Time, Clocks, and the Ordering of Events in a Distributed System

Leslie Lamport
Massachusetts Computer Associates, Inc.

Abstract

The concept of one event happening before another in a distributed system is examined, and is shown to define a partial ordering of the events. A distributed algorithm is given for synchronizing a system of logical clocks which can be used to totally order the events. The use of the total ordering is illustrated with a method for solving synchronization problems. The algorithm is then specialized for synchronizing physical clocks, and a bound is derived on how far out of synchrony the clocks can become.

Key Words and Phrases: distributed systems, computer networks, clock synchronization, multiprocess systems

CR Categories: 4.32, 5.29

Introduction

The concept of time is fundamental to our way of thinking. It is derived from the more basic concept of the order in which events occur. We say that something happened at 3:15 if it occurred after our clock read 3:15 and before it read 3:16. The concept of the temporal ordering of events pervades our thinking about systems. For example, in an airline reservation system we specify that a request for a reservation should be granted if it is made before the flight is filled. However, we will see that this concept must be carefully reexamined when considering events in a distributed system.

A distributed system consists of a collection of distinct processes which are spatially separated, and which communicate with one another by exchanging messages. A network of interconnected computers, such as the ARPA net, is a distributed system. A single computer can also be viewed as a distributed system in which the central control unit, the memory units, and the input-output channels are separate processes. A system is distributed if the message transmission delay is not negligible compared to the time between events in a single process.

We will concern ourselves primarily with systems of spatially separated computers. However, many of our remarks will apply more generally. In particular, a multiprocessing system on a single computer involves problems similar to those of a distributed system because of the unpredictable order in which certain events can occur.

In a distributed system, it is sometimes impossible to say that one of two events occurred first. The relation “happened before” is therefore only a partial ordering of the events in the system. We have found that problems often arise because people are not fully aware of this fact and its implications.

In this paper, we discuss the partial ordering defined by the “happened before” relation, and give a distributed algorithm for extending it to a consistent total ordering of all the events. This algorithm can provide a useful mechanism for implementing a distributed system. We illustrate its use with a simple method for solving synchronization problems. Unexpected, anomalous behavior can occur if the ordering obtained by this algorithm differs from that perceived by the user. This can be avoided by introducing real, physical clocks. We describe a simple method for synchronizing these clocks, and derive an upper bound on how far out of synchrony they can drift.

The Partial Ordering

Most people would probably say that an event a happened before an event b if a happened at an earlier time than b. They might justify this definition in terms of physical theories of time. However, if a system is to meet a specification correctly, then that specification must be given in terms of events observable within the system. If the specification is in terms of physical time, then the system must contain real clocks. Even if it does contain real clocks, there is still the problem that such clocks are not perfectly accurate and do not keep precise physical time. We will therefore define the “happened before” relation without using physical clocks.

We begin by defining our system more precisely. We assume that the system is composed of a collection of processes. Each process consists of a sequence of events. Depending upon the application, the execution of a subprogram on a computer could be one event, or the execution of a single machine instruction could be one event. We are assuming that the events of a process form a sequence, where a occurs before b in this sequence if a happens before b. In other words, a single process is defined to be a set of events with an a priori total ordering. This seems to be what is generally meant by a process.¹ It would be trivial to extend our definition to allow a process to split into distinct subprocesses, but we will not bother to do so.

We assume that sending or receiving a message is an event in a process. We can then define the “happened before” relation, denoted by ->, as follows.

Definition. The relation -> on the set of events of a system is the smallest relation satisfying the following three conditions: (1) If a and b are events in the same process, and a comes before b, then a -> b. (2) If a is the sending of a message by one process and b is the receipt of the same message by another process, then a -> b. (3) If a -> b and b -> c then a -> c. Two distinct events a and b are said to be concurrent if a -/-> b and b -/-> a.

We assume that a -/-> a for any event a. (Systems in which an event can happen before itself do not seem to be physically meaningful.) This implies that -> is an irreflexive partial ordering on the set of all events in the system.

