# Time, Clocks, and Leslie Lamport

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In the paper Time, Clocks, and the Ordering of Events in a Distributed System, Leslie Lamport defines the happens before relationship for distributed systems. This is a seminal work that is fundamental to the understanding of distributed systems. It addresses the simple yet profound problem of answering the question, “What happened when?” This is an overview of that paper and a summary of the key ideas.

## System Model

A distributed system is comprised of a group of processes that communicate by sending messages. Although the idea of a distributed system can be more general than that, for the scope of the paper the “nodes” are processes and the “edges” are messages. Other examples of distributed systems include cells in the body, people and the mail service, a multiprocessor computer system, etc. The term is a large umbrella.

An event in the system is the transmission or receipt of a message. A process, therefore, can be specified by the ordered collection of events that occur during its lifetime.

## Partial Order

This paper introduces the happens before relationship, one that can be used to define a partial order over a set of events. For every pair of events, it can be said that either (1) one events happens before the other, or (2) the events happen concurrently. In the latter case, it is impossible to tell which happened first because physical clocks cannot be trusted.

The happens before relationship for two events a and b is represented as a right-facing arrow ‘->’ and is defined by three characteristics as follows:

1. If event a comes before b in the same process, a -> b.
2. If event a is the sending of a message and b is the receiving of the same message, then a -> b.
3. If a -> b and b -> c, then a -> c.

These three rules give us a partial ordering over the set of events. The proof of this is left to the paper and as an exercise for the reader :)

## Logical Clocks

Now that we have a partial order over events, we can assign a logical time to each event. It is natural to write programs that rely on a clock, but in place of a physical time we can simply define some logical clocks based on our relationship.

Specifically, the paper defines a function C(e) which gives the logical time for event e in the distributed system. There is but one requirement for this function, that it satisfies the clock condition :

if a->b then C(a) < C(b)

This clock function can be defined by two rules:

• IR1. A process increments its clock before sending a message.
• IR2. A process receives a message, and then sets its clock to the maximum of:
• its current clock plus one
• the timestamp of the received message plus one

With these two rules, our clock function satisfies the clock condition. Again, left as an excercise to the reader to prove this.

## Total Order

Now that we have a partial ordering in terms of clock values (you can create scenarios where two events have the same value), we can define an arbitrary, deterministic _total ordering over all events. A trivial yet powerful method is to assign every process an identifier. When two events have the same clock value, they must have been executed on different processes. Therefore, we can order events by sorting again these events by their PID.

The paper does not define the ordering among processes, only that such an ordering exists; PIDs are a simple way to accomplish this.

The paper wraps up this section by using the total ordering property to solve distributed mutual exclusion. I don’t want to discuss it here because it reads more like a motivation for distributed total ordering than a core contribution of the paper.

## Physical Clocks

The method of total ordering can be further extended to synchronize distributed processes to some phyisical time, within some bound.

In our logical system, the clock condition holds for events occuring within the system but says nothing about external events. Indeed, the paper presents a scenario where an external event happens before another, but according to the system the inputs may happen concurrently or in reverse. So, the paper presents the strong clock condition, which is equivalent to the clock condition but also considers relevant external events to the system.

The paper presents two properties of a physical clock:

• PC1: There exists some constant k such that the physical clock rate for any process is one plus or minus k.
• PC2: There exists some constant e such that the absolute difference in clock rate between two processes is less than e.

The paper proceeds to present an algorithm for setting physical clocks which is similar to the rules IR1 and IR2. The proof for this algorithm is included in the paper. Basically, the observation is that given some minimal propagation time in between a message send and receive, we can bump the local clock to the maximum of the current time or the time the message was sent plus the minimal propagation time. In this way, messages passively synchronize the clocks of the nodes in the system.

# Conclusion

This paper presents a simple problem with a simple solution and profound impact. One of the core problems in distributed systems is consensus, and this paper presents a solution for consensus among a set of events. In distributed systems, even the mere problem of ordering events is difficult and non-trivial. The paper raises a red flag and warns us that even simple assumptions are dangerous within [distributed] systems.

From logical clocks we can build many other distributed system primitives, such as atomic broadcast, commit protocols, consensus algorithms, and much more!

At the time of writing, the paper has over 11,000 citations. This speaks to how influential the work is within the distributed systems community and how prolific the problem of time is to the field.

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