Time clocks and the ordering of events

A distributed system consists of a collection of distinct
processes which are spatially separated, and which com-
municate with one another by exchanging messages. A
network of interconnected computers, such as the ARPA
Operating R. Stockton Gaines net, is a distributed system. A single computer can also
Systems Editor be viewed as a distributed system in which the central
control unit, the memory units, and the input-output
Time, Clocks, and the channels are separate processes. A system is distributed
if the message transmission delay is not negligible com-
Ordering of Events in pared to the time between events in a single process.
We will concern ourselves primarily with systems of
a Distributed System spatially separated computers. However, many of our
remarks will apply more generally. In particular, a mul-
Leslie Lamport tiprocessing system on a single computer involves prob-
Massachusetts Computer Associates, Inc. lems 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
The concept of one event happening before another
say that one of two events occurred first. The relation
in a distributed system is examined, and is shown to
"happened before" is therefore only a partial ordering
define a partial ordering of the events. A distributed
of the events in the system. We have found that problems
algorithm is given for synchronizing a system of logical
often arise because people are not fully aware of this fact
clocks which can be used to totally order the events.
and its implications.
The use of the total ordering is illustrated with a
In this paper, we discuss the partial ordering defined
method for solving synchronization problems. The
by the "happened before" relation, and give a distributed
algorithm is then specialized for synchronizing physical
algorithm for extending it to a consistent total ordering
clocks, and a bound is derived on how far out of
of all the events. This algorithm can provide a useful
synchrony the clocks can become.
mechanism for implementing a distributed system. We
Key Words and Phrases: distributed systems,
illustrate its use with a simple method for solving syn-
computer networks, clock synchronization, multiprocess
chronization problems. Unexpected, anomalous behav-
systems
ior can occur if the ordering obtained by this algorithm
CR Categories: 4.32, 5.29
differs from that perceived by the user. This can be
avoided by introducing real, physical clocks. We describe
Introduction a simple method for synchronizing these clocks, and
derive an upper bound on how far out of synchrony they
The concept of time is fundamental to our way of can drift.
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 The Partial Ordering
and before it read 3:16. The concept of the temporal
ordering of events pervades our thinking about systems. Most people would probably say that an event a
For example, in an airline reservation system we specify happened before an event b if a happened at an earlier
that a request for a reservation should be granted if it is time than b. They might justify this definition in terms
made before the flight is filled. However, we will see that of physical theories of time. However, if a system is to
this concept must be carefully reexamined when consid- meet a specification correctly, then that specification
ering events in a distributed system. must be given in terms of events observable within the
General permission to make fair use in teaching or research of all system. If the specification is in terms of physical time,
or part of this material is granted to individual readers and to nonprofit
libraries acting for them provided that ACM's copyright notice is given then the system must contain real clocks. Even if it does
and that reference is made to the publication, to its date of issue, and contain real clocks, there is still the problem that such
to the fact that reprinting privileges were granted by permission o f the clocks are not perfectly accurate and do not keep precise
Association for Computing Machinery. To otherwise reprint a figure,
table, other substantial excerpt, or the entire work requires specific physical time. We will therefore define the "happened
permission as does republication, or systematic or multiple reproduc- before" relation without using physical clocks.
tion. We begin by defining our system more precisely. We
This work was supported by the Advanced Research Projects
Agency of the Department of Defense and Rome Air Development assume that the system is composed of a collection of
Center. It was monitored by Rome Air Development Center under processes. Each process consists of a sequence of events.
contract number F 30602-76-C-0094. Depending upon the application, the execution of a
Author's address: Computer Science Laboratory, SRI Interna-
tional, 333 Ravenswood Ave., Menlo Park CA 94025. subprogram on a computer could be one event, or the
© 1978 ACM 0001-0782/78/0700-0558 $00.75 execution of a single machine instruction could be one
558 Communications July 1978
of Volume 21
the ACM Number 7

Fig. 1. Fig. 2.