It is helpful to view this definition in terms of a “space-time diagram” such as Figure 1. The horizontal direction represents space, and the vertical direction represents time–later times being higher than earlier ones. The dots denote events, the vertical lines denote processes, and the wavy lines denote messages.² It is easy to see that a -> b means that one can go from a to b in the diagram by moving forward in time along process and message lines. For example, we have p1 -> r4 in Figure 1.

Figure 1. Space-time diagram. (See rendered source page image pages/page-02.png.)

Another way of viewing the definition is to say that a -> b means that it is possible for event a to causally affect event b. Two events are concurrent if neither can causally affect the other. For example, events p3 and q3 of Figure 1 are concurrent. Even though we have drawn the diagram to imply that q3 occurs at an earlier physical time than p3, process P cannot know what process Q did at q3 until it receives the message at p4. (Before event p4, P could at most know what Q was planning to do at q3.)

This definition will appear quite natural to the reader familiar with the invariant space-time formulation of special relativity, as described for example in [1] or the first chapter of [2]. In relativity, the ordering of events is defined in terms of messages that could be sent. However, we have taken the more pragmatic approach of only considering messages that actually are sent. We should be able to determine if a system performed correctly by knowing only those events which did occur, without knowing which events could have occurred.

Logical Clocks

We now introduce clocks into the system. We begin with an abstract point of view in which a clock is just a way of assigning a number to an event, where the number is thought of as the time at which the event occurred. More precisely, we define a clock Ci for each process Pi to be a function which assigns a number Ci(a) to any event a in that process. The entire system of clocks is represented by the function C which assigns to any event b the number C(b), where C(b) = Cj(b) if b is an event in process Pj. For now, we make no assumption about the relation of the numbers Ci(a) to physical time, so we can think of the clocks Ci as logical rather than physical clocks. They may be implemented by counters with no actual timing mechanism.

We now consider what it means for such a system of clocks to be correct. We cannot base our definition of correctness on physical time, since that would require introducing clocks which keep physical time. Our definition must be based on the order in which events occur. The strongest reasonable condition is that if an event a occurs before another event b, then a should happen at an earlier time than b. We state this condition more formally as follows.

Clock Condition. For any events a, b:

\[ a \to b \Longrightarrow C(a) < C(b) \]

Note that we cannot expect the converse condition to hold as well, since that would imply that any two concurrent events must occur at the same time. In Figure 1, p2 and p3 are both concurrent with q3, so this would mean that they both must occur at the same time as q3, which would contradict the Clock Condition because p2 -> p3.

It is easy to see from our definition of the relation -> that the Clock Condition is satisfied if the following two conditions hold.

C1. If a and b are events in process Pi, and a comes before b, then Ci(a) < Ci(b).

C2. If a is the sending of a message by process Pi and b is the receipt of that message by process Pj, then Ci(a) < Cj(b).

Let us consider the clocks in terms of a space-time diagram. We imagine that a process’ clock “ticks” through every number, with the ticks occurring between the process’ events. For example, if a and b are consecutive events in process Pi with Ci(a) = 4 and Ci(b) = 7, then clock ticks 5, 6, and 7 occur between the two events. We draw a dashed “tick line” through all the like-numbered ticks of the different processes. The space-time diagram of Figure 1 might then yield the picture in Figure 2. Condition C1 means that there must be a tick line between any two events on a process line, and condition C2 means that every message line must cross a tick line. From the pictorial meaning of ->, it is easy to see why these two conditions imply the Clock Condition.

Figure 2. Tick lines in a space-time diagram. (See rendered source page image pages/page-02.png.)

We can consider the tick lines to be the time coordinate lines of some Cartesian coordinate system on space-time. We can redraw Figure 2 to straighten these coordinate lines, thus obtaining Figure 3. Figure 3 is a valid alternate way of representing the same system of events as Figure 2. Without introducing the concept of physical time into the system (which requires introducing physical clocks), there is no way to decide which of these pictures is a better representation.

Figure 3. Redrawn space-time diagram with straightened coordinate lines. (See rendered source page image pages/page-03.png.)

