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. 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 time­ varying rate regions. with Jiaping Liu et al. ACM Sigmetrics 2007 Talk based on...

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 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 Mo-Walrand. α-fair allocations: 012 fairness efficiency PFMPD Maxmin 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 1.Modeling data networks 2.Fixed and convex rate regions (wired networks, wireless networks with centralized scheduling) 3.Arbitrary and fixed rate regions (wireless networks with distributed resource allocation) 4.Time-varying rate regions (wired networks with priority traffic, link failures..., wireless networks with fading / mobility)

7. Outline 1.Modeling data networks 2.Fixed and convex rate regions (wired networks, wireless networks with centralized scheduling) 3.Arbitrary and fixed rate regions (wireless networks with distributed resource allocation) 4.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. Performance metrics Users perceive performance at flow-level: the mean time to transfer documents Flow-level dynamics - Poisson arrivals of class-k flows: - Departures at rate (exp. flow sizes): Flows transferred in a finite time iff stability of the process of the numbers of flows Performance metric: capacity region The set of such that the system is stable at flow-level 1/(mean flow duration) 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 With fading / user mobility / variable interference: a time-varying rate region TDMA rate region

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

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

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

17. The math question K K x1x1 x2x2 xKxK solves 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 1.Modeling data networks 2.Fixed and convex rate regions (wired networks, wireless networks with centralized scheduling) 3.Arbitrary and fixed rate regions (wireless networks with distributed resource allocation) 4.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) 0 PF

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) 0 Maxmin

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) 0 α = 0.2

24. Outline 1.Modeling data networks 2.Fixed and convex rate regions (wired networks, wireless networks with centralized scheduling) 3.Arbitrary and fixed rate regions (wireless networks with distributed resource allocation) 4.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 α-fair allocations 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)

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

30. Fixed and arbitrary rate region 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 stable unstable ? ? ?

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

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!!! 012 PFMPD Maxmin Flow-level stability Min stab. regionMax stab. region Theorem 5(beta) There exists α 1, α 2 such that when the α-fair allocation achieves maximum (resp. minimum) stability if α α 2 ) α1α1 α2α2

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

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

35. Outline 1.Modeling data networks 2.Fixed and convex rate regions (wired networks, wireless networks with centralized scheduling) 3.Arbitrary and fixed rate regions (wireless networks with distributed resource allocation) 4.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!!! 012 PFMPD Maxmin Flow-level stability Min stab. regionMax stab. region Theorem 9(beta) There exists α 1, α 2 such that when the α-fair allocation achieves maximum (resp. minimum) stability if α α 2 ) α1α1 α2α2

39. Example 1: link failure

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

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

43. Conclusions Fixed and convex rate regions: wireless networks PFMPDMaxmin 0 12 Stability Flow throughput Non-convex or time-varying rate regions PFMPDMaxmin 0 12 Stability Maximum stabilityMinimum 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