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 CISS –

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)

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

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 Outline

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

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

Traffic demand Class-k flow arrivals: A Poisson process –Arrival intensity –Mean flow size –Traffic intensity

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

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 Flow rate in state x: This defines the realized resource allocation

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 Flow arrivalFlow departure

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

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 Outline

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

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

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 2

1 2 SNR = 10 dB Non-convex rate regions Without coordination, interference modifies the structure of the rate region Highly non-convex rate regions

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

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

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 Outline

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?

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 general flow size distributions Flow-level stability

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

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

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 Outline

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

Flow throuhghput in wired nets 1 A linear network 23 Flow throughput Short route Long route PF Max-min Performance is not very sensitive to the chosen utility function

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

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 Outline

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

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

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

α- fair allocations Corollary: There exists such that for all, the stability region of α-fair allocations is maximum and equal to 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

More classes Proposition: There exists such that for all, the stability region of α-fair allocations is maximum and equal to 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

Example 1 2 SNR = 10 dB

Convex rate regions: wired networks Conclusions 012 fairness efficiency PFMPD Maxmin Rules for the choice of the allocation PFMPD Maxmin 0 12 Stability Flow throughput

Convex rate regions: wireless networks Conclusions 012 fairness efficiency PFMPD Maxmin Rules for the choice of the allocation PFMPD Maxmin 0 12 Stability Flow throughput

Non-convex rate regions: wireless networks Conclusions 012 fairness efficiency PFMPD Maxmin Rules for the choice of the allocation PFMPD Maxmin 0 12 Stability Maximum stabilityMinimum stability

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.)