Time, Clocks and the Ordering of Events in a Distributed System - by Leslie Lamport

1. Time, Clocks and the Ordering of Events in a Distributed System - by Leslie Lamport

2. Outline • Partial Ordering • Logical Clocks • Total Ordering • External Factors • Physical Clocks • Conclusions

3. Partial Ordering • Partial Ordering : The smallest relation -> on the set of events of a system satisfying : • If a and b are events in the same process, and a comes before b, then a->b. • If a is the sending of a message by one process and b is the receipt of the same message by another process, then a->b. • If a->b and b->c then a->c. • Causality • If a -> b, then event a can affect event b. • I.e. b comes after (or depends on results of) event a.

4. Partial Ordering • Logical Concurrency (of Events) • Two events are concurrent if neither can causally affect the other. • I.e. a-->b and b-->a. • Note : • a-->a. We assume that it is impossible to construct systems where an event can happen before itself. • Hence -> is an irreflexive partial ordering on the set of all events in the system.

5. Partial Ordering

6. Examples • Time increases upward. • P1 and P2 are causal. (within a process) • P3 and Q3 are concurrent. (causally independent). [note that this is the case even though fig 1 seems to imply that Q3 occurs before P3] • P1 are R4 are causally dependent. • Only upon exchanging messages can we know regarding the ordering of events in the system.

7. Logical Clocks • At this point, we are only considering only Logical Clocks. • A clock is a way of assigning a number to an event. i.e. just a counter, no actual timing mechanism needed. • No relation of between the counter and physical time. • The “system” clock is the union of all clocks maintained by all the processes in the system.

8. Logical Clocks

9. Logical Clocks • Clock Condition :For any events a, b: • if a->b then C(a) < C(b). • Converse condition need NOT hold. • C1: If a and b are events in process Pi, and a comes before b, then Ci(a) < Ci(b). • C2 : If a is the sending of a message by process Pi and b is the receipt of that message by process Pk, then Ci(a) < Ck(b).

10. Logical Clocks • In terms of “ticks”, this means that : • tick line between events of a process. • tick line between messages across processes. • tick is NOT an event. • In terms of implementation, this means : • IR1 : Each process Pi increments Ci between any two successive events. • IR2 : If event a is the sending of a message m by process Pi then the message m contains a time-stamp Tm = Ci(a). Upon receiving a message m, process Pk sets Ck greater than or equal to its present value and greater than Tm.

11. Logical Clocks

12. Total Ordering • Extension of Partial Ordering. Break ties using the < relation between processes. This is a way of assigning priorities to processes. • Definition :If a is an event in process Pi and b is an event in process Pk, then a=>b if and only if either Ci(a) < Ck(b) or Ci(a) = Ck(b) and Pi < Pk • Ordering depends on the system of clocks Ci and is not unique. • Only the partial ordering is uniquely determined by the system of events.

13. Partial v/s Total Ordering by Examples

14. Total Ordering • Total ordering is important and useful in implementing a distributed system. • Mutex example. • Centralized scheduling will (may ?) not work. • Distributed Algorithm. • Problems : • algorithm requires active participation of all processes. (since to operate, each process needs to learn about operations of all processes). Failure of a single process will halt the system. • How to distinguish between pause and failure ?

15. External Factors • Messages external to the system can cause anomalies.[banking example]. • New System Á (contains all external factors). • New ordering operator : ® • Strong Clock Condition :For any events a, b in Á: • If a®b then C(a) < C(b). • The strong clock condition is not satisfied by our logical clocks. We turn to physical clocks...

16. Physical Clocks • But physical clocks are not exactly synchronized and run at different rates (although the rate differential is extremely small). • So, we need to adjust them ever so often to keep them “loosely synchronized” (I.e. keep them from drifting very far apart) atleast to a level that can allow the system to disambiguate ordering of events. • Above parameters related to m, the minimum time between events. • Specialized rules IR1 and IR2 for physical clocks.

17. Conclusions • Happened Before defines a Partial Ordering of events (arising from causal relationships). • Giving priorities to processes can convert this to a Total Ordering. • A centralized algorithm may not work. A Distributed algorithm has a “single point of failure” (process) ! • Even Total Ordering can break down due to events external to the system. • Physical clocks can help, but they must be “loosely synchronized” and must be reset forward when they drift far apart. • Minimum time between events defines the granularity of “drift far apart”.