Scheduling for moving vehicles with garantee Proportional Fairness between users Nga NGUYEN Supervisors: Olivier Brun, Balakrishna Prabhu Fr´ ejus Summer School 28 June 2018
Scheduling for moving vehicles with garanteeProportional Fairness between users
Nga NGUYEN
Supervisors: Olivier Brun, Balakrishna Prabhu
Frejus Summer School28 June 2018
Short Bio
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
1 Introduction
2 The Mathematical Model
3 Objective: Guarantee Proportional Fair and High Throughput
4 Some Existing Algorithms
5 Simulation
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Introduction
Current scheduling algorithms use current/past informationfor decision.
Connected vehicles technology will gives access to futureinformation (path prediction+ Signal-to-noise ratio (SNR)maps → predict future rate)
Can data on future improve efficiency of algorithms?
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
1 Introduction
2 The Mathematical Model
3 Objective: Guarantee Proportional Fair and High Throughput
4 Some Existing Algorithms
5 Simulation
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Mathematical Model: One Base Station
Figure: Drive-thru Internet systems.
For each time the base station (BS) allows at most onevehicle to allocate data.
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Objective: Guarantee Proportional Fair and HighThroughput
maximize C =K∑i=1
log
T∑j=1
αij rij
subject to
K∑i=1
αij = 1, αij ∈ {0, 1}
K is number of cars, T is number of time slots, αi ,j is theallocation, ri ,j is the rate.
log objective function stands for Proportional fairness betweenusers.
→ ri ,j is given, αi ,j is variable.
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
1 Introduction
2 The Mathematical Model
3 Objective: Guarantee Proportional Fair and High Throughput
4 Some Existing Algorithms
5 Simulation
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Some Existing Algorithms
Greedy. Current info only: choose the vehicle with bestcurrent rate.
PF-EXP[3]. Current + past info: choose user with best
current rate
total rate in the past
Used in 3G network; optimal in some cases (not necessarilytrue for road traffic)
Discrete Gradient [2]. Use current + past + future info:choose user with the best
current rate
total rate in the past + current + estimated future rate
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Projected Gradient: Upper bound- relaxed problem
Relax the integer constraints.
maximize C =K∑i=1
log
T∑j=1
αij rij
subject to
K∑i=1
αij = 1, αij ∈ [0, 1]
Benchmark algorithm for comparison
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Projected Gradient: Upper bound- relaxed problem
Relax the integer constraints.
maximize C =K∑i=1
log
T∑j=1
αij rij
subject to
K∑i=1
αij = 1, αij ∈ [0, 1]
Benchmark algorithm for comparison
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Formula for projected gradient
Update rule
Start with α(0) ∈ D where D is the feasible set.
α(n+1) = ΠD(α(n) + εn∇C (α(n))), with εn ∈ (0, 1) is learningrate at step n.
→ This update rule can approach the global optimal.In this below we compute the projection ΠD(α(n) + εn∇C (α(n))).
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Lemma
Given A,B two finite sets, A is nonempty set. Then there exists adecomposition B = B1 t B2 such that mean(A ∪ B1) ≤ i for everyi ∈ B1 and mean(A ∪ B1) > i for every i ∈ B2. With conventionthat boolean on empty set is always true.
Ex: A = {5},B = {1, 4, 6} then B1 = {6} and B2 = {1, 4}.
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Projected gradient
Fix α(n), denote ∇i ,jC = ∂C/∂αi ,j(α(n)).
Proposition
α(n+1) is given in the following formula: For each j , define
A(j) = {∇i ,jC |for i s.t α(n)i ,j > 0} and
B(j) = {∇i ,jC |for i s.t α(n)i ,j = 0}.
B1(j),B2(j) are defined as lemma 4.1 for two set A(j),B(j) andm := mean(A(j) ∩ B1(j)).Then
α(n+1)i ,j =
{α(n)i ,j if α
(n)i ,j = 0 and ∇i ,jC ∈ B2(j)
α(n)i ,j + εn(∇i ,jC − m) otherwise .
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Projected gradient
Proposition
If α∗ ∈ D and ∇C (α∗) = 0 then α∗ is the optimal value of therelax problem, where
∇C (α∗) =
{0 if α∗
i ,j = 0 and ∇i ,jC (α∗) ∈ B2(j)
∇i ,jC (α∗)− m otherwise
With this notation, α(n+1) = α(n) + εn∇C (α(n)). So if we doalgorithm until ∇C (α(n)) coverges to 0, it implies α(n) convergesto α∗ at which ∇C (α∗) = 0, i.e, α∗ is the optimal.→ We do recursion until ∇C (α(n)) coverges to 0.
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Simulations
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Moving vehicles and Measurement based
Figure: Rate in coverage range of one Base Station
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Moving vehicles and Measurement based
Figure: Rate in coverage range of one Base Station
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Michael J. Neely,Exploiting Mobility in Proportional Fair Cellular Scheduling:Measurements and Algorithms,IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 24,NO. 1, FEBRUARY 2016.
Robert Margolies et Al,Energy Optimal Control for Time Varying Wireless Networks,IEEE TRANSACTIONS ON INFORMATION THEORY, VOL.52, NO. 7, JULY 2006.
H. J. Kushner and P. A. Whiting,Asymptotic Properties of Proportional-Fair SharingAlgorithms: Extensions of the Algorithm,IEEE Transactions on Wireless Communications, VOL. 3, NO.4, JULY 2004
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users
Thank you for your attention!Any question?
Nga NGUYEN Scheduling for moving vehicles with garantee Proportional Fairness between users