a, CY ,Y cy c~

(9 (9 ~o (9 (9 ~)
O O U
~ o

P4' q7 r4

q6 -2 - - - q6 ;#.i
--
P3 q5 r3 P3' ~ ~~ ~~ _ ~ ~ - ~ Y _ r3

P2' r2

Pl ~ ql r1

event. We are assuming that the events of a process form the diagram by moving forward in time along process
a sequence, where a occurs before b in this sequence if and message lines. For example, we have p, --~ r4 in
a happens before b. In other words, a single process is Figure 1.
defined to be a set of events with an a priori total Another way of viewing the definition is to say that
ordering. This seems to be what is generally meant by a a --) b means that it is possible for event a to causally
process.~ It would be trivial to extend our definition to affect event b. Two events are concurrent if neither can
allow a process to split into distinct subprocesses, but we causally affect the other. For example, events pa and q:~
will not bother to do so. of Figure 1 are concurrent. Even though we have drawn
We assume that sending or receiving a message is an the diagram to imply that q3 occurs at an earlier physical
event in a process. We can then define the "happened time than 1)3, process P cannot know what process Q did
before" relation, denoted by "---~", as follows. at qa until it receives the message at p , (Before event p4,
Definition. The relation "---->"on the set of events of P could at most know what Q was planning to do at q:~.)
a system is the smallest relation satisfying the following This definition will appear quite natural to the reader
three conditions: (1) I f a and b are events in the same familiar with the invariant space-time formulation of
process, and a comes before b, then a ~ b. (2) I f a is the special relativity, as described for example in [1] or the
sending of a message by one process and b is the receipt first chapter of [2]. In relativity, the ordering of events is
o f the same message by another process, then a ~ b. (3) defined in terms of messages that could be sent. However,
I f a ~ b and b ~ c then a ---* c. Two distinct events a we have taken the more pragmatic approach of only
and b are said to be concurrent if a ~ b and b -/-* a. considering messages that actually are sent. We should
We assume that a ~ a for any event a. (Systems in be able to determine if a system performed correctly by
which an event can happen before itself do not seem to knowing only those events which did occur, without
be physically meaningful.) This implies that ~ is an knowing which events could have occurred.
irreflexive partial ordering on the set of all events in the
system.
It is helpful to view this definition in terms of a Logical Clocks
"space-time diagram" such as Figure 1. The horizontal
direction represents space, and the vertical direction We now introduce clocks into the system. We begin
represents t i m e - - l a t e r times being higher than earlier with an abstract point of view in which a clock is just a
ones. The dots denote events, the vertical lines denote way of assigning a number to an event, where the n u m b e r
processes, and the wavy lines denote messagesfl It is easy is thought of as the time at which the event occurred.
to see that a ~ b means that one can go from a to b in More precisely, we define a clock Ci for each process Pi
to be a function which assigns a number Ci(a) to any
' The choice of what constitutes an event affects the ordering of event a in that process. The entire system o f c l b c k s is
events in a process. For example, the receipt of a message might denote represented by the function C which assigns to any event
the setting of an interrupt bit in a computer, or the execution of a b the number C ( b ) , where C ( b ) = C/(b) i f b is an event
subprogram to handle that interrupt. Since interrupts need not be
handled in the order that they occur, this choice will affect the order- in process Pj. For now, we make no assumption about
ing of a process' message-receiving events. the relation of the numbers Ci(a) to physical time, so we
2 Observe that messages may be received out of order. We allow can think of the clocks Ci as logical rather than physical
the sending of several messages to be a single event, but for convenience
