CISS – 3.22.2006
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
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
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
Data networks at flow-level
• Wired networks
– Heyman-LakshmanNeidhardt'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
NETWORK
Class 1
Class 2
Class 3
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
• Flow rate in state x:
time
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
rate
• Network state process
time
Flow arrival
Flow departure
The capacity region
• First QoS requirement:
– Stability of process
Mean flow throughput
• Network capacity = max total traffic intensity compatible
with some QoS requirements
0
Resource
allocation
Flow-level
stability
Stationary
distribution
Performance
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
The rate region
• In state x, rates allocated to the different classes
• Rate region
• Wired networks
(0,1)
(1,1)
Rate region = a convex polytope
with facets orthogonal to some binary
vectors
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
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
SNR = 10 dB
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
SNR = 2 dB
Resource allocations
• An allocation chooses a point of the rate region in
each network state
• 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
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
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?
Flow-level stability
•
•
•
•
•
•
•
•
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
Maximum stability
• Consider an arbitrary rate region
Unstable
Proposition: The maximum stability region is the smallest convex
coordinate-convex set containing the rate region
This set is denoted by
Maximum stability
• Consider an arbitrary rate region
Stable
Proposition: The maximum stability region is the smallest convex
coordinate-convex set containing the rate region
This set is denoted by
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
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
Short route
2
3
Performance is not very
sensitive to the chosen
utility function
PF
Max-min
Long route
Flow throughput
1
Flow throughput
• A linear network
Flow throughput in wireless nets
• A cell with orthogonal transmissions
1
2
Flow throughput
PF
Performance is sensitive to the chosen utility function
Avoid max-min
Max-min
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
Two class networks
• A discrete rate region
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
Conclusions
• Rules for the choice of the allocation
efficiency
0
PF
MPD
1
2
Maxmin
fairness
• Convex rate regions: wired networks
0
Stability
Flow throughput
PF
MPD
1
2
Maxmin
Conclusions
• Rules for the choice of the allocation
efficiency
0
PF
MPD
1
2
Maxmin
fairness
• Convex rate regions: wireless networks
0
Stability
Flow throughput
PF
MPD
1
2
Maxmin
Conclusions
• Rules for the choice of the allocation
efficiency
0
PF
MPD
1
2
Maxmin
fairness
• Non-convex rate regions: wireless networks
0
PF
MPD
1
2
Stability
Maximum stability
Minimum stability
Maxmin
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.)