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CS 347: Parallel and Distributed Data Management Notes12: Time and Clocks

CS 347: Parallel and Distributed Data Management Notes12: Time and Clocks. Hector Garcia-Molina. Reference:. Leslie Lamport, Time, clocks, and the ordering of events in a distributed system, Communications of the ACM , Volume 21 , Issue 7 (July 1978), pages: 558 - 565.

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CS 347: Parallel and Distributed Data Management Notes12: Time and Clocks

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  1. CS 347: Parallel and DistributedData ManagementNotes12: Time and Clocks Hector Garcia-Molina Notes 12

  2. Reference: • Leslie Lamport, Time, clocks, and the ordering of events in a distributed system, Communications of the ACM, Volume 21 , Issue 7 (July 1978), pages: 558 - 565. • Available at:http://portal.acm.org/citation.cfm?id=359563 Notes 12

  3. Time and clocks • Time and clocks are fundamental concepts in distributed systems e.g.: timeouts identifying calls, transactions,… setting priorities versions of data ... Notes 12

  4. Questions • What is time? • How can clocks be implemented? Notes 12

  5. Ordering of events • More basic than time: event ordering event a happened event b 9 am before 10 am • Do not need physical time to order events event a happened before event b can affect Notes 12

  6. Ordering of events • “” is a partial ordering [ total ordering: for any two events a,b (ab) either a  b or b  a partial ordering: a,b can be concurrent ] Notes 12

  7. The model Process Process Process P Q R T I M E P4 Q7 R4 Q6 P3 R3 Q5 Q4 Q3 R2 P2 Q2 R1 P1 Q1 Notes 12

  8. Events within a process are totally ordered • Sending or receiving a message is an event • No assumptions about message transmission times E.g.: P1 P2 P1 Q2 P3, Q3 concurrent Notes 12

  9. Definition The relation  on the set of events of a system is the smallest relation satisfying the following 3 conditions: 1) if a,b events in same process, and a comes before b, then a  b 2) if a is sending a message and b is receipt of same message by a different process, then a  b 3) if a  b and b  c, than a  c • assume a  a • if a  b and b  a, than a,b are concurrent Notes 12

  10. Logical Clocks Process Process Process P Q R P4 Q7 R4 CR=9 Q6 R3 P3 Q5 CR=8 Q4 Q3 R2 P2 CR=7 T I M E Q2 CR=6 R1 P1 Q1 CR=5 Notes 12

  11. Overview • Logical Clocks • Physical Clocks • basic properties • synchronization scheme • synchronization with clock server • probabilistic synchronization Notes 12

  12. Logical Clock • Ci =logical clock (counter) at process i • C[b]= reading of Cj when event b occurs at process j • Clock condition for any events a,b IF a  b THEN C[a] < C[b] • No relationship to physical time Notes 12

  13. Clock Condition • If a  b THEN C[a] < C[b] • can be satisfied if the following conditions hold: If a and b are events in process i and a comes before b, then Ci[a] < Ci[b] If a is the sending of a message by process i and b is the receipt of that message by process j, then Ci[a] <Cj[b] C1 C2 Notes 12

  14. Process Process Process P Q R P4 Q7 R4 CQ=10 10>7 by C2 Q6 R3 P3 Q5 Q4 Q3 R2 P2 CR=7 T I M E Q2 7>5 by C1 R1 P1 Q1 CR=5 Notes 12

  15. To implement C1, C2: IR1 Each process i increments Ci between any two successive events IR2 - Let a be event “process i sends message to j” - Message contains timestamp Tm = Ci[a] - When message arrives at j (1) IF Tm > Cj THEN CjTm (2) event “arrival of message” takes place Notes 12

  16. Note: Ci[a] <Cj[b] a  b Note: a,b concurrent  Ci[a] = Cj[b] Note: Ci[a] =Cj[b]  a,b concurrent Notes 12

  17. Ordering Events Totally(breaking ties) Example: • A server wants to execute requests in order a is event that originated one request (at client) b is event that originates second request (at other client) • If C[a] < C[b], service a first (it could be that ab) • If C[a] = C[b], pick one (a,b concurrent) e.g., pick one with lower [ node#, process id ] Notes 12

  18. Total Ordering • Let “<” be a total ordering of the processes • Define total ordering of events “” a  b (a event in process i; b in j) if and only if (1) Ci[a] < Cj[b] or (2) Ci[a] = Cj[b] and i < j • “” not unique Notes 12

  19. Anomalous behavior server C=100 C=50 Reserve a Telephone Reserve a seat call seat Client A Client B 1 3 2 Notes 12

  20. C= 9am C= 9am Anomalous behavior server C=100 C=50 Reserve a Telephone Reserve a seat call seat Client A Client B 1 3 2 Notes 12

  21. Solutions (1) Include “telephone call” in “system” (2) Use perfect physical clocks (3) Use real physical clocks - may be “slightly off” - does not eliminate anomaly, but reduces its likelihood - useful for fault detection Notes 12

  22. Physical Clocks Ci(t) = clock reading at process i at physical time t Ci(t) time=t Notes 12

  23. Assume Ci(t) is a continuous, differentiable function except when clock is reset (message arrived) Ci(t) Ci+(x) Ci-(x)=lim Ci(x - ||) 0 Ci+(x)=lim Ci(x + ||) 0 Ci-(x) time=t x Notes 12