we will assume that the receipt of a single message does not coincide clocks. They m a y be implemented by counters with no
with the sending or receipt of any other message. actual timing mechanism.
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Fig. 3. condition C2 means that every message line must cross
CY n¢ a tick line. F r o m the pictorial meaning of--->, it is easy to
see why these two conditions imply the Clock Con-
8 8 8 dition.
c~! ~ ~iLql ~ .r 4 We can consider the tick lines to be the time coordi-
nate lines of some Cartesian coordinate system on space-
time. We can redraw Figure 2 to straighten these coor-
dinate 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.
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,
We now consider what it means for such a system of and the events represent certain actions during their
clocks to be correct. We cannot base our definition of execution. We will show how to introduce clocks into the
correctness on physical time, since that would require processes which satisfy the Clock Condition. Process Pi's
introducing clocks which keep physical time. Our defi- clock is represented by a register Ci, so that C~(a) is the
nition must be based on the order in which events occur. value contained by C~ during the event a. The value of
The strongest reasonable condition is that if an event a C~ will change between events, so changing Ci does not
occurs before another event b, then a should happen at itself constitute an event.
an earlier time than b. We state this condition more To guarantee that the system of clocks satisfies the
formally as follows. Clock Condition, we will insure that it satisfies conditions
C 1 and C2. Condition C 1 is simple; the processes need
Clock Condition. For any events a, b: only obey the following implementation rule:
if a---> b then C ( a ) < C(b).
IR1. Each process P~ increments Ci between any
Note that we cannot expect the converse condition to two successive events.
hold as well, since that would imply that any two con-
current events must occur at the same time. In Figure 1, To meet condition C2, we require that each message
p2 and p.~ are both concurrent with q3, so this would m contain a timestamp Tm which equals the time at which
mean that they both must occur at the same time as q.~, the message was sent. U p o n receiving a message time-
which would contradict the Clock Condition because p2 stamped Tin, a process must advance its clock to be later
-----> /93.
than Tin. More precisely, we have the following rule.
It is easy to see from our definition of the relation IR2. (a) I f event a is the sending of a message m
"---~" that the Clock Condition is satisfied if the following by process P~, then the message m contains a timestamp
two conditions hold. T m = Ci(a). (b) U p o n receiving a message m, process
C 1. I f a and b are events in process P~, and a comes Pi sets Ci greater than or equal to its present value and
before b, then Ci(a) < Ci(b). greater than Tin.
C2. If a is the sending of a message by process Pi In IR2(b) we consider the event which represents the
and b is the receipt of that message by process Pi, then receipt of the message m to occur after the setting of Ci.
Ci(a) < Ci(b). (This is just a notational nuisance, and is irrelevant in
Let us consider the clocks in terms of a space-time any actual implementation.) Obviously, IR2 insures that
diagram. We imagine that a process' clock "ticks" C2 is satisfied. Hence, the simple implementation rules
through every number, with the ticks occurring between I R l and IR2 imply that the Clock Condition is satisfied,
the process' events. For example, if a and b are consec- so they guarantee a correct system of logical clocks.
utive 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- Ordering the Events Totally
numbered ticks of the different processes. The space-
time diagram of Figure 1 might then yield the picture in We can use a system of clocks satisfying the Clock
Figure 2. Condition C 1 means that there must be a tick Condition to place a total ordering on the set of all
line between any two events on a process line, and system events. We simply order the events by the times
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the ACM Number 7

