Delay and Power-Optimal Control in Multi-Class Queueing Systems

0.51 MB·English

by  Chih-ping Li

1
Delay and Power-Optimal Control in Multi-Class
Queueing Systems
Chih-ping Li, Student Member, IEEE and Michael J. Neely, Senior Member, IEEE
Abstract—We consider optimizing average queueing delay 3) Under dynamic power allocation, minimizing average
and average power consumption in a nonpreemptive multi-class power consumption subject to delay constraints W ≤
n
M/G/1 queue with dynamic power control that affects instan-
d for all classes n.
1 taneous service rates. Four problems are studied: (1) satisfying n
4) Under dynamic power allocation, minimizing a separa-
1 per-class average delay constraints; (2) minimizing a separable
0 convex function of average delays subject to per-class delay bleconvexfunction(cid:80)Nn=1fn(Wn)ofaveragequeueing
2 constraints; (3) minimizing average power consumption subject delays(Wn)Nn=1 subjecttoanaveragepowerconstraint.
n toper-classdelayconstraints;(4)minimizingaseparableconvex These problems are presented with increasing complexity
functionofaveragedelayssubjecttoanaveragepowerconstraint.
a for the readers to gradually familiarize themselves with the
J Combining an achievable region approach in queueing systems
methodology we use to attack these problems.
and the Lyapunov optimization theory suitable for optimizing
4 dynamic systems with time average constraints, we propose a Each of the above problems is highly nontrivial, thus novel
1
unifiedframeworktosolvetheaboveproblems.Thesolutionsare yet simple approaches are needed. This paper provides such a

variants of dynamic cµ rules, and implement weighted priority frameworkbyconnectingtwopowerfulstochasticoptimization
]
C policies in every busy period, where weights are determined by theories:Theachievableregionapproachinqueueingsystems,
past queueing delays in all job classes. Our solutions require
O andtheLyapunovoptimizationtheoryinwirelessnetworks.In
limited statistical knowledge of arrivals and service times, and
. nostatisticalknowledgeisneededinthefirstproblem.Overall,we queueing systems, the achievable region approach that treats
h
provideanewsetoftoolsforstochasticoptimizationandcontrol optimal control problems as mathematical programming ones
t
a overmulti-classqueueingsystemswithtimeaverageconstraints. has been fruitful; see [1]–[4] for a detailed survey. In a
m
nonpreemptivemulti-classM/G/1queue,itisknownthatthe
[ collectionofallfeasibleaveragequeueingdelayvectorsforma

special polytope (a base of a polymatroid) with vertices being
2 I. INTRODUCTION
v the performance vectors of strict priority policies ([5], see
Stochastic scheduling over multi-class queueing systems
8 SectionIIIformoredetails).Asaresult,everyfeasibleaverage
has important applications such as CPU scheduling, request
7 queueingdelayvectorisattainablebyarandomizationofstrict
4 processing in web servers, and QoS provisioning to different
priority policies. Such randomization can be implemented in
2 types of traffic in a telecommunication network. In these sys-
a framed-based style, where a priority ordering is randomly
1. tems,powermanagementisincreasinglyimportantduetotheir deployed in every busy period using a probability distribution
0 massive energy consumption. To study this problem, in this
that is used in all busy periods (see Lemma 1 in Section III).
1 paperweconsiderasingle-servermulti-classqueueingsystem
Thisviewofthedelayperformanceregionisusefulinthefirst
1
whose instantaneous service rate is controllable by dynamic
: two delay control problems.
v power allocations. This is modeled as a nonpreemptive multi-
Inadditiontoqueueingdelay,whendynamicpowercontrol
i class M/G/1 queue with N job classes {1,...,N}, and the
X is part of the decision space, it is natural to consider dynamic
goal is to optimize average queueing delays of all job classes
r policies that allocate a fixed power in every busy period.
a andaveragepowerconsumptioninthisqueueingnetwork.We
The resulting joint power and delay performance region is
consider four delay and power control problems:
thenspannedbyframe-basedrandomizationsofpowercontrol
1) Designing a policy that yields average queueing delay and strict priority policies. We treat the last two delay and
WnofclassnsatisfyingWn ≤dnforallclasses,where power control problems as stochastic optimization over such
{d1,...,dN} are given feasible delay bounds. Here we a performance region (see Section VI-A for an example).
assume a fixed power allocation and no power control. Withtheabovecharacterizationofperformanceregions,we
2) Minimizingaseparableconvexfunction(cid:80)Nn=1fn(Wn) solve the four control problems using Lyapunov optimization
of average queueing delays (Wn)Nn=1 subject to delay theory. This theory is originally developed for stochastic
constraintsWn ≤dn forallclassesn;assumingafixed optimal control over time-slotted wireless networks [6], [7],
power allocation and no power control. later extended by [8], [9] that allow optimizing various per-
formanceobjectivessuchasaveragepower[10]orthroughput
Chih-ping Li (web: http://www-scf.usc.edu/∼chihpinl) and Michael J.
utility [11], and recently generalized to optimize dynamic
Neely (web: http://www-rcf.usc.edu/∼mjneely) are with the Department of
Electrical Engineering, University of Southern California, Los Angeles, CA systemsthathavearenewalstructure[12]–[15].TheLyapunov
90089,USA. optimization theory transforms time average constraints into
Thismaterialissupportedinpartbyoneormoreofthefollowing:theNSF
virtual queues that need to be stabilized. Using a Lyapunov
Career grant CCF-0747525, the Network Science Collaborative Technology
AlliancesponsoredbytheU.S.ArmyResearchLaboratory. drift argument, we construct frame-based policies to solve
2
the four control problems. The resulting policy is a sequence Minimizing a convex holding cost in a single-server multi-
of base policies implemented frame by frame, where the class queue is formulated as a restless bandit problem in [19],
collection of all base policies span the performance region [20], and Whittle’s index policies [21] are constructed as a
through time sharing or randomization. The base policy used heuristic solution. Work [22] proposes a generalized cµ rule
in each frame is chosen by minimizing a ratio of an expected to maximize a convex holding cost over a finite horizon in
“drift plus penalty” sum over the expected frame size, where a multi-class queue, and shows it is asymptotically optimal
theratioisafunctionofpastqueueingdelaysinalljobclasses. under heavy traffic. This paper provides a dynamic control
Inthispaperthebasepoliciesarenonpreemptivestrictpriority algorithmfortheminimizationofconvexfunctionsofaverage
policies with deterministic power allocations. delays. Especially, we consider additional time average power
Our methodology is as follows. By characterizing the per- and delay constraints, and our solutions require limited statis-
formance region using the collection of all randomizations of tical knowledge and have provable near-optimal performance.
basepolicies,foreachcontrolproblem,thereexistsanoptimal Thispaperalsoappliestopower-awareschedulingproblems
random mixture of base policies that solves the problem. in computer systems. These problems are widely studied in
Although the probability distribution that defines the optimal different contexts, where two main analytical tools are com-
random mixture is unknown, we construct a dynamic policy petitive analysis [23]–[27] and M/G/1-type queueing theory
usingLyapunovoptimizationtheory.Thispolicymakesgreedy (see[28]andreferencestherein),bothusedtooptimizemetrics
decisions in every frame, stabilizes all virtual queues (thus suchasaweightedsumofaveragepoweranddelay.Thispaper
satisfyingalltimeaverageconstraints),andyieldsnear-optimal presents a fundamentally different approach for more directed
performance. The existence of the optimal randomized policy controlover averagepower anddelays, andconsiders amulti-
is essential to prove these results. class setup with time average constraints.
