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Logical Clocks. Topics. Logical clocks Totally-Ordered Multicasting Vector timestamps. 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. 558-565. - PowerPoint PPT Presentation
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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
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
Assume that each process’s logical clock is set to 0
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.
Example: Totally Ordered Multicast
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
Example: Totally Ordered Multicast
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
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.
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.
Example Application: Bulletin Board
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.
Example Application: Bulletin Board
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.
Example Application: Bulletin Board
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.
Example Application: Bulletin Board
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]
Example Application: Bulletin Board
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
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