at which they occur. To break ties, we use any arbitrary clocks with'rules IR 1 and IR2, and use them to define a
total ordering < of the processes. More precisely, we total ordering ~ of all events. This provides a total
define a relation ~ as follows: if a is an event in process ordering of all request and release operations. With this
Pi and b is an event in process Pj, then a ~ b if and only ordering, finding a solution becomes a straightforward
if either (i) Ci{a) < Cj(b) or (ii) El(a) Cj(b) and Pi
----" exercise. It just involves making sure that each process
< Py. It is easy to see that this defines a total ordering, learns about all other processes' operations.
and that the Clock Condition implies that if To simplify the problem, we make some assumptions.
a ----> b then a ~ b. In other words, the relation ~ is a They are not essential, but they are introduced to avoid
way of completing the "happened before" partial order- distracting implementation details. We assume first o f all
ing to a total ordering, a that for any two processes P / a n d Pj, the messages sent
The ordering ~ depends upon the system of clocks from Pi to Pi are received in the same order as they are
Cz, and is not unique. Different choices of clocks which sent. Moreover, we assume that every message is even-
satisfy the Clock Condition yield different relations ~ . tually received. (These assumptions can be avoided by
Given any total ordering relation ~ which extends --->, introducing message numbers and message acknowledg-
there is a system of clocks satisfying the Clock Condition ment protocols.) We also assume that a process can send
which yields that relation. It is only the partial ordering messages directly to every other process.
which is uniquely determined by the system of events. Each process maintains its own request queue which
Being able to totally order the events can be very is never seen by any other process. We assume that the
useful in implementing a distributed system. In fact, the request queues initially contain the single message To:Po
reason for implementing a correct system of logical requests resource, where Po is the process initially granted
clocks is to obtain such a total ordering. We will illustrate the resource and To is less than the initial value of any
the use of this total ordering of events by solving the clock.
following version of the mutual exclusion problem. Con- The algorithm is then defined by the following five
sider a system composed of a fixed collection of processes rules. For convenience, the actions defined by each rule
which share a single resource. Only one process can use are assumed to form a single event.
the resource at a time, so the processes must synchronize 1. To request the resource, process Pi sends the mes-
themselves to avoid conflict. We wish to find an algo- sage TIn:P/requests resource to every other process, and
rithm for granting the resource to a process which satis- puts that message on its request queue, where T,~ is the
fies the following three conditions: (I) A process which timestamp of the message.
has been granted the resource must release it before it 2. When process Pj receives the message T,~:P~ re-
can be granted to another process. (II) Different requests quests resource, it places it on its request queue and sends
for the resource must be granted in the order in which a (timestamped) acknowledgment message to P~.'~
they are made. (III) If every process which is granted the 3. To release the resource, process P~ removes any
resource eventually releases it, then every request is Tm:Pi requests resource message from its request queue
eventually granted. and sends a (timestamped) Pi releases resource message
We assume that the resource is initially granted to to every other process.
exactly one process. 4. When process Pj receives a Pi releases resource
These are perfectly natural requirements. They pre- message, it removes any Tm:P~ requests resource message
cisely specify what it means for a solution to be correct/ from its request queue.
Observe how the conditions involve the ordering of 5. Process P/is granted the resource when the follow-
events. Condition II says nothing about which of two ing two conditions are satisfied: (i) There is a Tm:Pi
concurrently issued requests should be granted first. requests resource message in its request queue which is
It is important to realize that this is a nontrivial ordered before any other request in its queue by the
problem. Using a central scheduling process which grants relation ~ . (To define the relation " ~ " for messages,
requests in the order they are received will not work, we identify a message with the event of sending it.) (ii)
unless additional assumptions are made. To see this, let P~ has received a message from every other process time-
P0 be the scheduling process. Suppose P1 sends a request stamped later than Tin.~
to Po and then sends a message to P2. U p o n receiving the
Note that conditions (i) and (ii) of rule 5 are tested
latter message, Pe sends a request to Po. It is possible for locally by P~.
P2's request to reach P0 before Pl's request does. Condi-
It is easy to verify that the algorithm defined by these
tion II is then violated if P2's request is granted first.
rules satisfies conditions I-III. First of all, observe that
To solve the problem, we implement a system of
condition (ii) of rule 5, together with the assumption that
messages are received in order, guarantees that P~ has
;~The ordering < establishes a priority among the processes. If a learned about all requests which preceded its current
"fairer" method is desired, then < can be made a function of the clock
value. For example, if Ci(a) = C/b) andj < L then we can let a ~ b
ifj < C~(a) mod N --< i, and b ~ a otherwise; where N is the total '~This acknowledgment messageneed not be sent if Pj has already
number of processes. sent a messageto Pi timestamped later than T....
4 The term "eventually" should be made precise, but that would " If P, -< Pi, then Pi need only have receiveda messagetimestamped
require too long a diversion from our main topic. _>T,,, from P/.
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the ACM Number 7

