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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In Three Acts
How to allocate bandwidth to users?
How to model the network?
What criteria to use?
5A Physical View
Router : interconnect, where links meet.
Host : multi-user, endpoint of communication.
Link / Resource : bottleneck, each has finite capacity Cj.
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.
Extraneous elements have been removed.
Routes are just subsets of links / resources.
Represented by [Aji] : whether resource j is used by route i.
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.
10Maximize Overall Throughput
a® 0 : maximize throughput
a = 1 : proportional fairness
a® ¥ : max-min fairness
With ki = 1 / RTTi2,
a = 2 : TCP
Assume Ni(t) > 0.
Let L(N(t)) be a solution to the a-fair optimization.
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:
Look at slope:
By strong law of large numbers for renewal processes:
with probability 1.
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
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
If f is AC on I, the f is differentiable a.e. on I, and
What goes wrong in Lp ? L¥?
Let L be the set of strictly increasing Lipschitz continuous functions l mapping [0,¥) onto [0,¥),such that
(standard bounded metric)
For functions mapping to any Polish (complete, separable, metric) space.
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.
The resulting metric space is Polish.
from Gromoll & Williams