T. K. Srikanth and Sam Toueg. Optimal clock synchronization. J-ACM, 34(3):626--645, July 1987.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....synchronous systems. chronization [14, 18] Following an informal introduction in Section 2 and its formalization in Section 3, we show in Section 4 that a properly adapted implementation of the consistent broadcast primitive and hence the clock synchronization algorithm of Srikanth Toueg [21] can be analyzed under this model. Some conclusions in Section 5 eventually round off the paper. 2 Informal Overview We consider synchronous distributed systems consisting of n nodes, which are interconnected by a fully connected point to point or broadcast type network. The basic assumptions ....

....identically since we cannot assume identical information at different nodes, even in the faultless case. 4 Consistent Broadcasting In this section, we will prove that (a variant of) the nonauthenticated consistent broadcast primitive and hence the clock synchronization algorithms of [21] can be employed in distributed systems under a perception based hybrid fault model. Consistent broadcasting is implemented by means of two functions, broadcast and deliver, which can be used to disseminate a message (msg) throughout the system, in a way that guarantees properties P1 P3 defined ....

[Article contains additional citation context not shown here]

T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

....synchronous systems. chronization [16, 20] Following an informal introduction in Section 2 and its formalization in Section 3, we show in Section 4 that a properly adapted implementation of the consistent broadcast primitive and hence the clock synchronization algorithm of Srikanth Toueg [23] can be analyzed under this model. Section 5 is devoted to the analysis of our Fault Tolerant Interval convergence function [20] which generalizes Fault Tolerant Midpoint [7] under a comparable interval based fault model. Some conclusions in Section 6 eventually round off the paper. 2 Informal ....

....identically since we cannot assume identical information at different nodes, even in the faultless case. 4 Consistent Broadcasting In this section, we will prove that (a variant of) the nonauthenticated consistent broadcast primitive and hence the clock synchronization algorithms of [23] can be employed in distributed systems under a perception based hybrid fault model. Consistent broadcasting is implemented by means of two functions, broadcast and deliver, which can be used to disseminate a message (msg) throughout the system, in a way that guarantees properties P1 P3 defined ....

[Article contains additional citation context not shown here]

T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

....used in modern textbooks on very different subjects, like communication protocols and data base systems. The clock synchronization algorithms of Halpern Simons Strong Dolev [3] Lundelius Lynch [5] Lamport Melliar Smith [4] Mahaney Schneider [6] Cristian Aghili Strong [1] Srikant Toueg [7] (and the many variants derived from them) but also related theoretical work form a well established basis for applications that require synchronized clocks. So why wasting time and money by publishing a 1997 special issue devoted to global time in distributed systems at all Because enabling ....

T.K. Srikanth, S. Toueg. Optimal Clock Synchronization, Journal of the ACM, 34(3), July 1987, p. 626--645.

....in interval ( Q 5 5 Q (1 ) Q ) G 5 E RQ 3 1 E (1 ) u t) The constant is called the maximum discontinuity of a 2 clock. It accounts for the granularity and the adjustments of the clocks. Synchronization algorithms with = have an optimal clock drift rate [7]. A synchronization algorithm is correct when it satisfies the bounded deviation and clock drift requirements above. 4 Overview Clocks are synchronized in rounds, where correct processes approximately agree on the time when a round starts. At the end of each round, after reading all clocks, each ....

....Appendix that the deviation of correct clocks synchronized by the DFTM is bounded by o . Hence the maximum deviation is optimal, because we neglect summands of the form drift rate of correct clocks is limited by (see Appendix) and therefore the maximum drift rate of clocks is optimal [7]. We derive in the Appendix that the maximum discontinuity of the DFTM is bounded by ) o ) 7 Conclusion This paper proposes a new differential fault tolerant midpoint convergence function for fault tolerant internal clock synchronization in the presence of arbitrary process ....

T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, Jul 1987.

....this requirement can be expressed as (bounded deviation) For any processes correct at time Go U D , B CU tz , A second requirement is that the drift rate of a correct virtual clock be bounded by a constant 4vA . Synchronization algorithms for which f are called optimal [9] If denotes the maximum discontinuity of virtual clocks, this second requirement is expressed as: clock drift) For any process . m0]M2 [ G 94 z 7 9 , B Cp U j B =U : S p 4 z jr 9 , A clock synchronization algorithm is correct when it satisfies the bounded ....

