributed systems is an interesting and challenging problem. Much work has been done on bounding the possible deviation between clocks on different nodes (Dolev et al. 1986; Kopetz and Ochsenreiter 1987; Lamport and Melliar Smith 1985; Lundelius and Lynch 1984; Ramanathan et al. 1990a, 1990b; Volz et al. 1991; Welch and Lynch 1988) However, further work remains to be done in areas such as examining the performance of probabilistic clock synchronization algorithms (Arvind 1989; Arvind et al. 1991b; Cristian 1989) 38 AUTHOR NAME OR NAMES Multi Hop ....

Lundelius, J., and N. Lynch. 1984. An upper and lower bound for clock synchronization. Information and Control, 62(2/3):190--204.

....of broadcast messages and the ability to achieve certain goals in distributed systems. We have taken clock synchronization as a paradigm problem to study in this context because it has been so well studied, both in the presence of faults (see, for example, 1, 5, 9, 11, 13, 15] and without faults [8, 10, 12, 16]. The basic problem we consider is the same as that considered in [8, 12] we assume that each process has a clock that runs at the rate of real time, and the problem is to synchronize these clocks as tightly as possible. Of course, in practice, clocks drift apart. However, the drift is typically ....

.... We have taken clock synchronization as a paradigm problem to study in this context because it has been so well studied, both in the presence of faults (see, for example, 1, 5, 9, 11, 13, 15] and without faults [8, 10, 12, 16] The basic problem we consider is the same as that considered in [8, 12]: we assume that each process has a clock that runs at the rate of real time, and the problem is to synchronize these clocks as tightly as possible. Of course, in practice, clocks drift apart. However, the drift is typically small, and by ignoring it here, we can examine the impact of broadcasting ....

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J. Lundelius and N. Lynch, "An upper and lower bound for clock synchronization," Information and Control 62, 1984, pp. 190--204.

....of Synch on expiration of their own timers; but, since p j broadcasts synch at time t Gamma d and completes Synch at time t, it clearly follows that p j can only modify its local clock time at t and set it identical to that of p i . We remark that, by the results of Lundelius and Lynch in [7], an accuracy of (1 Gamma 1 n )u is achievable in the imperfect clocks model. This accuracy is slightly better than u, but we chose to present and use the slightly weaker one since it can be achieved by a much simpler algorithm than the one in [7] which might also be of independent interest. ....

....that, by the results of Lundelius and Lynch in [7] an accuracy of (1 Gamma 1 n )u is achievable in the imperfect clocks model. This accuracy is slightly better than u, but we chose to present and use the slightly weaker one since it can be achieved by a much simpler algorithm than the one in [7], which might also be of independent interest. Furthermore, the better accuracy achieved in [7] does not seem to considerably improve our subsequent results, if it improves them at all. We conclude this subsection with a simple observation encompassing the common knowledge acquired by the ....

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J. Lundelius and N. Lynch, "An Upper and Lower Bound for Clock Synchronization," Information and Control, Vol. 62, Nos. 2/3 (August/September 1984), pp. 190--204.

....Therefore in this execution q does not detect the failure of p within time B. Thus the protocol cannot be correct; this contradiction proves the lemma. Our lower bound proof uses the retiming techniques of shifting events in time and shrinking portions of executions that were used in [2] and [4]. The basic strategy of the proof is as follows. Beginning with an execution fi given by Lemma 4.1, we know that if p were to fail during the step at which it sends m 1 , then q would declare p faulty by time t 1 T . We would like to show that if the bound T guaranteed by the algorithm is ....

J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Control, Vol. 62, Nos. 2/3 (August /September 1984), pp. 190--204.

....speaking, p i slices each time period of 3u b into an interval of length 3u in which actions on a write request may not be initiated, followed by an interval of length b in which they may. Upon a W rite i (X; v) event and when in the 6 Although, by the results of Lundelius and Lynch in [11], an accuracy of u is not optimal, our synchronization algorithm is extremely simple and an accuracy of u suffices here. appropriate time interval, p i broadcasts an update(X; v) message and waits for an additional (1 Gamma fi)d time to set X i to v and issue Ack i (X) We now describe the ....

....the results in Section 3 are optimal up to a small additive number of multiples of u. The lower bound proof combines the use of various methods already common in the theory of distributed computing, namely, symmetry arguments and the technique of shifting executions (originally introduced in [11]) with a novel technique of augmenting executions to causally link them. The known lower bound of d Gamma u on message delay time is used in constructing the executions, to achieve non availability of knowledge to processes. As part of this proof, we derive a property of any sequentially ....

J. Lundelius and N. Lynch, "An Upper and Lower Bound for Clock Synchronization, " Information and Control, Vol. 62, No. 2/3, pp. 190--204, August/September 1984.

....signatures. The resynchronization algorithm is described in Section 3 and analyzed in Section 4. The worst case difference between logical clocks that is guaranteed by our algorithm is almost as small as possible, but a careful discussion of this property is beyond the scope of this paper (v. [DHS,HMM,LL]) We discuss issues related to initialization and joining in Section 5. In Section 6, we present a synchronous update service, which enables all correct processes to agree on which processes are currently joined; this service plays a key role in our join algorithm. The join algorithm is presented ....

....considers a model in which there exists a minimal bound on the time it takes a message to travel along a link. We have restricted our analysis to the simpler model based on A2. We leave it to the reader to verify that our results could also be obtained using the refined versions. As is shown in [HMM,LL], the tightness or precision of the synchronization need depend only on the uncertainty of message transmission and processing time, not on its upper bound. However, our experience suggests that for many practical environments, the uncertainty is essentially the upper bound, justifying this ....

