Content uploaded by Enzo Baccarelli
Author content
All content in this area was uploaded by Enzo Baccarelli
Content may be subject to copyright.
3294 IEEE TRA NSACTI ONS ON VEH ICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010
Conditionally Optimal Minimum-Delay Scheduling
for Bursty Traffic Over Fading Channels
Enzo Baccarelli, Nicola Cordeschi, and Mauro Biagi, Member, IEEE
Abstract—Next-generation wireless networks for personal com-
munication services should be designed to transfer delay-
sensitive bursty-traffic flows over energy-limited buffer-equipped
faded connections. In this application scenario, a still-open ques-
tion concerns the closed-form design of scheduling policies min-
imizing the average transfer delay under constraints on both
average and peak energies. Since, in this paper, both queue and
link states may assume finite, countable infinite, or even uncount-
able infinite values, we cannot resort to dynamic programming
to solve the aforementioned minimization problem. The key point
of the (somewhat) novel approach that we follow consists of the
minimization (on a per-step basis) of the queue length averaged
over the fading statistics and conditioned on the queue occupancy
at the previous step when two energy constraints are considered.
The first one is on the allowed peak energy, and the second one is
on the available average energy conditioned on the current queue
occupancy. The resulting optimal scheduler operates cross layer,
meaning that it allocates step-by-step energy on the basis of both
current queue and link states. We prove that, under the con-
sidered energy constraints, the scheduler retains two optimality
properties. First, its stability region is the maximal admissible one.
Second, the scheduler also minimizes the unconditional aver-
age queue length. The numerical tests that have been carried
out corroborate these optimality properties and give insight
about scheduler performance under heavy-tailed distributed input
traffic, such as that generated by variable-bit-rate (VBR) media
encoders.
Index Terms—Cross-layer scheduling, delay-sensitive bursty
traffic, fading channels, heavy-tailed traffic, queue management,
resource management.
I. INTRODUCTION AND GOALS
RECENTLY, there has been widespread interest in the
extension of delay-sensitive multimedia applications to
the mobile domain by exploiting wireless connections [1]. The
development of multimedia portable terminals (such as smart
phones, notebooks, and personal digital assistants) is triggering
the proliferation of mobile multimedia services. However, real-
time multimedia applications tend to be delay sensitive so that
wireless connections are able to provide only limited support
for multimedia. Currently, two of the major challenges in
Manuscript received March 6, 2009; revised July 31, 2009, December 15,
2009 and April 15, 2010; accepted April 24, 2010. Date of publication June 7,
2010; date of current version September 17, 2010. This work was supported
in part by the Italian National Project “Wireless multiplatfOrm mimo ac-
tive access netwoRks for QoS-demanding muLtimedia Delivery (WORLD)”
under Grant 2007R989S. The review of this paper was coordinated by
Prof. W. A. Krzymien.
The authors are with the INFOCOM Department, Università degli Studi
di Roma “La Sapienza,” 00184 Rome, Italy (e-mail: enzobac@infocom.
uniroma1.it; cordeschi@infocom.uniroma1.it; biagi@infocom.uniroma1.it).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVT.2010.2051824
supporting real-time mobile applications over wireless connec-
tions concern energy management at the mobile devices and
minimization of end-to-end average transfer delay [1]. Since
several multimedia services generate (possibly heavy-tailed
distributed) bursty traffic that tolerates delays (in the specific
application-dependent range), in principle, these features could
be exploited to design cross-layer adaptive resource-allocation
policies that jointly account for the states of both the physical
and data-link layers. The ultimate target, in a single-queue prob-
lem perspective, is to optimize the overall system performance
in terms of average transfer delay under constraints on the
available energy budget. This is the focus of this paper.
A. Tackled Problem
Specifically, after modeling the transmit node as a time-
slotted loss-free fluid GI/G/1 queuing system (see [2, ch. 6] and
[3, ch. 10] for a general introduction to this queuing model),
the problem we will examine deals with the closed-form cross-
layer (e.g., queue and link-state aware) design of the scheduler
performing optimal energy and rate allocation.
In our framework, the queue and link states are modeled as
random variables (RVs), which may be of discrete, continuous,
or even mixed type. This means that the number of values
allowed for queue and link states may be finite, countable
infinite, or even uncountable infinite, so that approaches based
on dynamic programming (DP) would lead to unimplementable
(due to infinite complexity) schedulers [5].
Therefore, since the main goal is to obtain a closed-form
expression for the single-user optimal scheduler, and, until now,
no closed-form analytical formulas have been available for
the probability density function (pdf) of the state of GI/G/1
queueing systems [2], [6], [7], the target that we pursue is
the minimization (on a per-slot basis) of the current queue
length averaged over the fading statistics and conditioned on
the queue occupancy measured at the previous slot. (In short,
the considered target is the minimization of the conditional
average queue state.) In our framework, this target must be
accomplished under two energy constraints. The first one is on
the available energy per slot averaged over the fading statistics
and conditioned on the current queue state. The second con-
straint that we consider is on the allowed peak energy per slot.
It is dictated by the emission limits typically imposed by the
physical layer of the considered wireless system and may arise
from spectral compatibility issues and/or from the maximum
throughput allowed by the adopted encoder/modulator [8].
Overall, the resulting constrained optimization problem that we
will tackle is an instance of optimal resource allocation for
0018-9545/$26.00 © 2010 IEEE
BACCARELLI et al.:MINIMUM-DELAY SCHEDULING FOR BURSTY TRAFFIC OVER FADING CHANNELS 3295
energy-limited faded connections under bursty delay-sensitive
input traffic.
B. Related Works and Main Contributions of this Paper
Optimized joint-energy/queue control policies derived by
exploiting the analytical tool of the Markov decision process
(MDP) and implemented via DP are presented, e.g., in [9]–
[13] and, more recently, in [14] and [15]. Specifically, in [9], a
discrete-state faded link has been considered, and the structural
properties of the scheduling policy that optimizes the energy-
versus-delay tradeoff were provided. For the same application
scenario, closed-form expressions for the optimal policy in
terms of signal-to-interference ratio have been presented in
[10]. References [11] and [13] resorted to the queuing theory
to develop optimized scheduling policies for real-time traffic
under hard constraints on the packet deadlines. However, all
these contributions do not account for constraints on the al-
lowed average and/or peak energies and focus on finite-state
Gilbert–Elliot models. In [14] and [15], the problem of optimal
transmission scheduling over time-correlated finite-state faded
channels has been modeled as a cross-layer MDP and then
iteratively solved by resorting to the DP tool. Furthermore,
some structural properties of the optimal scheduler that give
insight about the effects of the temporal correlation of the fad-
ing phenomena on the scheduler’s behavior have been derived
in [14] and [15]. In these contributions, the state of the fading
channel, the transmit queue, and the number of transmit rates
allowed by the scheduler have been assumed to be finite values.
Furthermore, no constraints on the average and/or peak energies
are taken into account.
Perhaps, works exploiting approaches that are closest to that
followed in this paper are those based on the analytical tool
of convex optimization and nonlinear programming. Collins
and Cruz [16] considered the effects of traffic burstiness for
two-state Gilbert–Elliot channels under a constraint on the
average delay and adopted a linear rate function to measure
the throughput conveyed by the system. Fu et al. [17] resorted
to the convex optimization tool for solving the dual problems
of maximizing the average number of transmitted information
units (IUs) under a limited energy budget and minimizing the
total energy needed to transfer an assigned number of IUs. Both
problems have been solved in [17] in the presence of hard-
deadline constraints. The dual problems of [17] differ from
those tackled in this paper, mainly because the queuing aspect
is not addressed and the pursued approach is not of cross layer.
Explicit deadlines for the packets to be transmitted are also con-
sidered in [18]–[22]. Specifically, Schugers et al. [18] analyzed
energy-efficient real-time packet scheduling for time-invariant
channels with uncoded M-ary quadrature amplitude modula-
tions. This work used a sufficient schedulability test to assure
no deadline misses. With the condition being only sufficient, it
is possible to reject more packets than necessary. A heuristic
algorithm has been proposed in [18] to carry out the energy
scheduling. In [19], a hard maximum delay constraint for each
packet was considered, and then, the connection between maxi-
mum delay scheduling and linear filtering for time-uncorrelated
input-traffic flows was established. Mangharam et al. [20]
focused on providing quality-of-service (QoS) support for mul-
tiple users with bursty MPEG-4 video transmissions over time-
varying channels. This work also used deadline miss rate as
QoS requirement and proposed an online greedy algorithm to
satisfy the constraints advanced by multiple clients. Zafer and
Modiano [21] developed a generalization of energy-efficient
packet scheduling based on the calculus approach. Specifically,
in [21], an offline generalization of the energy minimization
problem and an online scheduler for Poisson-distributed ar-
rivals have been presented. An offline algorithm has also been
proposed in [22] for Gaussian transmission channels with the
proof of optimality under individual packet delay. This al-
gorithm is a generalization of [11], with the group deadline
replaced by individual deadline. In the recent contribution [23],
a delay-scheduling problem where a single packet of Bbits
must be transmitted over a continuous-state block-faded chan-
nel by a hard deadline of T≥2slots has been considered. The
objective is the minimization of the (unconditional) average
total energy, and the considered rate function is the logarithmic
one. In [23], the optimal scheduler has been presented in closed
form for the case T=2, whereas closed-form expressions for
suboptimal schedulers have been presented for the (general)
case of T≥3. Since, in [23], the transmission of a single
packet has been considered, the effects of random arrivals have
not been considered at all. Furthermore, no constraints on the
average and/or peak energies are taken into account.