The reader may find it helpful to visualize a two-dimensional spatial network of processes, which yields a three-dimensional space-time diagram. Processes and messages are still represented by lines, but tick lines become two-dimensional surfaces.

Let us now assume that the processes are algorithms, and the events represent certain actions during their execution. We will show how to introduce clocks into the processes which satisfy the Clock Condition. Process Pi’s clock is represented by a register Ci, so that Ci(a) is the value contained by Ci during the event a. The value of Ci will change between events, so changing Ci does not itself constitute an event.

To guarantee that the system of clocks satisfies the Clock Condition, we will insure that it satisfies conditions C1 and C2. Condition C1 is simple; the processes need only obey the following implementation rule:

IR1. Each process Pi increments Ci between any two successive events.

To meet condition C2, we require that each message m contain a timestamp Tm which equals the time at which the message was sent. Upon receiving a message timestamped Tm, a process must advance its clock to be later than Tm. More precisely, we have the following rule.

IR2. (a) If event a is the sending of a message m by process Pi, then the message m contains a timestamp Tm = Ci(a). (b) Upon receiving a message m, process Pj sets Cj greater than or equal to its present value and greater than Tm.

In IR2(b) we consider the event which represents the receipt of the message m to occur after the setting of Cj. (This is just a notational nuisance, and is irrelevant in any actual implementation.) Obviously, IR2 insures that C2 is satisfied. Hence, the simple implementation rules IR1 and IR2 imply that the Clock Condition is satisfied, so they guarantee a correct system of logical clocks.

Ordering the Events Totally

We can use a system of clocks satisfying the Clock Condition to place a total ordering on the set of all system events. We simply order the events by the times at which they occur. To break ties, we use any arbitrary total ordering < of the processes. More precisely, we define a relation => as follows: if a is an event in process Pi and b is an event in process Pj, then a => b if and only if either (i) Ci(a) < Cj(b) or (ii) Ci(a) = Cj(b) and Pi < Pj. It is easy to see that this defines a total ordering, and that the Clock Condition implies that if a -> b then a => b. In other words, the relation => is a way of completing the “happened before” partial ordering to a total ordering.³

The ordering => depends upon the system of clocks Ci, and is not unique. Different choices of clocks which satisfy the Clock Condition yield different relations =>. Given any total ordering relation => which extends ->, there is a system of clocks satisfying the Clock Condition which yields that relation. It is only the partial ordering which is uniquely determined by the system of events.

Being able to totally order the events can be very useful in implementing a distributed system. In fact, the reason for implementing a correct system of logical clocks is to obtain such a total ordering. We will illustrate the use of this total ordering of events by solving the following version of the mutual exclusion problem. Consider a system composed of a fixed collection of processes which share a single resource. Only one process can use the resource at a time, so the processes must synchronize themselves to avoid conflict. We wish to find an algorithm for granting the resource to a process which satisfies the following three conditions: (I) A process which has been granted the resource must release it before it can be granted to another process. (II) Different requests for the resource must be granted in the order in which they are made. (III) If every process which is granted the resource eventually releases it, then every request is eventually granted.

We assume that the resource is initially granted to exactly one process.

These are perfectly natural requirements. They precisely specify what it means for a solution to be correct.⁴ Observe how the conditions involve the ordering of events. Condition II says nothing about which of two concurrently issued requests should be granted first.

It is important to realize that this is a nontrivial problem. Using a central scheduling process which grants requests in the order they are received will not work, unless additional assumptions are made. To see this, let P0 be the scheduling process. Suppose P1 sends a request to P0 and then sends a message to P2. Upon receiving the latter message, P2 sends a request to P0. It is possible for P2’s request to reach P0 before P1’s request does. Condition II is then violated if P2’s request is granted first.

To solve the problem, we implement a system of clocks with rules IR1 and IR2, and use them to define a total ordering => of all events. This provides a total ordering of all request and release operations. With this ordering, finding a solution becomes a straightforward exercise. It just involves making sure that each process learns about all other processes’ operations.

