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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: mazum@ecn.purdue.edu
Joint Work with Prof. Ness B. Shroff and Jang-Won Lee
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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
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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
Selection ratio of class 2 0.568 0.392 0.020
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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Numerical results (contd)
b1 2.5 3.5 4.5
Selection ratio of class 1 0.566 0.391 0.230
Selection ratio of class 2 0.288 0.389 0.484
Utility (Our)/Utility (Single) 4.196 3.852 3.525
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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Numerical results (contd)
Ratio of class 1 0.4 0.6 0.8
Selection ratio of class 1 0.849 0.653 0.499
Selection ratio of class 2 0.004 0 0
Utility (Our)/Utility (Single) 3.409 3.912 3.980
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
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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
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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
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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
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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
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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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Numerical results (contd)
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
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