request. Since rules 3 and 4 are the only ones which Anomalous Behavior
delete messages from the request queue, it is then easy to
see that condition I holds. Condition II follows from the Our resource scheduling algorithm ordered the re-
fact that the total ordering ~ extends the partial ordering quests according to the total ordering =*. This permits
---~. Rule 2 guarantees that after Pi requests the resource, the following type of "anomalous behavior." Consider a
condition (ii) of rule 5 will eventually hold. Rules 3 and nationwide system of interconnected computers. Suppose
4 imply that if each process which is granted the resource a person issues a request A on a computer A, and then
eventually releases it, then condition (i) of rule 5 will telephones a friend in another city to have him issue a
eventually hold, thus proving condition III. request B on a different computer B. It is quite possible
This is a distributed algorithm. Each process inde- for request B to receive a lower timestamp and be ordered
pendently follows these rules, and there is no central before request A. This can happen because the system
synchronizing process or central storage. This approach has no way of knowing that A actually preceded B, since
can be generalized to implement any desired synchroni- that precedence informatiori is based on messages exter-
zation for such a distributed multiprocess system. The nal to the system.
synchronization is specified in terms of a State Machine, Let us examine the source of the problem more
consisting of a set C of possible commands, a set S of closely. Let O° be the set of all system events. Let us
possible states, and a function e: C × S--~ S. The relation introduce a set of events which contains the events in b°
e(C, S) -- S' means that executing the command C with together with all other relevant external events, such as
the machine in state S causes the machine state to change the phone calls in our example. Let ~ denote the "hap-
to S'. In our example, the set C consists of all the pened before" relation for ~. In our example, we had A
commands Pi requests resource and P~ releases resource, B, but A-~ B. It is obvious that no algorithm based
and the state consists of a queue of waiting request entirely upon events in 0 °, and which does not relate
commands, where the request at the head of the queue those events in any way with the other events i n ~ , can
is the currently granted one. Executing a request com- guarantee that request A is ordered before request B.
mand adds the request to the tail of the queue, and There are two possible ways to avoid such anomalous
executing a release command removes a command from behavior. The first way is to explicitly introduce into the
'he queue. 7 system the necessary information about the ordering
Each process independently simulates the execution --~. In our example, the person issuing request A could
of the State Machine, using the commands issued by all receive the timestamp TA of that request from the system.
the processes. Synchronization is achieved because all When issuing request B, his friend could specify that B
processes order the commands according to their time- be given a timestamp later than TA. This gives the user
stamps (using the relation ~ ) , so each process uses the the responsibility for avoiding anomalous behavior.
same sequence of commands. A process can execute a The second approach is to construct a system of
command timestamped T when it has learned of all clocks which satisfies the following condition.
commands issued by all other processes with timestamps
Strong Clock Condition. For any events a, b in O°:
less than or equal to T. The precise algorithm is straight-
i f a --~ b then C(a} < C(b).
forward, and we will not bother to describe it.
This method allows one t o implement any desired This is stronger than the ordinary Clock Condition be-
form of multiprocess synchronization in a distributed cause ~ is a stronger relation than ---~. It is not in general
system. However, the resulting algorithm requires the satisfied by our logical clocks.
active participation of all the processes. A process must Let us identify ~ with some set of "real" events in
know all the commands issued by other processes, so physical space-time, and let ~ be the partial ordering of
that the failure of a single process will make it impossible events defined by special relativity. One of the mysteries
for any other process to execute State Machine com- of the universe is that it is possible to construct a system
mands, thereby halting the system. of physical clocks which, running quite independently of
The problem of failure is a difficult one, and it is one another, will satisfy the Strong Clock Condition. We
beyond the scope of this paper to discuss it in any detail. can therefore use physical clocks to eliminate anomalous
We will just observe that the entire concept of failure is behavior. We now turn our attention to such clocks.
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. Physical Clocks
A user can tell that a system has "crashed" only because
he has been waiting too long for a response. A method Let us introduce a physical time coordinate into our
which works despite the failure of individual processes space-time picture, and let Ci(t) denote the reading of
the clock Ci at physical time t.8 For mathematical con-
or communication lines is described in [3].
We will assume a Newtonian space-time. If the relative motion
7 If each process does not strictly alternate request and release of the clocks or gravitational effects are not negligible, then CM) must
commands, then executing a release command could delete zero, one, be deduced from the actual clock reading by transforming from proper
or more than one request from the queue. time to the arbitrarily chosen time coordinate.

