Scheduling in a random environment: stability and asymptotic optimality

0.54 MB·English

by  U. Ayesta

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Scheduling in a random environment:
∗
stability and asymptotic optimality
U. Ayesta1,2, M. Erausquin1,3, M. Jonckheere4, I.M. Verloop1
1BCAM – Basque Center for Applied Mathematics, Derio, Spain
2IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
3UPV/EHU, University of the Basque Country, Bilbao, Spain
4CONICET, Buenos Aires, Argentina
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Abstract—Weinvestigatetheschedulingofacommonresource of minimizing mean users’ delay, there arises a key tradeoff
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betweenseveralconcurrentuserswhenthefeasibletransmission in the design of scheduling mechanisms between making full

n rate of each user varies randomly over time. Time is slotted use of the opportunistic gains, and prioritizing users having
a and users arrive and depart upon service completion. This
small residual service sizes.
J may model for example the flow-level behavior of end-users
0 in a narrowband HDR wireless channel (CDMA 1xEV-DO). As Broadly speaking, researchers have explored scheduling in
3 performance criteria we consider the stability of the system and wirelesssystemsbothatthepacketlevelandattheflowlevel.
the mean delay experienced by the users. Given the complexity Inpacket-level modelsitis typicallyassumedthat thereexists
] of the problem we investigate the fluid-scaled system, which afinitenumberofpermanentusers.Thefocusofthescheduler
F allows to obtain important results and insights for the original
is on the number of packets in the queue of each user. We
P system: (1) We characterize for a large class of scheduling
. policies the stability conditions and identify a set of maximum refer for example to [34], [2], [33], [17], [3], [25], [29] for
s stable policies, giving in each time slot preference to users being thislineofresearch.Inaflow-levelmodelinstead,usersarrive
c
[ in their best possible channel condition. We find in particular randomly to the system and leave after receiving their finite-
that many opportunistic scheduling policies like Score-Based
sizedservicedemands.Thisallowstocapturetheperformance
1 [8], Proportionally Best [1] or Potential Improvement [4] are
as perceived by the end-users, see for example [8], [21], [9],
v stable under the maximum stability conditions, whereas the
4 opportunistic scheduler Relative-Best [9] or the cµ-rule are not. [26], [22], [1], [4], [30]. For surveys on flow-level modeling
9 (2) We show that choosing the right tie-breaking rule is crucial we refer to [24] and [10]. In [23], hybrid models are studied.
7 for the performance (e.g. average delay) as perceived by a user. The performance evaluation and optimization of wireless
5 Weprovethatapolicyisasymptoticallyoptimalifitismaximum networks at the flow level has proved to be extremely chal-
. stable and the tie-breaking rule gives priority to the user with
1 lenging. One of the most successful approaches has been the
the highest departure probability. We will refer to such tie-
0 breaking rule as myopic. (3) We derive the growth rates of the so-called time-scale separation argument (see [9], [11], [12],
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numberofusersinthesysteminoverloadsettingsundervarious [29], [1], [30]) where it is assumed that at the flow scale the
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policies, which give additional insights on the performance. (4) dynamicsofthechannelfluctuationscanbeaveragedout.Un-
:
v We conclude that simple priority-index policies with the myopic der this time-scale assumption, it was shown in [11] that any
i tie-breaking rule, are stable and asymptotically optimal. All our
X utility-based scheduling policy is stable in a flow-level model.
findings are validated with extensive numerical experiments.
The authors of [1] make the same assumption when they
r
a discuss rate-stability for priority-index policies. In the context
I. INTRODUCTION of optimal control, in [29], [30] scheduling mechanisms were
Next generation wireless networks are expected to support introduced and evaluated. In [4] optimal control is studied
awidevarietyofdataservices.Duetofadingandinterference withoutthetime-scaleseparationassumption.TheLagrangian-
effects, for each user, the quality of a downlink channel, and relaxation method allowed the authors of [4] to construct the
hence its transmission rate, fluctuates over time. This has Potential Improvement (PI) scheduling policy, which is opti-
triggered a large amount of work aiming at understanding the mal for a relaxed optimization problem. In addition, several
performanceofchannel-awareschedulingpolicies.Itisbynow otherpolicieshavebeenproposedandnumericallyinvestigated
accepted that so-called “opportunistic schedulers” have many in the literature, among others the Proportional Fair [14]
desirable properties (see for example [9]). A policy is called discipline, the Score-Based (SB) algorithm [8], the Relative
opportunistic if it takes advantage of the channel fluctuations Best (RB) scheduler [6] and Proportionally Best (PB) [1].
by serving a user whose channel condition is in a good state To sum up, without a time-scaling separation argument,
with respect to its own statistical behavior. With the objective which is a rather strong assumption, the performance of
opportunistic schedulers, regarding stability and performance
∗Research partially supported by grant MTM2010-17405 of the MICINN perceivedbytheusers,isnotwellunderstood.Inordertogain
(Spain) and grant PI2010-2 of the Basque Government (Department of
betterinsightintothelatterissue,inthispaperwewillstudya
Education and Research). Martin Erausquin’s PhD. is supported by grants
ECO2008-00777andFPUAP2008-02014,bothoftheMEC(Spain). flow-levelmodelwithoutthetime-scaleseparationassumption.
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More precisely, we assume that data users arrive randomly in II. MODELDESCRIPTION
time and have a finite amount of data to download. Time is
We consider a time-slotted system serving one user in each
slotted and the quality of the channel condition of each user
timeslot.ThismodelsforinstanceaCDMA1xEV-DOsystem
variespertimeslot.Ineverytimeslotatmostoneusermaybe
as explained in Remark 1. There are K classes of users, and
served. We are interested in stability and optimization of the
in each time slot the number of class-k users arriving to the
system. Given the complexity of the problem we first prove
system, A , follows an i.i.d. sequence of random variables,
convergence of the fluid-scaled system towards a unique fluid k
with E(A )=λ and E(A2)<∞. For each user the depar-
limit.Wenotethattheprecisecharacterizationofthefluidlimit k k k
ture probability varies over time as the quality of the channel
involves averaging phenomena of the scaled system which is
is changing from slot to slot. The quality of the channel (or
not grasped by the usual description of weak fluid limits.
state of the channel) for a class-k user is modeled as an i.i.d.