To simplify the problem, we make some assumptions. They are not essential, but they are introduced to avoid distracting implementation details. We assume first of all that for any two processes Pi and Pj, the messages sent from Pi to Pj are received in the same order as they are sent. Moreover, we assume that every message is eventually received. (These assumptions can be avoided by introducing message numbers and message acknowledgment protocols.) We also assume that a process can send messages directly to every other process.

Each process maintains its own request queue which is never seen by any other process. We assume that the request queues initially contain the single message T0:P0 requests resource, where P0 is the process initially granted the resource and T0 is less than the initial value of any clock.

The algorithm is then defined by the following five rules. For convenience, the actions defined by each rule are assumed to form a single event.

1. To request the resource, process Pi sends the message Tm:Pi requests resource to every other process, and puts that message on its request queue, where Tm is the timestamp of the message.

2. When process Pj receives the message Tm:Pi requests resource, it places it on its request queue and sends a timestamped acknowledgment message to Pi.⁵

3. To release the resource, process Pi removes any Tm:Pi requests resource message from its request queue and sends a timestamped Pi releases resource message to every other process.

4. When process Pj receives a Pi releases resource message, it removes any Tm:Pi requests resource message from its request queue.

5. Process Pi is granted the resource when the following two conditions are satisfied: (i) There is a Tm:Pi requests resource message in its request queue which is ordered before any other request in its queue by the relation =>. (To define the relation => for messages, we identify a message with the event of sending it.) (ii) Pi has received a message from every other process timestamped later than Tm.⁶

Note that conditions (i) and (ii) of rule 5 are tested locally by Pi.

It is easy to verify that the algorithm defined by these rules satisfies conditions I-III. First of all, observe that condition (ii) of rule 5, together with the assumption that messages are received in order, guarantees that Pi has learned about all requests which preceded its current request. Since rules 3 and 4 are the only ones which delete messages from the request queue, it is then easy to see that condition I holds. Condition II follows from the fact that the total ordering => extends the partial ordering ->. Rule 2 guarantees that after Pi requests the resource, condition (ii) of rule 5 will eventually hold. Rules 3 and 4 imply that if each process which is granted the resource eventually releases it, then condition (i) of rule 5 will eventually hold, thus proving condition III.

This is a distributed algorithm. Each process independently follows these rules, and there is no central synchronizing process or central storage. This approach can be generalized to implement any desired synchronization for such a distributed multiprocess system. The synchronization is specified in terms of a State Machine, consisting of a set C of possible commands, a set S of possible states, and a function e: C x S -> S. The relation e(C, S) -> S' means that executing the command C with the machine in state S causes the machine state to change to S'. In our example, the set C consists of all the commands Pi requests resource and Pi releases resource, and the state consists of a queue of waiting request commands, where the request at the head of the queue is the currently granted one. Executing a request command adds the request to the tail of the queue, and executing a release command removes a command from the queue.⁷

Each process independently simulates the execution of the State Machine, using the commands issued by all the processes. Synchronization is achieved because all processes order the commands according to their timestamps (using the relation =>), so each process uses the same sequence of commands. A process can execute a command timestamped T when it has learned of all commands issued by all other processes with timestamps less than or equal to T. The precise algorithm is straightforward, and we will not bother to describe it.

This method allows one to implement any desired form of multiprocess synchronization in a distributed system. However, the resulting algorithm requires the active participation of all the processes. A process must know all the commands issued by other processes, so that the failure of a single process will make it impossible for any other process to execute State Machine commands, thereby halting the system.

The problem of failure is a difficult one, and it is beyond the scope of this paper to discuss it in any detail. We will just observe that the entire concept of failure is only meaningful in the context of physical time. Without physical time, there is no way to distinguish a failed process from one which is just pausing between events. A user can tell that a system has “crashed” only because he has been waiting too long for a response. A method which works despite the failure of individual processes or communication lines is described in [3].