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venience, we assume that the clocks run continuously We now describe our algorithm for insuring that PC2
rather than in discrete "ticks." (A discrete clock can be holds. Let m be a message which is sent at physical time
thought of as a continuous one in which there is an error t and received at time t'. We define l,m ~ - t t - - I to be the
of up to ½ "tick" in reading it.) More precisely, we total delay of the message m. This delay will, of course,
assume that Ci(t) is a continuous, differentiable function not be known to the process which receives m. However,
of t except for isolated j u m p discontinuities where the we assume that the receiving process knows some mini-
clock is reset. Then dCg(t)/dt represents the rate at which m u m delay tzm >_ 0 such that ~£m ~ Pro. We call ~,, = I,m
the clock is running at time t. -- #m the unpredictable delay of the message.
In order for the clock Cg to be a true physical clock, We now specialize rules I R I and 2 for our physical
it must run at approximately the correct rate. That is, we clocks as follows:
must have d C i ( t ) / d t -~ 1 for all t. More precisely, we will IR 1'. For each i, if Pi does not receive a message at
assume that the following condition is satisfied: physical time t, then C/is differentiable at t and dCg(t)/dt
>0.
P C I . There exists a constant x << 1
IR2'. (a) If Pg sends a message m at physical time t,
such that for all i: [dCg(t)/dt - 1 [ < x.
then m contains a timestamp T m = C/(t). (b) Upon
For typical crystal controlled clocks, x _< 10-(~. receiving a message m at time t', process P/ sets C/(t')
It is not enough for the clocks individually to run at equal to m a x i m u m (Cj(t' - 0), Tm + /Zm). 9
approximately the correct rate. They must be synchro- Although the rules are formally specified in terms of
nized so that Cg(t) = C/(t) for all i,j, and t. More precisely, the physical time parameter, a process only needs to
there must be a sufficiently small constant e so that the know its own clock reading and the timestamps of mes-
following condition holds: sages it receives. For mathematical convenience, we are
PC2. For all i, j: [ C i ( t ) - Cy(t)[ < •. assuming that each event occurs at a precise instant of
physical time, and different events in the same process
If we consider vertical distance in Figure 2 to represent
occur at different times. These rules are then specializa-
physical time, then PC2 states that the variation in height
tions of rules IR1 and IR2, so our system of clocks
of a single tick line is less than E.
satisfies the Clock Condition. The fact that real events
Since two different clocks will never run at exactly
have a finite duration causes no difficulty in implement-
the same rate, they will tend to drift further and further
ing the algorithm. The only real concern in the imple-
apart. We must therefore devise an algorithm to insure
mentation is making sure that the discrete clock ticks are
that PC2 always holds. First, however, let us examine
frequent enough so C 1 is maintained.
how small x and • must be to prevent anomalous behav-
We now show that this clock synchronizing algorithm
ior. We must insure that the system 5e of relevant physical
can be used to satisfy condition PC2. We assume that
events satisfies the Strong Clock Condition. We assume
the system of processes is described by a directed graph
that our clocks satisfy the ordinary Clock Condition, so
in which an arc from process Pi to process P/represents
we need only require that the Strong Clock Condition
a communication line over which messages are sent
holds when a and b are events in 0 ° with a 4-> b.
directly from Pi to P/. We say that a message is sent over
Hence, we need only consider events occurring in differ-
this arc every T seconds if for any t, Pi sends at least one
ent processes.
message to P / b e t w e e n physical times t and t + -r. The
Let # be a number such that if event a occurs at
diameter of the directed graph is the smallest n u m b e r d
physical time t and event b in another process satisfies
such that for any pair of distinct processes P/, Pk, there
a ~ b, then b occurs later than physical time t + bt. In
is a path from P / t o P~ having at most d arcs.
other words,/~ is less than the shortest transmission time
In addition to establishing PC2, the following theo-
for interprocess messages. We can always choose # equal
rem bounds the length of time it can take the clocks to
to the shortest distance between processes divided by the
become synchronized when the system is first started.
speed of light. However, depending upon how messages
THEOREM. Assume a strongly connected graph of
in ~ are transmitted, # could be significantly larger.
processes with diameter d which always obeys rules IR 1'
To avoid anomalous behavior, we must make sure
and IR2'. Assume that for any message m, #m --< # for
that for any i, j, and t: Ci(t + #) - CAt) > 0. Combining
some constant g, and that for all t > to: (a) PC 1 holds.
this with PC I and 2 allows us to relate the required
(b) There are constants ~"and ~ such that every ~-seconds
smallness of x and ~ to the value of # as follows. We
a message with an unpredictable delay less than ~ is sent
assume that when a clock is reset, it is always set forward
over every arc. Then PC2 is satisfied with • = d(2x~- +
and never back. (Setting it back could cause C I to be
~) for all t > to + Td, where the approximations assume
violated.) PCI then implies that Cg(t + #) - Cg(t) > (1 # + ~<< z.
- x)#. Using PC2, it is then easy to deduce that Cg(t +
The proof of this theorem is surprisingly difficult,
#) - C/(t) > 0 if the following inequality holds:
and is given in the Appendix. There has been a great
E/(I - ~) _< ~. deal of work done on the problem of synchronizing
This inequality together with PC 1 and PC2 implies that physical clocks. We refer the reader to [4] for an intro-
anomalous behavior is impossible. :) C/(t' - 0) = lim C,(t' - 181).