The fluid-limit description allows us to obtain several im- sequence of random variables taking values in the finite set
portant results and insights for the original wireless system. N := {1,2,...,N }. For each time slot we let q denote
k k k,n
First of all, we characterize the maximum stability conditions the probability that a class-k user is in channel state n∈N .
k
(theweakestpossibleconditionsonthetrafficparameterssuch Associatedwithchannelstatenisadepartureprobabilityµ .
k,n
that there exists a scheduling policy that makes the system This can be used for instance to model a system in which the
positive recurrent) and show that the set of policies that are service requirements are geometric (see Remark 1). Without
stable under the maximum stability condition have a very loss of generality we assume that the channel conditions are
simple characterization: whenever there are users present that ordered such that 0 ≤ µ ≤ µ ≤ ··· ≤ µ ≤ 1, and
k,1 k,2 k,Nk
are currently in their best channel condition, only such users q µ (cid:54)=0, ∀k.Thechannelconditionofaclass-k user
k,Nk k,Nk
are served. These policies will be referred to as Best-Rate is independent of the channel conditions of all the other users
(BR)policies.Suchacharacterizationwaspreviouslygivenfor and of the channel quality history.
rate stability [1], but, to the best of our knowledge, stochastic Ineachtimeslott,ascheduler/policyf decideswhichuser
stability was still an open issue. is served. Because of the Markov property of the system, we
Second, for a large class of scheduling policies we deter- focusonpoliciesthatbasedecisionsonthecurrentnumberof
mine the exact stability conditions and conclude that many userspresentinthevariousclassesandontheircurrentchannel
known opportunistic scheduling policies like SB, PB, or PI states. For a given scheduling policy f, let Xkf(t) denote the
are stable under the maximum stability conditions, whereas numberofclass-kusersinthevariousclassesattimeslottand
the opportunistic scheduler RB or cµ-rule are not. Xf(t) = (X1f(t),...,XKf (t)). Since the channel conditions
are i.i.d. and independent of the process Xf(·), the process
Third, we demonstrate the importance for the choice of the Xf isMarkovand,inaddition,forthemodelingitissufficient
tie-breakingrulewhenthegoalistooptimizetheperformance.
to focus on the Markovian description in terms of the number
Until now, the literature proposed to break ties at random, of users in each class, Xf(·), instead of the number of users
see for example [6], [8], [9], [1]. We instead propose to
in each channel state.
give priority to the user with highest instantaneous departure
Let us introduce some more notation. We denote by |x| the
probabilitywhentherearemultipleusersintheirbestchannel
l norm of a vector x. The notation x ≤ y is used for the
1
conditions, which we refer to as the myopic tie-breaking
coordinate-wise ordering: x ≤ y ,∀i. Finally, we denote by
i i
rule. We prove that BR policies with the myopic tie-breaking
u.o.c. the uniform convergence on compact sets.
rule are asymptotically fluid optimal and our numerical ex-
Performance criteria: Our performance criteria are stability
periments further illustrate that the myopic tie-breaking rule
and long-run average number of users. We use the following
significantlyimprovestheperformance.Thisinturnshowsthat
definition for stability:
simple priority-index policies that balance opportunistic gains
withsize-basedinformation,willbebothmaximumstableand Definition 1. A scheduling policy f is stable if the process
asymptotically optimal. Xf is positive recurrent.
Fourth, our convergence result allows to compare the per- Because of the time-varying channel conditions the system
formance of the various policies in an overload setting. More is not work-conserving, and hence it depends strongly on the
precisely, we determine the growth rates of the number of employed scheduling policy whether the system can be made
users in the various classes and find that BR policies with a stable. We define the maximum stability conditions as the
myopic tie-breaking rule minimize the total growth rate. conditions on the traffic inputs such that there exists a policy
that can make the system stable. A maximum stable policy is
The paper is organized as follows. In Section II we present
apolicythatisstableunderthemaximumstabilityconditions.
the model. In Section III we introduce the scheduling policies
From the performance point of view it is therefore of crucial
of interest and define their tie-breaking rules. In Section IV
importance to design a scheduler that is maximum stable.
we derive fluid limits for a large class of policies. This allows
to obtain our stability results as presented in Section V. In Besidesstability,anotherimportantperformancemeasureis
Section VI we characterize asymptotically optimal policies, the long-run time-average holding cost,
both in normal regime and in overload, and discuss the
K T
importanceofthetie-breakingrule.InSectionVIIweperform limsup 1 (cid:88)(cid:88)c E(Xf(t)), (1)
numerical experiments to validate our theoretical findings. T→∞ T k k
k=1t=0
3
BRPpolicies
SB PB
BRpolicies
RB
PI Fluidoptimalpolicies
Maximumstablepolicies
Cµ PriorityIndexpolicies
Fig.1. Classificationofschedulers.
withc >0theholdingcostincurredpertimeslotforhavinga at most to one user at a time. As reported in the literature,
k
class-kuserinthesystem.Whenc =1,∀k,thisisequivalent in an OFDM system a user can experience a deep fading in
k
to minimizing the mean sojourn time (cf. Little’s law). one subcarrier, while on the same subcarrier another user
couldbeingoodcondition[32].Underi.i.d.assumptionsitis
Remark 1 (Modeling of a wireless data network). Our
expectedthatforthemetricsunderconsiderationinthispaper
model,eventhoughsimple,capturessomeofthekeyproperties
itisofnointeresttoservemultipleclassesinparallel,thatis,
of wireless communication systems. Time is slotted, as is the
in every slot all the sub-carriers will be assigned to only one
case in the CDMA 1xEV-DO [5] and the OFDM-based LTE
of the classes. However, serving multiple classes in parallel
systems [31]. The available transmission rate of each user
could be though of interest for other metrics like fairness, an
fluctuatesduetofadingeffects,andasaconsequence,itvaries
issue that even though not studied here, definitely deserves a
from one slot to another. We note that in real systems the
thorough investigation.
number of feasible transmission rates is finite (see [5]).
One may classify the users into different classes based on
III. POLICIES
their applications or traffic conditions for example. Let the
In this section we introduce scheduling policies that will
service requirement (in bits) of a class-k user be a geometric
random variable denoted by B , and let E(B ) denote its be used throughout the paper. Most of these policies are
k k
opportunistic(withtheexceptionofthecµ-rule),meaningthat
expectation. Let ∆ denote the amount of bits transferred
they take advantage of channel fluctuations by serving a user
in one slot under the current channel condition. Note that
whose channel condition is currently in a good state, in some
in practice ∆ will vary from slot to slot depending on the
sense, with respect to its own statistical behavior.
channel condition, and the allocation. The probability that
a user leaves the system is approximately P(b ≤ B ≤ We first introduce priority-index policies, which are very
k
b + ∆|B > b) ≈ ∆/E(B ), which does not depend on popular due to their simplicity from an implementation point
k k
of view. A priority-index policy is characterized by an index
the attained service b (memoryless property of the geometric
function that assigns an index to each user based solely on its
distribution).Thisexpressionbecomesasymptoticallyexactas
the ratio ∆/E(B ) goes to 0. Hence, this is the case if the class and its current state.
k
mean service requirement (in bits) of a user is very large Definition2(Priority-indexpolicy). Ineverytimeslot,auser
compared to the amount of bits that can be served in one that has the highest index among all users present is served.
