Short-Term Fairness and LongTerm QoS
Lei Ying
ECE dept, Iowa State University,
Joint work with Bo Tan, UIUC and R. Srikant,
UIUC
2016/3/23
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Resource allocation for the Internet

Resource allocation algorithm for the Internet are
designed to ensure fairness among users present in the
network
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Assume the number of users is fixed (static model)
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In reality, the users arrive, bringing in a certain amount of
work in the form of a file to be transferred, and depart
when the work is completed (connection-level model)
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Resource allocation for the Internet

The stability of the network when there are file arrivals
and departures has been studied in a number of papers
(Robert&Massoulie’98, Veciana et al’01,
Bonald&Massoulie’01, Lin et al’07)

The network is stochastically stable under the
proportional-fairness if

Connection-level performance beyond stability?
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Network and flow model

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
Consider a network with L links and
R routes
File arrivals of each type: Poisson,
rate r
File size of each type: Exponential,
parameter r
Capacity of each link = cl
The capacity of each link is divided
among the files using the link
A file departs after it has transferred
its data
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Resource allocation and backlog

nr(t): number of files of type r

xr(t): rate allocated to flows of type r at time t

Backlog is affected by the rate allocation

Backlog:
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Resource allocation and backlog

Proportionally-fair resource allocation on the backlog
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Proportionally-fairness can be implemented in a distributed
fashion

Support the maximum connection-level stability

Doesn’t maximize the departure rate at each time slot
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Line network example
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r= r=, cl=1
n1[t]=n2[t]=n3[t]) x1[t]=x2[t]=x3[t]=0.5 ) overall departure
rate is 1.5

x2[t]=x3[t]=1 ) overall departure rate is 2
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Long-term QoS

Goal: Study the impact of proportionally-fair resource allocation
on the backlog

Obtain an upper-bound on the backlog under proportional
fairness

Find the optimal resource allocation strategy to minimize the
backlog

Obtain a lower bound on the backlog under the optimal strategy

Compare the upper and lower bound in the heavy-traffic regime:
r r ! 1
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Long-term QoS: Line network

Optimal policies for a line network with two links were proposed
by Verloop et al’ 06.
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The delay-performance of the optimal policies and the
proportionally-fair policy were compared using simulations, and it
was shown that the gap is less than 20%.
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Optimal resource allocation: Star network
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If all the 3 file types are nonempty
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Serve each of them at rate 0.5
If only 2 file types are nonempty


Recall each link has capacity 1
Serve the file type with more
files at rate 1
If only 1 file type is non-empty
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Serve it at rate 1
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Intuition behind optimality
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x=(0.5,0.5,0.5) maximizes total
service rate,

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If only 2 file types are nonempty, serve the one with the
larger number of files


Feasible only when all file types
are non-empty.
This would increase the likelihood
that all file types are non-empty in
the future
Motivated by Verloop et al (2005)
for 2-link, 3-flow network
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Proof of optimality
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Use uniformization to convert to discrete-time
problem
Consider the objective
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Prove the optimality of the scheme for all N
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Use induction and dynamic programming
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Performance of the optimal scheme
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Largest 2 file types behave like a single queue: total
service rate for them = 1
2
m1(t)
1
m2(t)
m3(t)

Suggests the Lyapunov function:
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Optimal scheme vs proportional fairness
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Lower bound for optimal scheme:
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Heavy-traffic limit
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Performance of proportional fairness
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Lyapunov function
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E[W[t+1] – W[t] ] = 0 in steady-state
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Upper bound on steady-state backlog
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Compare with upper bound
upper bound / lower bound = 1.5
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Simulation results
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Upper bound for general networks
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Lyapunov function
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Upper bound for general networks
Upper bound for general networks
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Upper bound
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This result complements the work of Kang, Kelly, Lee,
Williams (2007)
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Their model assumes each link has a dedicated flow;
Letting the load due to local flows go to zero leads to a
heuristic upper bound
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Line network
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Our upper bound
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Upper bound by Kang, Kelly, Lee, Williams (2007)
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Star network
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Our upper bound
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Upper bound by Kang, Kelly,
Lee, Williams (2007)
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Summary

Derived an upper bound for general
networks, which linearly increases with the
number of routes in the network.

Tighter lower bound?
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