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duction to the subject. T h e methods described in the ti+l, to <-- t~, and that at time t[ process Pi sends a message
literature are useful for estimating the message delays to process Pi+l which is received at time ti+l with an
ktm and for adjusting the clock frequencies d C i / d t (for unpredictable delay less than 4. T h e n repeated applica-
clocks which permit such an adjustment). However, the tion o f the inequality (3) yields the following result for
requirement that clocks are never set backwards seems t >_ t n + l .
to distinguish our situation from ones previously studied,
Ct~t(t) --> Cl(tl') + (1 - ~)(t - tl') - n~. (4)
and we believe this t h e o r e m to be a new result.
F r o m PC1, I R I ' and 2' we deduce that
C l ( / l ' ) >" C l ( t l ) + (1 -- K)(tl' -- /1).
Conclusion
C o m b i n i n g this with (4) and using (2), we get
W e have seen that the concept o f " h a p p e n i n g before"
Cn+a(t) > C~(tl) + (1 - x)(t - t~) - n~ (5)
defines an invariant partial ordering of the events in a
distributed multiprocess system. W e described an algo- for t > t n + l .
rithm for extending that partial ordering to a s o m e w h a t F o r a n y two processes P and P', we can find a
arbitrary total ordering, and showed how this total or- sequence of processes P -- Po, P~ . . . . . Pn+~ = P', n _< d,
dering can be used to solve a simple synchronization with c o m m u n i c a t i o n arcs f r o m each Pi to Pi+~. By hy-
problem. A future p a p e r will show how this a p p r o a c h pothesis (b) we can find times ti, t[ with t[ - ti <- T and
can be extended to solve a n y synchronization problem. ti+l - t" <_ v, where v = # + 4. Hence, an inequality o f
T h e total ordering defined by the algorithm is some- the form (5) holds with n <_ d w h e n e v e r t >_ t~ + d('r +
what arbitrary. It can produce a n o m a l o u s b e h a v i o r if it v). F o r a n y i, j and any t, tl with tl > to and t ___ t~ + d(z
disagrees with the ordering perceived by the system's + v) we therefore have:
users. This can be prevented by the use of properly
Ci(t) _> Cj(ta) + (1 - x)(t - tx) - d~. (6)
synchronized physical clocks. O u r t h e o r e m showed h o w
closely the clocks can be synchronized. N o w let m be any message t i m e s t a m p e d Tin, a n d
In a distributed system, it is i m p o r t a n t to realize that suppose it is sent at time t and received at time t'. W e
the order in which events occur is only a partial ordering. pretend that m has a clock Cm which runs at a constant
W e believe that this idea is useful in understanding any rate such that C,~(t) = tm and Cm(t') = tm +/~m. T h e n
multiprocess system. It should help one to u n d e r s t a n d #0, ___t' - t implies that d C m / d t _< 1. Rule I R 2 ' (b) simply
the basic p r o b l e m s o f multiprocessing independently of sets Cj(t') to m a x i m u m (Cj(t' - 0), Cm(t')). Hence, clocks
the m e c h a n i s m s used to solve them. are reset only by setting t h e m equal to other clocks.
F o r a n y time tx >-- to + / ~ / ( 1 - ~), let Cx be the clock
Appendix having the largest value at time t~. Since all clocks run
at a rate less t h a n 1 + x, we have for all i and all t >_ tx:
Proof of the Theorem Ci(t) _< Cx(tx) + (1 + x)(t - tx). (7)
F o r a n y i and t, let us define C~t to be a clock which
is set equal to C~ at time t and runs at the same rate as W e n o w consider the following two cases: (i) Cx is the
Ci, but is never reset. In other words, clock Cq o f process Pq. (ii) Cx is the clock Cm o f a
message sent at time ta by process Pq. In case (i), (7)
simply becomes
C i t ( t ') = Ci(t) + [dCz(t)/dtldt (1)
El(t) -< Cq(tx) + (1 + x)(t - tx). (8i)
In case (ii), since Cm(tx) = Cq(tl) and d C m / d t _ 1, we
for all t' >_ t. Note that
have
Ci(t') >_ Cit(t ') for all t' >__ t. (2)
Cx(tx) <_ Cq(tl) + (tx - tO.
Suppose process P~ at time tl sends a message to
process Pz which is received at time t2 with an unpre- Hence, (7) yields
dictable delay _< ~, where to <- ta _< t2. T h e n for all t ___ t2 Ci(t) <-~ Cq(/1) + (1 + K)(t -- /1). (8ii)
we have:
Since tx >-- to + ~t/(1 -- X), we get
C~(t) >_ C~(t2) + (1 - x)(t - t2) [by (1) and P C I ]
> Cfftl) +/~m + (1 -- x)(t -- t2) [by I R 2 ' (b)] Cq(tx -- ~/(1 -- K)) <_ Cq(tx) - ~ [by P C I ]
= Cl(tl) + (1 - x)(t - tl) - [(t2 - tO - ~m] + x(t2 - t,)
___ Cm(tx) - / z [by choice o f m]
>-- Cl(tl) + (1 - x ) ( t - tl) - 4. -< Cm(tx) - (tx - t l ) # m / V m [tXm <-- r.t, tx - tl <_ v,,]
=Tm [by definition o f Cm]
Hence, with these assumptions, for all t >_ t2 we have: = Cq(tl) [by IR2'(a)].
C~(t) _> Cl(tl) + (1 - x)(t - / 1 ) - - 4" (3) Hence, Cq(tx - / L / ( 1 - x)) ___ Cq(tl), so tx - tl < - / x / ( l -
NOW suppose that for i = 1, ..., n we have t, _< t ~, < ~) and thus ll ~ to.