slot.Lets denotethetransmissionrate(inbitspersecond)
k,n
Priority-index policies might need to be augmented with a
of a class-k user when the channel state is n. For the CDMA
suitable tie-breaking rule. Such a rule refers to the strategy
1xEV-DO system, the amount of bits transferred in one slot is
adopted when there is a tie on the highest index value. A tie
∆ = s ·t . Hence, the departure probability of a class-k
k,n c
means that there are several users present having the highest
user under channel condition n can be approximated by
index value, but these users belong to different classes. In the
µ := sk,n·tc, (2) literature, most of the papers specify to break ties at random
k,n E(B ) (see for example [8], [6], [1]). We define the myopic tie-
k
breakingruleastherulethatamongtheuserswiththehighest
where t is the length of the slot (for example t =1.67ms in
c c index, it selects the one with highest value for c µ , ∀ k
the CDMA 1xEV-DO system). In Section VII we perform nu- k k,Nk
(the c ’s refer to the holding cost introduced in Section II.)
merical experiments with departure probabilities as obtained k
One of our main contributions will be that the choice for the
from a practical setting using (2).
tie-breaking rule is crucial for the performance of the system
Remark 2 (Modeling an OFDM system). A natural exten- and that the myopic tie-breaking rule is close to optimal (this
sion to our modeling framework will be to allow that in every will be further developed in Sections VI and Section VII).
slot multiple users can be served in parallel as it happens to In [8] the Score-Based (SB) policy is introduced. SB is
be in the OFDM-based (3GPP LTE) system. In such a system a priority-index policy where the index value of a class-k
there are M subcarriers and a subcarrier can be assigned user in state n is given by (cid:80)n q , and ties are broken
n˜=1 k,n˜
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at random. In [4] the Potential Improvement (PI) policy is the fluid-scaled processes Yf,r(t) := Xkf,r((cid:98)rt(cid:99)), t ≥ 0, k =
k r
introduced. PI is a priority-index policy with as index value 1,...,K, with Yr(0)=x(0). We can write
(cid:80)
c µ / q (µ −µ ),andthetie-breakingruleisthe
k k,n k,n˜ k,n˜ k,n
n˜>n 1(cid:88)(cid:98)rt(cid:99) 1 (cid:88)Nk
myopic tie-breaking rule. An important subset of the priority- Yf,r(t)=x (0)+ A (s)− S (Tf,r(rt)), (3)
index policies are the so-called weight-based policies. k k r k r k,n k,n
s=1 n=1
Definition 3 (Weight-based policy). A priority-index policy where Tf,r(t) is defined as the cumulative amount of time
withindexfunctionω µ .Hereω denotesaclassdependent k,n
k k,n k thatwasspentonservingclassk instatenduringtheinterval
weight.
(0,t] and S (t) denotes the total number of class-k users
k,n
Important examples of weight-based policies are: the cµ- that have been completed while receiving service for a total
rule (ω = c , with c the holding cost), Relative Best (RB) duration of time t when being in state n.
k k k
[6] (ω = 1/(cid:80)Nk q µ ), and Proportionally Best (PB) In order to derive stability and fluid optimality results,
k n=1 k,n k,n
[1] (ω = 1/µ ). For all these policies, ties are broken at we will be interested in limits of the fluid-scaled process.
k k,Nk
random. In Section IV-A we will characterize a generic description
It will be convenient to define the following two classes of of weak fluid limits (usually not unique) of Equation (3),
policies, which play an important role in the results on the following the same reasoning as in [15]. In Section IV-B we
stability analysis and asymptotic optimality. focus instead on a special class of policies for which we can
proveconvergenceinprobabilitytowardsauniquelimit,which
Definition 4 (Best Rate (BR) policies). The BR policies are
will be referred to as the strong fluid limit. (In [20] similar is
such that whenever there are users present that are currently
done but only for a subset of the state space.) We discuss the
in their best channel condition, i.e., in state N , such a user
k differencesbetweenweakandstrongfluidlimitsinmoredetail
is served.
in Remark 5, after having introduced formally both concepts.
Definition 5 (Best Rate Priority (BRP) policies). The BRP
policies are BR policies with a myopic tie-breaking rule.
A. Convergence towards weak fluid limits
As a consequence of our main results, we will obtain that
From (3), we obtain the following result that describes
the classes of policies BR and BRP have desirable properties:
the generic characterization of weak fluid limits for a given
In Section V we prove that any BR policy is stable under the
policyf.Thelemmawillallowtodeterminemaximumstable
maximumstabilityconditionsandinSectionVIwederivethat
policies (Theorem V.2) and to characterize asymptotically
BRP policies are asymptotically optimal.
optimal policies (Section IV).
In Figure 1 we have summarized the various (classes of)
policies. Note that SB, PB and PI are BR policies. This Lemma IV.1. For almost all sample paths ω and any se-
follows since the highest possible index value is 1 for SB quence rk, there exists a subsequence rkl such that for all
and PB, and ∞ for PI, and these indices can only be obtained k =1,2,...,K, n=1,2,...,Nk, and t≥0,
whenever a user is in its best possible channel condition. RB
and the traditional cµ rule (i.e., giving in each time slot strict lim Ykf,rkl(t)=ykf(t), u.o.c., and (4)
l→∞
preemptive priority to the user having the highest c µ )
k k,n Tf,rkl(t)
however do not belong to BR policies since, depending on lim k,n =τf (t), u.o.c.,
the set of parameters, the index value of a class-k user in l→∞ rkl k,n
state n(cid:54)=N might be larger than the index value of a class-l
k with (yf(·),τf (·)) a continuous function. In addition,
user in state Nl. Remark also that PI is the only BRP policy. k k,n
(cid:88)Nk
IV. FLUIDLIMITSANDCONVERGENCE ykf(t)=xk(0)+λkt− µk,nτkf,n(t), (5)
n=1
Inthissectionwestudyfluid-scalinglimitsforalargeclass
ofpolicies.Fluidscalingortime-spacescalings,corresponding yf(t) ≥ 0, τf (0) = 0, (cid:80) τf (t) ≤ t, and τf (·) are
k k,n k,n k,n k,n
to “zooming out” the trajectories, have been used extensively non-decreasing and Lipschitz continuous functions.
tostudystochasticprocesseswithcomplexdynamics[16].The
limiting processes are usually much simpler to describe while Proof: The proof follows similarly to that of [13, Proposi-
theyprovidecrucialinsightsonthebehaviorofthenon-scaled tion 4.12]. Define Ak(t) as the number of class-k users that
version of the process. In particular, the convergence results arrive in time slot t. We note that Ak(1),Ak(2),..., are in-
willallowustoprovecrucialresultsonstabilityandoptimality dependentanddistributedaccordingtoAk,withE(Ak)=λk.
