Designing data networks
A flow-level perspective
Alexandre Proutiere
Microsoft Research
Workshop on Mathematical Modeling and Analysis of
Computer Networks
June 2007, ENS, Paris
Talk based on ...
• 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 timevarying rate regions.
with Jiaping Liu et al. ACM Sigmetrics 2007
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?
Related work: the thru-fairness trade-off
• Mo-Walrand. α-fair allocations:
0
PF
MPD
1
2
Maxmin
fairness
efficiency
• 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
• 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
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!
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)
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)
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
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
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
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
1/(mean flow duration)
0
Performance metric: capacity region
The set of
such
that the system is stable at flow-level
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
Rate regions
• Wireless networks with centralized scheduling
a convex polytope
TDMA rate region
• With fading / user mobility / variable interference: a time-varying
rate region
Rate regions
• Wireless networks with distributed resource allocation, power
control / rate adaptation
a continuous non-convex rate region
1
2
SNR = 10 dB
Rate regions
• Wireless networks with distributed resource allocation, without
power control
a discrete rate region
1
2
The big picture
Packet level:
rate region, utility func.
Design
(choice of U)
Flow-level traffic
demand
Objective
Multi-class queue
with state-dependent
capacity
Capacity region
Flow-level performance
The math question
x1
x2
solves
K
xK
K
How to choose the utility function U such that the stability
region of the queuing system is maximized?
(or more generally some performance metrics?)
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)
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
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
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)
PF
0
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)
Maxmin
0
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)
α = 0.2
0
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)
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
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
Fixed and arbitrary rate region
• α-fair allocations
Fixed and arbitrary rate region
is the set of points in the rate
region actually scheduled by the αfair 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)
• α-fair allocations
Fixed and arbitrary rate region
stable
• α-fair allocations
Theorem 3
The capacity region of the α-fair allocation contains the smallest
coordinate-convex set containing
Fixed and arbitrary rate region
?
unstable
?
stable
?
• 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
Fixed and arbitrary rate region
unstable
stable
• α-fair allocations
Corollary 1
In case of continuous
, the capacity region of the α-fair
allocation is the smallest coordinate-convex set containing
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!!!
Theorem 5(beta)
There exists α1, α2 such that when the α-fair allocation achieves
maximum (resp. minimum) stability if α<α1 (resp. if α>α2)
α1
0
PF
1
α2
MPD
2
Flow-level
stability
Max stab. region
Min stab. region
Maxmin
Example: Shannon networks
• A network of interfering links with power control (no time
coordination)
• Link rates follow Shannon formula, e.g.
1
3
2
Example: Shannon networks
Theorem 6*
For α≥1, the α-fair allocation problem can be re-formulated as a
convex problem
Corollary
For α≥1, the α-fair allocation
achieves minimum stability
The gap between the minimum
and maximum capacity region
increases with interference
*Papandriopoulos et al., ICC 2006
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)
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
Time-varying rate region
• α-fair allocations
- In state x, the rate vector scheduled, when the rate region is
denoted by
• Capacity region
Theorem 8
The capacity region of the α-fair allocation is the smallest
coordinate-convex set containing
, is
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!!!
Theorem 9(beta)
There exists α1, α2 such that when the α-fair allocation achieves
maximum (resp. minimum) stability if α<α1 (resp. if α>α2)
α1
0
PF
1
α2
MPD
2
Flow-level
stability
Max stab. region
Min stab. region
Maxmin
Example 1: link failure
Example 1: link failure
Example 2: the downlink of cell. net.
Class 1
Class 2
TDMA rate regions
Example 2: the downlink of cell. net.
Conclusions
• Fixed and convex rate regions: wireless networks
0
PF
MPD
1
2
Maxmin
Stability
Flow throughput
• Non-convex or time-varying rate regions
0
PF
MPD
1
2
Stability
Maximum stability
Minimum stability
Maxmin
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