T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--

....functions, yielding a theory that corresponds to those of Shankar and Miner. As examples of the verification of concrete, published algorithms, the formal verification of an instance of an averaging algorithms (by Welch and Lynch [1] and of a non averaging algorithm (by Srikant and Toueg [9]) is exhibited. 1 Introduction Clock synchronization is one of the central elements of distributed dependable real time systems. Many mechanisms for realizing dependability properties in distributed real time systems rely on the fact that the different processes or computing nodes can be ....

....of algorithms that use an averaging function. The formalization of these clock synchronization algorithms takes a convergence function as a generic parameter; the underlying algorithm and its correctness argument remain fixed. In contrast, non averaging algorithms like the one presented in [9] use a different algorithm and do not rely on a convergence function. In this paper we report on the formal analysis of a broad class of clock synchronization algorithms. Starting from formalizations of the non averaging algorithm of Srikant and Toueg [9] and the averaging algorithm of Welch and ....

[Article contains additional citation context not shown here]

T. K. Srikanth and S. Toueg. Optimal clock synchronization. JACM, 34(3):626--645, July 1987.

....it is absent in some of the current systems (e.g. the Berkeley Motes [10] Moreover, GPS requires a clear sky view, and thus does not work inside buildings, underwater, or beneath dense foliage. There is a large literature on how to synchronize clocks in traditional networked systems (e.g. [3, 8, 15, 12]) among these, the Network Time Protocol [12] is the most widely deployed time synchronization algorithm and is notable for being scalable, self configuring, robust to failures, and thoroughly tested. NTP and the other traditional methods, despite their many differences, all achieve ....

SRIKANTH, T. K., AND TOUEG, S. Optimal clock synchronization. J-ACM 34, 3 (July 1987), 626--645.

....in the full version of the paper. 5. FUTURE DIRECTIONS Our results require that at most a third of the processors are faulty during each period. Previous clock synchronization protocols assuming authenticated channels (as we do) were able to require only a majority of non faulty processors [19, 27]. It is interesting to close this gap. In [10] there is another weaker requirement: only that subnetwork containing non faulty processors remain connected (but [10] also assumes signatures) It may be possible to prove a variant of this for our protocol, in particular it would be interesting to ....

T.K. Srikanth and S. Toueg, Optimal clock synchronization, JACM Vol. 34, No. 3, July 1987, pp. 626--645.

....phenomenon. Given these new challenges, are traditional time synchronization schemes the best choice for this new domain 3. TRADITIONAL NETWORK TIME SYNCHRONIZATION Over the years, many protocols have been designed for maintaining synchronization of physical clocks over computer networks [7, 15, 23, 30]. These protocols all have basic features in common: a simple connectionless messaging protocol; exchange of clock information between clients and one (or a few) servers; methods for mitigating the effects of nondeterminism in message delivery and processing; and an algorithm on the client for ....

T. K. Srikanth and Sam Toueg. Optimal clock synchronization. J-ACM, 34(3):626--645, July 1987.

....and will therefore be detected and excluded by the group membership algorithm. It is described as a push down stack in the TTP C specification [75] but this seems to be an error. Miner was translated to PVS and generalized (to admit nonaveraging algorithms such as that of Srikanth and Toueg [73] that do not conform to Schneider s treatment) by Schwier and von Henke [69] This treatment was then extended to the TTA algorithm by Pfeifer, Schwier and von Henke [45] The TTA algorithm is intended to operate in networks where there are at least four good clocks, and it is able to mask any ....

T. K. Srikanth and Sam Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987. 5

....phenomenon. Given these new challenges, are traditional time synchronization schemes the best choice for this new domain 3. TRADITIONAL NETWORK TIME SYNCHRONIZATION Over the years, many protocols have been designed for maintaining synchronization of physical clocks over computer networks [6, 14, 21, 27]. These protocols all have basic features in common: a simple connectionless messaging protocol; exchange of clock information between clients and one (or a few) servers; methods for mitigating the effects of nondeterminism in message delivery and processing; and an algorithm on the client for ....

T. K. Srikanth and Sam Toueg. Optimal clock synchronization. J-ACM, 34(3):626--645, July 1987.

....in sensor networks. In many sensor applications, relative relationships are often sufficient (e.g. to determine a target s speed) absolute time values can be unnecessary. Over the years, many protocols have been designed for maintaining synchronization of physical clocks over computer networks [9, 26, 5, 20]. These protocols all have basic features in common: a simple connectionless messaging protocol; exchange of clock information between clients and one (or a few) servers; methods for mitigating the effects of nondeterminism in message delivery and processing; and an algorithm on the client for ....

