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

Regions Alexandre ProutiereFrance Telecom R&D

ENS Paris

Joint work with T. Bonald

CISS – 3.22.2006

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 2Class 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 arrival Flow departure

The capacity region

• Network capacity = max total traffic intensity compatible with some QoS requirements

Mean fl

ow

th

roughput

0

• First QoS requirement: – Stability of process

PerformanceStationary distribution

Flow-levelstability

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

• Interfering links without sched. coordination

1

2SNR = 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

2SNR = 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 genera l 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 allocationachieves 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

2 3

Flow

th

roughp

ut

Flow

th

roughp

ut

Short route

Long route

PF

Max-min

Performance is not verysensitive to the chosenutility function

Flow throughput in wireless nets• A cell with orthogonal transmissions

Flow

th

roughp

ut

PF

Max-min

1

2

Performance is sensitive to the chosen utility functionAvoid 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 policyis 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 stabilityregion of any α-fair allocations is maximum

α-fair allocations

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

Corollary: There exists such that for all , the stabilityregion of α-fair allocations is minimum and equal to the smallestcoordinate-convex set containing the contour of

More classes

Proposition: There exists such that for all , the stabilityregion 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 quiteinefficient

Example1

2SNR = 10 dB

• Convex rate regions: wired networks

Conclusions

0 1 2fairness

efficiency

PF MPD Maxmin

• Rules for the choice of the allocation

PF MPD Maxmin

0 1 2

Stability

Flow throughput

• Convex rate regions: wireless networks

Conclusions

0 1 2fairness

efficiency

PF MPD Maxmin

• Rules for the choice of the allocation

PF MPD Maxmin

0 1 2

Stability

Flow throughput

• Non-convex rate regions: wireless networks

Conclusions

0 1 2fairness

efficiency

PF MPD Maxmin

• Rules for the choice of the allocation

PF MPD Maxmin

0 1 2

Stability

Maximum stability Minimum 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.)