  24. How do we enforce clock condition? If a  b then C(a) < C(b) Notes 12

  25. IR1 Logical clocks IR1’ Each process i increments Ci between any two successive events For each process i, if i does not receive a message at time t, then Ci is differentiable at t and dCi(t) dt > 0 a T I M E b process i Notes 12

  26. IR2 Logical clocks • Let a be event “process i sends message m to j” • m contains timestamp Tm = Ci[a] • when m arrives at j (1) IF Tm > Cj THEN Cj Tm (2) event “arrival of m” takes place Notes 12

  27. IR2’ • Let a be event “process i sends message m to j” at physical time t • m contains timestamp Tm = Ci(t) • Let m be minimum transmission delay for m • m arrives at process j at physical time t’ t’ > t + m • IF Tm + m>Cj-(t’)THEN Cj+ Tm+ m • After clock adjust, event “arrival of m” takes place optional Notes 12

  28. IF Tm + m>Cj-(t’)THEN Cj+ Tm+ m • t’ m T I M E m • t i j Notes 12

  29. Additional properties PC1 Clock drift There exists a contstant k <<1 such that for all processes i dCi(t) dt - 1 < k Notes 12

  30. dCi(t) dt - 1 < k y+1+k Ci For typical crystal controlled clocks, k < 10-6 k y+1 k y+1-k y time x x+1 Notes 12

  31. PC2 Clock synchronization for all i,j: | Ci(t) - Cj(t) | <  Notes 12

  32. PC2 can be satisfied if a message is sent on every link at least every  seconds diameter = d • • • • • • d=4 (not 3) Notes 12

  33. let m be the transmission time of message m m= minimum delay + unpredictable delay m = m + Zm Assume constant  such that for all m: Zm   Assume that for all m: m =  t2 • maximum transit time: + shortest transit time:  • t1 Notes 12

  34. Theorem PC2 holds for all t > to + d with   d(2k + ) (assuming  +  <<  ). Notes 12

  35. Argument for d=1 slow fast s f Worst case scenario: t1: m leaves f for s withTm = Cf(t1) t2: m arrives at s:Cs(t2) = Cf(t1) +  Cf(t2) - Cs(t2) is as large as possible,  t2+ : no communication for  sec; f is as fast as possible, s as slow as possible. Just before communication at t2+  clocks are as far apart as possible,  Notes 12

  36. clock reading k f   k  s t2 + time t2 So,  = 2 +  Now just need largest possible  Notes 12

  37. • Cs(t2) t2 + • Cf(t1) t1 s f Cf(t2) = Cf(t1) + (1+k)(+) Cs(t2) = Cf(t1) +   = (1+k)( + ) -  max  = k(+) +  Notes 12

  38.  =2k +  max  = k(+) +  Thus:  = 2 +  + k(+) Assuming that k<<1 and (+) << we get:   2 +  Notes 12

  39. Summary Physical clocks: • clock condition a  b  C(a) < C(b) • drift < k • | Ci(t) - Cj(t) | <  • can be implemented as discrete Notes 12

  40. Uses for Physical Clocks: (1) To order events (2) Timeouts Example: “Reply to my request by time t1” At time t2= (++)(1+) + t1 we can timeout my clock may drift Notes 12

  41. t2 = (++)(1+) + t1 max trans + t1 at remote clock  t1 on my clock my node other Notes 12

  42. Do physical clocks rule out anomalous behavior? No anomalous behavior if Cs(t2)>Cf (t1) Cs(t2) min message out of system Cs(t1)=Cf (t1) -  Cf (t1) s slow clock f fast clock Notes 12

  43. No anomalous behavior if Cs(t2)>Cf (t1) Worst possible scenario: • Smallest possible transmission time min • Cs, Cf as far apart as possible Cs(t1) = Cf (t1) -  • Smallest possible Cs(t2) = Cf (t1) -  + S.P.increment = Cf(t1) -  + min(1-k) Notes 12

  44. No A.B. if Cs(t2) > Cf(t1) I.e., if Cf(t1) -  + min(1-k) > Cf(t1)  1 - k min > Notes 12

  45. Using a clock service • With IR2’, a fast clock speeds up all clocks • Use instead a single, reliable clock service (eg. WWV), assume it is perfect “real time” (k=0) • IR2’ is now: Cj+(t’)Tm+min whenever timing message arrives • If max. transmission time from central clock is < + then e = 2kt + (1 + k) (Why? ) Notes 12

  46. Using a clock service - analysis 0  error at synchronization time, node j  T++ Cj=T+ slowest clock message T+ Cj=T+ fastest clock message time T server node j Notes 12

  47. Using a clock service - analysis - cont. fast clock server time t possible difference telapsed time Node that got fastest message 0  slow clock Node that got slowest message time  Notes 12

  48. Why can period bet+ ? slow m+ t +  t +t fast m t j server Notes 12

  49. k(t+) server time  telapsed time t kt Node that got fastest message 0 worst case for clock reset  Node that got slowest time   Notes 12

  50. k(t+) server time  t kt Node that got fastest message 0 worst case for clock reset  Node that got slowest time |Cs(t) - Cj(t)|  kt + x |Ck(t) - Cj(t)|  2kt + x(1+k) Notes 12

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