T. K. Srikanth and S. Toueg. Optimal clock synchronization. J-ACM, 34(3):626--645, July 1987.

....The complexity of protocol design is thus managed through a form of divide and conquer at each step, a designer focuses on a small number of additional attacks now permitted in the newly weakened system model. This idea of deriving a protocol is not new. For example, it has been proposed in [9, 26, 102, 81, 3, 4] as a way to obtain fault tolerant protocols for di#erent failure models. Prior work proposes mechanical transformations between protocols for di#erent system models. Although theoretically interesting, the mechanical transformations too often yielded unnecessarily complex and ine#cient protocols. ....

T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

....as the t 1 st reading, and the reading from another nonfaulty clock must be at least as great as the n t th; hence, the average of these two readings should be close to the middle of the spread of readings from good clocks. The most important event based algorithm is that of Srikanth and Toueg [ST87] it is attractive because it achieves optimal accuracy. Both averaging and event based algorithms require at least 3a 1 nodes to tolerate a arbitrary faults. 20 2.3.1 SAFEbus The SAFEbus bus is quad redundant (a pair of self checking pairs) and each of its four components comprises two data ....

T. K. Srikanth and Sam Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

....events may be needed to make this fault tolerant. The number of faulty nodes that can be tolerated, and the quality of synchronization that can be achieved, depend on the details of the algorithm, and on the fault hypothesis under which it operates. The event based algorithm of Srikanth and Toueg [21] is particularly attractive because it achieves optimal accuracy. 3 Fault Hypotheses and Fault Containment Units Safety critical aerospace functions are generally required to have failure rates less than 10 9 per hour [5] and an architecture that is intended to support several such functions ....

T. K. Srikanth and Sam Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

....in sensor networks. In many sensor applications, relative relationships are often sufficient (e.g. to determine a target s speed) absolute time values can be unnecessary. Over the years, many protocols have been designed for maintaining synchronization of physical clocks over computer networks [8, 25, 4, 19]. These protocols all have basic features in common: a simple connectionless messaging protocol; exchange of clock information between clients and one (or a few) servers; methods for mitigating the effects of nondeterminism in message delivery and processing; and an algorithm on the client for ....

T. K. Srikanth and Sam Toueg. Optimal clock synchronization. J-ACM, 34(3):626--645, July 1987.

....occurs in which a message apparently arrives before it is sent. Solutions exist in which logical clocks [ 9, 10 ] are kept, but these involve time stamping each message. Such systems are applicable to distributed systems or where debugging is a goal [ 11 ] Fault tolerant solutions [ 12, 13, 14 ] typically involve O(n 2 ) messages 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 ....

T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

....what tstart to use, which should be the list of entities involved (elist) and how to decide (decision) This seems to be a reasonable assumption. In fact, this problem is similar to the problem of initiation of a clock synchronization round, and has been solved in that context (see for instance [43, 50]) A malicious entity could try to corrupt the result of the service proposing a di erent list of entities, a di erent tstart or even a di erent decision parameter. This is not possible because the TTCB decides that a set of entities want to engage in the same run of the consensus precisely ....

T. K. Srikanth and S. Toueg. Optimal Clock Synchronization. Journal of the Association for Computing Machinery, 34(3):627-645, July 1987.

....systems and Byzantine agreement methodologies. Linear methods for digital telephone network synchronization are summarized in [16] while Byzantine methods for clock synchronization are summarized in [15] While reliable clock synchronization has been studied using agreement algorithms [15] [33], in practice it is not possible to distinguish the truechimer clocks, which maintain timekeeping accuracy to a previously published (and trusted) standard, from the falseticker clocks, which do not, on other than a statistical basis. In addition, the algorithms discussed in the literature do not ....

....inconsistent offsets in successive readings or to different readers. These algorithms use an agreement protocol involving successive rounds of readings, possibly relayed and possibly augmented by digital signatures. Examples include the fireworks algorithm of [12] and the optimum algorithm of [33]. However, these algorithms as described require an excessive burden of messages, especially when large numbers of clocks are involved, and require statistically awkward assumptions in order to certify correctness. While drawing upon the technology of agreement algorithms, the NTP peer selection ....

Srikanth, T.K., and S. Toueg. Optimal clock synchronization. JACM 34, 3 (July 1987), 626-645.

