Nonconvex Wireless

8/8/2019 Nonconvex Wireless

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Non-convex Optimization and Resource Allocationin Wireless Communication Networks

Ravi R. Mazumdar

School of Electrical and Computer Engineering

Purdue University

E-mail: [email protected]

Joint Work with Prof. Ness B. Shroff and Jang-Won Lee

ECE Purdue Universit . 1/


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OutlineIntroduction and non-convexity

Joint power and rate allocation for the downlink in (CDMA)wireless systems

Opportunistic power scheduling for the downlink inmulti-server wireless systems

Conclusion and future work



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MotivationTremendous growth in the number of users in communication

networksIncreasing demand on various services that can provide QoS

Scarce network resources

Need to efficiently design and engineer resource allocationschemes for heterogeneous services



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MotivationMost services are elastic

can adjust the amount of resource consumption to somedegree

By appropriately exploiting the elasticity of services

can maintain high efficiency and fairness

can alleviate congestion within the network

Need appropriate model for the elasticity

Utility

degree of users (services) satisfaction or performance byacquiring a certain amount of resource

different elasticity with different utility functions

example: expected throughput as a function of power

allocation in wireless system



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Total system utility

maximization

max

Mi=1

Ui(x)

s. t. gk(x) 0, k = 1, 2, , K

x X

If all Ui and gk are concave and X is a convex set,

convex optimization problem

can be solved by using standard techniquesOtherwise,

non-convex optimization problem

difficult to solve requiring a complex algorithm



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Non-convexity in

resource allocationIn general, three types of utility functions

concave: traditional data services on the Internetsigmoidal-like (S): many wireless services and real-time

services on the Internet

convex: some wireless services

Resource allocation

Utility

ConcaveSigmoidalConvex



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Non-convexity (contd)

0 200

1

f(

)

BPSK

DPSK

FSK

Packet transmission success probability



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Non-convexity (contd)Increasing demand for wireless and real-time services

non-concave utility functions becoming importantnon-convex optimization problem complex algorithm for a global optimum

Can we develop a simple algorithm for the approximation to theglobal optimum?



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Inefficiency of naive

approach11 users and 10 units of a resource

Utility function for each user: U(x)Approximate U(x) with concave function V(x)

With V(x), for each user,

x = 1011however, U(x) = 0

zero total system utility

By allocating one unit to 10users and zero to one user:

10 units of total systemutility 0 1 2

1

U(x)V(x)

x*

Need resource allocation algorithms taking into account theproperties of non-concave functions



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Dual approach (contd)Primal problem

maxM

i=1

Ui(x)

s. t. gk(x) 0,

k = 1, 2, ,Kx X

Dual problem

min Q()

s. t. 0,

Q() = maxxX

{

M

i=1

Ui(x) +

K

k=1

kgk(x)}

non-convex optimiza-tion

convex optimizationsimpler constraints

smaller dimension

May not guarantee the feasible and optimal primal solution



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Part I

Joint power and rate allocation for thedownlink in (CDMA) wireless systems



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Why joint power and rate

allocation?Power is fundamental radio resource

trade off between performance of each userVariable data rate

trade off between data rate and the probability of packettransmission success for a given power allocation

By jointly optimizing power and data rate allocation, thesystem performance can be further improved



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Related workOh and Wasserman [MOBICOM99]: Uplink power and ratecontrol for a single class system without constraint on themaximum data rate

if applied to downlink, single server transmission isoptimal

Bedekar et al. [GLOBECOM99] and Berggren et al. [JSAC01]:Downlink power and rate control without constraint on themaximum data rate

single server transmission is optimal



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Our workCDMA system that supports variable data rate by variablespreading gain

Downlink in a single cell

Snapshot of a time-slot

Constant Path gain and interference level during the

time-slot

Base-station has the total transmission power limit PT

Each user i has

Rmaxi : maximum data rate

fi: function for packet transmission success probability



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Signal to Interference and

Noise Ratio (SINR)SINR for user i

i(Ri, P) =WRi

Pi

(M

m=1 Pm Pi) + Ai

M: number of users in the cell

W: chip rate

: orthogonality factor

Pi: power allocation for user i

Ri: data rate of user iAi = Ii/Gi: transmission environment of user i

Ii: background noise and intercell interference at user iGi: path gain from the base-station to user i