Anomalous Behavior

Our resource scheduling algorithm ordered the requests according to the total ordering =>. This permits the following type of “anomalous behavior.” Consider a nationwide system of interconnected computers. Suppose a person issues a request A on a computer A, and then telephones a friend in another city to have him issue a request B on a different computer B. It is quite possible for request B to receive a lower timestamp and be ordered before request A. This can happen because the system has no way of knowing that A actually preceded B, since that precedence information is based on messages external to the system.

Let us examine the source of the problem more closely. Let S be the set of all system events. Let us introduce a set of events S* which contains the events in S together with all other relevant external events, such as the phone calls in our example. Let -->* denote the “happened before” relation for S*. In our example, we had A -->* B, but not A -> B. It is obvious that no algorithm based entirely upon events in S, and which does not relate those events in any way with the other events in S*, can guarantee that request A is ordered before request B.

There are two possible ways to avoid such anomalous behavior. The first way is to explicitly introduce into the system the necessary information about the ordering -->*. In our example, the person issuing request A could receive the timestamp TA of that request from the system. When issuing request B, his friend could specify that B be given a timestamp later than TA. This gives the user the responsibility for avoiding anomalous behavior.

The second approach is to construct a system of clocks which satisfies the following condition.

Strong Clock Condition. For any events a, b in S*:

\[ a \to^* b \Longrightarrow C(a) < C(b) \]

This is stronger than the ordinary Clock Condition because -->* is a stronger relation than ->. It is not in general satisfied by our logical clocks.

Let us identify S* with some set of “real” events in physical space-time, and let -->* be the partial ordering of events defined by special relativity. One of the mysteries of the universe is that it is possible to construct a system of physical clocks which, running quite independently of one another, will satisfy the Strong Clock Condition. We can therefore use physical clocks to eliminate anomalous behavior. We now turn our attention to such clocks.

Physical Clocks

Let us introduce a physical time coordinate into our space-time picture, and let Ci(t) denote the reading of the clock Ci at physical time t.⁸ For mathematical convenience, we assume that the clocks run continuously rather than in discrete “ticks.” (A discrete clock can be thought of as a continuous one in which there is an error of up to 1/2 “tick” in reading it.) More precisely, we assume that Ci(t) is a continuous, differentiable function of t except for isolated jump discontinuities where the clock is reset. Then dCi(t)/dt represents the rate at which the clock is running at time t.

In order for the clock Ci to be a true physical clock, it must run at approximately the correct rate. That is, we must have dCi(t)/dt ≈ 1 for all t. More precisely, we will assume that the following condition is satisfied:

PC1. There exists a constant κ << 1 such that for all i:

\[ \left|\frac{dC_i(t)}{dt} - 1\right| < \kappa \]

For typical crystal controlled clocks, κ <= 10^-6.

It is not enough for the clocks individually to run at approximately the correct rate. They must be synchronized so that Ci(t) ≈ Cj(t) for all i, j, and t. More precisely, there must be a sufficiently small constant ε so that the following condition holds:

PC2. For all i, j:

\[ \left|C_i(t) - C_j(t)\right| < \epsilon \]

If we consider vertical distance in Figure 2 to represent physical time, then PC2 states that the variation in height of a single tick line is less than ε.

Since two different clocks will never run at exactly the same rate, they will tend to drift further and further apart. We must therefore devise an algorithm to insure that PC2 always holds. First, however, let us examine how small κ and ε must be to prevent anomalous behavior. We must insure that the system S* of relevant physical events satisfies the Strong Clock Condition. We assume that our clocks satisfy the ordinary Clock Condition, so we need only require that the Strong Clock Condition holds when a and b are events in S with a -->* b. Hence, we need only consider events occurring in different processes.

Let μ be a number such that if event a occurs at physical time t and event b in another process satisfies a -->* b, then b occurs later than physical time t + μ. In other words, μ is less than the shortest transmission time for interprocess messages. We can always choose μ equal to the shortest distance between processes divided by the speed of light. However, depending upon how messages in S* are transmitted, μ could be significantly larger.