....can reconstruct the total ordering of any sequence of events that were causal. That is, if event A causes event B, the timestamp of A is less than the timestamp of B. Over the years, many protocols have been designed for maintaining synchronization of physical clocks over computer networks [GZ89, ST87, Cri89, Mil94] These protocols all have basic features in common: a simple connectionless messaging protocol; exchange of clock information between clients and one (or a few) servers; methods for mitigating the effects of nondeterminism in message delivery and processing; and an algorithm on the ....

T. K. Srikanth and Sam Toueg. Optimal clock synchronization. J-ACM, 34(3):626--645, July 1987.

....involve con dential data. Time and Clock Synchronisation In order to implement synchronous and partially synchronous protocols, we need time and clock synchronization primitives for those hosts with no local TTCB is available. Several clock synchronization protocols are well established today [105, 53, 93]. However, most if not all of them assume only benign faults. The MAFTIA model suggests to use topology aware clock synchronization, which can exploit the WAN of LANs network model and also tolerates malicious faults. More precisely: At the LAN level, a protocol tailored to local area networks ....

T. K. Srikanth and Sam Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626-645, July 1987.

....imprecision generated by the drift between clocks during the re synchronization interval T . However, since the drift, ae p in PC 2, is typically of the order of 10 Gamma6 s, the precision enhancement property of the convergence function is the relevant factor. 4 In any case, limited to ae p [23]. 2 3 The Approach taken The applications in view for real time systems in general require accuracy towards some real time reference. When a good external synchronization source is not available, algorithms that preserve accuracy, like the one in [23] become important. Additionally, those ....

....factor. 4 In any case, limited to ae p [23] 2 3 The Approach taken The applications in view for real time systems in general require accuracy towards some real time reference. When a good external synchronization source is not available, algorithms that preserve accuracy, like the one in [23], become important. Additionally, those systems oriented to distributed computer control require a precision better than that normally achieved with software based algorithms. In fact, a major limitation of all known software clock synchronization algorithms designed for arbitrary networks, is ....

[Article contains additional citation context not shown here]

T. K. Srikanth and Sam Toueg. Optimal Clock Synchronization. Journal of the Association for Computing Machinery, 34(3):627--645, July 1987.

....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 ....

Srikanth, T. K., and Toueg, S. Optimal clock synchronization. J. ACM 34, 3 (July 1987), 626--645.

....time, leading to a faulty local interval measurement, and an instantaneous backward correction additionally causes negative interval measurements. Continuous synchronization avoids these drawbacks by spreading the correction over the next synchronization period and applying it continuously. In [4,10], a general approach for achieving continuous correction in any round based clock synchronization protocol is shown. However, message losses in the wireless environment can lead for some systems to miss some rounds, thus making this basic approach not applicable. In [12] the authors show how a ....

T. K. Srikanth, and S. Toueg, "Optimal clock synchronization ", Journal of the ACM 34(3), July 1987, pp. 626-645.

....a rate and state synchronization algorithm: We see, that a CSA needs a good rate synchronization, but the CRA only little or even no state synchronization. This is also the key to have a separated analysis, putting a CSA on top of a CRA. Note that there is no conflict with the optimal result of [ST87], since the hardware drift is now affected by the CRA. Nicht vergessen die state correction ausweisen, um die unkorrigiert Clock zu bekommen Let us briefly compare these algorithms: S state is gut zugaenglich, bei R ist rate nicht (measurement) R hat grosse Zeitkonstanten, S hat kleine (eng, ....

T. K. Srikanth, S. Toueg, Optimal Clock Synchronization, Journal of ACM, Vol 34, No 3, July 1987, p. 626-645.

....37 1. OA provides the same, slightly suboptimal worst case traditional accuracy and drift as the FTM algorithm of [LWL88] The synchronized clocks drift away from realtime by at most the maximum drift ae max of the (worst) physical clock plus some smaller terms. Note that it has been proved in [ST87] that the worst case drift cannot be better than the drift of the physical clocks. Algorithms that are optimal in this respect have been provided in [ST87] FC95b] and [Sch97a] 2. It is important to note that accuracy intervals can grow much faster than traditional accuracy, which reveals the ....

....away from realtime by at most the maximum drift ae max of the (worst) physical clock plus some smaller terms. Note that it has been proved in [ST87] that the worst case drift cannot be better than the drift of the physical clocks. Algorithms that are optimal in this respect have been provided in [ST87] FC95b] and [Sch97a] 2. It is important to note that accuracy intervals can grow much faster than traditional accuracy, which reveals the major weakness of the orthogonal accuracy algorithm: Basically, traditional accuracy increases resp. decreases at most by the maximum enlargement of ....