In our policies for the four control problems, requests of Intherestofthepaper,thedetailedqueueingmodelisgiven
different classes are prioritized by a dynamic cµ rule [1] in Section II, followed by a summary of useful M/G/1 prop-
which,ineverybusyperiod,assignsprioritiesinthedecreasing erties in Section III. The four delay-power control problems
order of weights associated with each class. The weights of are solved in Section IV-VII, followed by simulation results.
all classes are updated at the end of every busy period by
simple queue-like rules (so that different priorities may be
II. QUEUEINGMODEL
assigned in different busy periods), which capture the running
We only consider queueing delay, not system delay (queue-
difference between the current and the desired performance.
ing plus service) in this paper. System delay can be eas-
The dynamic cµ rule in the first problem does not require any
ily incorporated since, in a nonpreemptive system, average
statistical knowledge of arrivals and service times. The policy
queueing and system delay differ only by a constant (the
for the second problem requires only the mean but not higher
average service time). We will use “delay” and “queueing
momentsofarrivalsandservicetimes.Inthelasttwoproblems
delay” interchangeably in the rest of the paper.
with dynamic power control, beside the dynamic cµ rules, a
Consider a single-server queueing system processing jobs
power level is allocated in every busy period by optimizing a
categorized into N classes. In each class n ∈ {1,2,...,N},
weighted sum of power and power-dependent average delays.
jobsarriveasaPoissonprocesswithrateλ .Eachclassnjob
The policies for the third and the last problem require the n
has size S . We assume S is i.i.d. in each class, independent
mean and the first two moments of arrivals and service times, n n
across classes, and that the first four moments of S are finite
respectively, because of dynamic power allocations. n
for all classes n. The system processes arrivals nonpreemp-
Ineachofthelastthreeproblems,ourpoliciesyieldperfor-
tively with instantaneous service rate µ(P(t)), where µ(·)
mance that is at most O(1/V) away from the optimal, where
is a concave, continuous, and nondecreasing function of the
V > 0 is a control parameter that can be chosen sufficiently
allocatedpowerP(t)(theconcavityofrate-powerrelationship
largetoyieldnear-optimalperformance.Thetradeoffofchoos-
isobservedincomputersystems[28]–[30]).Withineachclass,
ing large V values is the amount of time required to meet
arrivals are served in a first-in-first-out fashion. We consider
the time average constraints. In this paper we also propose
a frame-based system, where each frame consists of an idle
a proportional delay fairness criterion, in the same spirit
period and the following busy period. Let t be the start of
as the well-known rate proportional fairness [16] or utility k
the kth frame for each k ∈ Z+; the kth frame is [t ,t ).
proportional fairness [17], and show that the corresponding k k+1
Definet =0andassumethesystemisinitiallyempty.Define
delay objective functions are quadratic. Overall, since our 0
T (cid:44) t −t as the size of frame k. Let A denote the
policiesusedynamiccµruleswithweightsofsimpleupdates, k k+1 k n,k
set of class n arrivals in frame k. For each job i ∈ A , let
andrequirelimitedstatisticalknowledge,theyscalegracefully n,k
W(i) denote its queueing delay.
with the number of job classes and are suitable for online n,k
implementation. The control over this queueing system is power allocations
Intheliterature,work[18]characterizesmulti-classG/M/c and job scheduling across all classes. We restrict to the
queues that have polymatroidal performance regions, and followingframe-basedpoliciesthatarebothcausalandwork-
provides two numerical methods to minimize a separable conserving:1
convex function of average delays as an unconstrained static
1Causalitymeansthateverycontroldecisiondependsonlyonthecurrent
optimization problem. But in [18] it is unclear how to control
andpaststatesofthesystem;work-conservingmeansthattheserverisnever
the queueing system to achieve the optimal performance. idlewhenthereisstillworktodo.
3
In every frame k ∈ Z+, use a fixed power level which shows W is indeed the limiting average delay.2 The
n
P ∈[P ,P ]andanonpreemptivestrictpriority definition in (1) replaces lim by limsup to guarantee it is
k min max
policy π(k) for the duration of the busy period in well-defined.
that frame. The decisions are possibly random.
Inthesepolicies,P denotesthemaximumpowerallocation. III. PRELIMINARIES
max
We assume P is finite, but sufficiently large to ensure This section summarizes useful properties of a nonpreemp-
max
feasibility of the desired delay constraints. The minimum tivemulti-classM/G/1queue.Hereweassumeafixedpower
power P is chosen to be large enough so that the queue allocation P and a fixed service rate µ(P) (this is extended
min
is stable even if power P is used for all time. In particular, in Section VI). Let X (cid:44) S /µ(P) be the service time of a
min n n
for stability we need class n job. Define ρ (cid:44)λ E[X ]. Fix an arrival rate vector
n n n
(λ )N satisfying (cid:80)N ρ <1; the rate vector (λ )N is
(cid:88)N λ E[Sn] <1⇒µ(P )> (cid:88)N λ E[S ]. supnponr=ta1ble in the quenu=ei1ngnnetwork. n n=1
n=1 nµ(Pmin) min n=1 n n For each k ∈ Z+, let Ik and Bk denote the kth idle and
busy period, respectively; the frame size T = I +B . The
The strict priority rule π(k) = (π (k))N is represented by k k k
n n=1 distributionofB (andT )isfixedunderanywork-conserving
k k
a permutation of {1,...,N}, where class π (k) gets the nth
n policy,sincethesamplepathofunfinishedworkinthesystem
highest priority.
is independent of scheduling policies. Due to the memoryless
The motivation of focusing on the above frame-based poli- property of Poisson arrivals, we have E[I ] = 1/((cid:80)N λ )
cies is to simplify the control of the queueing system to k n=1 n
for all k. For the same reason, the system renews itself at
achieve complex performance objectives. Simulations in [31],
the start of each frame. Consequently, the frame size T , busy
k
however, suggest that this method may incur higher variance
periodB ,andtheper-framejobarrivals|A |ofclassn,are
k n,k
in performance than policies that take control actions based
alli.i.d.overk.Usingrenewalrewardtheory[32]withrenewal
on job occupancies in the queue. Yet, job-level scheduling
epochs defined at frame boundaries {t }∞ , we have:
seems difficult to attack problems considered in this paper. It k k=0
may involve solving high-dimensional (partially observable) E[T ]= E[Ik] = 1 (3)
Markovdecisionprocesseswithtimeaveragepoweranddelay k 1−(cid:80)N ρ (1−(cid:80)N ρ )(cid:80)N λ
n=1 n n=1 n n=1 n
constraints and convex holding costs. E[|A |]=λ E[T ], ∀n∈{1,...,N}, ∀k ∈Z+. (4)
n,k n k
It is useful to consider the randomized policy π that is
rand
A. Definition of Average Delay
definedbyagivenprobabilitydistributionoverallpossibleN!
