23/4/19 1

Short-Term Fairness and Long-Term QoS

Lei Ying

ECE dept, Iowa State University,

Joint work with Bo Tan, UIUC and R. Srikant, UIUC

23/4/19 2

Resource allocation for the Internet

Resource allocation algorithm for the Internet are designed to ensure fairness among users present in the network

Assume the number of users is fixed (static model)

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)

23/4/19 3

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?

23/4/19 4

Network and flow model

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

23/4/19 5

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:

23/4/19 6

Resource allocation and backlog Proportionally-fair resource allocation on the backlog

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

23/4/19 7

Line network example

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

23/4/19 8

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

23/4/19 9

Long-term QoS: Line network

Optimal policies for a line network with two links were proposed

by Verloop et al’ 06.

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%.

23/4/19 10

Optimal resource allocation: Star network If all the 3 file types are non-

empty Serve each of them at rate 0.5

If only 2 file types are non-empty Serve the file type with more

files at rate 1

If only 1 file type is non-empty Serve it at rate 1

Recall each link has capacity 1

23/4/19 11

Intuition behind optimality x=(0.5,0.5,0.5) maximizes total

service rate, Feasible only when all file types

are non-empty.

If only 2 file types are non-empty, serve the one with the larger number of files 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

23/4/19 12

Proof of optimality

Use uniformization to convert to discrete-time problem

Consider the objective

Prove the optimality of the scheme for all N Use induction and dynamic programming

23/4/19 13

Performance of the optimal scheme Largest 2 file types behave like a single queue: total

service rate for them = 1

Suggests the Lyapunov function:

m1(t)

m2(t)

2 1

m3(t)

23/4/19 14

Optimal scheme vs proportional fairness Lower bound for optimal scheme:

Heavy-traffic limit

23/4/19 15

Performance of proportional fairness Lyapunov function

E[W[t+1] – W[t] ] = 0 in steady-state

Upper bound on steady-state backlog

Compare with upper bound

upper bound / lower bound = 1.5

23/4/19 16

Simulation results

23/4/19 17

Upper bound for general networks Lyapunov function

Upper bound for general networks

23/4/19 19

Upper bound for general networks Upper bound

This result complements the work of Kang, Kelly, Lee, Williams (2007)

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

23/4/19 20

Line network

Our upper bound

Upper bound by Kang, Kelly, Lee, Williams (2007)

23/4/19 21

Star network Our upper bound

Upper bound by Kang, Kelly, Lee, Williams (2007)

23/4/19 22

Summary

Derived an upper bound for general networks, which linearly increases with the number of routes in the network.

Tighter lower bound?