The effects of automatic repeat request (ARQ) strategies
possibly implemented at the data-link layer are explicitly con-
sidered in [24], where the delay-versus-energy tradeoff of a
wireless file-fetch system is addressed by resorting to linear
energy and delay cost functions for characterizing the system
performance. However, in [24], no constraints on the available
average and peak energies are provided, and therefore, the
resulting optimization problem is, indeed, unconstrained. Thus,
from the outset, the main contributions of this work may be so
summarized.
1) First, we develop closed-form expressions for the optimal
scheduler of the tackled cross-layer single-queue opti-
mization problem, and we characterize the corresponding
steady-state performance via (tight) closed-form upper
and lower bounds.
2) Second, after developing a necessary and sufficient con-
dition for the strong stability of the optimal scheduler,
we prove that the corresponding stability region is the
maximal one under the considered energy constraints.
3) Third, we prove that, under the considered energy
constraints, the developed scheduler also minimizes the
unconditional average queue length.
4) Fourth, we present an “on-the-fly” implementation of the
optimal scheduler that require no a priori knowledge
about fading and input-traffic pdfs.
5) Finally, we test the actual performance of the opti-
mal scheduler on Rayleigh-faded channels under heavy-
tailed input-traffic streams, such as those generated by
variable-bit-rate (VBR) media encoders. Performance
comparisons with DP-based schedulers [9], [12], [16] are
also provided.
3296 IEEE TRA NSACTI ONS ON VEH ICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010
Fig. 1. Considered client–server system model.
C. Outline of the Paper
The rest of this paper is organized as follows: After intro-
ducing the system model and the problem setup in Section II,
the optimal scheduler and its basic structural properties are
presented in Section III. Section IV deals with the stability
region of the scheduler, whereas, in Section V, we prove the
aforementioned unconditional optimality of the scheduler under
constraints on the available average energy conditioned on the
queue occupancy. Then, we detail the expressions assumed
by the optimal scheduler for some rate functions of practical
interest (i.e., the α-powered and logarithmic ones), and in
Section VI, we test the actual performance and robustness
properties of the proposed scheduler on several application
scenarios. Some conclusive remarks are drawn in Section VII,
whereas analytical proofs of the main results are deferred to the
Appendices.
About the adopted notation, underlined letters denote
vectors, and scalar random variables are denoted by bold
characters, whereas their outcomes are indicated by the corre-
sponding unbolded symbols. E{.}is the expectation operator,
R+
0is the set of nonnegative real numbers, R+is the set of
strictly positive real numbers, Δ
=means “equal by definition,”
and [x]+indicates max{x, 0}.P(A)is the probability of the
event A,pσ(σ)is the pdf of the RV σ, and Eσ{ϕ(σ;s)}Δ
=
ϕ(σ;s)pσ(σ)dσ denotes the expectation of the bi-argumental
function ϕ(σ;s)carried out only over the pdf of the RV σ.
Finally, [f(x)]b
aindicates max{a; min{f(x); b}}, and log(.)is
the natural logarithm.
II. SYSTEM MODEL AND PROBLEM SETUP
By referring to the server–client system architecture shown
in Fig. 1 [1], we consider a discrete-time (i.e., slotted) GI/G/1
fluid queue model [2, ch. 6], [3, ch. 10], where the slot length
is unit, and slot tspans the (semi-open) interval [t, (t+ 1)),
t∈N+
0. The IUs to be transmitted arrive at the queue system
from a VBR encoder at the end of each slot and are buffered.
According to the general GI/G/1 queue model described, e.g.,
in [2, ch. 6] and [3, ch. 10], the arrival process a(t)∈R+
0,t≥
0, is assumed to be an independent and identically distributed
(i.i.d.) random sequence, with pdf pa(a)and average intensity
λΔ
=E{a(t)}(in IU per slot). The fading phenomena affecting
the wireless link are assumed to be constant (time invariant)
over each slot (the so-called block-fading model is assumed [8])
and are considered i.i.d. from slot to slot. Thus, the link state
σ(t)∈R+
0over slot tis modeled as a real nonnegative RV with
pdf pσ(σ). Furthermore, the link-state value σ(t)is assumed to
be perfectly known at the transmitter at the beginning of slot t,
so that, slot by slot, perfect link-state information is available
at the transmit node of Fig. 1. Let s(t)∈R+
0be the number
of IUs (i.e., the amount of fluid) buffered in the queue at the
beginning of slot t(see Fig. 1). Thus, after denoting by r(t)
(in IU per slot) the number of IUs transmitted over the tth slot,
because data are assumed to be arriving at the end of each slot,
as in [10], we can assume r(t)≤s(t)so that Lindley’s equation
[2, p. 275], [3, ch. 10]
s(t+1)=s(t)+a(t)−r(t),t≥0(1)
dictates the evolution of the (discrete-time) queue-length
process {s(t)∈R+
0,t≥0}. According to the seminal result of
Loynes [2], [6], under suitable stability conditions, we detail
in Section IV that there exists a unique stationary solution to
the recursion (1), and for any initial condition s(0), the queue
length process {s(t)}in (1) converges (in finite time) to this
stationary process [2], [6], [7]. Thus, unless otherwise stated,
in the following, we assume that the queue operates under
the stationary regime so that {s(t)}indicates the stationary
and ergodic solution to the recursion in (1), and ps(s)is the
corresponding steady-state pdf.
The cost to drain r(t)units of fluid at slot tis the amount
of energy E(t)(in Joules) required for their transmission. Thus,
we may assume that the corresponding number IU(t)of IUs
forwarded over the link of Fig. 1 over slot tdepends on both
E(t)and the link state σ(t)via the rate function R(.;.)adopted
to measure the goodput performance of the considered system
so that we can write
IU(t)Δ
=R(E(t); σ(t)) ,t≥1.(2)
The rate function R(.;.)in (2) is a real nonnegative function,
depending on two nonnegative real arguments E(.)and σ(.),
and it is measured in IU per slot. Roughly speaking, R(.;.)
summarizes the goodput performance of the considered system,
so that its behavior and analytical properties may depend on
several system parameters, such as the requested QoS, the
employed coding gain, the fading statistics, the performance
of the ARQ strategy possibly implemented at the data-link
layer, etc. Therefore, in the following, we limit to introduce
a few (quite mild) assumptions on R(.;.), which, in fact, are
retained by several rate functions of practical interest (see, e.g.,
[25, ch. 2 and 4]).
First, the rate function R(E;σ)is continuous on R+
0×
R+
0, and its first- and second-order derivatives are continuous
on R+×R+
0. Second, it vanishes at E=0 and σ=0, i.e.,
R(E=0;σ)≡R(E;σ=0)≡0. Third, it is nondecreasing
for both E≥0and σ≥0. Fourth, for any assigned σ=0,
the rate function is assumed to be strictly concave in the
Evariable, i.e., Rεε(E;σ)Δ
=∂2R(E;σ)/∂E2<0,forE>0
and σ=0. Finally, we assume that its first-order derivative
BACCARELLI et al.:MINIMUM-DELAY SCHEDULING FOR BURSTY TRAFFIC OVER FADING CHANNELS 3297
Rε(E;σ)Δ
=∂R(E;σ)/∂Eperformed with respect to the Ear-
gument is nondecreasing in the σargument for σ≥0.
Several relevant examples of rate functions of practical inter-
est [25, ch. 2, 4] meeting the aforementioned assumptions will
explicitly be considered in Section V-B.
Remark 1—About the Validity Limits of the Considered Sys-
tem Model: The previously introduced block-fading assump-
tion holds when the coherence time of the wireless connection
of Fig. 1 is longer than the time duration of a slot. Since
typical slot times for packet transmission are limited up to
1–2 ms in current third-generation/fourth-generation wireless
systems, then the block-fading assumption may be considered
met at medium/low terminal speeds [8], whereas the i.i.d.
assumption on the link-state random sequence {σ(t)}is well
met by time-division multiple access, frequency hopping, or
packet-based interleaved systems of practical interest, where
each transmitted IU is independently detected by the client
terminal. This assumption will be relaxed in Section VI, where
we test the performance robustness of the proposed scheduler
against fading-correlation effects possibly arising, e.g., in actual
mobile applications.
From a networking point of view, the system in Fig. 1
is assumed to manage a single end-to-end connection. When
multiple connections share the same bottleneck (wireless) link,
in principle, fair service disciplines (such as, e.g., the weight
fair queueing or time fairness [1, ch. 12], [26]) may be imple-
mented at the switching/multiplexing nodes to provide isolation
among competing flows so that the model and the analysis we
will develop can still be considered valid. The analysis of the
effects induced by the (possible) greedy (e.g., unfair) behavior
of concurrent flows is outside the scope of this paper.
A. Setup of the Optimization Problem
Since the average delay is related to the average queue
length by Little’s formula [2, p. 10], the goal that we will
pursue is the minimization of the average queue length. Specif-
ically, after indicating by x(t)Δ
=[σ(t),s(t)] ∈(R+
0)2the bi-
argumental state of the overall system in Fig. 1, the scheduling
problem that we focus on deals with the optimal design of the
number r(t)(in IU per slot) of IUs to be transmitted across
the connection over slot twhen x(t)is the current system state.
