Some Recent Progress in Adversarial Queuing Theory

1. Some Recent Progress in Adversarial Queuing Theory Matthew Andrews Bell Labs

2. Topics of Interest (Theory) • Adversarial Queuing Theory • Subject of this talk • Analysis of wireless networks • Resource allocation under time-varying network conditions • Complexity of network design • First non-trivial hardness results for edge-disjoint paths, congestion minimization, buy-at-bulk network design • Complexity of wavelength assignment in optical networks

3. Topics of Interest (Theory) • Network Utility Maximization (Kelly model) • Incorporation of scheduling dynamics into NUM • Introducing hard deadlines into NUM • Approximate solution to packing problems • Developing algorithms with running time O(1/ε) • (Garg-Konemann etc have running time O(1/ε2) )

4. Topics of Interest (Engineering) • Mobile Ad-Hoc Networking • Developing NUM-motivated stack for MANETs • Addressing MAC, scheduling, routing, admission control • Part of $4M/year DARPA-sponsored project • Joint with Lockheed-Martin, Stanford, Princeton, UIUC, UCSB • Wireless resource allocation • QoS-aware scheduling algorithm for high-speed wireless data • Implemented in Lucent’s 3G offerings

5. Topics of Interest (Engineering) • Wireless measurement • Built tool for analyzing multiple layers of wireless data stack • Human perception of data networks • Developed Mean Opinion Score for data networks • High-speed switch scheduling • Scheduling algorithm for control of Terabit optical switch fabric • Content Delivery and Distribution • Developed assignment algorithms for Lucent’s content delivery products

6. Adversarial Network Analysis S2 S1 S3 t2 t3 t1 • Networks of queues and sessions

7. Stability • Definitions • Stable: queuesizes remain bounded • Unstable: queuesizes are unbounded • We would like queues to be stable as long as no queue is overloaded Stable Total queuesize Time Total queuesize Unstable Time ??

8. Input process S2 0.45 0.45 S1 S3 0.45 t2 • Standard queuing assumption: • All routes are fixed • Packets injected according to stationary stochastic process (e.g. Poisson process) • This talk • What happens if routes vary in arbitrary manner? • What happens if packet injections are arbitrary and non-stationary? • Allows for worst-case analysis - “Adversarial Queueing Theory” t3 t1

9. Talk Outline • Part 0 • Description of 3 models in which adversary has varying degrees of control • Statement of results showing that the model matters • Parts 1&2 • Two recent results on adversarial queuing systems

10. Permanent sessions: Constant arrivals S2 0.45 0.45 S1 r = 0.9 S3 0.45 t2 t3 • Each session i has rate ri • Routes are specified • Injection rate into session i equalsri at all times (e.g. Poisson(ri )) • Sum of session rates passing through capacity C queue  rC • r = network load parameter • Interesting case is r  1 t1

11. Permanent sessions: Bounded arrivals S2 0.45 0.45 S1 r = 0.9 S3 0.45 t2 t3 t1 • Each session i has max rate ri • Injection rate into session i at mostri at all times • Sum of max session rates passing through capacity C queue  rC • Standard model in networking literature, e.g. • Fair Queueing literature (e.g. Parekh-Gallager), • Network calculus (e.g. Cruz, Le Boudec-Thiran book)

12. Permanent sessions: Bounded arrivals S2 0.45 0.2 S1 r = 0.9 S3 0.1 t2 t3 t1 • Each session i has max rate ri • Injection rate into session i at mostri at all times • Sum of max session rates passing through capacity C queue  rC • Standard model in networking literature, e.g. • Fair Queueing literature (e.g. Parekh-Gallager), • Network calculus (e.g. Cruz, Le Boudec-Thiran book)

13. Permanent sessions: Bounded arrivals S2 0.3 0.45 S1 r = 0.9 S3 0.45 t2 t3 t1 • Each session i has max rate ri • Injection rate into session i at mostri at all times • Sum of max session rates passing through capacity C queue  rC • Standard model in networking literature, e.g. • Fair Queueing literature (e.g. Parekh-Gallager), • Network calculus (e.g. Cruz, Le Boudec-Thiran book)

