Designing data networks A flow-level perspective

1. Designing data networks A flow-level perspective Alexandre Proutiere Microsoft Research Workshop on Mathematical Modeling and Analysis of Computer Networks June 2007, ENS, Paris

2. Talk based on ... • Flow-level stability of utility-based allocation in non-convex rate regions. • with Thomas Bonald. CISS 2006 • Capacity of wireless networks with intra- and inter-cell mobility. • with Sem Borst and Nidhi Hegde. Infocom 2006 • Flow­level stability of data networks with non­convex and time­varying rate regions. • with Jiaping Liu et al. ACM Sigmetrics 2007

3. Issues • Since Kelly 1997, resource allocation schemes in data nets are (proved to be) designed so as to maximize some network utility • Is it good idea? • How to choose the utility function? • Should this choice depend on the underlying network resources?

4. Related work: the thru-fairness trade-off • Mo-Walrand. α-fair allocations: Maxmin PF MPD fairness 0 1 2 efficiency • Tang-Wang-Low. Counter-intuitive throughput behaviors in networks under end-to-end control. IEEE/ACM ToN, 2006. • Wired nets: A fixed number of permanent TCP connections • The total long-term thru is not monotone in α • Radunovic-Le Boudec. Rate Performance Objectives of Multihop Wireless Networks, IEEE Trans. on Mobile Computing, 2004. • Wireless multihop nets: a fixed nb of TPC connections • PF outperforms Max-min • Qualcomm HDR. A PF scheduler

5. A flow-level analysis • Most of existing work on data networks assume a fixed population of TCP connections or flows • However users perceive performance at flow-level: durations of the connections • The instantaneous thru of the network is not a sufficient metric to design the networks (i.e., to choose the notion of utility) • Let’s adopt a flow-level approach: a dynamic population of flows!

6. Outline • Modeling data networks • Fixed and convex rate regions (wired networks, wireless networks with centralized scheduling) • Arbitrary and fixed rate regions (wireless networks with distributed resource allocation) • Time-varying rate regions (wired networks with priority traffic, link failures ..., wireless networks with fading / mobility)

7. Outline • Modeling data networks • Fixed and convex rate regions (wired networks, wireless networks with centralized scheduling) • Arbitrary and fixed rate regions (wireless networks with distributed resource allocation) • Time-varying rate regions (wired networks with priority traffic, link failures ..., wireless networks with fading / mobility)

8. Resource sharing in data networks • Network: a set of resources • Data flows classified according to the set of used resources • Flow-level network state: • Packet-level mechanisms • (TCP+scheduling) share • resources among flows total rate of class-k flows in state x

9. Rate region • Fix the network state (the population of flows) • The rate region of a network is the set of feasible long-term • rates • NB: Most often, the rate region does not depend on the network state

10. Resource sharing objectives • Congestion control and scheduling algorithms share resources, i.e., choose a point in the rate region depending on the network state • An optimization approach – Kelly 1997 • (TCP+sched) solves: • Why? Because • - TCP does so (Kelly) • Distributed implementation

11. Flows transferred in a finite time iff stability of the process • of the numbers of flows Performance metrics • Users perceive performance at flow-level: the mean time to • transfer documents 1/(mean flow duration) • Flow-level dynamics • Poisson arrivals of class-k flows: • Departures at rate (exp. flow sizes): Performance metric: capacity region The set of such that the system is stable at flow-level 0

12. Rate regions • Wired networks with fixed link capacities: • a convex polytope • Wired networks with priority traffic / link failure+multi-path • routing: a time-varying convex polytope

13. Rate regions • Wireless networks with centralized scheduling • a convex polytope TDMA rate region • With fading / user mobility / variable interference: a time-varying rate region

14. Rate regions • Wireless networks with distributed resource allocation, power control / rate adaptation • a continuous non-convex rate region 1 SNR = 10 dB 2

15. 1 2 Rate regions • Wireless networks with distributed resource allocation, without power control • a discrete rate region

16. Design (choice of U) Objective The big picture Packet level: rate region, utility func. Multi-class queue with state-dependent capacity Flow-level traffic demand Capacity region Flow-level performance

17. The math question x1 x2 K K solves xK How to choose the utility function U such that the stability region of the queuing system is maximized? (or more generally some performance metrics?)

18. Outline • Modeling data networks • Fixed and convex rate regions (wired networks, wireless networks with centralized scheduling) • Arbitrary and fixed rate regions (wireless networks with distributed resource allocation) • Time-varying rate regions (wired networks with priority traffic, link failures ..., wireless networks with fading / mobility)

19. Fixed convex rate region Theorem 1* Any α-fair allocation (α>0) achieves maximum stability, and the capacity region is the rate region *Bonald-Massoulie 2001 Tassiulas-Ephremides 1992 Bonald-Massoulie-Proutiere-Virtamo 2006

20. Fixed convex rate region Theorem 1 Any α-fair allocation (α>0) achieves maximum stability, and the capacity region is the rate region The choice of the utility function is not crucial for stability purposes! Optimization approaches to design data network mechanisms are a good idea

21. Vote for PF! • It is robust to traffic characterisitcs evolution • Massoulie: PF and BF are cloes to each other • It realizes a good fairness-efficiency trade-off in wired networks • Bonald-Roberts • It has to be chosen for wireless systems 1/(mean flow duration) PF 0