of schedulers for the stochastic system. Sinceµk,nistheprobabilityofcompletingaclass-kuserwhen
it receives service while being at state n, by the law of large
Thefluidscalingconsistsinstudyingasequenceofsystems
indexed by r, i.e., for a given policy f we let Xf,r(t) denote numbers, we obtain that, almost surely,
k
the number of class-k users at time t when the initial state
(cid:98)rt(cid:99)
equals Xr(0) = rx (0), k = 1,...,K, with r ∈ N, and 1(cid:88) 1
Xf,r(t) =k (Xf,r(t),k··· ,Xf,r(t)). We are then interested in rl→im∞r Ak(s)=λkt, and rl→im∞rSk,n(rt)=µk,nt. (6)
1 K s=1
5
SinceTf,r(t)denotesthecumulativeamountoftimespenton of U. Intuitively, this means that the drift vector-field has
k,n
serving class-k users in state n in time interval [0,t], we get limits when we make the number of users of some of the
classes go to infinity, and that we can interchange the order
Tf,r(rt) Tf,r(rs)
k,n − k,n ≤t−s, for every t≥s, of the coordinates when taking these limits. We assume in the
r r following that the drift vector has uniform limits, so that we
i.e.,Tf,r(t):=Tf,r(rt)/r isLipschitzcontinuous.Therefore, can define the asymptotic drifts δU :N|U| →RK as follows:
k,n k,n
by the Arzela-Ascoli theorem [27] we obtain that, for almost
δU(x ):= lim δ(x). (7)
every sample path w and for any subsequence rk, there exists U xk→∞, k∈Uc
a subsequence r of r such that lim Tf,rkl(rt)/r =
kl k l→∞ k,n Here, Uc corresponds to the “saturated” classes for which we
τf (t), u.o.c.. Now, using Equations (3) and (6), it follows
k,n let the number of users go to ∞. We define the stochastic
that liml→∞Ykf,rkl(t)=ykf(t), with yf(t) as given in (5). (cid:50) process XU as the U-dimensional stochastic process corre-
sponding to the original process seeing an infinite number of
We can now give our definition of weak fluid limits. usersofclassk ∈Uc andletπU denoteitsstationarymeasure
assuming it exists. We define the averaged drift vectors by
Definition 6. We call the processes τf(t) and yf(t) (as
obtained in Lemma IV.1) weak fluid limits for policy f. δ˜U = (cid:88) δU(x)πU(x). (8)
Note that, in general, these fluid limits can be different x∈N|U|
dependingonthesamplepathandthesubsequenceconsidered.
Finally, following [12] we say that a vector field v is partially
A policy is said to have a unique fluid limit if, for all sample
increasing if v (x) is increasing in x for all j (cid:54)= i. These
paths and all subsequences, the weak fluid limits coincide. i j
assumptions, which are crucial to prove the convergence
towards the unique strong fluid limit, are verified for many
B. Convergence towards a strong fluid limit
cases of interest, see the next lemma.
Inthissubsectionwewilldetermineuniquefluidlimitsfora
LemmaIV.2. ApriorityindexpolicyoraBRpolicywithnon-
specialclassofpolicies.Moreprecisely,wewillproveconver-
state dependent tie-breaking rule (i.e., independently of the
gence in probability towards a unique limit. The derivation of
numbers of users) induces a partially increasing drift vector
the strong fluid limit will prove to be very useful: It allows to
field with uniform limits.
calculatetheexactstabilityconditions(policydependent)(see
Theorem V.1 and the numerical Section VII). In addition, the
Proof: We prove the lemma for BR policies, the other case
exactcharacterizationofthestrongfluidlimitprovidescrucial
being similar. When increasing the number of users of one
insightsintotheperformanceofthesystem,suchasthegrowth
class only, the probability that this class has at least one user
ratesofthenumberofusersovertimeandmonotonicityresults
in its best possible state is increased. Hence, given that the
with respect to the tie-breaking rule.
tie-breaking rule does not depend on the number of users,
Obtaining exact fluid-limit characterizations will require to
the probability that this class is served is increased while the
deal with averaging phenomena: it may happen that one class
probability that a user of another class is served decreases.
of users reaches its stationary regime, i.e., is empty in the
This implies that the drift vector field is partially increasing.
fluidscaling,beforetheotherclassesdo.Inthiscase,thedrift
By the independence of the channel variations, the proba-
of the other classes needs to be averaged with the stationary bility that class i ∈ Uc has at least one user in its best state
dwiisltlriibnuvtoilovneoafvtehriasgceldasdsr.ifHtsenacsew,ailldebsecdriepfitinoendoifnth(8e)fl(uwide lriemfeirt Hise1nc−e,(w1h−enqit,hNei)nxuim, wbehresroefxcilaisss-tiheusneursm,bie∈r Uofc,cglarsosw-isulasregres,.
to this as second-vector fields, following [18]). the probability of having in each class in Uc at least one user
We focus on the class of policies that induce partially in its best state (and hence causing a drift δU(x )) converges
U
increasing drift vector fields with uniform limits. In order to
to1.Togetherwiththepropertythatthetie-breakingruledoes
describe this class of policies we need first to introduce the
not depend on the number of users, this implies that
driftfunctionsanddriftvectorfields.Forthestochasticprocess
Xf(t) associated with a policy f, we define the drift function δ(x)=δU(x ) (cid:89) (1−(1−q )xi)+o(1/|x|), (9)
by δf(x):=(δf(x),...,δf (x)), x∈NK, with U i,Ni
1 K i∈Uc
δif(x):=E(Xif(1)−xi|Xf(0)=x). as xk → ∞,k ∈ Uc, which in turn implies the uniform
convergence of δ(·) to δU(·) for any U. (cid:50)
(We will drop the superscript f when it is clear that we con-
siderauniquepolicy.)Wesaythatavectorfieldv :NK →RK
WenowstatethemainresultofthisSection,thedescriptionof
has uniform limits [12] if for any U ⊂{1,...K}, there exists
a function vU :N|U| →RK (constant when U =∅) such that the strong fluid limit. The proof can be found in Appendix A.