To avoid anomalous behavior, we must make sure that for any i, j, and t: Ci(t + μ) - Cj(t) > 0. Combining this with PC1 and PC2 allows us to relate the required smallness of κ and ε to the value of μ as follows. We assume that when a clock is reset, it is always set forward and never back. (Setting it back could cause C1 to be violated.) PC1 then implies that Ci(t + μ) - Ci(t) > (1 - κ)μ. Using PC2, it is then easy to deduce that Ci(t + μ) - Cj(t) > 0 if the following inequality holds:

\[ \frac{\epsilon}{1 - \kappa} \le \mu \]

This inequality together with PC1 and PC2 implies that anomalous behavior is impossible.

We now describe our algorithm for insuring that PC2 holds. Let m be a message which is sent at physical time t and received at time t'. We define νm = t' - t to be the total delay of the message m. This delay will, of course, not be known to the process which receives m. However, we assume that the receiving process knows some minimum delay μm >= 0 such that μm <= νm. We call ξm = νm - μm the unpredictable delay of the message.

We now specialize rules IR1 and IR2 for our physical clocks as follows:

IR1’. For each i, if Pi does not receive a message at physical time t, then Ci is differentiable at t and dCi(t)/dt > 0.

IR2’. (a) If Pi sends a message m at physical time t, then m contains a timestamp Tm = Ci(t). (b) Upon receiving a message m at time t', process Pj sets Cj(t') equal to maximum(Cj(t' - 0), Tm + μm).⁹

Although the rules are formally specified in terms of the physical time parameter, a process only needs to know its own clock reading and the timestamps of messages it receives. For mathematical convenience, we are assuming that each event occurs at a precise instant of physical time, and different events in the same process occur at different times. These rules are then specializations of rules IR1 and IR2, so our system of clocks satisfies the Clock Condition. The fact that real events have a finite duration causes no difficulty in implementing the algorithm. The only real concern in the implementation is making sure that the discrete clock ticks are frequent enough so C1 is maintained.

We now show that this clock synchronizing algorithm can be used to satisfy condition PC2. We assume that the system of processes is described by a directed graph in which an arc from process Pi to process Pj represents a communication line over which messages are sent directly from Pi to Pj. We say that a message is sent over this arc every τ seconds if for any t, Pi sends at least one message to Pj between physical times t and t + τ. The diameter of the directed graph is the smallest number d such that for any pair of distinct processes Pj, Pk, there is a path from Pj to Pk having at most d arcs.

In addition to establishing PC2, the following theorem bounds the length of time it can take the clocks to become synchronized when the system is first started.

THEOREM. Assume a strongly connected graph of processes with diameter d which always obeys rules IR1’ and IR2’. Assume that for any message m, μm <= μ for some constant μ, and that for all t >= t0: (a) PC1 holds. (b) There are constants τ and ξ such that every τ seconds a message with an unpredictable delay less than ξ is sent over every arc. Then PC2 is satisfied with ε ≈ d(2κτ + ξ) for all t >= t0 + τd, where the approximations assume μ + ξ << τ.

The proof of this theorem is surprisingly difficult, and is given in the Appendix. There has been a great deal of work done on the problem of synchronizing physical clocks. We refer the reader to [4] for an introduction to the subject. The methods described in the literature are useful for estimating the message delays μm and for adjusting the clock frequencies dCi/dt (for clocks which permit such an adjustment). However, the requirement that clocks are never set backwards seems to distinguish our situation from ones previously studied, and we believe this theorem to be a new result.

Conclusion

We have seen that the concept of “happening before” defines an invariant partial ordering of the events in a distributed multiprocess system. We described an algorithm for extending that partial ordering to a somewhat arbitrary total ordering, and showed how this total ordering can be used to solve a simple synchronization problem. A future paper will show how this approach can be extended to solve any synchronization problem.

The total ordering defined by the algorithm is somewhat arbitrary. It can produce anomalous behavior if it disagrees with the ordering perceived by the system’s users. This can be prevented by the use of properly synchronized physical clocks. Our theorem showed how closely the clocks can be synchronized.