The ultimate goal is the minimization of the conditional average
buffer length E{s(t+1)|s(t)}under constraints on both the
available conditional average energy per slot Eave (in Joules)
and the allowed peak energy Ep(in Joules). To properly state
the mentioned scheduling problem, let
E(t)≡ε(σ(t); r(t)) Δ
=R−1(σ(t); r(t)) ,t≥1(3)
be the energy requested to transmit r(t)IUs when the link state
is σ(t).1Therefore, the considered optimization problem may
1Equation (3) can directly be obtained by inverting the rate function in (2)
with respect to the Evariable. As a direct consequence of the aforementioned
properties assumed on R(.;.), it follows that the energy function is nondecreas-
ing and convex in the rvariable.
formally be stated as follows:
min
r(t)E{s(t+1)|s(t)}(4)
s.t.: E{ε(σ;r(t)) |s(t)}≤E
ave (4.1)
ε(σ(t); r(t)) ≤E
p(4.2)
where the number r(t)of delivered IUs that minimizes the
objective function in (4) is searched over the set of deterministic
stationary policies, depending on both the current queue and
link states, i.e.,
r(t)≡r(σ(t); s(t)) .(5)
Before proceeding, an explicative remark about the considered
scheduling problem is in order.
Remark 2—Some Considerations About the Considered
Optimization Problem: From a practical point of view, sev-
eral contributions (see, e.g., [3, ch. 13 and 17] and refer-
ences therein) support the utilization of the conditional average
queue’s backlog in (4), as an effective metric for characteriz-
ing the performance of queueing systems fed by heavy-tailed
distributed input traffics [7]. This is, indeed, the case of traffic
flows generated by VBR MPEG-type video encoders, which
typically output Pareto-like distributed coded streams [1, ch. 2
and 5], [3, ch. 12 and references therein]. Inspired by these
contributions, in (4), we minimize the expected queue length
s(t+1) at the next slot (t+1), given the number s(t)of
IUs buffered at the current slot t. From an analytical point of
view, the reason for considering the conditional expectation in
(4) in place of the—perhaps more conventional—unconditional
expectation [9], [10], [12] E{s(t)}Δ
=s
sps(s)ds is that closed-
form computation of the latter requires that the (steady-state)
pdf ps(s)of the queue state also be analytically available. In
turn, this requires solving the corresponding Lindley’s equation
[2], [6]
ps(s)=
s∗
σ
pa(a=s−s∗+r(σ;s∗)) pσ(σ)ps(s∗)dσ ds∗
which, unfortunately, resists a closed-form solution (see, e.g.,
[26] for some recent remarks on this question). For the same
reason, in (4.1), we consider a constraint on the available aver-
age energy conditioned on the current queue state s(t), in place
of the (more conventional) unconditional average constraint [9],
[10], [12] E{ε(σ;s)}≤E
ave.
About the optimal scheduler, for the problem at hand, it
can be proven (see, e.g., [9] and [10]) that an optimal deter-
ministic stationary policy not depending on the previous link
and buffer states always exists. Furthermore, since the arrival
process is assumed to be i.i.d., the stationary policy will not
directly depend on arrivals [9]; therefore, we can restrict the
attention to the policies of type (5). In Section V, we detail the
link among the (closed-form) solution ropt(.;.)of the condi-
tional optimization problem in (4)–(4.2) and the corresponding
(DP-based) unconditional optimization problem addressed,
e.g., in [9], [10], and [12].
3298 IEEE TRA NSACTI ONS ON VEH ICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010
B. System Comparison
About the framework that we considered, aside from mainly
theoretical contributions, the interesting work in [27] developed
a control strategy for the problems of minimizing energy con-
sumption in a wireless network with adaptive transmission rates
under weak stability and the related problem of maximizing
the average throughput subject to peak and average energy
constraints by dropping all new arrivals exceeding a parameter-
controlled threshold to make the system strongly stable. In this
work, a somewhat novel Lyapunov drift technique is developed,
which enables stability and performance optimization to be
simultaneously pursued. This technique extends the Lyapunov
methods and bridges the gap between the convex optimization
theory and stochastic queuing control problems. However, [27]
made a set of assumption that does not allow applying its results
to our framework. Specifically, it assumed a finite number
of channel-state vectors and a compact set for the acceptable
power vectors. These assumptions are fundamental in [27] and
cannot easily be removed. In fact, without the assumption of
finite channel-state space, the basic problem in [27, Th. 1] can-
not be formulated, and without compactness, the result stated
in [27, App. A] about the existence of limiting probabilities
and the following minimum power statement can no longer be
proven. Conversely, in our work, both queue and link states may
assume finite, countable infinite, or uncountable infinite values;
the peak energy constraints in charge for compactness can be
relaxed, and our framework and optimality statements still hold.
Furthermore, since the schedulers in [27] are not derived by
direct solution of the minimum energy and maximal throughput
optimization problems but via an application of the novel
Lyapunov condition of [27, Lemma 1], the author claims that
the performance bounds provided by [27, Th. 2 and 3] are
fundamental to prove the “almost” optimality results of sched-
ulers, meaning that schedulers can perform arbitrarily close
to the optimum average performances. In this regard, when
the adopted rate function and/or the average quadratic value
of input are unbounded, the constant Bin [27, eq. (4)] and
(consequent) performance bounds are not finite. Furthermore,
nothing can be said about the (weak) stability and average
power consumption of the proposed scheduler for the average-
energy-control algorithm in [27, Sec. III] and the average power
constraint and average throughput performances for the average
energy-constraint problem [27, Sec. IV], respectively. Due to
the preceding argumentations, the controller in [27] is not able
to work with multimedia bursty traffic, as, e.g., the heavy-
tailed traffic generated by video encoders, which is usually
hard to cope with. On the other hand, in the following, we
show that, even if the upper bounds in (25) and (26) for the
average steady-state buffer length are no longer useful and we
are no longer able to prove strong stability for these kinds of
unbounded inputs, our scheduler continues to provide weak
stability by meeting, at the same time, the energy constraints
(see Remark 3 and Appendix C for the proofs). Furthermore,
by numerical evaluation, in Section VI, we show that, at least
in the simulation scenarios that we consider, our scheduler
continues to be, in fact, strongly stable. Finally, [27] also
proposed an interesting mechanism for considering the average
power constraint via a virtual power-queue model for maximum
throughput maximization with admission control (this simpli-
fies the problem and makes it easier to provide system stability),
by adding the hypothesis on the finiteness of the maximum
first derivative of the rate function with respect to power for
any possible channel state. This is not true, e.g., when the rate
function is a logarithmic or a linear function and the channel
state σpresents an unbounded support (e.g., Rayleigh channel).
In the following, we do not assume finite channel states, finite
maximum rate functions, or first-derivative rate function values.
III. OPTIMAL SCHEDULER MINIMIZING THE
CONDITIONAL AVERAGE QUEUE LENGTH
We now recast in an equivalent (but simpler) form the opti-
mization problem in (4). Specifically, by exploiting Lindley’s
equation in (1), we arrive at Proposition 1.
Proposition 1: The constrained optimization problem in
(4)–(4.2) may be stated in the following equivalent form:
max
r(·)E{r(σ(t); s(t)) |s(t)}(6)
s.t.: E{ε(σ(t); r(σ(t); s(t)) |s(t))}≤E
ave (6.1)
0≤r(σ(t); s(t)) ≤rp(σ(t); s(t)) (6.2)
where rp(σ(t); s(t)) Δ
= min{s(t),R(σ(t); Ep)}is the allowed
peak throughput when the system state is x(t)≡[σ(t),s(t)].
Proof: From Lindley’s equation in (1), we derive the
following expression for the average buffer length conditioned
on s(t):E{s(t+1)|s(t)}=E
σ{u(t)+λ}=E
σ{u(t)}+λ,
where λΔ
=E{a(t)}, and u(t)Δ
=(s(t)−r(t)) denotes the
amount of data to still be served at slot t. The last relationship
allows us to rewrite the objective function (4) as in (6). Thus,
after expressing the constraint in (4.2) on the peak energy as a
function of the scheduled rate r(.)and recalling the assumption
r(t)≤s(t), we arrive at
r(σ(t); s(t)) ≤min s(t),ε
−1(σ;Ep)≡
≡min {s(t),R(σ;Ep)}Δ
=rp(s(t); σ(t)) (7)
which indeed proves (6.2).
Therefore, since, by inspection, it may be recognized that
the optimization problem in (6)–(6.2) is indeed an instance
of convex constrained optimization problem [28], the resulting
solution ropt(.;.)may be computed in closed form, as detailed
by Proposition 2 (see Appendix A for the proof).
Proposition 2: Under the previously reported assumptions,
the optimal scheduler ropt(σ(t); s(t)) solution of the con-
strained problem in (6)–(6.2) is given by
ropt(σ(t); s(t))≡rp(σ(t); s(t)) ,for s(t)<s1
ε−1
rσ;1
μ(s(t)) rp(σ,s(t))
0,for s(t)≥s1
(8)
where ε−1
r(.;.)denotes the inverse function of the first r
derivative of ε(.;.). Furthermore, the threshold s1in (8) marks
the “low-occupation” buffer region from the “high-occupation”
BACCARELLI et al.:MINIMUM-DELAY SCHEDULING FOR BURSTY TRAFFIC OVER FADING CHANNELS 3299
buffer region, and it is computed by solving the following
algebraic equation:
σ
ε(σ;rp(σ;s1)) pσ(σ)dσ =Eave.(9)
Finally, μ(s(t)) in (8) is the optimal value of the dual variable
of the optimization problem, and it may be computed by solving
the following (functional) equation:
σ
εσ;ε−1
rσ;1
μ(s(t))rp(σ;s(t))
0pσ(σ)dσ =Eave
(10)
for s(t)≥s1.