14. Temporary sessions (Adversarial Queueing Model – AQM) S2 0 0.9 S1 r = 0.9 S3 0 t2 t3 t1 • Session injection rates vary over time • Sum of current session rates passing through capacity C queue  rC • Model introduced by Borodin et al. 1996

15. Temporary sessions (Adversarial Queueing Model – AQM) S2 0.9 0 S1 r = 0.9 S3 0.9 t2 t3 t1 • Session injection rates vary over time • Sum of current session rates passing through capacity C queue  rC • Model introduced by Borodin et al. 1996

16. Stability • A scheduling algorithm is stable if queues remain bounded whenever r < 1 Stable Total queuesize Time Total queuesize Unstable Time

17. The Model Matters Perm. sessions w/ bounded inj. Perm. sessions w/ const. inj. Temp. sessions unstable unstable stable FIFO AAFKLL 1996 A 2000 Bramson 1996 Generalized Processor Sharing unstable stable stable A-Zhang 2000 Parekh-Gallager “Longest In System” stable stable stable AAFKLL 1996

18. Talk Outline • Part 1: An adversary is bad for FIFO • Stochastic model: FIFO scheduling is stable as long as network is not overloaded • Adversarial model: FIFO can be unstable at arbitrarily small network loads • Part 2: An adversary is OK for some schedulers • “Max-Weight” scheduling algorithm stable in stochastically generated dynamic graphs (Tassiulas-Ephremides) • This talk: “Max-Weight” scheduling algorithm stable in adversarially generated dynamic graphs • However, different proof is required

19. S1 t1 S2 t2 S3 t3 Part 1: FIFO Queue • FIFO queue serves packets in order received • We consider idealized model in which packet size  0 • Study rate at which data arrives at/departs from queue

20. FIFO stability Perm. sessions w/ bounded inj. Perm. sessions w/ const. inj. FIFO Temp. sessions Stable for all r<1 ? No No Yes AAFKLL 1996 A 2000 Bramson 1996 Stable for any r>0 ? Yes No ?? Bramson 1996 Bhattacharjee-Goel 2003 • What happens at small network loads in permanent sessions model with bounded arrivals? • Is there some network load below which FIFO is stable? • No!!!!

21. FIFO stability Perm. sessions w/ bounded inj. Perm. sessions w/ const. inj. FIFO Temp. sessions Stable for all r<1 ? No No Yes AAFKLL 1996 A 2000 Bramson 1996 Stable for any r>0 ? Yes No No Bramson 1996 Bhattacharjee-Goel 2003 A 2007 • What happens at small network loads in permanent sessions model with bounded arrivals? • Is there some network load below which FIFO is stable? • No!!!!

22. FIFO instability S2 0.45 0.45 S1 r = 0.9 S3 0.45 t2 t3 t1 • Thm: For all e> 0, there exists network and injection pattern s.t. • Network load is r <e w.r.t. perm sessions model with bounded arrivals • The network is unstable when FIFO is used

23. FIFO instability S2 0.45 0.2 S1 r = 0.9 S3 0.1 t2 t3 t1 • Thm: For all e> 0, there exists network and injection pattern s.t. • Network load is r <e w.r.t. perm sessions model with bounded arrivals • The network is unstable when FIFO is used

24. FIFO instability S2 0.3 0.45 S1 r = 0.9 S3 0.45 t2 t3 t1 • Thm: For all e> 0, there exists network and injection pattern s.t. • Network load is r <e w.r.t. perm sessions model with bounded arrivals • The network is unstable when FIFO is used

25. Transfer Process • How can we move fluid from one FIFO queue to another? Injection rate e Existing fluid of height 1

26. Transfer Process • How can we move fluid from one FIFO queue to another? Arrival rate 1 / 1 + e Injection rate e Total arrival rate 1 + e Departure rate e / 1 + e Departure rate 1 Departure rate 1 / 1 + e

27. Transfer Process • How can we move fluid from one FIFO queue to another? Arrival rate 1 / 1 + e Injection rate e Total arrival rate 1 + e Departure rate e / 1 + e Departure rate 1 Departure rate 1 / 1 + e