The average delay under policies we propose later may not priority orderings. Specifically, policy π randomly selects
rand
have well-defined limits. Thus, inspired by [13], we define prioritiesatthebeginningofeverynewframeaccordingtothis
(cid:104) (cid:105) distribution,andimplementsthecorrespondingnonpreemptive
E (cid:80)K−1(cid:80) W(i)
W (cid:44)limsup k=0 i∈An,k n,k (1) priority rule for the duration of the frame. Again by renewal
n K→∞ E(cid:104)(cid:80)K−1|A |(cid:105) rewardtheory,theaveragequeueingdelays(Wn)Nn=1rendered
k=0 n,k by a π policy satisfy in each frame k ∈Z+:
rand
as the average delay of class n ∈ {1,...,N}, where |A |  
is the number of class n arrivals during frame k. We onn,lky E (cid:88) Wn(i,k)=E(cid:20)(cid:90) tk+1Qn(t)dt(cid:21)=λnWnE[Tk],
consider delay sampled at frame boundaries for simplicity. To i∈An,k tk
verify (1), note that the running average delay of class n jobs (5)
up to time tK is equal to where we recall that W(i) represents only the queueing delay
n,k
(cid:80)K−1(cid:80) W(i) 1 (cid:80)K−1(cid:80) W(i) (not including service time), and Qn(t) denotes the number
k=0 i∈An,k n,k = K k=0 i∈An,k n,k. of class n jobs waiting in the queue (not including that in the
(cid:80)Kk=−01|An,k| K1 (cid:80)Kk=−01|An,k| server) at time t.
Next we summarize useful properties of the performance
Define
region of average queueing delay vectors (W )N in a
n n=1
K−1 K−1
wav (cid:44) lim 1 (cid:88) (cid:88) W(i), aav (cid:44) lim 1 (cid:88) |A |. nonpreemptive multi-class M/G/1 queue. For these results
n K→∞K n,k n K→∞K n,k we refer readers to [1], [5], [33] for a detailed introduction.
k=0 i∈An,k k=0 Define the value x (cid:44)ρ W for each class n∈{1,...,N},
n n n
Ifbothlimitswavandaavexist,theratiowav/aavisthelimiting anddenotebyΩtheperformanceregionofthevector(x )N .
n n n n n n=1
average delay for class n. In this case, we get The set Ω is a special polytope called (a base of) a polyma-
(cid:104) (cid:105) troid [34]. An important property of the polymatroid Ω is:
lim E 1 (cid:80)K−1(cid:80) W(i)
Wn = K→∞ K (cid:104) k=0 i∈An,k (cid:105)n,k 2Thesecondequalityin(2),wherewepassthelimitintotheexpectation,
(cid:104) limK→∞E K1 (cid:80)Kk=−01|An,k| (cid:105) (2) cstaantebdeapsrfoovlelodwbsy.Laetge{nXerna}li∞nze=d1Laenbdes{gYune}’s∞nd=o1mbineattwedocsoeqnuveerngceenscoeftrhaenodroemm
= E limK→∞ K1 (cid:80)Kk=−01(cid:80)i∈An,kWn(i,k) = wnav, v(2a)riaFbolressosmucehrathnadto:m(1v)ar0iab≤les|XXna|n≤d YY,nXwnit→h pXrobaabnidlitYyn1→forYalwl inth;
E(cid:104)limK→∞ K1 (cid:80)Kk=−01|An,k|(cid:105) aanv plirmobna→bi∞lityE1[X; (n3])=limEn[→X∞].TEh[eYdne]ta=ilsEa[rYe]om<it∞ted.fTohrebnreEvi[tXy.] is finite and
4
(1) Each vertex of Ω is the performance vector of a strict discrete-time virtual delay queue {Z }∞ where Z is
n,k k=0 n,k+1
nonpreemptive priority rule; (2) Conversely, the performance updated at frame boundary t following the equation
k+1
vector of each strict nonpreemptive priority rule is a vertex  
ovefrΩtic.eIsnoofthΩeranwdotrhdes,sethteorfesitsricatonnoen-ptore-oemneptmivaepppriniogribtyetrwueleesn. Zn,k+1 =maxZn,k+ (cid:88) (cid:16)Wn(i,k) −dn(cid:17), 0. (8)
As a result, every feasible performance vector (x )N ∈ Ω, i∈An,k
orequivalentlyeveryfeasiblequeueingdelayvectorn(nW=n1)Nn=1, Assume Zn,0 = 0 for all n. In (8), the delays Wn(i,k) and
isattainedbyarandomizationofstrictnonpreemptivepriority constant dn can viewed as arrivals and service of the queue
policies.Forcompleteness,weformalizethelastknownresult {Zn,k}∞k=0, respectively. If this queue is stabilized, we know
in the next lemma. that the average arrival rate to the queue (being the per-frame
average sum of class n delays (cid:80) W(i)) is less than or
Lemma 1. In a nonpreemptive multi-class M/G/1 queue, n∈An,k n,k
equaltotheaverageservicerate(beingthevalued multiplied
n
define
by the average number of class n arrivals per frame), from
W (cid:44)(cid:8)(Wn)Nn=1 |(ρnWn)Nn=1 ∈Ω(cid:9) which we infer Wn ≤dn. This is formalized below.
Definition 1. We say queue {Z }∞ is mean rate stable if
as the performance region [5] of average queueing delays. n,k k=0
lim E[Z ]/K =0.
Then: K→∞ n,K
1) The performance vector (Wn)Nn=1 of each frame-based Lemma 3. If queue {Zn,k}∞k=0 is mean rate stable, then
randomized policy πrand is in the delay region W. Wn ≤dn.
2) Conversely, every vector (W )N in the delay region
n n=1 Proof of Lemma 3: From (8) we get
W is the performance vector of a π policy.
rand Z ≥Z −d |A |+ (cid:88) W(i).
n,k+1 n,k n n,k n,k
Proof of Lemma 1: Given in Appendix A.
i∈An,k
Optimizing a linear function over the polymatroidal region
Summingtheaboveoverk ∈{0,...,K−1},usingZ =0,
n,0
Ω will be useful. The solution is the following cµ rule:
and taking expectation yields
Lemma 2 (The cµ rule [1], [33]). In a nonpreemptive multi- (cid:34)K−1 (cid:35) K−1 
class M/G/1 queue, define xn (cid:44) ρnWn and consider the E[Zn,K]≥−dnE (cid:88) |An,k| +E(cid:88) (cid:88) Wn(i,k).
linear program:
k=0 k=0 i∈An,k
(cid:104) (cid:105)
N Dividing the above by E (cid:80)K−1|A | yields
(cid:88) k=0 n,k
minimize: c x (6)
n n (cid:104) (cid:105)
subject to: (nx=n1)Nn=1 ∈Ω (7) E(cid:104)(cid:80)EK[Z−n1,|KA] |(cid:105) ≥ E (cid:80)EKk(cid:104)=(cid:80)−01K(cid:80)−i1∈|AAn,kW|(cid:105)n(i,k) −dn.