A. Basic Properties of the Optimal Scheduler
The optimal scheduler in (8) retains some basic properties
that characterize its actual behavior in faded environments.
Since, in our framework, both link and queue states may
assume even uncountable infinite values and the expectations in
(4)–(4.1) are conditioned on the queue occupancy, these basic
properties are indeed formally different from the corresponding
properties reported, e.g., in [9], [12], and [16], and neither
may be derived by exploiting the (usual) tool of the MDP (see
Appendix B). In this respect, we anticipate that the concavity
property and the limit behavior presented by Propositions 3
and 4are not retained by the schedulers derived in [9], [12],
and [16].
In detail, as a consequence of the monotonic property of rate
function R(.;.)(see Section II), it can directly be viewed that,
for any assigned value sof the queue length, ropt(σ;s)is not
decreasing over σ≥0, i.e.,
ropt(σ;s)≥ropt (σ;s),for σ>σ
.(11)
Furthermore, after indicating by
IU(s)Δ
=E
σropt(σ;s),s≥0(in IU per slot)(12)
the optimal number of transmitted IUs averaged over the link-
state pdf when the queue state is s,Proposition 3 holds (see
Appendix B for the proof):
Proposition 3: IU(s)in (12) is nondecreasing and concave
for any s≥0. Furthermore, it is strictly increasing for s≤s1
and strictly concave for s>s
1.
Finally, in Appendix B, it is also proven that the behavior of
μ(s(t)) in (10) as a function of the queue state s(t)is captured
by Proposition 4.
Proposition 4: μ(s(t)) in (10) is not decreasing for s(t)≥0
and vanishes for s(t)<s
1while approaching a finite strictly
positive value μ∞for large s(t)values, i.e.,
lim
s(t)→∞ μ(s(t))=μ∞,with 0<μ
∞<+∞.(13)
B. On Complexity and the Adaptive Implementation of the
Optimal Scheduler
About the implementation complexity, we note that μ(s(t))
in (10) and the threshold s1in (9) may be computed offline on
the basis of the system parameters and pdf pσ(σ)of the link
state. Thus, the online implementation of the optimal scheduler
ropt(σ;s(t)) in (8) may be accomplished via a simple three-
way memoryless threshold detector, whose input is the value
assumed (slot by slot) by the system state x≡[σ;s].This
means that the optimal scheduler that we derived may actually
be implemented without resorting to cumbersome DP-based
iterative algorithms.
Furthermore, a key property of the optimal scheduler in (8)
is that it does not require any aprioriknowledge about the
actual pdf pa(a)of the input traffic,2whereas it utilizes
the pdf pσ(σ)of the link state only for the computation of the
expectations at the left-hand side of (9) and (10). Therefore,
since, under the assumed ergodic operating conditions, these
expectations converge to the corresponding sample averages,
the approximations s1and μ(s(t)) of s1and μ(s(t)) in (8) may
be computed by solving the following window-based sample
equations:
1
W
t
j=t−W+1
ε(σ(j); rp(σ(j); s1))=Eave (14)
1
W
t
j=t−W+1
εσ(j); ε−1
rσ(j); 1
μ(s(t)) rp(σ(j);s(t))
0
=Eave (15)
instead of (9) and (10), where, due to ergodicity and i.i.d.
assumption on channel states, we have limW→∞ s1=s1and
limW→∞ μ(s(t)) = μ(s(t)). We conclude that, by using the
sliding-window sample averages reported at the left-hand side
of (14) and (15), the optimal scheduler may be implemented on
the fly, e.g., without any aprioriknowledge about pσ(σ).In
fact, online implementation only requires that a measurement
of the actual link state σ(t)be available on the transmit node of
Fig. 1. Numerical plots of Section VI-D support the conclusion
that (short) observation windows Wof about five to ten slot
times suffice for the convergence of the previously reported
sample averages. Hence, the on-the-fly implementation of the
proposed scheduler also allows it to quickly track apriori
unknown time variations of the link and/or traffic pdfs.
IV. STABILITY AND STABILITY REGION
OF THE OPTIMAL SCHEDULER
In this section, we will present some structural properties
of the previously derived optimal scheduler, which, from an
application perspective, play a key role. In fact, they point
out that, under suitable operating conditions, the scheduler
achieving the minimization of the conditional expectation in
(4) retains the maximal stability region; more so, it is also able
2We anticipate that its performance depends on the actual pdf of the input
traffic (see Section VI).
3300 IEEE TRA NSACTI ONS ON VEH ICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010
to attain the minimum of the corresponding unconditional ex-
pectation. About the stability region of the proposed scheduler
[i.e., the maximal value of λguaranteeing a stable behavior of
the scheduler in (5)], it is expected that the maximal λvalue
depends on the available average energy Eave so that, roughly
speaking, bigger Eave values give rise to larger stability regions.
To gain insight about the actual relationship among the stability
region and the available average energy, we recall that, by
definition, a queue system is strongly stable when the following
inequality is met [29]:
lim
t→∞ sup 1
t
t−1
n=0
E{s(n)}<∞.(16)
In turn, a classic result based on the Lyapunov-drift tool [29
and references therein] guarantees that (16) holds when a lower
bounded function L(s(t)), a buffer-state value s, and a constant
β>0exist such that, for all slot times t, three conditions are
met [29].
1) For any s(t)≤s, there exists t0≥1such that P{s(t+
t0)=0|s(t)}>0.
2) E{L(s(t+ 1))|s(t)}<∞, for any s(t)≥0.
3) E{L(s(t+ 1)) −L(s(t))|s(t)}≤−βs(t), for any
s(t)>s.
Thus, after defining
r∞(σ)Δ
= lim
s→∞ ropt(σ;s)(17)
which is the limit throughput generated by the optimal sched-
uler in (8) at the saturation, we prove, in Appendix C, the
following necessary and sufficient condition for the strong
stability of our scheduler:
Proposition 5—(Strong-Stability Condition): Let us assume
input-traffic flows with finite second-order moments, e.g.,
Ea2(t)<∞.(18)
Thus, the scheduler in (8) is strongly stable if and only if the
following condition is met:
E{r∞(σ)}>λ. (19)
Before proceeding, some explicative remarks about the
meaning and actual applications of the stability condition in
(19) can be of interest.
Remark 3—On the Weak Stability of the Proposed Scheduler:
We point out that, when the input traffic fails to meet (18)
(e.g., when E{a2}is unbounded), condition (19) becomes
necessary and sufficient for the weak stability of our scheduler
in the sense of [29], i.e., in the vanishing overflow sense:3
for any ε>0, a buffer-state value Balways exists such that
the probability limt→∞ P{s(t)>B}is less than ε(see Ap-
pendix C for the proof). From an application point of view,
heavy-tailed distributed traffics are relevant examples of input
flows with unbounded second moments [3]. Therefore, to be
3It is equivalent to the existence of the steady-state probability distribution
for countable infinite buffer-state space.
able to guarantee at least weak stability, it is fundamental to
manage these kinds of multimedia sources (see Section VI-B
for some performance results under heavy-tailed distributed
input traffic).
Remark 4—Design of the Stability Region: From a statistical
point of view, the expectation in (19) represents the average
throughput generated by our scheduler when it operates at the
saturation [see (17)]. Thus, condition (19) differs from the
more classic condition reported, e.g., in [30, Lemmas 3.6 and
4.4], where E{r∞(σ)}is replaced by the average throughput
μΔ
=E{ropt(σ;s)}. The reason leading to resorting to the
average limit throughput in (19), in place of the average one
μ, is, in our framework, that the optimal rate in (8) is not
guaranteed to be limited for any value (σ, s)assumedbythe
system state.4Thus, in our application scenario, the admissible
condition on the server process reported in [30, Def. 3.5] is not
guaranteed to hold. Furthermore, even when there is an upper
limit on the sustainable rate (a finite number of modulation
or coding schemes supported and limited quantized channel
states), μcannot directly be known; therefore, condition (19)
has to be preferred in any case to allow proper system design.
Specifically, an exploitation of (8) allows us to recast condition
(19) in the following equivalent form:
σ
εσ;ε−1
rσ;1
μ∞R(σ;Ep)
0pσ(σ)dσ > λ (20)
where the limit value μ∞may be computed from (10) as the
solution of the following algebraic equation:
σ
εσ;ε−1
rσ;1
μ∞R(σ;Ep)
0pσ(σ)dσ =Eave.(21)
From a system perspective, (20) and (21) represent a set of
algebraic equations in the system parameters Ep,Eave, and λ
that allows us to design the stability region of the considered
queuing system [see Section V-B for some application exam-
ples of (20) and (21)].
A. Maximal Stability Region
About the stability region, we point out that, generally, many
schedulers address the (somewhat easier) problem of allowing
the maximal stability region in the weak sense without address-
ing the problem of average buffer-state minimization. In fact,
schedulers maximizing the stability region are not guaranteed
to minimize the average queue length [30]. On the other hand,
in principle, schedulers that meet both constraints (4.1) and
(4.2) and are stable even for λvalues that are larger than
E{r∞(σ)}but not minimizing the objective in (4) could exist.