SINR is a function of power and rate allocation



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Packet transmission

success probability: fi

fi is an increasing function of i

For a given Ri, ifM

m=1 Pm = PT, fi isconcave function,

S function, or

convex function

of its own power allocation Pi

0 10

1

P

f(P)

BPSK

DPSK

FSK



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Problem formulation

(A) maxPi,Ri

Mi=1

Rifi(i(Ri, P))

s. t.M

i=1 Pi PT

0 Pi PT, i0 Ri R

maxi , i V

Ri = Ri , i V

V: a subset of users that have variable data rateRifi(i(Ri, P)): expected throughput of user i

Goal: Obtaining power and rate allocation that maximizes the

expected total system throughput with constraints on the totaltransmission power limit of the base-station and the maximum datarate of each user



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Optimal rate allocationTo maximize the expected total system throughput, thebase-station must transmit at the maximum power limit

Redefine SINR for user i as

i(Ri, Pi)=

W

Ri

PiPT Pi + Ai

=W

Ri

Pi

Mj=1 Pj Pi + Ai

= i(Ri, P)

For a given power allocation Pi, the optimal rate of user i,

Ri (Pi) =

WPii (PTPi+Ai) , if i V Pi

Rmaxi

i (PT+Ai)

W+Rmaxi

i

Rmaxi , if i V, Pi >Rmaxi

i (PT+Ai)W+Rmax

ii

Ri , if i V,

where i = arg max1{1

fi()}.



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Equivalent power

allocation problem

(B) max

Mi=1

Ui(Pi)

s.t.M

i=1 Pi PT

0 Pi PT, i,

Ui(Pi) =

Wi

PiPTPi+Ai

fi(i ), if i V, Pi

Rmaxi

i (PT+Ai)W+Rmax

ii

Rmaxi

fi(i(Rmax

i

, Pi)), if i V, Pi >Rmaxi

i (PT+Ai)

W+Rmax

i

i

Ri f(i(Ri , Pi)), if i V

Ui(Pi) is a convex, concave, or S function of Pi.



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Power allocationAmount of power maximizing net utility

Pi() = argmax0PiPT

{Ui(Pi) Pi}

Maximum willingness to pay per unit power

maxi = min{ 0 | max0PPT

{Ui(P) P} = 0}, i

unique for each user i

if > maxi , then Pi() = 0

if < maxi , then Pi() > 0



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Power allocation (contd)Assume that max1

max2

maxM

User selectionSelect users from 1 to K that satisfies

K = max1jM

{

j

i=1

Pi(max

j

) PT}

Users are selected in a decreasing order of maxi

Power allocation

Find such thatK

i=1 Pi() = PT

Allocate power to each selected user i as Pi()

Optimal power allocation for the selected users



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OptimalityP: our power allocation

Po

: optimal power allocationIfM

i=1 Ui(i(Poi )) as M ,

Mi=1

Ui(i(P

i

))Mi=1 Ui(i(P

oi ))

1, as M

Our power allocation is

asymptotically optimala good approximation of the optimal power allocation witha large number of small users



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Multiple access strategyIf

Rmax

i Ai

PTW

i , i,

single server transmission is optimal

when users have high maximum data rate or are

experiencing poor transmission environmentwhen there is no constraint on the maximum data rate

W: chip rate

i

: constant that depends on fi

Ai = Ii/Gi: transmission environment of user i

Ii: intercell interference and background noise at user i

Gi: path gain from the base-station to user i


M l i l


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Multiple access strategy

(contd)IfM

i=1 Pi(maxM ) PT, selecting all users is optimal

If P1(max2 ) PT, selecting only user 1 is optimalOtherwise, selecting a subset of users can be optimal

Condition for optimal multiple access strategy depends on

time-varying parameters such asnumber of users

type of users (utility functions)

channel condition of users

Static multiple access strategy could be inefficient

Need dynamic multiple access strategy (dynamic multi-server

transmission)