28. Transfer Process • How can we move fluid from one FIFO queue to another? Arrival rate 1 / 1 + e Injection rate e Total arrival rate 1 + e Departure rate e / 1 + e Departure rate 1 Departure rate 1 / 1 + e

29. Transfer Process • How can we move fluid from one FIFO queue to another? Injection rate e

30. Transfer Process • How can we move fluid from one FIFO queue to another? Injection rate 0

31. Transfer Process • How can we move fluid from one FIFO queue to another? Injection rate 0

32. Transfer Process • How can we move fluid from one FIFO queue to another? Injection rate 0

33. Repeated Transfer Process • Can repeat transfer process indefinitely

34. Repeated Transfer Process • Can repeat transfer process indefinitely

35. Repeated Transfer Process • Can repeat transfer process indefinitely

36. Repeated Transfer Process • Can repeat transfer process indefinitely • Two problems: • Want amount of fluid to grow! • Sessions are of finite length: some fluid eventually departs

37. Transfer Process Revisited • Replace departing fluid Injection rate e Injection rate r Departure rate < r

38. Transfer Process Revisited • Replace departing fluid Injection rate e Injection rate r Departure rate < r

39. Transfer Process Revisited • Replace departing fluid Injection rate e Injection rate r Departure rate < r

40. Transfer Process Revisited Works but network load r >> 1 • Replace departing fluid Injection rate e Injection rate r Departure rate < r

41. Parallel Transfers Arrival rate ~ r / 1 - e Injection rate r 2 3 1 4 Departure rate ~ r / 1 - e • Transfer process between queues 1 & 3 • Transfer process between queues 2 & 4 • Newly injected traffic passes through queues 1&2 • Queues 1&2 are decreasing: hence rate amplification!!!

42. Parallel Transfers Arrival rate ~ r / 1 - e Injection rate r 2 3 1 4 Departure rate ~ r / 1 - e • For a sufficiently large number of parallel transfers • Can make arriving traffic have higher rate than departing traffic • Instability!!!

43. Part 2: Max-Weight stability • Routing and scheduling in wireless networks • Focus on special case of adversarial dynamic graphs • Edges appear and disappear over time • This process is controlled by an adversary • Study performance of Max-Weight-Tassiulas-Ephremides-Differential-Backlog-Backpressure algorithm • Joint with Kyomin Jung (MIT) • To appear STOC 2007

44. The Model • Adversary controls packet arrivals and edge appearances • Algorithm decides which packet should be sent along an edge when it appears • Unlike previous model, routes are not specified. Algorithm is allowed to choose routes • Easy to adapt results to fixed routing case

45. The Model • : Number of times edge e appears in time window W • adversary: For any time window W of size the adversary can associate with each packet a simple path from source to destination s.t. each edge is used by those paths at most times.

46. The Model • : Per-destination queue for destination d at vertex v • : Size of

47. Main Algorithm • At each time, for each open edge (v,u), choose d such that is maximized where is a given constant. • Send one packet over the edge from to .

48. MaxWeight Algorithm • Max-Weight is very well studied • L. Tassiulas, A. Ephremides, IEEE Transactions on Information theory 1992. • B. Awerbuch, T. Leighton, STOC 1994. • + 100s of other papers • Provides adaptive routing and scheduling • Fully distributed

49. Dynamics of the Algorithm • The algorithm can be understood so that the following potential function decreases as much as possible under the edge-constraint. Where is the height of the packet. • Our question: Is the algorithm stable for any adversary? i.e. is the queue size bounded over time?

50. Previous Work on Adversarial Model • Stability of the Max-Weight for static graph, multi-commodity case when . [W. Aiello, E. Kushilevitz, R. Ostrovsky, A. Rosen, STOC 1998] Main idea: They show that if queue size becomes large enough, the potential function decreases at each time. • Stability for dynamic graph, single-commodity case, when . [E. Anshelevich, D. Kempe, J. Kleinberg, STOC 2002] Main idea: They use max-flow min-cut thm for single commodity case.