Logical Clocks

Logical Clocks

Topics

Logical clocksTotally-Ordered MulticastingVector timestamps

ReadingsVan Steen and Tanenbaum: 5.2Coulouris: 10.4L. Lamport, “Time, Clocks and the Ordering of

Events in Distributed Systems,” Communications of the ACM, Vol. 21, No. 7, July 1978, pp. 558-565.

C.J. Fidge, “Timestamps in Message-Passing Systems that Preserve the Partial Ordering”, Proceedings of the 11th Australian Computer Science Conference, Brisbane, pp. 56-66, February 1988.

Ordering of Events

For many applications, it is sufficient to be able to agree on the order that events occur and not the actual time of occurrence.

It is possible to use a logical clock to unambiguously order events

May be totally unrelated to real time.Lamport showed this is possible (1978).

The Happened-Before Relation Lamport’s algorithm synchronizes logical clocks

and is based on the happened-before relation: a b is read as “a happened before b”

The definition of the happened-before relation: If a and b are events in the same process and a occurs before b, then

a b For any message m, send(m) send(m) rcv(m), where send(m) is the

event of sending the message and rcv(m) is event of receiving it. If a, b and c are events such that a b and b c then a c

The Happened-Before RelationIf two events, x and y, happen in different

processes that do not exchange messages , then x y is not true, but neither is y x

The happened-before relation is sometimes referred to as causality.

ExampleSay in process P1 you have

a code segment as follows:

1.1 x = 5;

1.2 y = 10*x;

1.3 send(y,P2);

Say in process P2 you have a code segment as follows:

2.1 a=8;

2.2 b=20*a;

2.3 rcv(y,P1);

2.4 b = b+y;

Let’s say that you start P1 and P2 at the same time. You know that 1.1 occurs before 1.2 which occurs before 1.3; You know that 2.1 occurs before 2.2 which occurs before 2.3 which is before 2.4.You do not know if 1.1 occurs before 2.1 or if 2.1 occurs before 1.1.You do know that 1.3 occurs before 2.3 and 2.4

Example Continuing from the example on the previous page –

The order of actual occurrence of operations is often not consistent from execution to execution. For example: Execution 1 (order of occurrence): 1.1, 1.2, 1.3, 2.1, 2.2, 2.3, 2.4 Execution 2 (order of occurrence): 2.1,2.2,2.3,1.3, 2.3,2.4 Execution 3 (order of occurrence) 1.1, 2.1, 2.2, 1.2, 1.3, 2.3, 2.4

We can say that 1.1 “happens before” 2.3, but not that 1.1 “happens before” 2.2 or that 2.2 “happens before” 1.1.

Note that the above executions provide the same result.

Lamport’s Algorithm

We need a way of measuring time such that for every event a, we can assign it a time value C(a) on which all processes agree on the following: The clock time C must monotonically increase i.e., always

go forward. If a b then C(a) < C(b)

Each process, p, maintains a local counter Cp

The counter is adjusted based on the rules presented on the next page.

Lamport’s Algorithm

1. Cp is incremented before each event is issued at process p: Cp = Cp + 1

2. When p sends a message m, it piggybacks on m the value t=Cp

3. On receiving (m,t), process q computes Cq = max(Cq,t) and then applies the first rule before timestamping the event rcv(m).

Example

a

b

P1 P2 P3

c

d

e

f

g

h

i

j

k

l

Assume that each process’s logical clock is set to 0

Example

a

b

P1 P2 P3

c

d

e

f

g

h

i

j

k

l


1 112

323

4 4

5

6

3

Example

From the timing diagram on the previous slide, what can you say about the following events? Between a and b: a b Between b and f: b f Between e and k: concurrent Between c and h: concurrent Between k and h: k h

Total Order

A timestamp of 1 is associated with events a, e, j in processes P1, P2, P3 respectively.

A timestamp of 2 is associated with events b, k in processes P1, P3 respectively.

The times may be the same but the events are distinct.

We would like to create a total order of all events i.e. for an event a, b we would like to say that either a b or b a

Total Order

Create total order by attaching a process number to an event.

