Short-Term Fairness and Long-Term QoS
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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)
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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
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
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
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.
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 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
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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
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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
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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)
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Optimal scheme vs proportional fairness Lower bound for optimal scheme:
Heavy-traffic limit
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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
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Simulation results
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Upper bound for general networks Lyapunov function
Upper bound for general networks
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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
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Line network
Our upper bound
Upper bound by Kang, Kelly, Lee, Williams (2007)
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Star network Our upper bound
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?