k=0 n,k k=0 n,k
wherec arenonnegativeconstants.Weassume(cid:80)N ρ <1
forstabnility,andthatsecondmomentsE(cid:2)X2(cid:3)ofsenr=vi1centimes Taking a limsup as K →∞ and using (1) yields
n E[Z ] K
arefiniteforallclassesn.Theoptimalsolutionto(6)-(7)isa W ≤d +limsup n,K .
strict nonpreemptive priority policy that assigns priorities in n n K→∞ K E(cid:104)(cid:80)K−1|A |(cid:105)
k=0 n,k
the decreasing order of c . That says, if c ≥c ≥...≥c ,
n 1 2 N Using E[|A |]=λ E[T ]≥λ E[I ]=λ E[I ], we get
then class 1 gets the highest priority, class 2 gets the second n,k n k n k n 0
hqiugehueesitngprdioerlaityy,Wan∗ndosfocolans.sInntihsis case, the optimal average Wn ≤dn+ λnE1[I0]Kl→im∞E[ZKn,K] =dn
by mean rate stability of Z .
∗ R n,k
W = ,
n (1−(cid:80)n−1ρ )(1−(cid:80)n ρ ) A. Delay Feasible Policy
k=0 k k=0 k
The following policy stabilizes every {Z }∞ queue in
where ρ (cid:44)0 and R(cid:44) 1(cid:80)N λ E(cid:2)X2(cid:3). n,k k=0
0 2 n=1 n n the mean rate stable sense and thus achieves Wn ≤dn for all
classes n.
Delay Feasible (DelayFeas) Policy:
IV. ACHIEVINGDELAYCONSTRAINTS
• In every frame k ∈ Z+, update Zn,k by (8) and serve
Thefirstproblemweconsideristoconstructaframe-based jobs using nonpreemptive strict priorities assigned in the
policy that yields average delays satisfying W ≤ d for all decreasing order of Z ; ties are broken arbitrarily.
n n n,k
classesn∈{1,...,N},wheredn >0aregivenconstants.We We note that the DelayFeas policy does not require any statis-
assume a fixed power allocation and that the delay constraints tical knowledge of job arrivals and service times. Intuitively,
are feasible. each Z queue tracks the amount of past queueing delays
n,k
Our solution relies on tracking the running difference be- in class n exceeding the desired delay bound d (see (8)),
n
tween past queueing delays for each class n and the desired and the DelayFeas policy gives priorities to classes that more
delay bound d . For each class n∈{1,...,N}, we define a severely violate their delay constraints.
n
5
B. Motivation of the DelayFeas Policy busyperiod),thevalueE[T ]isthesameforalljobscheduling
k
The structure of the DelayFeas policy follows a Lyapunov policies. Then our desired policy, in every frame k, chooses a
dcorinftstaarngtusmθennt>.D0efifnoervaellctcolraZsskes(cid:44)n(,Zwne,k)dNne=fi1n.eFtohresoqmuaedfirantiitce joovberscahleldufeliansgibtloemdienliamyizveecthtoerms e(tWricn(cid:80),k)NnNn==11θ.nIZf nw,keλnchWoons,ke
Lyapunov function θn = E[Xn] for all classes n,3 the desired policy minimizes
(cid:80)N Z λ E[X ]W ineveryframek.Fromlemma2,
N n=1 n,k n n n,k
1 (cid:88) this is achieved by the priority service rule defined by the
L(Z )(cid:44) θ Z2
k 2 n n,k DelayFeas policy.
n=1
asaweightedscalarmeasureofqueuesizes(Z )N .Define
n,k n=1
the one-frame Lyapunov drift
C. Performance of the DelayFeas Policy
∆(Z )(cid:44)E[L(Z )−L(Z )|Z ]
k k+1 k k Theorem 1. For every collection of feasible delay bounds
as the conditional expected difference of L(Zk) over a frame. {d1,...,dN}, the DelayFeas policy yields average delays
Taking square of (8) and using (max[a,0])2 ≤a2 yields satisfying Wn ≤dn for all classes n∈{1,...,N}.
 2 Proof of Theorem 1: It suffices to show that the
Zn2,k+1 ≤Zn,k+ (cid:88) (cid:16)Wn(i,k) −dn(cid:17) . (9) DelayFeaspolicyyieldsmeanratestabilityforallZn,k queues
byLemma3.ByLemma1,thereexistsarandomizedpriority
i∈An,k
policy π∗ (introduced in Section III) that yields average
Multiplying (9) by θ /2, summing over n∈{1,...,N}, and ra∗nd ∗
n delays W satisfying W ≤ d for all classes n. Since the
taking conditional expectation on Z , we get n n n
k DelayFeaspolicyminimizesthelasttermof(12)ineachframe
 2  (underθ =E[X ]foralln),comparingtheDelayFeaspolicy
N n n
∆(Zk)≤ 12 (cid:88)θnE (cid:88) (cid:16)Wn(i,k) −dn(cid:17) |Zk with the πr∗and policy yields, in every frame k,
n=1 i∈An,k N N
+(cid:88)N θnZn,kE (cid:88) (cid:16)Wn(i,k) −dn(cid:17)|Zk. n(cid:88)=1θnZn,kλnWDn,eklayFeas ≤n(cid:88)=1θnZn,kλnW∗n.
n=1 i∈An,k Itfollowsthat(12)undertheDelayFeaspolicyisfurtherupper
(10)
bounded by
Lemma7inAppendixBshowsthatthesecondtermof(10)is
bounded by a finite constant C >0. It leads to the following N
∆(Z )≤C+E[T ](cid:88)θ Z λ (WDelayFeas−d )
Lyapunov drift inequality: k k n n,k n n,k n
  n=1
∆(Zk)≤C+(cid:88)N θnZn,kE (cid:88) (cid:16)Wn(i,k) −dn(cid:17)|Zk. ≤C+E[Tk](cid:88)N θnZn,kλn(W∗n−dn)≤C.
n=1 i∈An,k n=1
(11)
Over all frame-based policies, we are interested in the one Taking expectation, summing over k ∈ {0,...,K −1}, and
that, in each frame k after observing Z , minimizes the right noting L(Z )=0, we get
k 0
side of (11). Recall that our policy on frame k chooses which
nonpreemptiveprioritiestouseduringtheframe.Toshowthat E[L(Z )]= 1 (cid:88)N θ E(cid:2)Z2 (cid:3)≤KC.
this is exactly the DelayFeas policy, we simplify (11). Under K 2 n n,K
a frame-based policy, we have by renewal reward theory n=1
  It follows that E(cid:2)Z2 (cid:3) ≤ 2KC/θ for all classes n. Since
E (cid:88) Wn(i,k) |Zk=λnWn,kE[Tk], Zn,K ≥0, we get n,K n
i∈An,k (cid:114)
(cid:104) (cid:105) (cid:112)
where Wn,k denotes the long-term average delay of class n 0≤E[Zn,K]≤ E Zn2,K ≤ 2KC/θn.
if the control in frame k is repeated in every frame. Together
with E[|An,k|]=λnE[Tk], inequality (11) is re-written as Dividing the above by K and passing K →∞ yields
(cid:32) N (cid:33)
(cid:88) E[Z ]
∆(Zk)≤ C−E[Tk] θnZn,kλndn lim n,K =0, ∀n∈{1,...,N},
K→∞ K
n=1 (12)
N
+E[T ](cid:88)θ Z λ W . and all Zn,k queues are mean rate stable.
k n n,k n n,k
n=1
Because in this section we do not have dynamic power 3WenotethatthemeanservicetimeE[Xn]asavalueofθnisonlyneeded
in the arguments constructing the DelayFeas policy. The DelayFeas policy
allocation (so that power is fixed to the same value in every itselfdoesnotneedtheknowledgeofE[Xn].
