Designing data networks A flow-level perspective

transcript

Alexandre Proutiere Microsoft Research

Workshop on Mathematical Modeling and Analysis of Computer Networks

June 2007, ENS, Paris

• 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

• Flow level stability of data networks with non convex and time varying rate regions.with Jiaping Liu et al. ACM Sigmetrics 2007

Talk based on ...

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

• 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

• Mo-Walrand. α-fair allocations:

0 1 2fairness

efficiency

PF MPD Maxmin

• 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

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

Performance metric: capacity regionThe set of such that the system is stable at flow-level

1/(mean flow duration)

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

• With fading / user mobility / variable interference: a time-varying rate region

TDMA rate region

Rate regions

• Wireless networks with distributed resource allocation, power control / rate adaptation a continuous non-convex rate region

2SNR = 10 dB

Rate regions

• Wireless networks with distributed resource allocation, without power control a discrete rate region

The big picture

Flow-level traffic demand

Multi-class queuewith state-dependent capacity

Packet level: rate region, utility func.

Capacity regionFlow-level performanceObjective

Design(choice of U)

The math question

solves

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

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 1Any 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 evolutionMassoulie: PF and BF are cloes to each other

• It realizes a good fairness-efficiency trade-off in wired networksBonald-Roberts

• It has to be chosen for wireless systems

Vote for PF!

• It is robust to traffic characteristics evolutionMassoulie: PF and BF are close to each other

Maxmin

Vote for PF!

• It is robust to traffic characterisitcs evolutionMassoulie: PF and BF are cloes to each other

α = 0.2

Outline

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

• 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

• α-fair allocations

is the set of points in the rate region actually scheduled by the α-fair 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)

Theorem 3The capacity region of the α-fair allocation contains the smallest coordinate-convex set containing

stable

Theorem 3The capacity region of the α-fair allocation contains the smallest coordinate-convex set containing

Theorem 4The system under the α-fair allocation is unstable when ρ belongs to

stable

unstable

Corollary 1In case of continuous , the capacity region of the α-fair allocation is the smallest coordinate-convex set containing

stable

unstable

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!!!

PF MPD Maxmin

Flow-levelstability

Min stab. regionMax stab. region

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 α2

Example: Shannon networks

• A network of interfering links with power control (no time coordination)• Link rates follow Shannon formula, e.g.

Example: Shannon networks

Theorem 6*For α≥1, the α-fair allocation problem can be re-formulated as a convex problem

*Papandriopoulos et al., ICC 2006

CorollaryFor α≥1, the α-fair allocation achieves minimum stability

The gap between the minimum and maximum capacity region increases with interference

Outline

Time-varying rate region

• Model: a convex rate region with stationary ergodic variations

• Maximum capacity region

Theorem 7There 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 , is denoted by

• Capacity region

Theorem 8The 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!!!

PF MPD Maxmin

Flow-levelstability

Min stab. regionMax stab. region

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 α2

Example 1: link failure

Example 2: the downlink of cell. net.

TDMA rate regions

Class 1 Class 2

Example 2: the downlink of cell. net.

Conclusions

• Fixed and convex rate regions: wireless networks

PF MPD Maxmin

Stability

Flow throughput

• Non-convex or time-varying rate regions

PF MPD Maxmin

Stability

Maximum stability Minimum stability

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

Designing data networks A flow-level perspective

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