In a distributed system, it is important to realize that the order in which events occur is only a partial ordering. We believe that this idea is useful in understanding any multiprocess system. It should help one to understand the basic problems of multiprocessing independently of the mechanisms used to solve them.

Appendix

Proof of the Theorem

For any i and t, let us define Ci^t to be a clock which is set equal to Ci at time t and runs at the same rate as Ci, but is never reset. In other words,

\[ C_i^t(t') = C_i(t) + \int_t^{t'} [dC_i(u)/du]du \tag{1} \]

for all t' >= t. Note that

\[ C_i(t') \ge C_i^t(t') \quad \text{for all } t' \ge t. \tag{2} \]

Suppose process P1 at time t1 sends a message to process P2 which is received at time t2 with an unpredictable delay <= ξ, where t0 <= t1 <= t2. Then for all t >= t2 we have:

\[ \begin{aligned} C_2^{t_2}(t) &\ge C_2^{t_2}(t_2) + (1-\kappa)(t-t_2) && \text{[by (1) and PC1]} \\ &\ge C_1(t_1) + \mu_m + (1-\kappa)(t-t_2) && \text{[by IR2' (b)]} \\ &= C_1(t_1) + (1-\kappa)(t-t_1) - [(t_2-t_1)-\mu_m] + \kappa(t_2-t_1) \\ &\ge C_1(t_1) + (1-\kappa)(t-t_1) - \xi. \end{aligned} \]

Hence, with these assumptions, for all t >= t2 we have:

\[ C_2^{t_2}(t) \ge C_1(t_1) + (1-\kappa)(t-t_1) - \xi. \tag{3} \]

Now suppose that for i = 1, ..., n we have ti <= ti' < ti+1, t0 <= t1, and that at time ti' process Pi sends a message to process Pi+1 which is received at time ti+1 with an unpredictable delay less than ξ. Then repeated application of the inequality (3) yields the following result for t >= tn+1.

\[ C_{n+1}^{t_{n+1}}(t) \ge C_1(t_1') + (1-\kappa)(t-t_1') - n\xi. \tag{4} \]

From PC1, IR1’ and IR2’ we deduce that

\[ C_1(t_1') \ge C_1(t_1) + (1-\kappa)(t_1'-t_1). \]

Combining this with (4) and using (2), we get

\[ C_{n+1}(t) \ge C_1(t_1) + (1-\kappa)(t-t_1) - n\xi \tag{5} \]

for t >= tn+1.

For any two processes P and P', we can find a sequence of processes P = P0, P1, ..., Pn+1 = P', n <= d, with communication arcs from each Pi to Pi+1. By hypothesis (b) we can find times ti, ti' with ti' - ti <= τ and ti+1 - ti' <= ν, where ν = μ + ξ. Hence, an inequality of the form (5) holds with n <= d whenever t >= t1 + d(τ + ν). For any i, j and any t, t1 with t1 >= t0 and t >= t1 + d(τ + ν) we therefore have:

\[ C_i(t) \ge C_j(t_1) + (1-\kappa)(t-t_1) - d\xi. \tag{6} \]

Now let m be any message timestamped Tm, and suppose it is sent at time t and received at time t'. We pretend that m has a clock Cm which runs at a constant rate such that Cm(t) = Tm and Cm(t') = Tm + μm. Then μm <= t' - t implies that dCm/dt <= 1. Rule IR2’ (b) simply sets Cj(t') to maximum(Cj(t' - 0), Cm(t')). Hence, clocks are reset only by setting them equal to other clocks.

