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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
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
• 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.)