Flow-level Stability of Utility-based Allocations for Non-convex Rate Regions

1. CISS – 3.22.2006 Flow-level Stability of Utility-based Allocations for Non-convex Rate Regions Alexandre Proutiere France Telecom R&D ENS Paris Joint work with T. Bonald

2. Scope • Performance evaluation of data networks at flow-level • What is the mean time to transfer a document? • Wireless networks: rate region is non-convex • How do usual utility-based allocations perform? • How should we choose the network utility? Is Proportional fairness a good objective? 1 2 (Aloha)

3. Outline • Flow-level models for data networks • Rate regions and utility-based resource allocations • Flow-level stability • The case of convex rate regions • The case of non-convex rate regions

4. Outline • Flow-level models for data networks • Rate regions and utility-based resource allocations • Flow-level stability • The case of convex rate regions • The case of non-convex rate regions

5. Data networks at flow-level • Wireless networks • Telatar-Gallager'95 • Stamatelos-Koukoulidis-'97 • Borst'03 • Borst-Bonald-Hegde-P.'03… • Lin-Shroff'05 • Srikant'05 • …. • Wired networks • Heyman-Lakshman-Neidhardt'97 • Massoulie-Roberts'98 • Bonald-P.'03 • Kelly-Williams'04 • Key-Massoulie • …

6. Class 1 Class 2 Class 3 Data networks • Network: a set of resources • Notion of flow class: require the use of the same resources NETWORK

7. Traffic demand • Class-k flow arrivals: A Poisson process • Arrival intensity • Mean flow size • Traffic intensity

8. Performance metrics • The mean time to transfer a flow • … or the mean flow throughput

9. Flow rate in state x: This defines the realized resource allocation Packet-level dynamics • Fix the numbers of flows of each class • Network state • The instanteneous rate of a flow depends on: • its class • the access rate • TCP • the scheduling policy • … rate time

10. Flow arrival Flow departure Flow-level dynamics • Time-scale separation assumption • Flow rates converge instantaneously when the network state changes • Random numbers of active flows • Flows initiated by users • … cease upon completion • Network state process rate time

11. First QoS requirement: • Stability of process Mean flow throughput Resource allocation Flow-level stability Stationary distribution Performance The capacity region • Network capacity = max total traffic intensity compatible with some QoS requirements 0

12. Outline • Flow-level models for data networks • Rate regions and utility-based resource allocations • Flow-level stability • The case of convex rate regions • The case of non-convex rate regions

13. Rate region • Wired networks (0,1) (1,1) Rate region = a convex polytope with facets orthogonal to some binary vectors The rate region • In state x, rates allocated to the different classes

14. A single cell network (no interference) 1 2 Convex rate region in wireless networks • In case of wireless networks with coordination, interference is avoided • The rate region is still convex

15. Interfering links without sched. coordination 1 2 Non-convex rate regions • Without coordination, interference modifies the structure of the rate region • Highly non-convex rate regions

16. Non-convex rate regions • Without coordination, interference modifies the structure of the rate region • Highly non-convex rate regions • Interfering links without sched. coordination 1 SNR = 10 dB 2

17. Non-convex rate regions • Without coordination, interference modifies the structure of the rate region • Highly non-convex rate regions • Interfering links without sched. coordination 1 SNR = 2 dB 2

18. α-fair allocations Dynamic network state Static network state Resource allocations • An allocation chooses a point of the rate region in each network state • Utility-based allocations • ↑ : realized in a distributed way • ↓ : do not maximize utility in a dynamic setting

19. Outline • Flow-level models for data networks • Rate regions and utility-based resource allocations • Flow-level stability • The case of convex rate regions • The case of non-convex rate regions

20. Issues • With a given allocation, what traffic intensities the network can support? i.e., what is the flow-level stability region? • How does the non-convexity of the rate region impact the capacity region?

21. Flow-level stability • De Veciana-Lee-Konstantopoulos'99 Wired networks, stability of max-min • Bonald-Massoulie'01 - Wired networks, Stability of any α fair allocations • Yeh'03 – Wired networks, other utility functions • Bonald-Massoulie-P.-Virtamo'06 – Stability of α fair allocations on any convex rate regions • Borst'03 – Stability of opportunistic schedulers in wireless networks • Lin-Shroff-Srikant'05, – Stability in absence of the time-scale separation assumption • Borst-Jonckheere'06 – Stability with state-dependent rate regions • Massoulie'06 – Stability of PF with genera l flow size distributions

22. Proposition: The maximum stability region is the smallest convex coordinate-convex set containing the rate region This set is denoted by Unstable Maximum stability • Consider an arbitrary rate region

23. Maximum stability • Consider an arbitrary rate region Stable Proposition: The maximum stability region is the smallest convex coordinate-convex set containing the rate region This set is denoted by

24. Outline • Flow-level models for data networks • Rate regions and utility-based resource allocations • Flow-level stability • The case of convex rate regions • The case of non-convex rate regions

25. Proposition: In case of convex rate regions, any α-fair allocation achieves maximum stability Stability for convex rate regions In particular, for convex rate regions, the capacity region does not depend on the chosen utility function

26. PF Max-min Flow throuhghput in wired nets Short route • A linear network Flow throughput 1 2 3 Long route Performance is not very sensitive to the chosen utility function Flow throughput

27. Flow throughput in wireless nets • A cell with orthogonal transmissions PF Flow throughput Max-min 1 2 Performance is sensitive to the chosen utility function Avoid max-min

28. Outline • Flow-level models for data networks • Rate regions and utility-based resource allocations • Flow-level stability • The case of convex rate regions • The case of non-convex rate regions

29. Monotone cone policies: a set of cones (i) • scheduled when • and are scheduled on the axis • Any of the two points or is scheduled when provided and Two class networks • A discrete rate region

30. Two class networks Proposition: The stability region of a monotone cone policy is the smallest coordinate-convex set containing the contour of the set of scheduled points

31. Corollary: If the rate region has a convex structure, the stability region of any α-fair allocations is maximum α-fair allocations • They are montone cone policies • Directions of the switching line between and

32. Corollary: There exists such that for all , the stability region of α-fair allocations is minimum and equal to the smallest coordinate-convex set containing the contour of α-fair allocations Corollary: There exists such that for all , the stability region of α-fair allocations is maximum and equal to

33. Proposition: For , the stability region depends on detailed traffic characteristics Proposition: When the rate region is strictly not convex, PF never achieves maximum stability and can be quite inefficient More classes Proposition: There exists such that for all , the stability region of α-fair allocations is maximum and equal to

34. Example 1 SNR = 10 dB 2

35. Conclusions • Rules for the choice of the allocation • Convex rate regions: wired networks Maxmin PF MPD fairness 0 1 2 efficiency Maxmin PF MPD 0 1 2 Stability Flow throughput

36. Conclusions • Rules for the choice of the allocation • Convex rate regions: wireless networks Maxmin PF MPD fairness 0 1 2 efficiency Maxmin PF MPD 0 1 2 Stability Flow throughput

37. Maximum stability Minimum stability Conclusions • Rules for the choice of the allocation • Non-convex rate regions: wireless networks Maxmin PF MPD fairness 0 1 2 efficiency Maxmin PF MPD 0 1 2 Stability

38. Conclusions • For non-convex rate regions, max-min or PF may not be convenient choices • When the utility function is well chosen, the stability is maximized as if the rate region were convexified • Next step: designing distributed random algorithms to max this utility • Example: decentralized power control scheme (e.g. Bambos et al.)