Network%20Bandwidth%20Allocation

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Network Bandwidth Allocation

• (and Stability)
• In Three Acts

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Problem Statement
How to allocate bandwidth to users? How to model
the network? What criteria to use?
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Act I
Modeling
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A Physical View
Router interconnect, where links meet. Host
multi-user, endpoint of communication. Link /
Resource bottleneck, each has finite capacity
Cj.
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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.
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Simplification
Extraneous elements have been removed.
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Abstraction
Routes are just subsets of links /
resources. Represented by Aji whether
resource j is used by route i. Capacity
constraint
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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.
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Act II
PerformanceCriteria
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Allocation Efficiency

• An allocation L is feasible if capacity
  constraint satisfied.
• A feasible allocation L is efficient if we dont
  have m ³ L for any other feasible m.
• Defined at a point in time, regardless of usage.

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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.

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Maximize Overall Throughput

• That is, max
• No unique allocation.
• Could get unexpected results.

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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.

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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.

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a-Fair Allocations
Maximize
Subject to
With ki1, a 0 maximize throughput a
1 proportional fairness a
max-min fairness With ki 1 / RTTi2, a 2
TCP
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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
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Properties of a-Fair Allocations
Assume Ni(t) gt 0. Let L(N(t)) be a solution to
the a-fair optimization.

• The optimal L exists and is unique.
• Its positive L gt 0.
• Scale invariance L(rN) L(N), for r gt 0.
• Continuity L is continuous in N.
• System is stable when

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Act III
Fluids Formalities
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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 gt 0
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Fluid Limit Visual
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Fluid Limit Math
Look at slope
By strong law of large numbers for renewal
processes
with probability 1.
Thus
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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
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Fluid Analysis is Easier
Definition
A complex function f is absolutely continuous on
Ia,b if for every e gt 0 there is a d gt 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
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Visualizing Fluid Flow
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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

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Towards a Formal Framework

• Interested in stochastic processes with samples
  paths in DÂ0, ), the space of right continuous
  real functions having left limits.

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

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Why we need a better metric.


What goes wrong in Lp ? L ?
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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.
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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.
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Fluid Limit Theorem
from Gromoll Williams
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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.

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Papers
2000
1995
2005
Dai
Gromoll, Williams
Bonald, Massoulié
Kelly, Maulloo, Tan
Kelly
Kelly, Williams
Mo, Walrand
Massoulié