1. Outline Theoretical Foundations
Fundamental limitations of distributed systems
Logical clocks
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2. Distributed Systems
A distributed system is a collection of independent computers that appears to its users as a single coherent system
Independent computers mean that they do not share memory or clock
The computers communicate with each other by exchanging messages over a communication network
The messages are delivered after an arbitrary transmission delay
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3. Inherent Limitations of a Distributed System
Absence of a global clock
In a centralized system, time is unambiguous
In a distributed system, there exists no system wide common clock
In other words, the notion of global time does not exist
Impact of the absence of global time
Difficult to reason about temporal order of events
Makes it harder to collect up-to-date information on the state of the entire system
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4. Absence of Global Time
When each machine has its own clock, an event that occurred after another event may nevertheless be assigned an earlier time.
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5. Inherent Limitations of a Distributed System
Absence of shared memory
An up-to-date state of the entire system is not available to any individual process
This information, however, is necessary to reason about the system’s behavior, debugging, recovering from failures
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6. Absence of Shared Memory – cont.
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7. Two Approaches for a Global Clock
First approach is to physically synchronize the clocks on different computers
Synchronize them to a common server
Synchronize them to an average of the clocks
Second approach is to establish what is known as logical clock
They can be used to reason about the temporal ordering of events
But they are not related to the physical clock
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8. Physical Clocks
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9. Physical Clocks – cont.
TAI seconds are of constant length, unlike solar seconds. Leap seconds are introduced when necessary to keep in phase with the sun.
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10. Clock Synchronization Algorithms
The relation between clock time and UTC when clocks tick at different rates.
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11. Cristian's Algorithm Getting the current time from a time server.
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12. The Berkeley Algorithm
The time daemon asks all the other machines for their clock values
The machines answer
The time daemon tells everyone how to adjust their clock
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13. Logical Clocks
There are technical issues with the clock synchronization approaches
Due to unpredictable message transmission delays, two processes can observe a global clock value at different instants
The physical clocks can drift from the physical time and thus we cannot have a system of perfectly synchronized clocks
For many purposes, it is sufficient that all machines agree on the same time
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14. Lamport’s Logical Clocks
For a wide of algorithms, what matters is the internal consistency of clocks, not whether they are close to the real time
For these algorithms, the clocks are often called logical locks
Lamport proposed a scheme to order events in a distributed system using logical clocks
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15. Lamport’s Logical Clocks – cont.
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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16. Lamport’s Logical Clocks – cont.
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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17. Lamport’s Logical Clocks – cont.
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18. Lamport’s Logical Clocks – cont.
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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19. 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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20. 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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21. An Example
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22. Clocks with Different Rates
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23. 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
Where is any arbitrary relation that totally orders the processes to break ties
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24. Example: Totally-Ordered Multicasting
Updating a replicated database and leaving it in an inconsistent state.
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25. 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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26. A Limitation of Lamport’s Clocks
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27. 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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28. Vector Clocks – cont.
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29. Vector Clocks – cont.
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30. Vector Clocks – cont. Assertion
At any instant,
Events a and b are casually related if ta < tb or tb < ta. Otherwise, these events are concurrent
In a system of vector clocks,
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