Proposition 6 proves the (not obvious) property that this is not
the case, and the proposed scheduler in (8) retains the maximal
admissible stability region.
4Just as a simple example, we anticipate that the (quite-common) logarithmic
and α-powered rate functions of Section V-B give rise to throughput values that
grow unbounded for large σoutcomes.
BACCARELLI et al.:MINIMUM-DELAY SCHEDULING FOR BURSTY TRAFFIC OVER FADING CHANNELS 3301
Proposition 6: Under constraints (4.1) and (4.2), the sched-
uler in (8) achieves the maximal stability region.
Proof: The proof is by contradiction. Thus, let us as-
sume that, given an input traffic with average intensity
λ>
E{r∞(σ)}, there exists a strongly stable scheduler r(σ(t); s(t))
that meets both constraints in (4.1) and (4.2). Therefore, since
strong stability requires that μΔ
=E{r(σ;
s)}≥
λ[31], it fol-
lows that (at least) a buffer-state value smust exist such that
Eσ{r(σ;s)}≥
λ. Hence, the nondecreasing property of the
optimal scheduler (see Proposition 3) would lead to
Eσ{r(σ;s)}≥
λ>E{r∞(σ)}≥Eσropt(σ;s).
Therefore, since, by design, the scheduler in (8) maximizes the
conditional average throughput [see (6)], the preceding chain of
inequalities constitutes a contradiction.
V. C ONDITIONAL-VERSUS-UNCONDITIONAL
QUEUE-LENGTH MINIMIZATION
AND PERFORMANCE BOUNDS
Previous works [9], [12], [14]–[16] on the optimal schedul-
ing over fading channels tackle the problem of the constrained
minimization of the unconditional average queue length, which
is formally defined as follows:
min
r(t)lim
t→∞ sup 1
t
t−1
n=0
E{s(n)}(22)
s.t.: lim
t→∞ sup 1
t
t−1
n=0
E{E(n)}≤E
ave (22.1)
E(t)≤E
pfor any t≥0.(22.2)
Since the preceding problem retains the structure of a con-
strained MDP, the resulting optimal scheduler is memoryless
and time invariant [10], [14], [15]. Unfortunately, this prob-
lem resists a closed-form solution. Thus, the resulting opti-
mal scheduler may be computed and implemented (only) via
DP-based iterative algorithms, which, in turn, limit the numbers
of link and queue states to be both finite (see, e.g., [14] and [15]
for some recent contributions on this topic). Therefore, it may
be of theoretic, as well as practical, interest to investigate about
conditions (if they exist) under which the scheduler in (8) also
minimizes the unconditional expectation in (22). Toward this
end, we begin to note that the constrained optimization problem
in (22)–(22.2) differs from that we consider in (4)–(4.2) under
two aspects. First, the objective function in (22) is the (steady-
state) queue length averaged over both link and queue states,
whereas that in (4) is averaged only over the link state. Second,
the energy constraint in (22.1) is averaged over both the link
and queue states, whereas that in (4.1) is averaged over only
the link state (e.g., the expectation in (4.1) is conditioned
on the queue state). However, in Appendix D, we prove that
when the unconditional expectation in (22.1) is replaced by
the conditional expectation in (4.1), the scheduler solution of
the resulting optimization problem (22)–(22.2) and (4.1) is still
given by (8). Specifically, Proposition 7 holds.
Proposition 7—On the Optimality of the Scheduler in (8):
Under the energy constraints (4.1) and (4.2), the scheduler
minimizing the objective function in (4) also minimizes the
objective function in (22), i.e.,
arg min
r(.){E{s(t+1)|s(t)}} ≡
≡arg min
r(.)lim sup
t→∞
1
t
t−1
n=0
E{s(n)}.
Before proceeding, three main points about the meaning/
implications of Proposition 7 are in order.
First, from a formal point of view, Proposition 7 points out a
property of the scheduler in (8) that stems from the structural
properties already presented in Sections III-A and IV (see
Appendix D, where we detail the proof of Proposition 7). This
is the reason we introduced the scheduler of (8) as the optimal
solution of the optimization problem in (4)–(4.2), and then,
we proved its optimality, even in the framework addressed by
Proposition 7.
Second, we stress that the scheduler of (8) is no longer
optimal when both the conditional expectations in (4) and (4.1)
are replaced by the corresponding unconditional expectations.
As even recently pointed out, e.g., in [14], [15], and [23],
currently, only iterative DP-based solutions are available for the
optimization problem in (22)–(22.2), whose actual implemen-
tations require finite system states.
Third, since the energy constraint in (22.1) is weaker than
that in (4.1), we expect that performances (in terms of un-
conditional average queue length) of the scheduler solution of
the optimization problem in (22)–(22.2) outperform that of our
scheduler in (8), and the curves of Section VI-E confirm, in-
deed, this conclusion. However, we anticipate that these curves
also show that the corresponding performances loss remains
limited up to 10%. This is in agreement with the property
stated by Proposition 7, which, roughly speaking, allows us to
argue that the scheduler in (8) is nearly optimal, even in the
unconditional average sense.
A. Performance Evaluation of the Optimal Scheduler in (8)
Although the optimal scheduler in (8) is in closed form, its
average performance resists closed-form evaluation. However,
from the structural properties described by Proposition 3,we
can derive closed-form lower and upper bounds on the resulting
steady-state unconditional average queue length, as detailed by
Proposition 8 (see [28] for the proof).
Proposition 8—On the Steady-State Performance of the Op-
timal Scheduler in (8): Let ˜sbe the queue-length solution of
the following algebraic equation:
Eσropt(σ;˜s)=λ. (23)
Thus, the steady-state unconditional average queue length of
the optimal scheduler in (8) is lower bounded by s, i.e.,
lim
t→∞ Esopt(t)≥˜s. (24)
3302 IEEE TRA NSACTI ONS ON VEH ICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010
Furthermore, after introducing the dummy positions
AΔ
= lims→∞{E{a2}+E
σ{(ropt(σ;s))2}} and GΔ
=
lims→∞{Eσ{ropt (σ;s)}−λ}, the steady-state average
queue length of the scheduler in (8) may be upper bounded
as in
lim
t→∞ Esopt(t)≤1+ 4λ
Gmax sub;2A
G+2A
G
(25)
where sub is the solution of the following algebraic equation:
Eσropt(σ;sub )=λ+G
2.(26)
B. Characterization of the Scheduler’s Expression to Some
Rate Functions of Practical Interest
To gain insight into the performance improvements actually
offered by the optimal scheduling policy in (8), let us specialize
the previously presented (general) results to some rate functions
commonly employed to characterize the performance of queue
systems [25, ch. 2 and 4].
By definition, the α-powered rate function equates [13], [17],
[31] R(σ, E)Δ
=(σE)α,0<α≤1so that the corresponding
energy function is given by
ε(r;σ)= 1
σr1/α.(27)
At the transport layer of the ISO/OSI protocol stack, α-powered
rate functions are sometimes utilized to measure the goodput
sustained by transport links [17], whereas at the physical layer,
they are employed to capture the transmission capacity of
spread-spectrum code-division-multiple-access-like communi-
cation systems working at low signal-to-noise ratios (SNRs)
[8]. In this case, the general expression in (8) for the optimal
scheduler specializes to
ropt(σ;s)=⎧
⎨
⎩
min {s, (σEp)α},for s<s1
minασ
μ(s)α
1−α,s,(σEp)α
,for s≥s1.
(28)
In particular, for Ep=+∞, from (20) and (21), we obtain the
following (simple) relationship for the queue stability:
Eave >λ1/α
Eσα/(1−α)(1−α)/α (29)
which, in turn, may directly be employed for planning/testing
the stability region of the underlying queue system.
The logarithmic rate function
R(σ, E)Δ
= log(1 + σE)(in nats per slot) (30)
is commonly employed to measure the so-called Shannon
capacity of transmission systems impaired by Gaussian-
distributed additive noise-plus-interference disturbance [8]. In
Fig. 2. Behavior (in log scale) of ropt(s;σ)in (31) at Ep=23(J).
this case, the general relationship (8) for the optimal scheduler
reduces to
ropt(σ;s)=
=min {s, log(1 + σEp)},for s<s
1
min log σ
μ(s),s,log(1 + σEp)+,for s≥s1.
(31)
VI. PERFORMANCE TESTS AND COMPARISONS
In this section, we test the actual performance and robust-
ness properties of the optimal scheduler in (8) for different
application scenarios of practical interest. In the simulated
environments, the link state σ(.)represents the instantaneous
(e.g., fading affected) SNR measured at the receiver side of
Fig. 1, when the energy radiated by the transmit node over a
slot time is unit. The link state σ(.)behavior is modeled as
a central chi-squared RV with 4◦of freedom, e.g., pσ(σ)=
σexp(−σ),σ≥0, which captures the statistical behavior of
Rayleigh-faded multiple-antenna systems equipped with two
receive antennas [8]. Each numerical curve that we report in
the following figures has been obtained by averaging (via the
Monte Carlo method) 104independent sample paths obtained
by implementing the overall faded-impaired queue system of
Fig. 1. The rate function adopted for testing the performance of
the simulated systems is the logarithmic one.