T. K. Srikanth and Sam Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

....resources commonly found in many parallel systems, namely the nodes clocks, and a message passing network. ICV algorithms use simpler communication services than other categories of fault tolerant internal clock synchronization algorithms, which rely on services such as message authentication [6, 16], atomic broadcast [2] or agreement protocols [8, algorithms COM and CSM] Clock dissemination protocols used by ICV algorithms can stagger clock synchronization messages (hereby referred to simply as messages) over time to reduce network contention [3] and they can also limit the number of TSPs ....

....algorithms. With that goal in mind, we selected a subset of interesting algorithms. For more thorough surveys, the reader is referred to [14, 17] Our discussion focuses on network based algorithms (i.e. algorithms which distribute clock information via the existing system s network) [2, 6, 8, 11, 16]. These differ fundamentally from hardware based algorithms, which use dedicated clock distribution networks (e.g. 15] In addition, our discussion on network based algorithms does not include algorithms which assume that a subset of the system s TSPs can provide a reference time (e.g. as in ....

[Article contains additional citation context not shown here]

T. K. Srikanth and S. Toueg, "Optimal Clock Synchronization, " J. ACM, Vol. 34, No. 3, Jul. 1987, pp. 626-645.

....resources commonly found in many parallel systems, namely the nodes clocks, and a message passing network. ICV algorithms use simpler communication services than other categories of fault tolerant internal clock synchronization algorithms, which rely on services such as message authentication [9, 32], atomic broadcast [3, 36] or agreement protocols [12, algorithms COM and CSM] Communication protocols for ICV algorithms can use simpler broadcast or multicast primitives, in which a TSP disseminates its clock value to a number of destination TSPs using clock synchronization messages (hereby ....

....algorithms. With that goal in mind, we selected a subset of interesting algorithms. For more thorough surveys, the reader is referred to [1, 26, 29, 34] Our discussion focuses on network based algorithms (i.e. algorithms which distribute clock information via the existing system s network) [3, 9, 12, 20, 21, 25, 32, 36]. These differ fundamentally from hardware based algorithms, which use dedicated clock distribution networks [31, 35] In addition, our discussion on network based algorithms does not include algorithms which assume that a subset of the system s TSPs can provide a reference time. For example, ....

[Article contains additional citation context not shown here]

T. K. Srikanth and S. Toueg, "Optimal Clock Synchronization," Journal of the ACM, Vol. 34, No. 3, July 1987, pp. 626-645.

....either by hardware or by software. Hardware clock synchronization [SR87, KO87] achieves very tight synchronization through the use of special synchronization hardware at each processor, and uses a separate network solely for clock signals. In contrast, software clock synchronization algorithms [CAS86, HSSD84, ST87, LL88, LMS85, VCR97, PB95, GZ89, MS85, FC95a, FC97, OS94, Arv94, Cri89] use standard communication networks and send synchronization messages to get the clocks synchronized. They do not require specific hardware, but do not provide synchronization as tight as hardware algorithms. Software clock synchronization algorithms decompose themselves in deterministic, ....

....they introduce logical clocks . The value of a logical clock at real time t, denoted L p i (t) is determined by adding an adjustment term, denoted A p i (t) to the local hardware clock H p i (t) The adjustment term A p i (t) can either be a discrete value changed at each resynchronization [SC90, ST87] or a linear function of time [SC90, Cri89] In order to implement external synchronization algorithms, some processors use the GPS (Global Positioning System) or radio receivers to obtain the time signal broadcast by a standard source of time, as the UTC (Universal Time Coordinated) Such ....

[Article contains additional citation context not shown here]

T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

....functions, yielding a theory that corresponds to those of Shankar and Miner. As examples of the verification of concrete, published algorithms, the formal verification of an instance of an averaging algorithms (by Welch and Lynch [3] and of a non averaging algorithm (by Srikanth and Toueg [14]) is discussed. 1 Introduction Clock synchronization is one of the central elements of distributed dependable real time systems. Many mechanisms for realizing dependability properties in distributed real time systems rely on the fact that the different processes or computing nodes can be ....

....of algorithms that use an averaging function. The formalization of these clock synchronization algorithms takes a convergence function as a generic parameter; the underlying algorithm and its correctness argument remain fixed. In contrast, non averaging algorithms like the one presented in [14] do not fit Miner s and Shankar s theories because they use a different algorithm and do not rely on a convergence function. In this paper, we report on the formal analysis of a broader class of clock synchronization algorithms than those formally analyzed before. Our original starting point was ....