J. Lundelius and N. Lynch, "An upper and lower bound for clock synchronization," Information and Control 62, 1984, pp. 190-204.

....synchronization algorithm should not deteriorate the excellent quality of external synchronization. In fact, a major limitation of all known software clock synchronization algorithms designed for arbitrary networks, is that precision is limited either by the variance of the message delivery delay[22], or worse, by its upper bound[35] This problem may be minimized with hardware support, either by implementing clock synchronization exclusively by hardware[13, 19] or by using hybrid schemes[27, 17] which attempt at reducing that variance. In large scale systems, the distance, added to the very ....

Jennifer Lundelius and Nancy Lynch. An upper and lower bound for clock synchronization. Information and Control, 62:190--204, 1984.

....bounds with a less precise dependency on the timing uncertainty than we obtain here. The work presented here is part of an emerging study of the real time behavior of distributed systems. Other work in this area includes the extensive literature on clock synchronization algorithms, for example, [10, 16, 20, 21, 30, 32], and on specific problems, such as mutual exclusion [3] synchronization [4, 29] leader election [6] and failure detection [18, 27] The rest of this paper is organized as follows. In Section 2, we present the definitions of the model and of the failure types we address. We next present two ....

Lundelius, J., and Lynch, N. An upper and lower bound for clock synchronization. Information and Control 62, 2/3 (1984), 190--204.

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J. Lundelius-Welch and N. A. Lynch. An upper and lower bound for clock synchronization. Information and Control, 62:190--204, 1984.

....B, respectively. Therefore in this execution q does not detect the failure of p within time B. This is a contradiction on the assumed protocol. Our lower bound proof uses the retiming techniques of shifting events in time and shrinking portions of executions that were developed in [AL89] and [LL84]. Theorem 5.2 In a system with links of capacity and delay d, no correct timeout protocol can guarantee failures to be detected within less than time min(2Cd d= C d= Cd d) Proof: Let T = min(2Cd d= C d= Cd d) For contradiction, assume the existence of a protocol that guarantees a ....

J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Control, Vol. 62, Nos. 2/3 (August/September 1984), pp. 190-- 204.

....variants is justified by the wide spectrum of applications. One of the popular variants studied theoretically is internal synchronization in the case where all clocks in the system are assumed to run exactly at the rate of real time (we call such clocks ift fire hereafter) Lundelius and Lynch [19] consider the case in which there is a communication link between each pair of processorsEand message latency bounds are identical for all links in the system. For this caseEthey present a synchronization algorithm In this thesis, numbers range over R U o, o unless explicitly indicated ....

....range over R U o, o unless explicitly indicated otherwise. Square brackets are used to denote intervals, including the case of infinite intervals. that gives optimal tightness in the worst possible scenario allowable by the system speci fications. Halpern et al. 13] generalized the results of [19] to networks whose underlying topology is arbitraryFand whose message latency bounds may be different for each link. The main idea in the analysis of [13] is to formulate the problem as a linear program; solv ing this programFthey find the worst case scenarioFand an algorithm is presented so that ....

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J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Computation1'62(2-3):190 2041.

....initial state q 0;i and a distinguished fail state. A configuration is a vector C = q 1 ; q n ) where q i is the local state of p i ; denote state i (C) q i . The initial configuration is the vector (q 0;1 ; q 0;n ) Processes communicate by sending messages 4 See [13, 22, 25, 27, 37, 39], for example. 5 These definitions could be expressed in terms of the general timed automaton model described in [1] and [29] however, we choose here to present the definitions directly, in order to avoid the intervening layer of definitions. 5 (taken from some alphabet M) to each other. A ....

Lundelius, J., and Lynch, N. An upper and lower bound for clock synchronization. Information and Control 62, 2/3 (August/September 1984), 190--204.

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J Lundelius and N Lynch. An upper and lower bound for clock synchronization. Information and Control, Vol. 62, Nos. 2/3, September 1984.

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L. Lundelius and N. Lynch. An Upper and Lower Bound for Clock Synchronization. Inform. Control, 62:190--204, 1984.

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J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Control, 62(2/3):190--204, Aug./Sept. 1984.

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Jennifer Lundelius and Nancy Lynch. An upper and lower bound for clock synchronization. Information and Control, 62(2/3), pages 190--204, August/September 1984.

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J. Lundelius, and N. Lynch, An Upper and Lower Bound for Clock Synchronization,  Information and Control, Vol. 62, pp. 190-205, Aug/Sep. 1984.

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J.Lundelius and N.Lynch, An Upper and Lower Bound for Clock Synchronization, Information and Control, 1984, Vol. 62, pp. 190--204.

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J. Lundelius and N. Lynch, "An Upper and Lower Bound for Clock Synchronization," Information and Control, No. 62, 1984, pp. 190-204. 8

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#LL84# J. Lundelius and N. Lynch, #An Upper and Lower Bound for Clock Synchronization, " Information and Control, 62:190#

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J. Lundelius and N. Lynch. An upper and lower bound for clock synchronization. Information and Control, Vol. 62, Nos. 2/3 (1984), pp. 190--204.

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J. Lundelius, N. Lynch, An Upper and Lower Bound for Clock Synchronization, Inf. & Control 62, 1984, 190-204.

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