lim sup |v(x)−vU(x )|=0, Theorem IV.3. For a given policy f inducing a partially
U
R→∞x∈NK:|xUc|>R increasing drift vector field with uniform limits, we have
where x denotes the restriction of the vector x to indices
U lim P( sup |Yf,r(s)−yf(s)|≥(cid:15))=0, for all (cid:15)>0,
in the subclass U and Uc denotes the complementary set r→∞ 0≤s≤t
6
with yf(t) a piece-wise linear function that can be described and if T <∞, then U ={1,...,k−1} and
k−1 k−1
recursively as follows. Let U =∅ and T =0. Then we have,
0 0 k−1
dykf(t) =δ˜f,Ul, t∈[Tf,Tf ], (10) δ˜Uk−1 =(0,...,0,λk−µk,Nk(1−(cid:88)j=1 µjλ,Njj),...,λK). (15)
dt k l l+1
yf(Tf) Proof: Using Lemma IV.2, the drift δ(·) associated to a BR
with Tf =Tf + min k l , (11) policy is partially increasing with uniform limits, hence the
l+1 l k∈Ulc,δ˜kUl<0 −δ˜kf,Ul strong fluid limit is given by Theorem IV.3. When U = ∅,
yf(Tf) there are infinitely many users of each class and hence there
and U =U ∪ argmin k l , (12)
l+1 l k∈Ulc,δ˜kUl<0 −δ˜kf,Ul i(s13a)lw.NayosteathcalatsTs-11<us∞eriinfaitnsdboensltystiafteµ,1λ,wN11hi<ch1di(rseecetldyeifimnpitliioens
and if there exists no k ∈Uc with δ˜f,Ul <0, then Tf =∞. of T1 in Theorem IV.3). In this case the process X{1} is
l k l+1
ergodic. For T ≤ t ≤ T , we can simplify the asymptotic
1 2
We can now give our definition of the strong fluid limit. drift δ˜{1} using the specific properties of the policy and rate
Definition 7. For a given policy f inducing a partially conservation arguments: let A1,x1 be the event that class 1 is
increasing drift vector field with uniform limits, we call the servedandtherearex1 class-1userswhenalltheotherclasses
process yf(t) (as obtained in Theorem IV.3) the strong fluid are saturated.With a slight abuseof notation, let usdenote by
limit for the policy f. π{1}(A1,x1)theprobabilityofeventA1,x1 underthestationary
distribution of X{1}. Since X{1} is ergodic, by rate stability
Remark 3 (Calculation of the averaged drifts). Theorem IV.3 we have the following rate-conservation equation, see (46),
characterizes the strong fluid limit as a piece-wise linear
(cid:88)
function with slopes δ˜U. In practice, the calculations of these π{1}(A1,x1)µ1,N1 =λ1,
slopes involve: x1
•• dcaerlcivuilnagtinthgethaesysmtapttiootnicardyridfitsstr(isbeueti(o7n)s),of XU, w{1h}ic(hsogiivnepsatrhtiactu(cid:80)larxt1hπer{e1}is(Astc1i,lxl1a)n=infi1n−iteµa1λ,mN11ou.nStinocfeclUas1s-=2
• averaging the asymptotic drifts with these stationary users which are exclusively served when there are no class-
distributions (see (8)). 1 users in their best state) class 2 receives service at rate
Forinstance,assumeK =2,N1 =2,N2 =1,andaBernoulli µ2,N2(cid:80)x1π{1}(Ac1,x1)=µ2,N2(1−µ1λ,N11) which gives (14).
arrival process. Consider the policy that gives priority to the ConsidernowthecasewhereU :={1,...,k−1}(assuming
best class-1 user present in the system and otherwise (i.e., as before that (cid:80)k−1 λj < 1). Let A be the event that
when there is no class 1) serves a class-2 user. Then class j ∈U is serjv=e1dµajn,Ndjthere are x ujs,exrUs of class i, i∈U.
i
By rate-conservation arguments, see (46), we obtain
δ∅ =(λ −µ ,λ ),
1 1,N1 2 (cid:88)
δ{1}(x1)=(λ1−s1(x1),λ2−µ2,N21(x1=0)), xU πU(Aj,xU)µj,Nj =λj, j ∈U.
Twhitehsp1r(oxc1e)ss=Xµ{11,N}1(is1−a(11−-dqi1m,Ne1n)sxio1n)+alµM1,Na1r−ko1v(1c−hqa1i,nN1w)xit1h. tNhoattin(cid:80)g thaπtUth(e∪setsAAj,xU)a=re d(cid:80)isjoints,λj∀ j. ∈SiUnc,ethcilsasims pklieiss
stationary distribution xU j∈U j,xU j∈U µj,Nj
only served when no class-i users are being served, i ∈
π{1}(x )=C(cid:89)x1 λ1(1−s1(j−1)), U(cid:80), theπrUe(∪is a cAlass-k)).dHepeanrctue,rewweiothbtapirnobEaqbuialittiyonµ(k1,N5)k.(1 −(cid:50)
1 (1−λ1)s1(j) xU j∈U j,xU
j=1
Remark 4. For all BR policies where the scheduler chooses
where C is a normalization constant. The average drift can withprobabilityαU toserveclasskwhenasubsetU ofclasses
k
now be computed using (8). has at least one user in its best channel condition, the fluid
limit in the interior of the orthant has a drift given by:
In the specific case of BR policies with a priority-type tie-
breaking rule, we can in fact explicitly derive the strong fluid δ∅ = (λ −α{1,...,K}µ ,λ −α{1...,K}µ ,
1 1 1,N1 2 2 2,N2
limit by making use of rate-conservation arguments. This will ··· ,λ −α{1,...,K}µ ). (16)
prove to be very useful to obtain fluid optimality statements. K K K,NK
However, in general we cannot explicitly derive the second-
Proposition IV.4. Consider a BR policy with a priority-type
vector fields. An exception is the case of two classes. Then,
tie-breaking rule. Let us reorder the classes according to the
usingtherate-conservationargumentasinthepreviouspropo-
priority ordering. The averaged drift vectors are
sition, we obtain (assuming w.l.o.g. that class 1 empties first)
δ˜∅ =(λ1−µ1,N1,λ2,...,λK), (13) δ˜{1} = (0,λ2−(1− µλ1 )µ2,N2).
if T <∞, then 1,N1
1
Remark 5 (Weak and strong fluid limits). Though a quite
(cid:18) (cid:19)
λ subtle technical point, it is worth emphasizing the concep-
δ˜{1} =(0,λ −µ 1− 1 ,...,λ ), (14)
2 2,N2 µ K tual difference between the notion of weak fluid limits and
1,N1
7
the notion of strong fluid limit, introduced in Sections IV-A of time serving users that are not in their best states, and
and IV-B, respectively. Note that “weak” versus “strong” are therefore not maximum stable. For an example, we refer
refers to accumulation points versus unique limit. The names to Section VII where we numerically obtain the stability
do however not take into account the mode of convergence. conditionsforRBandthecµ-rule,makinguseofTheoremV.1.
Weak limits are a powerful tool for stability if one can Remark6(Overload). When(cid:80)K λ /µ >1thesystem
characterize that they all vanish after a finite amount of time, k=1 k k,Nk
issaidtobeinoverload.Thatis,theredoesnotexistanypolicy
as will be used in Theorem V.2 for the set of BR policies.
thatcanmakethesystemstable.TheoremIV.3ishoweverstill
However, in general weak limits might not capture the precise
applicable,providinguswiththeratesatwhichthenumberof
asymptoticbehavioroftheprocess(see[13]).Onthecontrary,
usersinthedifferentclassesgrow:givenx(0)=0,thegrowth
whenhavingauniquestrongfluidlimit,theasymptoticbehav-
rateofthenumberofusersovertimeisgivenbyXf,r(r)/r =
iorofthescaledprocessiscompletelydescribed,allowingfor
Yf,r(1), which in the limit is equal to δf,U. This gives us
exampletoobtainthepolicy-dependentstabilityconditionsfor
a mean to compare the performance of various policies in
a large class of policies (see Theorem V.1).
overload (see as well Sections VI and VII).