of Volume 21
the ACM Number 7

Letting tl = tx in case (i)~ we c a n c o m b i n e (8i) a n d
(8ii) to d e d u c e that for a n y t, tx with t _> tx _> to + # /
(1 - x) there is a process Pq a n d a time tl with tx - # /
(1 - i¢) ~ tl "< tx such t h a t for all i:

Ci(t) _< Cq(t~) + (1 + x)(t - h). (9) Programming J.J. Homing
Languages Editor
C h o o s i n g t a n d tx with t _ tx + d ( r + ~,), we c a n c o m b i n e
(6) a n d (9) to c o n c l u d e that there exists a tl a n d a process
Pq such that for all i:
Shallow Binding in
Cq(t 0 + (1 - x)(t - tl) - d~ -< Ci(t) (10) Lisp 1.5
<: Cq(tl) + (1 + tc)(t - t]) Henry G. Baker, Jr.
Letting t = t~ + d ( r + u), we get Massachusetts Institute of Technology
d(r+p)__.t-tl_<d(r+v)+#/(1-x).
C o m b i n i n g this with (10), we get Shallow binding is a scheme which allows the value
of a variable to be accessed in a bounded amount of
Cq(tl) + (t - tl) - x d ( r + p) - d~ -< Ci(t) _< Cq(tl) (11)
computation. An elegant model for shallow binding in
+ (t - tl) + x[d(r + u) + / z / ( 1 - x)] Lisp 1.5 is presented in which context-switching is an
environment tree transformation called rerooting.
U s i n g the h y p o t h e s e s that x << 1 a n d / t _< ~, << r, we can
Rerooting is completely general and reversible, and is
rewrite (11) as the following a p p r o x i m a t e inequality.
optional in the sense that a Lisp 1.5 interpreter will
Cq(tl) + (t - tl) - d(tg'r + ~) ~.< Ci(t) (12) operate correctly whether or not rerooting is in-
voked on every context change. Since rerooting leaves
~' Cq(/1) + (l -- tl) -I- dKr.
assoc [ v, a] invariant, for all variables v and all
Since this h o l d s for all i, we get environments a, the programmer can have access to a
rerooting primitive, shallow[], which gives him dynamic
[Ci(t) - CAt)[ ~<d(2xr + O, control over whether accesses are shallow or deep, and
a n d this h o l d s for all t > to + dr. [] which affects only the speed of execution of a program,
N o t e that r e l a t i o n (11) o f the p r o o f yields an exact not its semantics. In addition, multiple processes can be
u p p e r b o u n d for [Ci(t) - Cj(t) l in case the a s s u m p t i o n active in the same environment structure, so long as
/~ + ~ << r is invalid. A n e x a m i n a t i o n o f the p r o o f rerooting is an indivisible operation. Finally, the
suggests a simple m e t h o d for r a p i d l y initializing the concept of rerooting is shown to combine the concept of
clocks, o r r e s y n c h r o n i z i n g t h e m i f t h e y s h o u l d go out o f shallow binding in Lisp with Dijkstra's display for Algol
s y n c h r o n y for a n y reason. E a c h process sends a message and hence is a general model for shallow binding.
w h i c h is r e l a y e d to every o t h e r process. T h e p r o c e d u r e Key Words and Phrases: Lisp 1.5, environment
c a n be i n i t i a t e d b y a n y process, a n d requires less t h a n trees, FUNARG'S, shallow binding, deep binding,
2 d ~ + ~) seconds to effect the s y n c h r o n i z a t i o n , a s s u m i n g multiprogramming, Algol display
e a c h o f the messages has a n u n p r e d i c t a b l e d e l a y less CR Categories: 4.13, 4.22, 4.32
t h a n ~.

A c k n o w l e d g m e n t . T h e use o f t i m e s t a m p s to o r d e r
operations, a n d the concept o f a n o m a l o u s b e h a v i o r are
d u e to P a u l J o h n s o n a n d R o b e r t T h o m a s .

Received March 1976; revised October 1977

References General permission to make fair use in teaching or research of all
1. Schwartz, J.T. Relativity in lllustrations. New York U. Press, New or part of this material is granted to individual readers and to nonprofit
York, 1962. libraries acting for them provided that ACM's copyright notice is given
2. Taylor, E.F., and Wheeler, J.A. Space-Time Physics, W.H. and that reference is made to the publication, to its date of issue, and
Freeman, San Francisco, 1966. to the fact that reprinting privileges were granted by permission of the
3. Lamport, L. The implementation of reliable distributed Association for Computing Machinery. To otherwise reprint a figure,
multiprocess systems. To appear in Computer Networks. table, other substantial excerpt, or the entire work requires specific
4. Ellingson, C, and Kulpinski, R.J. Dissemination of system-time. permission as does republication, or systematic or multiple reproduc-
1EEE Trans. Comm. Com-23, 5 (May 1973), 605-624. tion.
This research was supported by the Advanced Research Projects
Agency of the Department of Defense and was monitored by the Office
of Naval Research under contract number N00014-75-C-0522.
Author's present address: Computer Science Department, Univer-
sity of Rochester, Rochester, NY 14627.
© 1978 ACM 0001-0782/78/0700-0565 $00.75

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Time clocks and the ordering of events

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