1. Logical Clocks (2)

2. Topics
Logical clocks
Totally-Ordered Multicasting
Vector timestamps

3. Readings Van Steen and Tanenbaum: 5.2 Coulouris: 10.4
L. Lamport, “Time, Clocks and the Ordering of Events in Distributed Systems,” Communications of the ACM, Vol. 21, No. 7, July 1978, pp
C.J. Fidge, “Timestamps in Message-Passing Systems that Preserve the Partial Ordering”, Proceedings of the 11th Australian Computer Science Conference, Brisbane, pp , February 1988.

4. 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

5. Problems with Lamport Clocks
Why is this important?
For the banking example, it doesn’t matter what order operations are executed at each replica.
It only matters that all replicas execute the operations in the same order. This is total order.
Causal order is used when a message received by a process can potentially affect any subsequent message sent by that process. Those messages should be received in that order at all processes. Unrelated messages may be delivered in any order.
Do Lamport timestamps support causal order? NO

6. Example Application:Bulletin Board
The Internet’s electronic bulletin board (or Google’s Groups or any chat service)
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.
Should we be doing this?

7. 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 (more on this later)
Ordering:
total (makes the numbers the same at all sites)
Bulletin board:
os.interesting
causal (makes replies come after original message)
Item
From
Subject 23
A.Hanlon
Mach
FIFO (gives sender order) 24
G.Joseph
Microkernels 25
A.Hanlon
Re: Microkernels 26
T.L’Heureux
RPC performance
Figure 11.13
Colouris 27
M.Walker
Re: Mach
end •

8. 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.
The receipt of an article precedes the posting of a reaction.
The receipt of the reaction to an article should always follow the receipt of the article.

9. Example Application: Bulletin Board
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.

10. 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.

11. Vector Timestamps Properties of vector timestamps
vi [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

12. Vector Timestamps A vector clock is maintained as follows:
Initially all clock values are set to the smallest value (e.g., 0).
The local clock value is incremented at least once before each event of interest (in our case this will be a send event) in a process, q i.e., vq[q] = vq[q] +1
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] );

13. Vector Timestamp For two vector timestamps, va and vb
va  vb if there exists an i such that va[i]  vb[i]
va ≤ vb if for all i va[i] ≤ vb[i]
va < vb if for all i va[i] ≤ vb[i] AND va is not equal to vb
Events a and b are causally related if va < vb or vb< va .
Vector timestamps can be used to guarantee causal message delivery.

14. Example Application: Bulletin Board
Each process Pi has an array Vi where Vi[j] denotes the number of events that process Pi knows have taken place for Pj.
In this application “events” refers to the sending of a message.
Thus if Vi[j] = 6 then Pi knows that Pj has sent 6 messages.

15. Example Application: Bulletin Board
Let Vi be piggybacked on the message sent by process Pi to process Pj ; When process Pj receives the message, then Pj does the following:
For k = 1 to n do
Vj[k] = max(Vi[k], Vj [k] );
Thus if process i knows that process k sent 5 messages (Vi [k] =5 and Vj[k] = 3) then process j didn’t know about the latest two messages sent by process k.

16. Example Application: Bulletin Board
When process i sends a message, it does the following:
Vi[i] = Vi[i] + 1;
This is basically saying that the number of messages process i has sent is incremented by one.

17. Example Application: Bulletin Board
When a process Pi posts (sends) an article, a, it multicasts that article with timestamp 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 .

18. Example Application: Bulletin Board
Message r (from Pj ) is delivered to Pk iff the following conditions are met:
Vj[j] = Vk[j]+1
This condition is satisfied if r is the next message that Pk was expecting from process Pj
Assume that Vk[j]=5. This means that Pk knows Pj sent 5 messages and that it should be waiting for the 6th message.
If Vj[j] = 8 then messages 6,7 are lost or not yet arrived.
Vj[i] ≤ Vk[i] for all i not equal to j
This condition is satisfied if Pk has seen at least as many messages as seen by Pj when it sent message r.
Assume for some i that Vj[i] = 2 and Vk[i] =1 and thus Vj[i] > Vk[i] for some i. This indicates that Pi sent a message that was received by Pj but not Pk

19. Example Application: Bulletin Board
[0,0,0] P1
[0,0,0] P3
[0,0,0]
Post a
[1,0,0] a
[1,0,0] e
[1,0,0] c
[1,0,1]
r: Reply a d g b
[1,0,1]
Note that if two events happen without any message, then we can't say anything about their relative occurrence in time.
In this example a <= b <= c <= d <= f, but we can say little about e other than e <= f. We say that events such as a and e are concurrent.
Note also that “happened before” does not demonstrate causality. There is merely the potential for causality.
[1,0,1]
Message a arrives at P2 before the reply r from P3 does

20. Example Application: Bulletin Board
[0,0,0] P1
[0,0,0] P3
[0,0,0]
Post a
[1,0,0] a d
[1,0,0] g
[1,0,1]
r: Reply a
Buffered b
Note that if two events happen without any message, then we can't say anything about their relative occurrence in time.
In this example a <= b <= c <= d <= f, but we can say little about e other than e <= f. We say that events such as a and e are concurrent.
Note also that “happened before” does not demonstrate causality. There is merely the potential for causality. c
[1,0,1]
[1,0,0]
Deliver r
The message a arrives at P2 after the reply from P3; The reply is
not delivered right away.

21. 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.