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Network Bandwidth Allocation. (and Stability) In Three Acts. Problem Statement. How to allocate bandwidth to users? How to model the network? What criteria to use?. Act I. Modeling. Host 1. Host 2. Host 3. Host 4. Host 5. A Physical View.

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Network bandwidth allocation

Network Bandwidth Allocation

(and Stability)

In Three Acts


Problem statement
Problem Statement

How to allocate bandwidth to users?

How to model the network?

What criteria to use?


Act i
Act I

Modeling


A physical view

Host

1

Host

2

Host

3

Host

4

Host

5

A Physical View

Router : interconnect, where links meet.

Host : multi-user, endpoint of communication.

Link / Resource : bottleneck, each has finite capacity Cj.


System usage

Host

1

Host

2

Host

3

Host

4

Host

5

System Usage

Route : static path through network, supporting Ni(t) flows with Li(N(t)) allocated bandwidth.

Flows / Users : transfer documents of different sizes, evenly split allocated bandwidth along route. Dynamic. Not directed.


Simplification
Simplification

Extraneous elements have been removed.


Abstraction
Abstraction

Routes are just subsets of links / resources.

Represented by [Aji] : whether resource j is used by route i.

Capacity constraint:


Stochastic behavior

220

020

120

130

110

121

Stochastic Behavior

Model N(t) as a Markov process with countable state space.

Poisson user arrivals at rate ni.

Exponential document sizes with parameter mi.

Define traffic intensity ri = ni / mi.


Act ii
Act II

PerformanceCriteria


Allocation efficiency
Allocation Efficiency

  • An allocation L is feasible if capacity constraint satisfied.

  • A feasible allocation L is efficient if we don’t have m³L for any other feasible m.

  • Defined at a point in time, regardless of usage.


Stability
Stability

  • Stable « Markov chain positive recurrent.

  • Returns to each state with probability 1 in finite mean time.

  • Necessary, but not sufficient condition:

  • How tight this is gives us an idea of utilization.

  • Does not uniquely specify allocation.


Maximize overall throughput

10

10

10

Maximize Overall Throughput

  • That is, max

  • No unique allocation.

  • Could get unexpected results.


Max min fairness

12

12

Max-Min Fairness

  • Increase allocation for each user, unless doing so requires a corresponding decrease for a user of equal or lower bandwidth to satisfy the capacity constraints.

  • Uniquely determined.

  • Greedy algorithm. Not distributed.


Proportional fairness
Proportional Fairness

  • L is proportionally fair if for any other feasible allocation L* we have:

  • Same as maximizing:

  • Interpret as utility function.

  • Distributed algorithms known.


A fair allocations
a-Fair Allocations

Maximize

Subject to

With ki=1,

a® 0 : maximize throughput

a = 1 : proportional fairness

a® ¥ : max-min fairness

With ki = 1 / RTTi2,

a = 2 : TCP


Tcp bias
TCP Bias

  • Congestion window based on additive increase / multiplicative decrease mechanism.

  • Increase for each ACK received, once every Round Trip Time.

  • Timeouts based on RTT.

  • Bias against long RTT.

RTT

timeout


Properties of a fair allocations
Properties of a-Fair Allocations

Assume Ni(t) > 0.

Let L(N(t)) be a solution to the a-fair optimization.

  • The optimal L exists and is unique.

  • It’s positive: L > 0.

  • Scale invariance: L(rN) = L(N), for r > 0.

  • Continuity: L is continuous in N.

  • System is stable when


Act iii
Act III

Fluids &Formalities


Fluid models
Fluid Models

Decompose into non-decreasing processes:

Ni(0) : initial condition

Ei(t) : new arrival process

Ti(t) : cumulative bandwidth allocated

Si(t) : service process

Consider a sequence indexed by r > 0:



Fluid limit math
Fluid Limit : Math

Look at slope:

By strong law of large numbers for renewal processes:

Thus

with probability 1.


Fluid model solution
Fluid Model Solution

A fluid model solution is an absolutely continuous function

so that at each regular point t and each route i

and for each resource j


Fluid analysis is easier
Fluid Analysis is Easier

Definition

A complex function f is absolutely continuous on I=[a,b] if for every e > 0 there is a d > 0 such that

for any n and any disjoint collection of segments (a1,b1),…,(an,bn) in I whose lengths satisfy

Theorem

If f is AC on I, the f is differentiable a.e. on I, and



For stability
For Stability

  • If fluid system empties in finite time, then system is stable.

  • Show that

  • In general, what happens as t ®¥ when some of the resources are saturated?

  • We approach the invariant manifold, aka the set of invariant states


Towards a formal framework
Towards a Formal Framework

  • Interested in stochastic processes with samples paths in DÂ[0, ¥), the space of right continuous real functions having left limits.

  • Well behaved. At most countably many points of discontinuity.


Why we need a better metric
Why we need a better metric.

What goes wrong in Lp ? L¥?


Skorohod topology
Skorohod Topology

Let L be the set of strictly increasing Lipschitz continuous functions l mapping [0,¥) onto [0,¥),such that

Put

(standard bounded metric)

For functions mapping to any Polish (complete, separable, metric) space.


Prohorov metric
Prohorov Metric

Let (S,d) be a metric space,

B(S) the s-algebra of Borel subsets of S,

P(S) family of Borel probability measures on S.

Define

The resulting metric space is Polish.


Fluid limit theorem
Fluid Limit Theorem

from Gromoll & Williams


Outline of proof
Outline of Proof

  • Apply functional law of large numbers to load processes.

  • Derive dynamic equations for state and bounds.

  • State contained in compact set with probability 1 in limit.

  • State oscillations die down with probability 1 in limit.

  • Sequence is C-tight.

  • Weak limit points are fluid solutions with probability 1.


Papers
Papers

1995

2000

2005

Dai

Bonald,

Massoulié

Gromoll,

Williams

Kelly,

Maulloo,

Tan

Kelly

Kelly,

Williams

Mo,

Walrand

Massoulié


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