6
V. MINIMIZINGDELAYPENALTYFUNCTIONS (as seen in Section IV), the Yn,k queues are useful to achieve
the optimal delay vector (W∗)N .
Generalizing the first delay feasibility problem, next we n n=1
Delay Fairness (DelayFair) Policy:
optimize a separable penalty function of average delays. For
each class n, let fn(·) be a nondecreasing, nonnegative, con- 1) In the kth frame for each k ∈ Z+, after observing Zk
tinuous, and convex function of average delay Wn. Consider and Yk, use nonpreemptive strict priorities assigned in
the constrained penalty minimization problem the decreasing order of (Zn,k + Yn,k)/E[Sn], where
E[S ] is the mean size of a class n job. Ties are broken
n
N
(cid:88) arbitrarily.
minimize: f (W ) (13)
n n 2) Attheendofthekthframe,computeZ andY
n,k+1 n,k+1
n=1 forallclassesnby(8)and(17),respectively,wherer
n,k
subject to: W ≤d , ∀n∈{1,...,N}. (14)
n n is the solution to the convex program:
We assume that a constant power is allocated in all frames, minimize: Vf (r )−Y λ r
n n,k n,k n n,k
and that constraints (14) are feasible. The goal is to construct
a frame-based policy that solves (13)-(14). Let (W∗)N be subject to: 0≤rn,k ≤dn,
n n=1
the optimal solution to (13)-(14), attained by a randomized where V >0 is a predefined control parameter.
priority policy π∗ (by Lemma 1). While the DelayFeas policy in Section IV does not require
rand
any statistical knowledge of arrivals and service times, the
DelayFair policy needs the mean but not higher moments of
A. Delay Proportional Fairness
arrivals and service times for all classes n.
One interesting penalty function is the one that attains
Intheexampleofdelayproportionalfairnesswithquadratic
proportional fairness. We say a delay vector (W∗n)Nn=1 is penalty functions f (W ) = 1c W2 for all classes n, the
delay proportional fair if it is optimal under the quadratic n n 2 n n
second step of the DelayFair policy solves:
penalty function f (W ) = 1c W2 for each class n, where
n n 2 n n (cid:18)1 (cid:19)
cn > 0 are given constants. The intuition is two-fold. First, minimize: 2V cn rn2,k−Yn,kλnrn,k
underthequadraticpenaltyfunctions,anyfeasibledelayvector
(Wn)Nn=1 necessarily satisfies subject to: 0≤rn,k ≤dn.
(cid:104) (cid:105)
(cid:88)N f(cid:48)(W∗)(W −W∗)= (cid:88)c (W −W∗)W∗ ≥0, The solution is rn∗,k =min dn,YVn,kcλnn .
n n n n n n n n
n=1 n=1
(15) C. Motivation of the DelayFair Policy
which is analogous to the rate proportional fair [16] criterion The DelayFair policy follows a Lyapunov drift argument
similar to that in Section IV. Define Z (cid:44) (Z )N and
n(cid:88)N=1cnxnx−∗nx∗n ≤0, (16) Y21k(cid:80)(cid:44)Nn=(Y1(nZ,kn2),Nnk=+1.YDn2e,kfi)neantdhethLeyoapnue-nforavmfueknLcytiaopnunLno(,vkZdknr,=iYf1tk)(cid:44)
where (x )N is any feasible rate vector and (x∗)N is ∆(Z ,Y )(cid:44)E[L(Z ,Y )−L(Z ,Y )|Z ,Y ].
n n=1 n n=1 k k k+1 k+1 k k k k
the optimal rate vector. Second, rate proportional fairness,
Taking square of (17) yields
when deviating from the optimal solution, yields the aggre-
gate change of proportional rates less than or equal to zero  2
(see (16)); it penalizes large rates to increase. When delay Yn2,k+1 ≤Yn,k+ (cid:88) (cid:16)Wn(i,k) −rn,k(cid:17) . (18)
proportional fairness deviates from the optimal solution, the
i∈An,k
aggregatechangeofproportionaldelaysisalwaysnonnegative
Summing (9) and (18) over all classes n ∈ {1,...,N},
(see (15)); small delays are penalized for trying to improve.
dividing the result by 2, and taking conditional expectation
on Z and Y , we get
k k
B. Delay Fairness Policy
N
(cid:88)
In addition to having the {Zn,k}∞k=0 queues updated by (8) ∆(Zk,Yk)≤C− Zn,kdnE[|An,k||Zk,Yk]
for all classes n, we setup new discrete-time virtual queues n=1
{Y }∞ forallclassesn,whereY isupdatedatframe N
bounn,kdakr=y0t as: n,k+1 −(cid:88)Yn,kE[rn,k|An,k||Zk,Yk] (19)
k+1
n=1
   
Yn,k+1 =maxYn,k+ (cid:88) (cid:16)Wn(i,k) −rn,k(cid:17),0, (17) +(cid:88)N (Zn,k+Yn,k)E (cid:88) Wn(i,k) |Zk,Yk,
i∈An,k n=1 i∈An,k
where r ∈[0,d ] are auxiliary variables chosen at time t where C > 0 is a finite constant, different from that used
n,k n k
independent of frame size T and the number |A | of class in Section IV-B, upper bounding the sum of all (Z ,Y )-
k n,k k k
narrivalsinframek.AssumeY =0foralln.Whereasthe independent terms. This constant exists using arguments sim-
n,0
Z queues are useful to enforce delay constraints W ≤d ilar to those in Lemma 7 of Appendix B.
n,k n n
7
N
V (cid:80)AdNdingEt[of b(orth )siTdes|Zof ,(Y19)],thwehewreeigVhte>d p0eniasltya tperrme- +E[Tk](cid:88)(cid:16)Vfn(W∗n)−Yn,kλnW∗n(cid:17)
n=1 n n,k k k k n=1
defined control parameter, and evaluating the result under a
N
frame-based policy (similar as the analysis in Section IV-C), ≤C+VE[T ](cid:88)f (W∗). (21)
we get the following Lyapunov drift plus penalty inequality: k n n
n=1
(cid:88)N Removing the second term of (21) yields
∆(Z ,Y )+V E[f (r )T |Z ,Y ]
k k n n,k k k k
N
(cid:32) nN=1 (cid:33) ∆(Zk,Yk)≤C+VE[Tk](cid:88)fn(W∗n)≤C+VD, (22)
(cid:88)
≤ C−E[T ] Z λ d n=1
k n,k n n
n=1 (20) where D (cid:44)E[Tk](cid:80)Nn=1fn(W∗n) is a finite constant. Taking
(cid:88)N expectation of (22), summing over k ∈ {0,...,K −1}, and
+E[Tk] E[V fn(rn,k)−Yn,kλnrn,k |Zk,Yk] noting L(Z ,Y )=0 yields E[L(Z ,Y )]≤K(C+VD).