V. STABILITYANALYSIS VI. ASYMPTOTICFLUIDOPTIMALITY
The derivation of the weak and strong fluid limits in
Besides stability, another important performance measure
Section IV allows us to conclude for stochastic stability.
concerns the long-run average holding cost as given in (1).
The next Theorem derives the stability conditions for any
Deriving an optimal policy with respect to this criterion is
policy having a partially increasing drift with uniform limits,
difficult and the size of the state space makes the problem
using the strong fluid limit as obtained in Theorem IV.3.
intractable.Forthisreasonweintroducearelateddeterministic
TheoremV.1. Apolicyf inducingapartiallyincreasingdrift controlproblem,whichallowsustoprovethatanyBRPpolicy
vector field with uniform limits is stable if Tf <∞ for all l, is asymptotically optimal for the original stochastic system.
l
where Tf is given by Theorem IV.3. This emphasizes the important role of the tie-breaking rule in
l
order to achieve efficient performance of the system.
Proof: If Tlf < ∞ for all l, the strong fluid limit described We study the following deterministic fluid control model,
in Theorem IV.3 is equal to 0 for t large enough, i.e., Yf,r(t) whicharisesfromtheoriginalstochasticmodelbyonlytaking
converges in probability to 0 for t large enough. In addition, into account the mean drifts, i.e.,
the random variable Yf,r(t) is uniformly integrable. This can
k K
be seen by the fact that Ykf,r(t) can be upper bounded by min(cid:88)ckxuk(t), for all t≥0, subject to (18)
x (0) plus the users that have arrived until time (cid:98)rt(cid:99) divided u
k k=1
by r, which is uniform integrable, see [15, Lemma 4.5].
(cid:88)Nk (cid:90) t
Theconvergenceinprobabilityto0andtheuniformintegra- xu(t)=x (0)+λ t− µ u (v)dv, (19)
bility together imply that limr→∞E(Ykf,r(t))=0, for t large k k k n=1 k,n 0 k,n
enough, ∀ k, see [7, Theorem 3.5]. Using an extended Foster- xu(t)≥0, k =1,...,K, (20)
k
Lyapunovcriterionasexpressedin[18]or[28,Corollary9.8],
this implies the positive recurrence of Xf(·). (cid:50) (cid:88)K (cid:88)Nk
u (v)≤1, u (v)≥0, ∀ k,n,v ≥0, (21)
k,n k,n
k=1n=1
In the following theorem we state the maximum stability
andthecontrolfunctionsu (v)beingintegrable.Herexu(t)
condition, and prove that any BR policy achieves maximum k,n k
represents the amount of fluid in class k under control u(·).
stability.Theproofisbasedontheweakfluidlimitcharacteri-
We remark that though in general the fluid limit of a policy
zationasgiveninSectionIVandcanbefoundinAppendixB.
does depend on the distributions of the random environments
Theorem V.2. The maximum stability condition is (i.e.,theq ’s),thesedonotappearintheaboveequationsof
k,n
the fluid control model. This is because the fluid trajectory
K
(cid:88) λk <1. (17) xk(t) should be interpreted as a limit of the fluid-scaled
k=1µk,Nk process. Hence, when xk(t) > 0 this implies that there are
infinitely many class-k users so that with probability 1 there
In addition, any BR policy is maximum stable.
are class-k users in each of the channel state conditions (this
Condition(17)wasrecognizedasthemaximalratestability being independent of the exact values of the q >0’s).
k,n
conditionin[1]andasthemaximumstabilityconditionunder Anoptimalcontrolu∗(·)isderivedinthefollowinglemma.
a time-scale separation assumption in [9].
LemmaVI.1. Assumec µ ≥c µ ≥...≥c µ .
We note that SB, PI and PB are stable under the maximum 1 1,N1 2 2,N2 K K,NK
The fluid control u∗(·) that solves the fluid control problem is
stability conditions (they belong to the class of BR policies).
as follows. Let l=argmin{k :x (t)>0}. Then
The intuition behind Theorem V.2 is that asymptotically the k
scyosntseemrvinugndseyrstaemBRwhpeoreliccylasbsehkavheassadsepaartculraessipcraolbawboilrikty- u∗k,Nk(t)= µλk , for k <l, u∗l,Nl(t)=1−(cid:88)l−1 µλi ,
µ . On the contrary, other policies, including RB and the k,Nk i=1 i,Ni
k,Nk
cµ-rule, spend (at the fluid scale) a non-negligible fraction and u∗ (t)=0 otherwise.
k,n
8
Proof: Let us denote wu(t) = xu(t)/µ . First, we show that the lower bound holds in probability as well, i.e.,
k k k,Nk
that for any feasible control u(·), we have
K K
(cid:88)j wu∗(t)≤(cid:88)j wu(t), for all t≥0, j =1,...,K. (22) P(k(cid:88)=1ckYkf,r(t)−k(cid:88)=1ckxuk∗(t)≥0)→1, for all t≥0.
k k (24)
k=1 k=1
We define a policy to be asymptotically fluid optimal
If (cid:80)j wu∗(t) = 0, then (22) trivially holds. Now assume when the lower bound is obtained in probability, i.e.,
(cid:80)jk=1k=w1ku∗(kt) > 0. By definition of u∗(·) this implies that limr→∞P(|(cid:80)Kk=1ck(Ykf,r(s)−xu∗(s))| ≥ (cid:15)) = 0, ∀(cid:15) > 0.
(cid:80)j wu∗(s) > 0, for all s ∈ [0,t], since once all these The following Theorem characterizes a class of policies that
k=1 k
classes empty under u∗(t), they will remain empty. Since is asymptotically fluid optimal.