For any time tx >= t0 + μ/(1 - κ), let Cx be the clock having the largest value at time tx. Since all clocks run at a rate less than 1 + κ, we have for all i and all t >= tx:

\[ C_i(t) \le C_x(t_x) + (1+\kappa)(t-t_x). \tag{7} \]

We now consider the following two cases: (i) Cx is the clock Cq of process Pq. (ii) Cx is the clock Cm of a message sent at time t1 by process Pq. In case (i), (7) simply becomes

\[ C_i(t) \le C_q(t_x) + (1+\kappa)(t-t_x). \tag{8i} \]

In case (ii), since Cm(t1) = Cq(t1) and dCm/dt <= 1, we have

\[ C_x(t_x) \le C_q(t_1) + (t_x-t_1). \]

Hence, (7) yields

\[ C_i(t) \le C_q(t_1) + (1+\kappa)(t-t_1). \tag{8ii} \]

Since tx >= t0 + μ/(1 - κ), we get

\[ \begin{aligned} C_q(t_x-\mu/(1-\kappa)) &\le C_q(t_x)-\mu && \text{[by PC1]}\\ &\le C_m(t_x)-\mu && \text{[by choice of }m\text{]}\\ &\le C_m(t_x)-(t_x-t_1)\mu_m/\nu_m && [\mu_m \le \mu,\ t_x-t_1 \le \nu_m]\\ &= T_m && \text{[by definition of }C_m\text{]}\\ &= C_q(t_1) && \text{[by IR2'(a)]}. \end{aligned} \]

Hence, Cq(tx - μ/(1 - κ)) <= Cq(t1), so tx - t1 <= μ/(1 - κ) and thus t1 >= t0.

Letting t1 = tx in case (i), we can combine (8i) and (8ii) to deduce that for any t, tx with t >= tx >= t0 + μ/(1 - κ) there is a process Pq and a time t1 with tx - μ/(1 - κ) <= t1 <= tx such that for all i:

\[ C_i(t) \le C_q(t_1) + (1+\kappa)(t-t_1). \tag{9} \]

Choosing t and tx with t >= tx + d(τ + ν), we can combine (6) and (9) to conclude that there exists a t1 and a process Pq such that for all i:

\[ C_q(t_1) + (1-\kappa)(t-t_1) - d\xi \le C_i(t) \le C_q(t_1) + (1+\kappa)(t-t_1). \tag{10} \]

Letting t = tx + d(τ + ν), we get

\[ d(\tau+\nu) \le t-t_1 \le d(\tau+\nu) + \mu/(1-\kappa). \]

Combining this with (10), we get

\[ \begin{aligned} C_q(t_1) + (t-t_1) - \kappa d(\tau+\nu) - d\xi &\le C_i(t) \\ &\le C_q(t_1) + (t-t_1) + \kappa[d(\tau+\nu)+\mu/(1-\kappa)]. \end{aligned} \tag{11} \]

Using the hypotheses that κ << 1 and μ <= ν << τ, we can rewrite (11) as the following approximate inequality.

\[ C_q(t_1) + (t-t_1) - d(\kappa\tau+\xi) \lesssim C_i(t) \lesssim C_q(t_1) + (t-t_1) + d\kappa\tau. \tag{12} \]

Since this holds for all i, we get

\[ |C_i(t) - C_j(t)| \lesssim d(2\kappa\tau+\xi), \]

and this holds for all t >= t0 + dτ.

Note that relation (11) of the proof yields an exact upper bound for |Ci(t) - Cj(t)| in case the assumption μ + ξ << τ is invalid. An examination of the proof suggests a simple method for rapidly initializing the clocks, or resynchronizing them if they should go out of synchrony for any reason. Each process sends a message which is relayed to every other process. The procedure can be initiated by any process, and requires less than 2d(μ + ξ) seconds to effect the synchronization, assuming each of the messages has an unpredictable delay less than ξ.

Acknowledgment

The use of timestamps to order operations, and the concept of anomalous behavior are due to Paul Johnson and Robert Thomas.

Received March 1976; revised October 1977

References

1. Schwartz, J.T. Relativity in Illustrations. New York U. Press, New York, 1962.
2. Taylor, E.F., and Wheeler, J.A. Space-Time Physics, W.H. Freeman, San Francisco, 1966.
3. Lamport, L. The implementation of reliable distributed multiprocess systems. To appear in Computer Networks.
4. Ellingson, C., and Kulpinski, R.J. Dissemination of system-time. IEEE Trans. Comm. Com-23, 5 (May 1973), 605-624.