0 0 K K
n=1 It follows that, for each class n queue {Z }∞ , we have
N n,k k=0
+E[Tk]n(cid:88)=1(Zn,k+Yn,k)λnWn,k. 0≤ E[Zn,K] ≤(cid:118)(cid:117)(cid:117)(cid:116)E(cid:104)Zn2,K(cid:105)
Weareinterestedinminimizingtherightsideof(20)inevery K K2 (23)
frame k over all frame-based policies and (possibly random) (cid:114)2E[L(Z ,Y )] (cid:114)2C 2VD
choices of r . Recall that in this section a constant power is ≤ k K ≤ + .
n,k K2 K K
allocated in all frames so that the value E[T ] is fixed under
k Passing K → ∞ proves that queue {Z }∞ is mean rate
any work-conserving policy. The first and second step of the n,k k=0
stableforallclassesn.ThusconstraintsW ≤d aresatisfied
DelayFair policy minimizes the last (by Lemma 2) and the n n
by Lemma 3. Similarly, the {Y }∞ queues are mean rate
second-to-last term of (20), respectively. n,k k=0
stable for all classes n.
Next, taking expectation of (21), summing over k ∈
D. Performance of the DelayFair Policy
{0,...,K −1}, dividing by V, and noting L(Z ,Y ) = 0
0 0
Theorem 2. Given any feasible delay bounds {d ,...,d },
1 N yields
theDelayFairpolicyyieldsaveragedelayssatisfyingW ≤d
for all classes n ∈ {1,...,N}, and attains averagendelany E[L(ZK,YK)] +(cid:88)N E(cid:34)K(cid:88)−1f (r )T (cid:35)
penalty satisfying V n n,k k
N E(cid:104)(cid:80)K−1(cid:80) W(i)(cid:105) (cid:34)nK=−11 (cid:35)k=0N
liKm→s∞up(cid:88)fn Ek(cid:104)=(cid:80)0K−i1∈|AAn,k |(cid:105)n,k  ≤ KVC +E (cid:88) Tk (cid:88)fn(W∗n).
n=1 k=0 n,k k=0 n=1
(cid:104) (cid:105)
≤ C(cid:80)Nn=1λn +(cid:88)N f (W∗), Removing the first term and dividing by E (cid:80)Kk=−01Tk yields
V n n
(cid:104) (cid:105)
n=1 N E (cid:80)K−1f (r )T N
where V > 0 is a predefined control parameter and C > 0 (cid:88) k=0 n n,k k ≤ KC +(cid:88)f (W∗)
a finite constant. By choosing V sufficiently large, we attain E(cid:104)(cid:80)K−1T (cid:105) VE(cid:104)(cid:80)K−1T (cid:105) n n
arbitrarily close to the optimal delay penalty (cid:80)N f (W∗). n=1 k=0 k k=0 k n=1
We remark that the tradeoff of choosing a lanr=g1e Vn valnue (≤a) C(cid:80)NnV=1λn +(cid:88)N fn(W∗n), (24)
is the amount of time required for virtual queues {Zn,k}∞k=0 n=1
tahnedn{eYxtn,pkr}o∞ko=f)0, tthoatapisp,rtohaechtimmeearenqurairteedsftoabritlhiteyv(isretueal(2q3u)euine where (a) follows E[Tk] ≥ E[Ik] = 1/((cid:80)Nn=1λn). By [14,
Lemma 7.6] and convexity of f (·), we get
n
backlogs to be negligible with respect to the time horizon.
Proof of Theorem 2: Consider the optimal randomized N E(cid:104)(cid:80)K−1f (r )T (cid:105) N E(cid:104)(cid:80)K−1r T (cid:105)
pno.lSicinycπer∗atnhdetDhaetlayyieFladisroppotliimcyalmdienliamyiszWes∗nth≤erdignhtfosridaellocfla(s2s0e)s, n(cid:88)=1 Ek(cid:104)=(cid:80)0 kK=n−01Tn,kk(cid:105) k ≥n(cid:88)=1fn E(cid:104)(cid:80)k=Kk0=−01nT,kk(cid:105)k .
comparingtheDelayFair policywiththepolicyπ∗ andwith (25)
rand
the genie decision r∗ = W∗ for all classes n and frames Combining (24)(25) and taking a limsup as K →∞ yields
n,k n
k, inequality (20) under the DelayFair policy is further upper N E(cid:104)(cid:80)K−1r T (cid:105)
bounded by (cid:88) k=0 n,k k
limsup fn (cid:104) (cid:105) 
(cid:88)N K→∞ n=1 E (cid:80)Kk=−01Tk
∆(Z ,Y )+V E[f (r )T |Z ,Y ]
k k n=1 n n,k k k k ≤ C(cid:80)Nn=1λn +(cid:88)N f (W∗).
N N V n n
≤C−E[T ](cid:88)Z λ d +E[T ](cid:88)(Z +Y )λ W∗ n=1
k n,k n n k n,k n,k n n
The next lemma, proved in Appendix C, completes the proof.
n=1 n=1
8
Lemma 4. If queues {Y }∞ are mean rate stable for all The inequalities in W(P) show that the minimum delay for
n,k k=0
classes n, then each class is attained when it has priority over the other. The
N E(cid:104)(cid:80)K−1(cid:80) W(i)(cid:105) equality in W(P) follows the M/G/1 conservation law [35].
limsup(cid:88)fn k(cid:104)=0 i∈An,k (cid:105)n,k  Using the above parameters, we get
K→≤∞liKmn→=s∞u1pn(cid:88)N=1fnEE(cid:104)E(cid:80)(cid:80)(cid:104)Kk(cid:80)=Kk−=0−Kk10=1|−A0r1nnT,,kkkT|(cid:105)k(cid:105). W(P)=(W1,W2)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W1WW+122W≥≥2PP=((PPP22−−(P126))−3) .
VI. DELAY-CONSTRAINEDOPTIMALPOWERCONTROL
Fig.1showsthecollectionofdelayregionsW(P)fordifferent
In this section we incorporate dynamic power control into
values of P ∈ [4,10]. This joint region contains all feasible
the queueing system. As mentioned in Section II, we fo-
delay vectors under constant power allocations. Fig. 2 shows
cus on frame-based policies that allocate a constant power
P ∈ [P ,P ] over the duration of the kth busy period
k min max 0.7
(we assume zero power is allocated when the system is idle).
