Scheduling for moving vehicles with garantee Proportional

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


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?


1 Introduction




5 Simulation


Mathematical Model: One Base Station

Figure: Drive-thru Internet systems.

For each time the base station (BS) allows at most onevehicle to allocate data.


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.


1 Introduction




5 Simulation


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


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


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


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))).


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}.


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 .


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.


Simulations


Moving vehicles and Measurement based

Figure: Rate in coverage range of one Base Station


Moving vehicles and Measurement based

Figure: Rate in coverage range of one Base Station


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


Thank you for your attention!Any question?


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Scheduling for moving vehicles with garantee Proportional