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User selection strategyIf all users are homogeneous, selecting users according totransmission environment is optimal

higher priority to a user in a better transmissionenvironment

However, if users are heterogeneous, no simple optimal user

selection strategyOur user selection strategy provides a simple and unifiedselection strategy for heterogeneoususers



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User selection strategyUser i is called more efficient than user j if

Ui(i(P)) Uj(j(P)), P

More efficient user has a higher priority to be selected

When other conditions are the same, user i has a higherpriority to be selected than user j if

Rmaxi > Rmaxj (maximum data rate),

fi() > fj(), (transmission scheme), or

Ai < Aj (transmission environment)

Our user selection strategy provides a simple and efficient selectionstrategy for heterogeneous users



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Numerical resultsModel path gain considering distance loss and log-normallydistributed slow shadowing

Two classes of users, for a user in class i,

fi() = ci{1

1 + eai(bi) di}

Compare with the single-server system

BS BS

BS BS BS

BSBSBS

BS



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Numerical results (contd)

Rmax1 1562.5 6250 25000

Selection ratio of class 1 0.501 0.388 0.198


Utility (Our)/Utility (Single) 3.415 3.854 1.016

f1 = f2

Rmax1 = Rmax2 (R

max2 = 6250)

Selection ratio of class i: the ratio of the number of selectedusers to the number of users in class i



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b1 2.5 3.5 4.5




Rmax1 = Rmax2

f1 = f2 (a1 = a2, b2 = 3.5)

If bi < bj , then fi() fj(),

class i has a more efficient transmission scheme thanclass j



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Ratio of class 1 0.4 0.6 0.8


Selection ratio of class 2 0.004 0 0


Rmax1 = Rmax2

f1 = f2

Class 1: inner regionClass 2: outer region



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Part II

Opportunistic power scheduling for thedownlink in multi-server wireless

systems


Why opportunistic


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Why opportunistic

scheduling?Trade-off between efficiency and fairness due to

multi-class users

time-varying and location-dependent channel condition

Our previous problem

high system efficiency

however, unfair to some (inefficient) users

Fairness

achieved by an appropriate scheduling scheme

Opportunistic scheduling considering each usersdelay tolerance

fairness or performance constraint

time-varying channel condition


Si l


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Single-server vs.

Multi-serverSingle-server scheduling

Only one user can be scheduled in a time-slot

In every time-slot, must decidewhich user must be selected

Multi-server scheduling

Multiple users can be scheduled in a time-slotIn every time-slot, must decide

how many and which users must be selectedhow much power is allocated to each selected user

Most work studied single-server scheduling

However, single-server scheduling can be inefficient

Need dynamic multi-server scheduling



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Related workSingle-server scheduling

Qualcomms HDR: proportional fairness

Borst and Whiting [INFOCOM01]: constraint on utilitybased fairness

Liu, Chong, and Shroff [JSAC01,COMNET03]: constraints

on minimum performance, and utility and resource basedfairness

Multi-server scheduling

Kulkarni and Rosenberg [MSWiM03]: static multi-server

scheduling with independent interfacesLiu and Knightly [INFOCOM03]: dynamic multi-serverscheduling with constraint on utility based fairnessassuming orthogonality among users and linear

relationship between data rate and power allocation



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Our workDynamic multi-server scheduling for downlink in a single cell

Allow users to interfere with each otherPT is total transmission power at the base-station

Utility function Ui for user i: convex, concave, or "S" function

In each time-slot, system is in one of the states{

1, 2,

, S}corresponds to channel conditions of all users

stationary stochastic process with Prob{state s} = s

time-varying channel condition of each user is modeled as

a discrete state stationary stochastic processRequirement for each user

resource based fairness

utility based fairness

minimum performance



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SINR and utility functionSINR for user i when system is in state s

s,i(Ps,i) =NiPs,i

(PT Ps,i) + As,i

Define

Us,i(Ps,i)= Ui(s,i(Ps,i))

The utility function varies randomly according to thechannel condition


Problem formulation with


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Problem formulation with

minimum performance

(C) maxPs,i

Mi=1

E{Ui}(=

Mi=1

Ss=1

sUs,i(Ps,i))

s. t. E{Ui}(=S

s=1

sUs,i(Ps,i)) Ci, i = 1, 2, , M

Mi=1

Ps,i PT, s = 1, 2, , S

0 Ps,i PT, s, i

Goal: Obtaining power scheduling that maximizes the expected totalsystem utility with constraints on the minimum expected utility foreach userand the total transmission power limit for the base-station