0.6
Here,interestingquantitiessuchasframesizeT ,busyperiod
k
0.5
B , the set A of per-frame class n arrivals, and queueing
k n,k
delayW(i) areallfunctionsofpowerP .Similartothedelay W20.4
definitionn,k(1), we define the average pokwer consumption delay 0.3
E(cid:104)(cid:80)K−1P B (P )(cid:105) 0.2
k=0 k k k
P (cid:44)limsup , (26) 0.1
(cid:104) (cid:105)
K→∞ E (cid:80)Kk=−01Tk(Pk) 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
delay W1
where B (P ) and T (P ) emphasize the power dependence
k k k k
ofB andT .ItiseasytoshowthatbothB (P )andT (P ) Fig.1. ThecollectionofaveragedelayregionsW(P)fordifferentpower
k k k k k k levelsP ∈[4,10].
aredecreasinginP .Thegoalistosolvethedelay-constrained
k
power minimization problem:
minimize: P (27)
10
subject to: W ≤d , ∀n∈{1,...,N} (28)
n n
9
over frame-based power control and nonpreemptive priority
8
policies.
wer P 7
A. Power-Delay Performance Region po 6
Every frame-based power control and nonpreemptive prior- 5
ity policy can be viewed as a timing sharing or randomiza-
4
tion of stationary policies that make the same deterministic 0
0.2
decision in every frame. Using this point of view, next we 0.4
0.6
greivsueltainngexfarmomplefroafmthe-ebjaosiendt ppoowliecri-edse.laCyopnesridfoerrmaanctweor-ecgliaosns delay W1(P) 0.8 1 0 0.1 0.2 d0e.3lay W02.(4P) 0.5 0.6 0.7
nonpreemptive M/G/1 queue with parameters:
• λE1(cid:2)S=2(cid:3)1=, λ22. µ=(P2), =E[SP1.] F=orEea[cSh2]cl=assEn(cid:2)S∈22(cid:3){1=,2}1,, F(Pig,.W21.(P),TWhe2(aPu)g)m.ented performance region of power-delay vectors
1
the service time X has mean E[X ] = E[S ]/P and
n n n the associated augmented performance region of power-delay
secondmomentsE(cid:2)X2(cid:3)=E(cid:2)S2(cid:3)/P2.Forstability,we
must have λ E[X ]+nλ E[X ]n< 1 ⇒ P > 3. In this vectors (P,W1(P),W2(P)); its projection onto the delay
1 1 2 2 plane is Fig. 1. After timing sharing or randomization, the
example, let [4,10] be the feasible power region.
performance region of all frame-based power control and
Under a constant power allocation P, let W(P) denote the nonpreemptive priority policies is the convex hull of Fig. 2.
set of achievable queueing delay vectors (W1,W2). Define Theproblem(27)-(28)isviewedastochasticoptimizationover
ρn (cid:44)λnE[Xn] and R(cid:44) 21(cid:80)2n=1λnE(cid:2)Xn2(cid:3). Then we have such a convexified power-delay performance region.
 (cid:12) R 
 (cid:12)(cid:12)(cid:12) Wn ≥ 1−ρn, n∈{1,2} B. Dynamic Power Control Policy
W(P)=(W1,W2)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n(cid:88)=21ρnWn = 1(ρ−1+ρ1ρ−2)ρR2 . andWaesssuemtuepZthne,0s=am0efovrirtaulallcldaeslsaeys nqu.eWueesreZpnr,ekseanst ainst(r8ic)t,
(29) nonpreemptive priority policy by a permutation (π )N of
n n=1
9
{1,...,N}, where π denotes the job class that gets the nth Lemma 5 next shows that the minimizer of (34) is a de-
n
highest priority. terministic power allocation and strict nonpreemptive priority
Dynamic Power Control (DynPower) Policy: policy.Specifically,wemayconsidereachp∈P inLemma5
1) Inthekthframeforeachk ∈Z+,usethenonpreemptive denotes a deterministic power allocation and strict priority
strict priority rule (π )N that assigns priorities in policy, and random variable P denotes a randomized power
n n=1
the decreasing order of Z /E[S ]; ties are broken control and priority policy.
n,k n
arbitrarily.
Lemma 5. Let P be a continuous random variable with state
2) Allocate a fixed power P in frame k, where P is the
k k space P. Let G and H be two random variables that depend
solution to the following minimization of a weighted
on P such that, for each p ∈ P, G(p) and H(p) are well-
sum of power and average delays:
defined random variables. Define
(cid:32) N (cid:33)
minimize: V n(cid:88)=1λnE[Sn] µ(PPkk) (30) p∗ (cid:44)argminp∈P EE[[HG((pp))]], U∗ (cid:44) EE[[HG((pp∗∗))]].
N
(cid:88)
+ Zπn,kλπnWπn(Pk) Then E[G] ≥U∗ regardless of the distribution of P.
n=1 E[H]
subject to: Pk ∈[Pmin,Pmax]. (31) Proof: For each p∈P, we have E[G(p)] ≥U∗. Then
E[H(p)]
The value W (P ), given later in (37), is the average
delay of classπnπnkunder the priority rule (πn)Nn=1 and E[G] = EP [E[G(p)]] ≥ EP [U∗E[H(p)]] =U∗,
power allocation P . E[H] E [E[H(p)]] E [E[H(p)]]
k P P
3) Update queues Z for all classes n ∈ {1,...,N}
n,k
by (8) at every frame boundary. which is independent of the distribution of P.
TheaboveDynPower policyrequirestheknowledgeofarrival Under a fixed power allocation Pk and a strict nonpreemp-
tive priority rule, (34) is equal to
rates and the first two moments of job sizes for all classes
n (see (37)). We can remove its dependence on the second
VE[P B (P )]+(cid:80)N Z λ (W (P )−d )E[T (P )]
moments of job sizes, so that it only depends on the mean of k k k n=1 n,k n n,k k n k k
arrivals and job sizes; see Appendix D for details. E[Tk(Pk)]
C. Motivation of the DynPower Policy =VPk(cid:80)Nn=µ1(λPnE)[Sn] +(cid:88)N Zn,kλn(Wn,k(Pk)−dn),
k
We construct the Lyapunov drift argument. Define the n=1
(35)
Lyapunov function L(Z )= 1(cid:80)N Z2 and the one-frame
k 2 n=1 n,k
Lyapunov drift ∆(Zk)=E[L(Zk+1)−L(Zk)|Zk]. Similar where by renewal theory
as the derivation in Section IV-B, we have the Lyapunov drift
inequality: N   EE[[BTk((PPk))]] = (cid:88)N ρn(Pk)= (cid:88)N λnEµ([PSn)]
∆(Zk)≤C+(cid:88)Zn,kE (cid:88) (cid:16)Wn(i,k) −dn(cid:17)|Zk. k k n=1 n=1 k
n=1 i∈An,k and power-dependent terms are written as functions of Pk. It
(32)
follows that our desired policy in every frame k minimizes
Adding the weighted energy VE[P B (P )|Z ] to both
k k k k
sides of (32), where V >0 is a control parameter, yields (cid:32) N (cid:33) N
V (cid:88)λ E[S ] Pk +(cid:88)Z λ W (P ) (36)
∆(Zk)+VE[PkBk(Pk)|Zk]≤C+Φ(Zk), (33) n n µ(Pk) n,k n n,k k
n=1 n=1
where
(cid:34) overconstantpowerallocationsPk ∈[Pmin,Pmax]andnonpre-
Φ(Z )(cid:44)E VP B (P ) emptive strict priority rules.
