The network: mesh

1. Routing without Flow ControlCostas BuschRensselaer Polytechnic InstituteMaurice HerlihyBrown UniversityRoger WattenhoferMicrosoft Research

2. The network: mesh • Discrete time • Bi-directional links • At most one packet per link direction

3. Dynamic Routing: Packets are injected continuously destination

4. A new packet can be injected when there is a free link: A link direction is empty

5. Most dynamic routing algorithms use flow control: Don’t utilize all the free links Disadvantage: Network is under-utilized

6. Our Routing Algorithm: • No flow control • Utilizes all the free links Advantage: Network is fully-utilized

7. Features of our algorithm: • Dynamic • Hot potato • Optimal delivery time: • Injection time guaranty:

8. Talk Outline • The Algorithm • Time Analysis • Stability • Future Work

9. Hot-Potato Routing: • Nodes are buffer-less • Packets are immediately forwarded

10. Conflicts

11. Conflicts Conflict

12. Conflicts Deflected

13. Priorities: high low Packet states: Running Excited Active Sleeping

14. Sleeping packet destination Random destination

15. Sleeping packet destination Follows a path to destination

16. Sleeping packet becomes Active with probability

17. Active packet Follows a greedy path

18. Active packet Follows a greedy path

19. Active packet A conflict situation

20. Active packet Conflict A conflict situation

21. Active packet Deflected A conflict situation

22. Active packet becomes Excited with probability Deflected A conflict situation

23. Excited packet Follows a one-bend path

24. Excited packet becomes Running Follows a one-bend path

25. Running packet Follows a one-bend path

26. Talk Outline • The Algorithm • Time Analysis • Stability • Future Work

27. Good condition for a column: at most non-sleeping packets with destination in the column

28. Expected delivery time for one packet: (when the destination column is in good condition)

29. In expected time steps becomes active We will show: An active packet is delivered in expected time steps Initially a packet is sleeping

30. Interrupting a one-bend path Excited Time 1

31. Interrupting a one-bend path Running Time 2

32. Interrupting a one-bend path Excited Running Time 2

33. Interrupting a one-bend path Running Running Time 3

34. Interrupting a one-bend path Running conflict Running Time 4

35. Interrupting a one-bend path deflected Active Running Time 5

36. No interruption probability: Excitement probability Number of non-sleeping packets with destinations in same column

37. No interruption probability: constant (when the destination column is in good condition)

38. Expected number of deflections until success: Expected delivery time for an active packet: Probability of success after a deflection:

39. Talk Outline • The Algorithm • Time Analysis • Stability • Future Work

40. Divide time in time periods: Examine the condition of a column

41. 1 time period Good condition Bad condition

42. 1 time period Good condition Bad condition 4n time periods

43. 1 time period Good condition Bad condition

44. Proof Outline In a time period: • At most new non-sleeping • packets are generated with • destinations in the column • At least non-sleeping packets • are delivered (if )

45. Good condition Bad condition 4n time periods

46. Proof Outline In a time period: • At most new non-sleeping • packets are generated with • destinations in the column • At least non-sleeping packets • are delivered

47. Consequences: • Most of the time, the columns • are in good condition • Each packet is delivered in • expected time

48. Talk Outline • The Algorithm • Time Analysis • Stability • Future Work

49. Arbitrary network topologies • De-randomization: • Determistic destinations • No randomized algorithm • Small number of packets