Problem with minimum


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Problem with minimum

performance (contd)Main difficulties

Feasibility

assume that the system has call admission controlensuring a feasible solution

Non-convexity

dual approach

No knowledge for the underlying probability distribution a priori

stochastic subgradient algorithm



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Power schedulingIn each time-slot n, power is allocated to users by solving the dual of

(E) maxMi=1

Umps(n),i

((n), Ps(n),i)

s. t.

M

i=1

Ps(n),i PT

0 Ps(n),i PT, i = 1, 2, , M

Umps(n),i((n), Ps(n),i)

= (1 + (n)i )Us(n),i(Ps(n),i)

Similar to our previous problem


Power scheduling


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Power scheduling

(Contd)The utility function (i) is adjusted to guarantee the minimumperformance constraint by using a stochastic subgradient

algorithm

(n+1)i = [

(n)i

(n)v(n)i ]

+, i

v(n)i = Us(n),i(P

s(n),i

((n))) Ci

stochastic subgradient of the dual

Ps(n),i

((n)) is power allocation of user i in time-slot n

(n) converges to that solves the dual problem



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FeasibilityAlways satisfies the constraint on total transmission power limit

If

Q

M 0 as M , theniHU

i Ci

M 0 as M

Q: expected number of users with the same channelconditions

H: set of users whose performance constraints are notsatisfied

Ui : expected utility of user i in our power scheduling inour power scheduling

Asymptotically feasible on average

Increase in the randomness of the system improves thedegree of users satisfaction



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OptimalityIfM

i=1 Uoi and

QM

0 as M , then

Mi=1 U

iM

i=1 Uoi

1 and 0 as M

Uo

i : expected utility of user i in optimal power schedulingIf the above conditions are satisfied and Ui Ci, i, then

Mi=1 U

i

Mi=1 U

oi 1 as M

Asymptotically optimal



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Numerical resultsThe same cellular model as our previous problem

Four users and each user i

same sigmoid utility function Ui (Ui(0) = 0 and Ui() = 1)

same performance constraint Ci = 0.59

distance from the base-station to user i: di

d1 < d2 < d3 < d4Performance comparison with

Non-opportunistic scheduling

Greedy scheduling



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Numerical results (contd)Comparison of average utilities (104 time-slots)

User 1 2 3 4 TotalNon-opportunistic 0.590 0.590 0.590 0.590 2.360

Greedy 0.973 0.964 0.796 0.168 2.901

Our opportunistic 0.951 0.736 0.591 0.591 2.869



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2 4 6 8 101

1.5

2

2.5

3

3.5

4

Ratioofaver

agetotalsystemutilities

Ratio of average total system utility of our opportunistic powerscheduling to that of non-opportunistic power scheduling: standard deviation of each users channel condition



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ConclusionUtility framework

suitable for resource allocation with multi-media and data

services

a useful tool for resource allocation in the nextgenerations of communication networks

non-convex optimization problems in many cases

Dual approach provides

efficient solution in many cases

simple algorithm that can be easily implemented with a

(distributed) network protocol



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Conclusion (contd)In wireless systems

Single server transmission is optimal only when all users havehigh data rate

In general, need dynamic multiple access (dynamicmulti-server system)

Trade-off between efficiency and fairnessOpportunistic scheduling achieves both of them

Randomness of the system could be beneficial to efficient andfair resource allocation, if appropriately exploited



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Conclusion (contd)Other problems

Pricing based base-station assignment

considers both transmission environment of the user andcongestion level of the base-station

Congestion control on the Internet

algorithms for concave utility functions cause instabilityand congestion in the presence of real-time services withnon-concave utility functions

self-regulating property stabilizes the system and

alleviates congestion



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Future workScheduling considering

user dynamics

non-stationary environment

delay or short-term fairness constraints

Resource allocation considering upper layer protocols (e.g.,

TCP)Resource allocation for uplink and multi-cellular system


Documents

Nonconvex Wireless