Pi timestamps event e with Ci (e).i

We then say that Ci(a).i happens before Cj(b).j iff: Ci(a) < Cj(a); or

Ci(a) = Cj(b) and i < j

Example (total order)

a

b

P1 P2 P3

c

d

e

f

g

h

i

j

k

l


1.1 1.21.32.1

3.22.33.1

4.1 4.2

5.2

6.2

3.3

Example: Totally-Ordered MulticastApplication of Lamport timestamps (with total

order)Scenario

Replicated accounts in New York(NY) and San Francisco(SF)

Two transactions occur at the same time and multicast Current balance: $1,000 Add $100 at SF Add interest of 1% at NY If not done in the same order at each site then one

site will record a total amount of $1,111 and the other records $1,110.

Example: Totally-Ordered Multicasting

Updating a replicated database and leaving it in an inconsistent state.

Example: Totally-Ordered Multicasting

We must ensure that the two update operations are performed in the same order at each copy.

Although it makes a difference whether the deposit is processed before the interest update or the other way around, it does matter which order is followed from the point of view of consistency.

We need totally-ordered multicast, that is a multicast operation by which all messages are delivered in the same order to each receiver. NOTE: Multicast refers to the sender sending a message to a collection

of receivers.

Example: Totally Ordered MulticastAlgorithm

Update message is timestamped with sender’s logical time

Update message is multicast (including sender itself) When message is received

It is put into local queue Ordered according to timestamp, Multicast acknowledgement

Example:Totally Ordered Multicast

Message is delivered to applications only when It is at head of queue It has been acknowledged by all involved processes Pi sends an acknowledgement to Pj if

Pi has not made an update request

Pi’s identifier is less than Pj’s identifier

Pi’s update has been processed;

Lamport algorithm (extended for total order) ensures total ordering of events

Example: Totally Ordered Multicast

On the next slide m corresponds to “Add $100” and n corresponds to “Add interest of 1%”.

When sending an update message (e.g., m, n) the message will include the timestamp generated with the update was issued.


San Francisco (P1)

1.1

2.1

3.1

5.1

New York (P2)

1.2

2.2

3.2

4.2

Issue m

Send m

Recv n

Issue n

Send n

Recv m

Send ack(m)

6.1Send ack(n)

Recv ack(m)

5.2 Recv ack(n)

Process m


When P1 issues the update message (m) the timestamp associated with it is 1.1

When P2 issues the update message (n) the timestamp associated with it is 2.1

At both P1’s queue and P2’s queue the update messages are ordered such that m is before n.

A Note on Ordering and Consistency

The previous examples assumes that messages are received in the order they were delivered and the message passing is reliable.

There are different definitions of consistency.We will study these issues in more detail.

Problems with Lamport Clocks

Lamport timestamps do not capture causality.With Lamport’s clocks, one cannot directly

compare the timestamps of two events to determine their precedence relationship. If C(a) < C(b) is not true then a b is also not true. Knowing that C(a) < C(b) is true does not allow us to

conclude that a b is true. Example: In the first timing diagram, C(e) = 1 and C(b) =

2; thus C(e) < C(b) but it is not the case that e b

Problem with Lamport Clocks

The main problem is that a simple integer clock cannot order both events within a process and events in different processes.

C. Fidge developed an algorithm that overcomes this problem.

Fidge’s clock is represented as a vector [v1,v2,…,vn] with an integer clock value for each process (vi contains the clock value of process i). This is a vector timestamp.

Fidge’s Algorithm

Properties of vector timestampsvi [i] is the number of events that have

occurred so far at Pi

If vi [j] = k then Pi knows that k events have occurred at Pj

Fidge’s Algorithm

The Fidge’s logical clock is maintained as follows:

1. Initially all clock values are set to the smallest value (e.g., 0).

2. The local clock value is incremented at least once before each primitive event in a process i.e., vi[i] = vi[i] +1

3. The current value of the entire logical clock vector is delivered to the receiver for every outgoing message.

4. Values in the timestamp vectors are never decremented.

Fidge’s Algorithm

5. Upon receiving a message, the receiver sets the value of each entry in its local timestamp vector to the maximum of the two corresponding values in the local vector and in the remote vector received.

