Outline
• Theoretical Foundations
– Fundamental limitations of distributed systems
– Logical clocks
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Lamport’s Logical Clocks – review
• Definitions
– Happened before relation
• Happened before relation () captures the causal
dependencies between events
• It is defined as follows
– a  b, if a and b are events in the same process and a
occurred before b.
– a  b, if a is the event of sending a message m in a process
and b is the event of receipt of the same message m by
another process
– If a  b and b  c, then a  c, i.e., “” is transitive
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Lamport’s Logical Clocks – review
• Definitions – continued
– Causally related events
• Event a causally affects event b if a  b
– Concurrent events
• Two distinct events a and b are said to be concurrent
(denoted by a || b) if a  b and b  a
• For any two events, either a  b, b  a, or a || b
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Lamport’s Logical Clocks – review
• Logical clocks
– There is a clock at each process Pi in the system
• Which is a function that assigns a number to any event
a, called the timestamp of event a at Pi
• The numbers assigned by the system of the clocks
have no relation to physical time
• The logical clocks take monotonically increasing
values and can be implemented as counters
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Lamport’s Logical Clocks – cont.
• Conditions satisfied by the system of clocks
– For any two events, if a  b, then C(a) < C(b)
– [C1] For any two events a and b in a process Pi,
if a occurs before b, then
Ci(a) < Ci(b)
– [C2] If a is the event of sending a message m in
process Pi and b is the event of receiving the
same message m at process Pj, then
Ci(a) < Cj(b)
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Lamport’s Logical Clocks – cont.
• Implementation rules
– [IR1] Clock Ci is incremented between any two
successive events in process Pi
Ci := Ci + d ( d > 0)
– [IR2] If event a is the sending of message m by
process Pi, then message m is assigned a
timestamp tm = Ci(a). On receiving the same
message m by process Pj, Cj is set to
Cj := max(Cj, tm + d)
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An Example
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Total Ordering Using Lamport’s Clocks
• If a is any event at process Pi and b is any
event at process Pj, then a => b if and only if
either
Ci (a)  C j (b) or
Ci (a)  C j (b) and Pi  Pj
– Where  is any arbitrary relation that totally
orders the processes to break ties
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Example: Totally-Ordered Multicasting
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A Limitation of Lamport’s Clocks
• In Lamport’s system of logical clocks
– If a  b, then C(a) < C(b)
– The reverse if not necessarily true if the events
have occurred on different processes
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A Limitation of Lamport’s Clocks
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Vector Clocks
• Implementation rules
– [IR1] Clock Ci is incremented between any two
successive events in process Pi
Ci[i] := Ci[i] + d ( d > 0)
– [IR2] If event a is the sending of message m by
process Pi, then message m is assigned a
timestamp tm = Ci(a). On receiving the same
message m by process Pj, Cj is set to
Cj[k] := max(Cj[k], tm[k])
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Vector Clocks – cont.
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Vector Clocks – cont.
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Vector Clocks – cont.
• Assertion
– At any instant,
i, j : Ci [i]  C j [i]
• Events a and b are casually related if ta < tb or tb
< ta. Otherwise, these events are concurrent
• In a system of vector clocks,
a  b iff t  t
a
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Causal Ordering of Messages
• The causal ordering of messages try to
maintain the same causal relationship that
holds among “message send” events with the
corresponding “message receive” events
– In other words, if Send(M1) -> Send(M2), then
Receive(M1) -> Receive(M2)
– This is different from causal ordering of events
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Causal Ordering of Messages – cont.
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Causal Ordering of Messages – cont.
• The basic idea
– It is very simple
– Deliver a message only when no causality
constraints are violated
– Otherwise, the message is not delivered
immediately but is buffered until all the
preceding messages are delivered
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Birman-Schiper-Stephenson Protocol
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Schiper-Eggli-Sando Protocol
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Schiper-Eggli-Sando Protocol – cont.
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Schiper-Eggli-Sando Protocol – cont.
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