k k k k
To further simplify, for each fixed power level P , by
k
N (cid:35) Lemma 2, the cµ rule that assigns priorities in the decreasing
+(cid:88)Z (cid:88) (W(i) −d )|Z . order of Z /E[S ] minimizes the second term of (36)
n,k n,k n k n,k n
n=1 i∈An,k (note that minimizing a linear function over strict priority
rules is equivalent to minimizing over all randomized priority
Weareinterestedintheframe-basedpolicythat,ineachframe
k, allocates power and assigns priorities to minimize the ratio rules, since a vertex of the performance polytope attains the
minimum). This strict priority policy is optimal regardless
Φ(Z )
E[T (P )k|Z ]. (34) of the value of Pk, and thus is overall optimal; priority
k k k assignmentandpowercontrolaredecoupled.Werepresentthe
NotethatframesizeT (P )dependsonZ becausethepower optimal priority policy by (π )N , recalling that π denotes
k k k n n=1 n
allocation that affects T (P ) may be Z -dependent. For any thejobclassthatgetsthenthhighestpriority.Underpriorities
k k k
given power allocation P , T (P ) is independent of Z . (π )N and a fixed power allocation P , the average delay
k k k k n n=1 k
10
W (P ) for class π is equal to Since E[T (P )] is decreasing in P and, under a fixed
πn k n k k k
power allocation, is independent of scheduling policies, we
W (P )= 12(cid:80)Nn=1λnE(cid:2)Xn2(cid:3) get E[Tk(Pk)]≤E[T0(Pmin)] and
πn k (1−(cid:80)n−1 ρ )(1−(cid:80)n ρ )
m=0 πm m=0 πm K−1
= 12(cid:80)Nn=1λnE(cid:2)Sn2(cid:3) , E[L(ZK)]+V (cid:88) E[PkBk(Pk)]
(µ(P )−(cid:80)n−1 ρˆ )(µ(P )−(cid:80)n ρˆ ) k=0
k m=0 πm k m=0 πm(37) ≤K(C+VP∗E[T0(Pmin)]).
where ρˆ (cid:44) λ E[S ] if m ≥ 1 and 0 if m = 0. The Removing the second term and dividing by K2 yields
πm πm πm
above discussions lead to the DynPower policy. E[L(Z )] C+VP∗E[T (P )]
K ≤ 0 min .
K2 K
D. Performance of the DynPower Policy
Combining it with
Theorem3. LetP∗ betheoptimalaveragepoweroftheprob- (cid:118)
(cid:117) (cid:104) (cid:105)
Wlemn(≤27d)-n(2f8o)r.TalhlecDlaysnsePsowne∈rp{o1l,ic.y..a,cNhi}evaensddealtatayicnosnasvtreariangtes 0≤ E[Zn,K] ≤(cid:117)(cid:116)E Zn2,K ≤(cid:114)2E[L(ZK)]
K K2 K2
power P satisfying
C(cid:80)N λ and passing K → ∞ proves that queue {Zn,k}∞k=0 is mean
P ≤ n=1 n +P∗, rate stable for all classes n. Thus W ≤ d for all n by
V n n
Lemma 3.
where C > 0 is a finite constant and V > 0 a predefined Further, removing the first term in (38) and dividing the
control parameter. result by VE(cid:104)(cid:80)K−1T (P )(cid:105) yields
k=0 k k
Proof of Theorem 3: As discussed in Section VI-A, the
(cid:104) (cid:105)
power-delayperformanceregioninthisproblemisspannedby E (cid:80)K−1P B (P )
k=0 k k k C K
stationary power control and nonpreemptive priority policies (cid:104) (cid:105) ≤ (cid:104) (cid:105) +P∗
that use the same (possibly random) decision in every frame. E (cid:80)Kk=−01Tk(Pk) V E (cid:80)kK=−01Tk(Pk)
Let π∗ denote one such policy that yields the optimal average (a) C(cid:80)N λ
power P∗ with feasible delays W∗ ≤ d for all classes n. ≤ n=1 n +P∗,
n n V
Let P∗ be its power allocation in frame k. Since policy π∗
k
makes i.i.d. decisions over frames, by renewal reward theory where (a) uses E[T (P )] ≥ E[I ] = 1/((cid:80)N λ ). Passing
k k k n=1 n
we have K →∞ completes the proof.
E[P∗B(P∗)]
P∗ = k k .
E[T(P∗)]
k VII. OPTIMIZINGDELAYPENALTIESWITHAVERAGE
Then the ratio Φ(Zk) under policy π∗ (see the left side POWERCONSTRAINT
E[Tk(Pk)|Zk]
of (35)) is equal to
The fourth problem we consider is to, over frame-based
V E[Pk∗B(Pk∗)] +(cid:88)N Z λ (cid:16)W∗ −d (cid:17)≤VP∗. paoswepearracbolnetrcoolnavnedxfnuonncptiroenemopftdiveelayprvioercittoyrsp(oWlicie)sN, misnuimbjiezcet
E[T(P∗)] n,k n n n n n=1
k n=1 to an average power constraint:
Since the DynPower policy minimizes Φ(Zk) over N
frame-basedpolicies,includingtheoptimalpEo[lTikc(yPπk)∗|Z,tkh]eratio minimize: (cid:88)fn(Wn) (39)
Φ(Zk) under the DynPower policy satisfies n=1
E[Tk(Pk)|Zk] subject to: P ≤P . (40)
const
Φ(Z )
k ≤VP∗ ⇒Φ(Z )≤VP∗E[T (P )|Z ].
E[Tk(Pk)|Zk] k k k k The value P is defined in (26) and Pconst > 0 is a given
feasible bound. The penalty functions f (·) are assumed
n
Using this bound in (33) yields
nondecreasing, nonnegative, continuous, and convex for all
∆(Z )+VE[P B (P )|Z ]≤C+VP∗E[T (P )|Z ]. classes n. Power allocation in every busy period takes values
k k k k k k k k
in [P ,P ], and no power is allocated when the system is
min max
Taking expectation, summing over k ∈ {0,...,K −1}, and idle. In this problem, the region of feasible power-delay vec-
noting L(Z )=0 yields tors(P,W ,...,W )iscomplicatedbecausefeasibledelays
0 1 N
(W )N are indirectly decided by the power constraint (40).
K−1 n n=1
E[L(Z )]+V (cid:88) E[P B (P )] Using the same methodology as in the previous three prob-
K k k k
lems, we construct a frame-based policy to solve (39)-(40).
k=0(cid:34)K−1 (cid:35) (38) We setup the virtual delay queue {Yn,k}∞k=0 for each class
≤KC+VP∗E (cid:88) T (P ) . n ∈ {1,...,N} as in (17), in which the auxiliary variable
k k
r takes values in [0,Rmax] for some Rmax > 0 sufficiently
k=0 n,k n n