Let vq be piggybacked on the message sent by process q to process p; We then have: For i = 1 to n do

vp[i] = max(vp[i], vq [i] );

vp[p] = vp[p] + 1;

Fidge’s Algorithm

For two vector timestamps, Ta and Tb

Ta is not equal to Tb if there exists an i such that Ta[i] is not equal to Tb[i]

Ta <= Tb if for all i Ta[i] <= Tb[i]

Ta < Tb if for all i Ta[i] < = Tb[i] AND Ta is not equal to Tb

Events a and b are causally related if Ta < Tb or Tb< Ta .

Example

P2

a

b

P1

c

d

e

f

g

h

i

P3

j

k

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Example

P2

a

b

P1

c

d

e

f

g

h

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P3

j

k

l

[1,0,0]

[2,0,0]

[3,0,0]

[4,0,0]

[0,1,0]

[2,2,0]

[2,3,2]

[2,4,2]

[4,5,2]

[0,0,1]

[0,0,2]

[0,0,3]

Example Application:Bulletin Board

The Internet’s electronic bulletin board service (network news)

Users (processes) join specific groups (discussion groups).

Postings, whether they are articles or reactions, are multicast to all group members.

Could use a totally-ordered multicasting scheme.

Display from a Bulletin Board Program

Users run bulletin board applications which multicast messages One multicast group per topic (e.g. os.interesting) Require reliable multicast - so that all members receive messages Ordering:

Bulletin board: os.interesting

Item From Subject

23 A.Hanlon Mach

24 G.Joseph Microkernels

25 A.Hanlon Re: Microkernels

26 T.L’Heureux RPC performance

27 M.Walker Re: Mach

endFigure 11.13

total (makes the numbers the same at all sites)

FIFO (gives sender order

causal (makes replies come after original message)

•

Example Application: Bulletin Board

A totally-ordered multicasting scheme does not imply that if message B is delivered after message A, that B is a reaction to A.

Totally-ordered multicasting is too strong in this case.

The receipt of an article causally precedes the posting of a reaction. The receipt of the reaction to an article should always follow the receipt of the article.


If we look at the bulletin board example, it is allowed to have items 26 and 27 in different order at different sites.

Items 25 and 26 may be in different order at different sites.


Vector timestamps can be used to guarantee causal message delivery. A slight variation of Fidge’s algorithm is used.

Each process Pi has an array Vi where Vi[j] denotes the number of events that process Pi knows have taken place.

Vector timestamps are assumed to be updated only when posting or receiving articles i.e., when a message is sent or received. Incrementing a component is only done during sending.


When a process Pi posts an article, it multicasts that article as a message with the vector timestamp. Let’s calls this message a. Assume that the value of the timestamp is Vi

Process Pj posts a reaction. Let’s call this message r. Assume that the value of the timestamp is Vj

Note that Vj > Vi

Message r may arrive at Pk before message a.


Pk will postpone delivery of r to the display of the bulletin board until all messages that causally precede r have been received as well.

Message r is delivered iff the following conditions are met: Vj[j] = Vk[j]+1

This states that r is the next message that Pk was expecting from process Pj

Vj[i] <= Vk[i] for all i not equal to j This states that Pk has not seen any messages that were

not seen by Pj when it sent message r.


P2

a

P1

c

d

P3

e

g

[1,0,0]

[1,0,0][1,0,0]

[1,0,1]

[1,0,1]

Post a

r: Reply a

Message a arrives at P2 before the reply r from P3 does

b

[1,0,1]

[0,0,0] [0,0,0] [0,0,0]


P2

a

P1 P3

d

g

[1,0,0]

[1,0,0]

[1,0,1]

Post a

r: Reply a

Buffered

c[1,0,0]

The message a arrives at P2 after the reply from P3; The reply is not delivered right away.

b

[1,0,1]

[0,0,0] [0,0,0] [0,0,0]

Deliver r

Summary

No notion of a globally shared clock.Local (physical) clocks must be synchronized

based on algorithms that take into account network latency.

Knowing the absolute time is not necessary.Logical clocks can be used for ordering

purposes.

Logical Clocks

Documents

b ylets

order of occurrence

clock time c

actual time of occurrence

process p1

process p2

event rcvm

real time