u∗(t) gives full priority to classes 1 until j over classes
Theorem VI.3. Any BRP policy is asymptotically fluid opti-
j+1 until K, we have that (cid:80)j (cid:82)tu∗ (v)dv =t. Hence,
(cid:80)(cid:80)jkjk==11(cid:82)(cid:80)0tnNu=k∗k1,N(cid:82)k0t(vµµ)kkd,N,vnku=k,nt(v≥)dkv(cid:80)=,1wjk=h0i1c(cid:80)hk,iNNnm=kkp1li(cid:82)e0tsu(k2,2n)(,vs)idnvce≥ mPopratoli.omfa:lWcoenhtraovleadsxdukde∗tr(itv)e=dinλkL−emum∗k,NakV(It.)1µ.kT,Nhkis,dwriitfhtcuo∗i(n·c)idthees
with the drift of the strong fluid limit yBRP(·), see Propo-
j j
(cid:88)wu(t)−(cid:88)wu∗(t) sition IV.4, hence yBRP(t) = xu∗(t). Together with The-
k k orem IV.3, we then obtain that lim (cid:80)K c YBRP,r(t)
k=1 k=1 r→∞ k=1 k k
=(cid:88)j (cid:90) tu∗ (v)dv−(cid:88)j (cid:88)Nk (cid:90) t µk,n u (v)dv ≥0. cisonavsyermgpestoitnicparlolybaflbuiliidtyotpoti(cid:80)maKkl=.1ckxuk∗(t),i.e.,anyBRpolic(cid:50)y
k,Nk µ k,n
k=1 0 k=1n=1 0 k,Nk
It can be checked that the above implies that any BRP policy
The minimization term (cid:80)Kk=1cknuk(t) can be written as minimizes liminf E((cid:82)∞(cid:80)c Yf,r(t)dt). Unfortunately,
r→∞ 0 k k
K this does not give any performance guarantee in terms of the
(cid:88)
c µ wu(t)=(c µ −c µ )wu(t) long-runtime-averageholdingcostasinEquation(1).Numer-
k k,Nk k 1 1,N1 2 2,N2 1
icalexperimentsreportedinSectionVIIindicatehoweverthat
k=1
+(c µ −c µ )(wu(t)+wu(t))+··· BRP policies significantly outperform all other policies.
2 2,N2 3 3,N3 1 2
Notethattheoptimalityresultsdescribedinthissectionalso
K−1
(cid:88)
+(c µ −c µ ) wu(t) applyinoverloadsystems.Wehavethefollowingcorollaryfor
K−1 K−1,NK−1 K K,NK k
the total growth rate.
k=1
K
(cid:88) Corollary VI.4. Any BRP policy minimizes the growth rate
+c µ wu(t).
K K,NK k of the total cost, i.e., for all (cid:15)>0,
k=1
Together with (22) and cjµj,Nj−cj+1µj+1,Nj+1 ≥0, ∀ j, we lim P(cid:32)(cid:80)Kk=1ckXkr,f(r) − (cid:80)Kk=1ckXkr,BRP(r) ≥−(cid:15)(cid:33)=1.
obtain that (cid:80)K c µ wu(t) is minimized by u∗(·). (cid:50) r→∞ r r
k=1 k k,Nk k
Proof: Combining (24) with the asymptotic optimality of a
The optimal fluid cost serves as a lower bound for the fluid-
BRP policy, the statement is immediate. (cid:50)
scaled cost of the stochastic network, see the lemma below.
Lemma VI.2. For any policy f and for almost all sample To the best of our knowledge, the only policy studied in the
paths, we have literature that belongs to BRP, and hence is both maximum
stable and asymptotically optimal, is PI. We recall that PI
K K
liminf(cid:88)c Yf,r(t)≥(cid:88)c xu∗(t), for all t≥0. (23) was derived in [4] as the solution of a relaxed optimization
r→∞ k k k k problem. SB and PB will as well become asymptotically
k=1 k=1
optimal when the myopic tie-breaking rule would be applied,
Proof: Lemma IV.1 states that for almost all sample paths ω
showing the importance of the tie-breaking rule.
it holds that liminf Yf,r(t) = yf(t), with yf(t) a
r→∞ k k k
weak fluid limit for policy f (this follows by considering Remark 7. From Theorem VI.3 we conclude that the myopic
the subsequence r corresponding to the liminf-sequence in tie-breakingruleiscrucialinordertoobtainanasymptotically
l
Lemma IV.1). Note that a weak fluid limit is an admissible fluid optimal scheduling policy. In view of equation (2), we
trajectory for the fluid control problem. Hence, note that under this myopic rule, higher priority is given
to users with smaller service requirements, E(B ). Hence,
k
liminf(cid:88)K c Yf,r(t)=(cid:88)K c yf(t)≥(cid:88)K c xu∗(t). BRP policies appropriately mix size-based information with
r→∞ k k k k k k achieving opportunistic gains. This is in agreement with the
k=1 k=1 k=1 findings of [30] where the authors investigate the tradeoff
This concludes the proof. (cid:50) betweenprioritizingsmallusersandopportunisticscheduling:
Theyshowthat iftheopportunisticcapacityisupperbounded
Since (23) holds almost surely, it follows by Fatou’s lemma and increases as 1−ax, with a ∈ [0,1) and x the number
9
Channel state (n) 1 2 3 4 5 6 7 8 9 10 11
Transmission rate (kb/s) in CDMA 38.4 76.8 102.6 153.6 204.8 307.2 614.4 921.6 1228.8 1843.2 2457.6
Probabilities in CDMA 0.00 0.01 0.04 0.08 0.15 0.24 0.18 0.09 0.12 0.05 0.04
q 0 0 0.05 0 0.23 0 0.42 0 0.21 0 0.09
1,n
q 0 0 0.15 0 0.33 0 0.52 0 0 0 0
2,n
µ 0 0 0.017 0 0.033 0 0.1 0 0.2 0 0.4
1,n
µ 0 0 0.017 0 0.033 0 0.1 0 0 0 0
2,n
TABLEI
TRANSMISSIONRATESANDCHANNELCONDITIONPROBABILITIESINTHECDMA1XEV-DOWIRELESSNETWORK,ASREPORTEDIN[5].
of users, then a significant improvement of performance can a) Fluid limit: We first illustrate how the scaled process
be achieved by exploiting information on the service time converges to the fluid limit. We take r = 10000, Yr(0) =
requirement. In our model the capacity has this behavior, X(0)/r = (1,1) and plot the scaled processes Yr(t), Yr(t),
1 2
see for instance Equation (9), and as will be observed in and Yr(t)+Yr(t) for different policies, see Figure 2. In this
1 2
thenumericalresults,exploitingsize-basedinformationindeed simulation we set λ =0.14, so λ /µ =0.35.
1 1 1,N1
allows to obtain significant improvements. Wedescribethefluidlimityf(t)asdefinedinTheoremIV.3.