[Article contains additional citation context not shown here]

T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journ. of the ACM, 34(3):626--645, July 1987.

.... ISO hierarchy: Tan88, Com91, CS91, CS93, ANSA91a, ANSA91b, ANSA89, CD90, CDK94, XTP95] Pros and Cons of layered architectures: CT87, RST88, RST89, Ous90, AP93, KP93, KC94, BD95] Reliable stream communication: Rit84, Jac88, Tan88, Com91, CS91, CS93, CDK94] Failure Models and Classification: [Lam78b, Lam84, Ske82b, FLP85, ST87, CD90, Mar90, Cri91a, CT91, CHT92, GR93, SM94]. Kenneth P. Birman Building Secure and Reliable Network Applications 44 44 2. Communication Technologies Historically, it has rarely been necessary to understand details of the hardware components from which a computing system was constructed if one merely wishes to develop software for it. ....

....that are specific to distributed protocols and systems, since a full treatment of the topic would merit a book of its own. 20.1 Clock Synchronization Clock synchronization is an example of a topic that until the recent past represented an important area for distributed systems research [LAM84, LM85, Mar84, LM85, KO87, ST87, Cri89, CF94, VR92, CM96], overviews of the field can be found in [SWL90] and [Lis93] The introduction of the global positioning system, in the early 1990 s, greatly changed the situation. As recently as five years ago, a textbook such as this would have treated the problem in considerable detail, to the benefit of the ....

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T. K. Srikanth and Sam Toueg. Optimal Clock Synchronization. Journal of the ACM 34:3 (July 1987), 626645.

....and consistency operations are performed using two layered protocols. Each node participates in both the cluster level and inter cluster level convergence and consistency functions. The architecture supports a variety of synchronization primitives including those described in [3] 4] 12] [15], 11] We assume algorithm variants of those used in the MAFT[9] Virtually all existing convergence algorithms are based on fully connected structures, in which each non faulty participant derives the same set of clock values for all other nodes. The achievable synchronization skew for an N ....

T. K. Srikanth and S. Toueg, "Optimal clock synchronization," Journal of ACM, vol. 34, July 1987.

....loosely on max , which allows us to have a separated analysis, putting the state part on top of the rate one. Based on this observation we claim that the rate part of our state rate algorithm from Definition 9 can be combined with any one for state synchronization, e.g. the optimal algorithm by [18], the a posteriori agreement algorithm by [20] the probabilistic algorithm by [2] or the framework by [13] It only requires a rate adjustable clock and the possibility to exchange rate intervals within rounds to extend those approaches, since they all rely on the ordinary hardware drift rate ....

T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

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T. K. Srikanth and Sam Toueg. Optimal clock synchronization. J-ACM, 34(3):626--645, July 1987.

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Srikanth, T.K., Toueg, S.: Optimal Clock Synchronization. Journal of the ACM 34 (1987) 626--645

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T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626-- 645, July 1987.

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T. K. Srikanth and Sam Toueg. Optimal clock synchronization. J. ACM, 34(3):626--645, 1987.

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T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, 1987.

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T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626645, July 1987.

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T. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, 1987.

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T. K. Srikanth and Sam Toueg. Optimal clock synchronization. J-ACM, 34(3):626--645, July 1987.

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T. K. Srikanth and Sam Toueg. Optimal clock synchronization. J. ACM, 34(3):626--645, 1987.

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Srikanth, T., and S. Toueg. "Optimal Clock Synchronization", JACM 34, No. 3, pgs. 626-645, July 1987.

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T. K. Srikanth and S. Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

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T. K. Srikanth and Sam Toueg. Optimal clock synchronization. Journal of the ACM, 34(3), pages 626--645, July 1987.

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T. K. Srikanth and Sam Toueg. Optimal clock synchronization. Journal of the ACM, 34(3):626--645, July 1987.

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T. K. Srikanth and S. Toueg, "Optimal clock synchronization," J-ACM, vol. 34, no. 3, pp. 625--645, 1987.

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T. Srikanth and S. Toueg. Optimal clock synchronization. Jom"i,al of tt, c ACiFI, 34(3):625 645, July 1987. 13

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T. K. Srikanth and S. Toueg, "Optimal Clock Synchronization," Journal of the ACM, Vol. 34, No. 3, 1987, pp. 626-645.

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Srikanth T.K. and Toueg S., "Optimal Clock Synchronization", Journal of ACM, 34/3, pp. 626-- 645, July 1987.

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