A. Behavior of the Optimal Scheduling Policy
Fig. 2 reports the behavior (in log scale) of the scheduler in
(31) for σ≥0and some values of the queue length sand the
available average energy Eave . An examination of the plots of
Fig. 2 leads to four main conclusions. First, it is not guaranteed
that the optimal scheduler is work conserving, meaning that
it refrains to transmit when the channel state σis low (e.g.,
the fading is deep). Second, the scheduler’s behavior tries to
become (more) work conserving (e.g., the lower envelopes of
the plots of Fig. 2 tend to shift on the left-hand side) when
the available average energy increases. Third, at fixed s,the
BACCARELLI et al.:MINIMUM-DELAY SCHEDULING FOR BURSTY TRAFFIC OVER FADING CHANNELS 3303
Fig. 3. Behavior of the s1threshold defined by (9) for the application scenario
of Section V-A.
rate of increment of ropt(σ;s)with σincreases for increasing
values of Eave. In turn, this implies that higher values of Eave
allow ropt(σ;s)to faster approach its limit values s. Fourth, the
optimal scheduler tends to (fully) empty the queue only when
the channel state σis high (e.g., the fading is light). Finally, in
Fig. 3, we report the corresponding behavior of the s1threshold
defined by (9), showing, as for a fixed channel statistics, that
the low/high occupation buffer region is strictly related to the
system energy parameters Eave and Ep.
B. Performance of the Optimal Scheduler Under Heavy-Tailed
Distributed Input Traffic
In this section, we focus on the (average) performance of
the optimal scheduler in terms of unconditional average queue
length at the steady state, e.g., sΔ
= limt→∞ 1/t t−1
i=0 E{s(t)}.
The cumulative distribution function (CDF) Fa(a)that we
consider for the input arrivals is the following heavy-tailed
Pareto-type one [3]:
Fa(a)Δ
=P(a≤a)≡1−a
ω−β,ω≤a≤∞;β>1.
(32)
As we can see from (32), the tail of the considered distribution
becomes heavier as βdecreases toward 1, meaning that smaller
βvalues correspond to heavier distribution tails. As it is known,
Pareto-like CDFs well capture the heavy-tailed behavior of the
distribution of traffic flows generated by video encoders and
Web applications [3], [7]. The (so-called) location parameter
ωin (32) fixes the smallest value allowed the input arrivals.
Hence, for fixed β, larger ωvalues give rise to bigger values
of the corresponding statistical means λΔ
=E{a(t)}.Inthetest
that have been carried out, we tuned ωon the basis of the
requested λ.
The (numerically evaluated) plots of Fig. 4 give the behavior
of the average queue length sversus the average intensity of
the input traffic λfor some values of β. Interestingly enough,
the plots of Fig. 4 give insight about the effects of the β
parameter in (32) (e.g., the effect of the heavy-tailed behavior
Fig. 4. Numerically evaluated behavior of the average queue length sof the
optimal scheduler for the application scenario of Section VI-B.
of the distribution of the input traffic) on the average increment
of the queue length of the optimal scheduler. Specifically,
an examination of the limit behavior of the plots of Fig. 4
show that, at s= 300 nat, the average input-traffic intensity λ
sustained by the scheduler decreases from 2.78 to 2.5 nat/slot
(e.g., λreduces by about 11%) when βpasses from β=2.1to
β=1.5. These plots also unveil that, in any case, the effect of β
on the average queue length sis not so substantial in the region
of low offered traffic (e.g., we say for λvalues below 48%–50%
of the corresponding maximal sustainable ones), whereas this
effect becomes noticeable for values of λaround 75%–80% of
the corresponding maximal ones.
C. Performance Robustness Against Link-Estimation Errors
To test the degrading effects induced on the average queue
length sof the optimal scheduler by errors possibly impairing
the measured link states, we have perturbed the actual link-
state sequence {σ(t)}via a stationary zero-mean unit-variance
Gaussian noise sequence {n(t)}. Thus, the perturbed link-state
sequence {σ(t)}is (slot-by-slot) generated according to the
following relationship:
σ(t)=√σ(t)+1−E{σ2(t)}n(t)(33)
where the parameter ∈[0,1] is set to control the desired
average squared measurement error affecting σ(t). Specifically,
(/(1 −)) plays the role of normalized SNR affecting the
link measurement σ(t)in (33) so that =1 is the error-
free case, whereas lower values correspond to noisier state
measurement. Thus, after implementing the optimal scheduler
in (31) for the same application scenario of Section VI-B, with
βin (32) equal to 1.9, we have replaced {σ(t)}by the perturbed
sequence {σ(t)}at the input of the optimal scheduler. The
resulting values for the unconditional average queue length s
we have obtained are reported in Fig. 5 for ranging from
0.8 to 1. An examination of these plots shows that the average
increment in the queue length induced by errors in the state
3304 IEEE TRA NSACTI ONS ON VEH ICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010
Fig. 5. Average queue-length-versus-λfor some values of the link-
measurement error parameter for the application scenario of Section V-C.
Fig. 6. Average queue-length-versus-λfor different values of the observation
window Wfor the application scenario of Section VI-D.
measurements is quite negligible in the medium/low region of
the offered input traffic (e.g., for values of λbelow 75%–80%
of the maximal sustainable ones). Furthermore, even when the
average queue length sis limited up to 400–500 nat and the
values of the ratio (/(1 −)) as low as 5.6 (e.g., 7.48 dB)
are considered, the corresponding reduction in the sustained
average input-traffic intensity λwith respect to the error-free
case remains, indeed, bounded up to 5%.
D. Convergence Behavior of the On-the-Fly Implementation
of the Optimal Scheduler and Performance Robustness Against
Correlated Fading
To test the convergence behavior of the on-the-fly implemen-
tation of the scheduler (see Section III-B), we have numerically
evaluated the average queue length sexperienced by the op-
timal scheduler for the application scenario of Section III-B
when the βexponent in (32) is set to 1.9. Fig. 6 reports
the resulting plots of the optimal scheduler (implemented on
the basis of the actual pσ(σ)and pa(a)pdfs) and those of
the corresponding “on-the-fly” versions for values Wof the
observation window of five and ten slot times. The first set of
plots marked as —, −o−, and −∇−in Fig. 6 refers to the afore-
mentioned application scenario affected by time-uncorrelated
fading. As anticipated in Section III-B, these plots support the
performance robustness of on-the-fly implementation of the
proposed scheduler, even for observation windows Was small
as five slot times. Specifically, at λ=2.4nat, the performance
gap among the optimal curve (marked as - in Fig. 6) and that
of the corresponding on-the-fly implementation with W=5is
limited up to 16 nat.
Finally, the plot marked as −♦− in Fig. 6 reports the
performance of the on-the-fly version of the proposed scheduler
when the fading is Rayleigh distributed and exhibits a temporal
autocorrelation sequence {R(n),n=0,1,2,...}given by the
zeroth-order Bessel function of the first kind. As it is known,
the latter well describes the fading correlation experienced by
mobile radio channels [8], and its expression equates
R(n)=J0(2πfDTsn),n=0,1,... (34)
where fD(in hertz) is the Doppler frequency, and TS(in
seconds) is the slot duration. The performance plot marked by
−♦− in Fig. 6 refers to the case of
R(1) = J0(2πfDTs)≡0.95 (35)
which may be considered quite representative of mobiles mov-
ing at low speeds over urban/suburban areas [8]. A comparison
of the curves marked as −o−and −♦− in Fig. 6 supports
the conclusion that the performance of the proposed scheduler
is quite robust against impairing effects induced by fading
correlation. Specifically, even for correlation coefficients as
high as 0.95, the experienced performance loss is limited up
to 70–90 nat, even for observation windows as small as ten slot
times and intensity traffic λon the order of 90% of the maximal
sustainable one [e.g., at λvalues around 2.5 nat/slot].
E. Performance Comparison With DP-Based Schedulers
To allow the implementation of the DP algorithm, in this
section, we model the link state σ(t)as a discrete RV, which
may assume the Nσ=16 outcomes {σi=0.1+0.5(i−1),
i=1,...,16}according to the following probability
distribution:
P(σ=σi)≡1
Cσiexp (−σi),i=1,...,16 (36)
with C≡1.9956. The input arrivals are still Pareto distributed
with β=2.2[see (32)], and the considered energy constraints
are Eave =5J and Ep= 100 J. For the considered system, we
have numerically computed the optimal scheduler solution of
the constrained optimization problem in (22)–(22.2) by resort-
ing to the DP algorithm. The obtained performance (expressed
in terms of average unconditional queue length sDP at the
steady state) is reported in Fig. 7. On the same figure, we
also report the corresponding average queue length sof the
BACCARELLI et al.:MINIMUM-DELAY SCHEDULING FOR BURSTY TRAFFIC OVER FADING CHANNELS 3305
Fig. 7. Average unconditional queue-length-versus-λfor (−−−)the DP-
based optimal scheduler and (−)the proposed one for the application scenario
of Section VI-E.
scheduler in (31). Since the latter is the solution of the con-
strained optimization problem in (4)–(4.2) and the constraint in
(4.1) is stronger than that in (22.1) (see Section V), it may be
expected that the DP-based scheduler outperforms the proposed
scheduler (e.g., sDP < s), and the performance plots of Fig. 7
indeed corroborate this expectation. However, an examination
of these plots leads to the conclusion that the performance gap
indeed remains limited up to 8%–10% at medium-low λand
vanishes for λ→0. This is in agreement with the optimality
property previously reported in Proposition 7.