When both classes are saturated, i.e., U =∅, the drift is
VII. NUMERICALEXPERIMENTS δf,∅ =(λ −αfµ ,λ −(1−αf)µ ), (25)
1 1,N1 2 2,N2
We consider a CDMA 1xEV-DO system with two classes see Remark 4. Here αf is a random tie-breaking rule, i.e., in
of users (K = 2). Time is slotted, with the length of one caseofatie,αf istheprobabilitythatclass1isfavouredover
slot being t = 1.67ms. In each time slot, one new class-k
c class 2. For our set of parameters, the best class-1 user under
user arrives with probability λ . We choose 10.257 kb as the
k the cµ-rule and RB is always preferred over the best class-
expectedservicerequirementofbothaclass-1andclass-2user. 2 user, i.e., there occur no ties, hence one can set αf = 1
Associated to the state of the channel, we have transmission
in (25) for f = cµ,RB. For PI, SB and PB we do have
rates (kb/s), see Table I (taken from [5]). We assume that ties, and we set αPI = 1 (since PI applies the myopic tie-
class-1usershavefivepossibletransmissionrateswhileclass- breakingrule)andαSB =αPB =1/2(sinceSBandPBapply
2 users have three. The corresponding probabilities (q )
k,n a random tie-breaking rule). In Table II we present the so-
are given in Table I. In addition, applying equation (2) we obtainedvaluesforδf,∅.Fromthedriftsitisclearthatunderall
calculate the departure probabilities (µ ). We fix λ =0.05,
k,n 2 policies class 1 empties before class 2. The moment that this
sinoteλre2s/tµed2,Nin2 m=in0im.5i.ziWngetsheet ecx1pe=ctecd2to=tal1,nusombtheratowf eusaerres hTaPpIpe=nsT, cTµ1f=, cTanRBbe<deTriSvBed=froTmPBT,hseeoereamlsoIVF.3igaunred2saati)s.fies
in the system, see Equation (1). In addition, we compare the 1For Tf1 < t1≤ Tf,1the dri1ft of class 1 is 0, whereas
performance of the policies SB, RB, PI, PB and the cµ rule, 1 2
the drift of class 2 is going to depend on the policy. From
which were introduced in Section III.
Proposition IV.4 we have that for all BR policies (e.g. PI, SB
Before presenting the numerical results we first summarize
and PB) δf,{1} = (0,λ −µ (1−λ /µ )). For the cµ
the main conclusions that we will make in this section: 2 2,N2 1 1,N1
ruleandRBwecalculatethedriftnumericallyusingRemark3.
• We prove that not all policies obtain maximum stability. In particular we observe that these drifts are positive for the
Moreprecisely,wecalculatethestabilityconditionsunder latter two policies, which implies instability of the system, as
RBandthecµ-ruleandobservethatthesearemuchmore can be seen in Figure 2 b). We observe that for t ≤ Tf the
1
stringentthanthestabilityconditionforBRpolicies(e.g. number of class-2 users increases under policies PI, cµ and
SB, PB and PI). RB,whileforSBandPB,thedriftofclass-2usersisnegative.
• The drifts of the fluid limit, δU, which are calculated A direct consequence of the drift function is that SB, PB,
numerically (and in some cases theoretically), provide and PI (in fact all BR policies) empty the system at the
veryimportantinsightsontheperformance.Inparticular, same time (under the maximum stability condition), i.e., Tf
2
insightfulmonotonicityresultsinthetie-breakingruleare is the same (this can be seen directly from Equation (45) for
obtained with respect to the performance of the system. example). However, the performance of a policy will depend
In addition, the drift analysis allows to show that PI on the order in which classes are served. In the fluid limit,
outperformsallotherpoliciesintermsofthegrowthrates. this is fully determined by the choice of the tie-breaking rule.
• Our simulations illustrate that the tie-breaking rule has Notethat,ascanbeseenfromFigure2c),PI,(andhenceany
a very big impact on the performance of the system BR policy with the myopic tie-breaking rule) minimizes the
and we find that combining opportunistic scheduling total number of users at any moment in time.
with the myopic tie-breaking rule gives optimal mean b) Stability region: We now vary the value of λ from
1
performance, as was also suggested by the asymptotic 0.004 to 0.196, and as a consequence we have that ρ :=
fluid optimality result of BRP policies in Section VI. λ /µ +λ /µ varies from 0.51 to 0.99. The policies
1 1,N1 2 2,N2
10
Fig.2. (a)Scalednumberofclass-1users,(b)Scalednumberofclass-2users,(c)Scaledtotalnumberofusers
Fig. 3. (a) Mean number of users and stability thresholds, (b) PI under different tie-breaking rules: relative degradation (in %) over PI with α = 1, (c)
Scaledtotalnumberofusersinoverload,ρ=1.1.
δf, δf, 1 In order to investigate this issue in more depth, we simulate
∅ { }
f Class1 Class2 Class1 Class2 PI under different random tie-breaking rules, i.e., we let the
probability α vary from 0 until 1 (Recall that the parameter
PI -0.26 0.05 0 -0.015
cµ-rule -0.26 0.05 0 0.0096 α is the probability, in case of a tie, that class 1 is favoured
PB/SB -0.06 0 0 -0.015 over class 2). We emphasize that PI as defined in [4] uses
RB -0.26 0.05 0 0.0004 by default the myopic tie-breaking rule, i.e., αPI = 1. In
TABLEII Figure 3 b) we plot the relative degradation (in terms of the
DRIFTOFTHEFLUIDLIMIT. meannumberofusers)overPIaswevaryα.Weobservethat
the degradation in the mean performance is decreasing as α
increases.Inthefluid-scalingsystemasimilarobservationcan
δf, δf, 1
∅ { } bemade:from(25)andRemark4itfollowsthatthedriftofthe
f Class1 Class2 Class1 Class2 total number of users of the strong fluid limit (as obtained in
PI -0.16 0.05 0 0.01 Theorem IV.3) is decreasing (or constant) in α, for all time t.
cµ-rule -0.16 0.05 0 0.036 The results show that the myopic tie-breaking rule, which
PB/SB 0.04 0 - -
wasproventobeasymptoticallyoptimalinthefluidlimit(see
RB -0.16 0.05 0 0.029
TheoremVI.3),isinpracticeindeedoptimalwhenminimizing
TABLEIII
themeannumberofusers.Inaddition,therelativedegradation
DRIFTOFTHEFLUIDLIMITINOVERLOADSYSTEM.
ofthetie-breakingrulewithα=1/2(comparedtothemyopic
tie-breaking rule αPI = 1) can be very large. For example,
for ρ=0.8 the degradation is 29% and for ρ=0.9 it is 45%.
PI,PBandSBbelongtotheBRpolicies,andarehencestable d) Overload: In Figure 3 c) we plot a trajectory of the
when ρ < 1. For the cµ-rule and RB the stability condition total scaled number of users Yr(t) + Yr(t) (for r = 100)
1 2
canbecalculated(numerically)bysettingδf,{1} equaltozero when λ =0.240 and ρ=1.1, so all policies are unstable. In
2 1
and following the steps described in Remark 3. In particular, TableIIIwegivethevaluesforthedrifts(growthrates)ofthe
thecµ-ruleisstableifandonlyifρ<0.79andRBisstableif fluid limit yf(t) in this overload setting. In this example, the
andonlyifρ<0.84.InFigure3a)weplotthemeannumber worstperformanceisunderSBandthebestperformanceisfor
ofusersfordifferentvaluesofρandweobservethatthemean PI.ThisincontrarytothestableregimewhereSBismaximum
numberofusers(andhencethemeandelay)forthesepolicies stable with a performance strictly better than the cµ-rule and
grows to infinity as the load approaches the critical value. RB, see Figure 3 a). This implies that the performance of
c) Impact of Tie-Breaking rule: We study the impact this policy can differ very much between stable and overload
of the tie-breaking rule on the performance of the system. regimes. In addition, we note that the total growth rate of the