22. Vote for PF! • It is robust to traffic characteristics evolution • Massoulie: PF and BF are close to each other • It realizes a good fairness-efficiency trade-off in wired networks • Bonald-Roberts • It has to be chosen for wireless systems 1/(mean flow duration) Maxmin 0

23. Vote for PF! • It is robust to traffic characterisitcs evolution • Massoulie: PF and BF are cloes to each other • It realizes a good fairness-efficiency trade-off in wired networks • Bonald-Roberts • It has to be chosen for wireless systems 1/(mean flow duration) α = 0.2 0

24. Outline • Modeling data networks • Fixed and convex rate regions (wired networks, wireless networks with centralized scheduling) • Arbitrary and fixed rate regions (wireless networks with distributed resource allocation) • Time-varying rate regions (wired networks with priority traffic, link failures ..., wireless networks with fading / mobility)

25. Fixed and arbitrary rate region • Networks with 2 flow classes: the stability region of cone • policies (e.g. α-fair allocations) is known • Bonald-Proutiere 2006 • Networks with more flow classes: impossible to characterize the • stability region of usual allocations • - Stability of Aloha systems, Szpankowski, Anantharam,… • - More results in Borst-Jonckheere 2006 • - This talk: exhaustive analysis of α-fair allocations

26. Fixed and arbitrary rate region • Maximum capacity region Theorem 2* There exists an allocation stabilizing the network if and only if the traffic intensity vector ρ belongs to the smallest coordinate-convex, convex set containing the rate region *Tassiulas-Ephremides 1992

27. Fixed and arbitrary rate region • α-fair allocations

28. Fixed and arbitrary rate region is the set of points in the rate region actually scheduled by the α-fair allocation (i.e., the set of points a in the rate region such that there exists a state x for which a maximizes α-fairness) • α-fair allocations

29. Fixed and arbitrary rate region stable • α-fair allocations Theorem 3 The capacity region of the α-fair allocation contains the smallest coordinate-convex set containing

30. Fixed and arbitrary rate region unstable ? ? ? stable • α-fair allocations Theorem 3 The capacity region of the α-fair allocation contains the smallest coordinate-convex set containing Theorem 4 The system under the α-fair allocation is unstable when ρ belongs to

31. Fixed and arbitrary rate region unstable stable • α-fair allocations Corollary 1 In case of continuous , the capacity region of the α-fair allocation is the smallest coordinate-convex set containing

32. Efficiency vs fairness • The flow-level stability region depends on the chosen utility function • Stability decreases with the fairness parameter α • Max-min fairness is always the worse allocation!!! Theorem 5(beta) There exists α1, α2 such that when the α-fair allocation achieves maximum (resp. minimum) stability if α<α1 (resp. if α>α2) Maxmin PF MPD α1 α2 0 1 2 Flow-level stability Max stab. region Min stab. region

33. Example: Shannon networks • A network of interfering links with power control (no time coordination) • Link rates follow Shannon formula, e.g. 1 3 2

34. Example: Shannon networks Theorem 6* For α≥1, the α-fair allocation problem can be re-formulated as a convex problem Corollary For α≥1, the α-fair allocation achieves minimum stability The gap between the minimum and maximum capacity region increases with interference *Papandriopoulos et al., ICC 2006

35. Outline • Modeling data networks • Fixed and convex rate regions (wired networks, wireless networks with centralized scheduling) • Arbitrary and fixed rate regions (wireless networks with distributed resource allocation) • Time-varying rate regions (wired networks with priority traffic, link failures ..., wireless networks with fading / mobility)

36. Time-varying rate region • Model: a convex rate region with stationary ergodic variations • Maximum capacity region Theorem 7 There exists an allocation stabilizing the network if and only if the traffic intensity vector ρ belongs to

37. Time-varying rate region • α-fair allocations • In state x, the rate vector scheduled, when the rate region is , is • denoted by • Capacity region Theorem 8 The capacity region of the α-fair allocation is the smallest coordinate-convex set containing

38. Efficiency vs fairness • The flow-level stability region depends on the chosen utility function • Stability decreases with the fairness parameter α • Max-min fairness is always the worse allocation!!! Theorem 9(beta) There exists α1, α2 such that when the α-fair allocation achieves maximum (resp. minimum) stability if α<α1 (resp. if α>α2) Maxmin PF MPD α1 α2 0 1 2 Flow-level stability Max stab. region Min stab. region

39. Example 1: link failure

40. Example 1: link failure

41. Example 2: the downlink of cell. net. Class 1 Class 2 TDMA rate regions

42. Example 2: the downlink of cell. net.

43. Maximum stability Minimum stability Conclusions • Fixed and convex rate regions: wireless networks PF MPD Maxmin 0 1 2 Stability Flow throughput • Non-convex or time-varying rate regions PF MPD Maxmin 0 1 2 Stability

44. Conclusions • Instantaneous fairness has a price in terms of stability • Maxmin is always the worse allocation • PF is also the worse in Shannon networks • Does stability has a cost in terms of fairness (mean flow durations?) ... May be not ... • The stability / performance is higly impacted by the underlying rate region structure and its variations: there is no unique objective garanteeing performance all the time • We need to tune α to adapt to the network structure • .... • Utility based allocations are interesting but we have to change the notion of utility as the net evolves