VII. CONCLUSION
Overall, the main features of the proposed scheduler are the
following: 1) It is of closed form; 2) it allocates energy and rate
by working in a cross-layer fashion; 3) its stability region is
maximal; and 4) it may be implemented on the fly. Furthermore,
under suitable conditions on the allowed average energy, the
proposed scheduler is optimal, even in the sense of minimizing
the unconditional average length of the underlying GI/G/1
queue system. The numerical tests that have been carried out
support the robustness of the performance of the proposed
scheduler against several system impairments, such as link-
estimation errors and time-correlated fading.
APPENDIX A
PROOF O F PROPOSITION 2
Let us consider the optimization problem in (6)–(6.2)
at s(t)=0. Since, at s(t)=0, the peak throughput
rp(σ(t); s(t)) Δ
= min{s(t),R(σ(t); Ep)}strictly satisfies
the energy constraint in (6.1), due to the continuity of the
problem, an interval of buffer values [0,s
1]exists such that,
for any s(t)∈[0,s
1], the peak throughput still satisfies the
constraint in (6.1).5We conclude that, for s(t)∈[0,s
1],the
5This conclusion arises from the application of the Permanence-Sign Theo-
rem to the function φ(s(t)) Δ
=E
σ{ε(σ;rp(σ;s(t)))}−E
ave [28].
peak-throughput policy rp(σ(t); s(t)) is the optimal one. If
the peak-throughput scheduler were the optimal one for any
s(t)≥0, for any s(t), it would satisfy the constraint in (6.1) so
that we would have
lim
s(t)→∞ Eσ{ε(σ(t); rp(σ(t); s(t)))}
=E
σ{ε(σ;R(σ;Ep))}
=E
σ{Ep}
=Ep≤E
ave
arriving at a contradiction. Hence, the peak throughput cannot
be the optimal scheduler for any s(t)≥0, and the threshold
value s1must be finite. From the monotonic property of the
function ε(.;.), it follows that it can be computed as the unique
solution of the algebraic equation in (9).
Let us now derive the expression for the optimal scheduler
ropt(σ(t); s(t)) in (8) for s(t)>s
1. Since it may be viewed that
strong duality holds [28], the optimum scheduler may be ob-
tained via an application of the standard Karush–Khun–Tucker
optimality conditions. Thus, the resulting Lagrangean function
takes on the following form6:
L(r(σ;s(t)) ,μ(s(t))) Δ
=[s(t)−r(σ;s(t))]
!" #
u(σ;s(t))
pσ(σ)dσ
+μ(s(t)) ε(σ;r(s(t); σ)) pσ(σ)dσ −E
ave.
Hence, by imposing the Euler’s condition to the
preceding reported Lagrangean function, we have
εr(σ(t); r(σ(t); s(t))) = 1/μ(s(t)), where εr(.;.)is the
first derivative of the energy function with respect to the
rvariable, which allows us to obtain the unconstrained
solution r∗(σ(t); s(t)) = ε−1
r(σ, (1/μ(s(t)))) of the problem.
Thus, since the Lagrangean function is concave, it may be
recognized that the solution of the maximization problem
max0≤r(.;.)≤rp(·)L(r(σ(t); s(t)),μ(s(t))) is the projection of
r∗(σ(t); s(t)) onto the underlying definition set, i.e.,
ropt (σ(t); s(t))≡ε−1
rσ(t); 1
μ(s(t))rp(·)
0
for s(t)≥s1.
This completes the proof of (8) for s(t)≥s1. Finally, by
imposing the complementary condition [28] of the constraint
in (6.1), we arrive at μ(s(t))[E{ε(.;.)}−E
ave]=0, which, in
turn, leads to (10) for the computation of μ(s(t)).
APPENDIX B
PROOF S OF PROPOSITIONS 3AND 4
About the nondecreasing behavior of IU(s), the constraint in
(6.2) guarantees that, for any assigned link state σ, the solution
ropt(σ;s∗)at s∗is also admissible for any s>s
∗, but it could
no longer be optimal. Thus, directly from the definition in (12),
6We choose to solve the afforded problem as the minimization of
Eσ{u(σ(t); s(t))}. Obviously, it is equivalent to solve it as the maximization
of Eσ{r(σ(t); s(t))}.
3306 IEEE TRA NSACTI ONS ON VEH ICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010
we conclude that IU(s)≥IU(s∗)for s>s
∗. Furthermore, for
s≤s1,IU(s)in (12) is strictly increasing. In fact, for any
s≤s1, an interval Iof link-state values such that r(σ;s)≡
rp(σ;s)≡smust exist; otherwise, due to the constraint
on the average energy, we would have Eσ{ε(σ;r(σ;s))}≡
Eσ{ε(σ;R(σ;Ep))}≡E
p>Eave. Thus, for any s ≥s,we
have rp(σ;s)≥rp(σ;s), and since the preceding inequal-
ity must be strict for any σin I, we conclude that
IU(s)>IU(s).
About the concavity of IU(s)in (12), for any assigned s,
s ∈R+
0, and θ∈[0,1], let us pose sΔ
=θs+(1−θ)s, and
let r(.;.)be the scheduler so defined as
r(σ;s)Δ
=r(σ;θs+(1−θ)s)
Δ
=θropt(σ;s)+(1−θ)ropt(σ;s).(B.1)
Now, due to the constraint in (6.1) and the strictly convex
behavior of ε(σ;r)in (3), we have that
Eσ{ε(σ;r(σ;s))}≡
≡Eσε$σ;θropt(σ;s)+(1−θ)ropt(σ;s)%(a)
<
<θEσε$σ;ropt (σ;s)%+(1−θ)×
×Eσε$σ;ropt(σs)%(b)
≤θEave +(1−θ)Eave ≡Eave
(B.2)
where (a)stems from the convex property of ε(.;.),
whereas (b)accounts for the constraint in (6.1). Furthermore,
by jointly exploiting the definitory relationship in (B.1)
and the constraint in (6.2), we may limit r(.;.)as in
r(σ;s)≤θmin{s;R(σ;Ep)}+(1 −θ) min{s;R(σ;Ep)}≤
min{θs+(1−θ)s;R(σ;Ep)}, which, in turn, leads to
0≤r(σ;s)≤min {s;R(σ;Ep)}Δ
=rp(σ;s).(B.3)
Hence, since (B.2) and (B.3) prove that the scheduler in
(B.1) is admissible for the constrained optimization problem in
(6)–(6.2), the following inequality must hold:
Eσ{r(σ;s)}≤Eσropt(σ;s).(B.4)
Since the latter may be rewritten as in [see (B.1)]
θEσropt(σ;s)+(1−θ)Eσropt(σ;s)
≤Eσropt (σ;θs+(1−θ)s)
this proves the concavity of IU(s)in (12). Furthermore,
when sand s are both larger than s1, then we have
Eσ{ropt(σ;s)}=Eave , whereas, due to (B.2), we also have
Eσ{r(σ;s)}<Eave. Therefore, (B.4) must hold as a strict
inequality; otherwise, we could increase r(σ;s)to have
Eσ{r(σ;s)}>Eσ{ropt(σ;s)}, which constitutes a contradic-
tion. This completes the proof of Proposition 3.
About the behavior of μ(s(t)) pointed out by Proposition 4,
when the peak throughput is the optimal solution of problem
(6) (e.g., for s<s
1), we have μ(s(t)) = 0.Fors≥s1,by
resorting to the monotonic properties of ε(.;.)and εr(.;.),we
conclude that the solution of (10) always exists for any (pos-
sibly unbounded) s. About the monotonic property of μ(s(t)),
we begin to rewrite the expression in (8) retained by the optimal
scheduler for s>s
1in the following equivalent form:
ropt (σ(t); s(t))
≡min s(t); R(σ(t); Ep);ε−1
rσ(t); 1
μ(s(t))+
.(B.5)
Thus, let s(1) ≥s1and s(2) ≥s1be two buffer values and
μ(1) ≡μ(s(1))and μ(2) ≡μ(s(2) )be the corresponding val-
ues assumed by the μ-multiplier. Furthermore, let us sup-
pose s(2) >s
(1) and μ(2) <μ
(1). Since ε−1
r(σ(t); 1/μ(s(t)))
is not decreasing in 1/μ(s(t)), from (B.5), we deduce that
r(σ;s(2))≥r(σ;s(1) )for σ≥0and that a σinterval Imust
exist such that the preceding inequality is strict for any σin I.
Therefore, with (10) being met at s=s(1), it can no longer be
satisfied for s=s(2), and this constitutes a contradiction. The
proof of Proposition 4 is now complete.
APPENDIX C
PROOF OF PROPOSITION 5
For proving Proposition 5, we proceed to check that all the
stability conditions 1, 2, and 3 previously reported in Section IV
are met.
To begin with, since the probability P{s(t+1)<s(t)}is
strictly positive for any s(t)≤s, condition 1 is met.
To check condition 2, let us consider the following quadratic-
type Lyapunov function: L(s(t)) = s2(t). Since the following
developments hold:
Es2(t+1)|s(t)=s2(t)+E{a2}+Eσr2(σ,s(t))+2s(t)λ
−2s(t)Eσ{r(σ,s(t))}−2λEσ{r(σ,s(t))}(C.1)
we conclude that the previously reported conditional expecta-
tion is finite for any finite s(t)so that condition 2 of Section IV
is also met.
Now, by introducing (C.1) into the expression for
condition 2, the latter may equivalently be rewritten as
Eσ{r(σ,s(t))}−λ≥
E(a−r(σ,s(t)))2|s(t)
2s(t)+β
2
(C.2)
where E{(a−r(σ,s(t)))2|s(t)}=E{a2}+Eσ{r2(σ,s(t))}−
2λEσ{r(σ,s(t))}. Hence, the next step consists of proving that
(19) implies (C.2) so that condition (19) is sufficient for strong
stability. To this end, we note that the increasing monotonic
behavior of Eσ{r(σ,s)}in the svariable and the boundness
of the limit lims→∞ Eσ{r(σ,s)}≡Eσ{r∞(σ)}<+∞imply
that there exist a β>0and a ˜ssuch that, for any s>˜s,the
inequality (C.2) is guaranteed to hold when condition (19)
is met.
About the necessity of the condition in (19), we proceed
by contradiction, and then, we assume that Eσ{r∞(σ)}<λ.
With Eσ{ropt(σ,s)}being nondecreasing in s, this implies
BACCARELLI et al.:MINIMUM-DELAY SCHEDULING FOR BURSTY TRAFFIC OVER FADING CHANNELS 3307
that Eσ{ropt(σ,s)}<λ for any s≥0. Thus, by exploiting
Lindley’s equation in (1), we deduce that an α>0such that
E{s(t+1)−s(t)|s(t)}>αfor any s(t)must exist. The last
inequality leads to the following lower bound: E{s(t+1)}≥
α+E{s(t)}for any t≥0. In turn, the latter allows us to arrive
at the following limit:
lim
t→∞ E{s(t+1)}≥lim
t→∞ [(t+1)α]+E{s0}≡∞.(C.3)
Since the preceding limit is unbounded, regardless of the pdf of
the starting state s0, the necessity of condition (19) is proven.
As a final point, we note that, when E{a2}is unbounded, the
quadratic Lyapunov function in (C.1) no longer meets strong-
stability conditions. Nevertheless, by exploiting the same argu-
ments already reported in the first part of this Appendix, it can
be verified that the linear Lyapunov function L(s(t)) = s(t)
satisfies both conditions 1 and 2 and the following condition:
E{L(s(t+ 1)) −L(s(t)) |s(t)}≤−βfor any s(t)>˜s
(C.4)
which suffices to guarantee weak stability [29]. We can also
check that (19) is also necessary for the weak stability. In fact,
let us assume, by contradiction, that E{r∞(σ)}<λand that
the system is stable. From the nondecreasing property stated in
Proposition 3,itfollowsthatEσ{ropt(σ,s)}<λfor any sso
that, regardless of the actual steady-state pdf p(s), we have that
μ=E
s{Eσ{ropt(σ,s)}} <λso that the necessary condition
μ≥λfor the weak stability [31] is not met.
APPENDIX D
OPTIMALITY OF THE SCHEDULER OF (8)
Let ropt(σ;s)and r(σ;s)be the transmission rate in (8) of
the optimal scheduler and that of any other scheduler meeting
both energy constraints in (4.1) and (4.2). Let assume both
schedulers to be stable. Furthermore, let us indicate by σΔ
=
[σ(0),σ(1),...]and aΔ
=[a(0),a(1),...]the outcomes of the
link-state and arrival random sequences, respectively. Thus,
on the basis of Lindley’s equation in (1), it is straightforward
to conclude that, even under the same outcomes σand afor
the link states and input arrivals, the corresponding outcomes
sopt Δ
=[sopt(0),s
opt(1),...]and sΔ
=[s(0),s(1),...]of the
queue states generated by the schedulers are, in general, dif-
ferent, e.g., sopt =s. However, under the previously assumed
ergodic operating conditions, for the unconditional average
steady-state queue length E{s}for both schedulers, we can
write
E{s}≡lim
t→∞
1
t
t
i=1
s(i)(a)
≡lim
t→∞
1
t
t
i=1
t−1
k=0
(a(k)−r(k))
≡&lim
t→∞
t−1
k=0
(a(k)−r(k)) t−k
t'(D.1)
where (a)stems from the following telescopic relationship:
s(i)=i−1
k=0(a(k)−r(k)). Now, since the limit in (D.1) must
exist finite for both schedulers, even the corresponding limit
difference, e.g.,
lim
t→∞ &t−1
k=0 $r(s(k); σ(k)) −ropt$sopt (k); σ(k)%%t−k
t'
(D.2)
must be finite. Thus, after posing (
IU(s)Δ
=E
σ{r(σ;s)}and
IUopt(sopt)Δ
=E
σ{ropt(σ;sopt )}, from the outset, it follows
that even the expectation of (D.2) over the link-state pdf, i.e.,
lim
t→∞ &t−1
k=0 (
IU (s(k)) −IUopt $sopt(k)%t−k
t'(D.3)
must give rise to a finite limit (see Appendix E for the proof
of this claim). As a consequence, since the series in (D.3) must
converge, its argument must be vanishing, i.e.,
lim
k→∞ (
IU (s(k)) −IUopt $sopt(k)%=0.(D.4)
Furthermore, with IUopt(s)being nondecreasing (see
Proposition 3) and, by design, IUopt(s)≥(
IU(s)for any fixed
s, we conclude that a necessary condition for meeting (D.4) is
that the following inequality holds:
lim
k→∞ s(k)≥lim
k→∞ sopt(k).(D.5)
Therefore, since the system is assumed to operate under the
ergodic regime, (D.5) implies that
E{
s}≡lim
t→∞
1
t
t−1
k=0 s(k)≥lim
t→∞
1
t
t−1
k=0
sopt(k)≡E{sopt }.
(D.6)
Overall, the inequality in (D.6) proves that, under the consid-
ered operating conditions, the scheduler in (8) also minimizes
the unconditional average queue length at steady state.
APPENDIX E
CONVERGENCE OF (D.3) TO A FINITE LIMIT
About the convergence of (D.2) and (D.3), we note that
the term ((t−k)/t)is definitely unessential. Therefore, af-
ter introducing the dummy positions a(s(k); sopt(k); σ(k)) Δ
=
r(s(k); σ(k)) −ropt(sopt (k); σ(k)) and A(s(k); sopt(k)) Δ
=
(
IU(s(k)) −IUopt (sopt(k)), we have to prove that the
convergence of
∞
k=0
a$s(k); sopt(k); σ(k)%(E.1)
to a finite limit also guarantees the existence of a finite limit for
the following infinite sum:
∞
k=0
A$s(k); sopt(k)%.(E.2)
3308 IEEE TRA NSACTI ONS ON VEH ICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010
Toward this end, we note that, for any fixed pair (s(k); sopt(k))
of buffer occupations, by definition, A(s(k); sopt (k)) equates
the average value of the random variable a(s(k); sopt (k); σ)
done with respect to the current channel state σ(i.e.,
A(s(k); sopt(k)) Δ
=E
σ{a(s(k); sopt(k); σ)}). Furthermore,
for any fixed pair (s1;s2)and any sequence σΔ
=
[σ(0),σ(1),...]of link-state outcomes, the ergodic
operating condition guarantees that we also have
A(s1;s2) = limt→∞(1/t)t
i=1 a(s1;s2;σ(i)) so that we
can write
∀ε∗>0∃N∈Ns.t. ∀n>N
)))))
1
n
n
i=1
a(s1;s2;σ(i)) −A(s1;s2))))))<ε
∗.(E.3)
Now, an application of the necessary and sufficient Cauchy
convergence condition to (E.1) allows us to write
∀ε1>0∃n1∈Ns.t. ∀n>n
1∀p∈N
)))))
n+p
k=n+1
a$s(k); sopt(k); σ(k)%)))))<ε
1.(E.4)
Let us rewrite now (E.4) for a different εvalue, i.e.,
∀ε2>0∃n2∈Ns.t. ∀m>n
2∀q∈N
)))))
m+q
i=m+1
a$s(i); sopt(i); σ(i)%)))))<ε
2.(E.5)
Obviously, we can choose m>max{n+p;n2}so that each
term in (E.4) is different from all terms in (E.5). Hence, from
(E.4) and (E.5), we have that, for any pand q
)))))
n+p
k=n+1
a$s(k); sopt(k); σ(k)%+
+
m+q
i=m+1
a$s(i); sopt(i); σ(i)%)))))<ε
1+ε2.(E.6)
Now, due to the stability and ergodic operating conditions
of both schedulers, for any pair (s(k); sopt(k)),k=n+
1,...,n+ppresent in the first summation of (E.6), it is
possible to choose a value of qthat is large enough so that we
are able, for any N, to extract from the second summation in
(E.6) Nterms that are related to the same pair (s(k); sopt(k))
of buffer states and incorporate them in the first sum of (E.6). So
doing, (E.6) can be rewritten in the following equivalent form:
)))))n+p
k=n+1
i∈Ik
a$s(k); sopt(k); σ(i)%
+
i∈I
a$s(i); sopt(i); σ(i)%)))))<ε
1+ε2.(E.7)
In (E.7), the index sets Ik,k=n+1,...,n+p(with cardi-
nality |Ik|=N+1), and I(with cardinality |I| =q−Np)
form a partition of {n+1,...,n+p}∪{m+1,...,m+q}.
Thus, since |x−y|≥||x|−|y||, for any xand y,from(E.7),
we derive the following inequality:
))))))))))
n+p
k=n+1
(N+1)1
N+1
i∈Ik
a$s(k); sopt(k); σ(i)%)))))
−)))))
i∈I
a$s(i); sopt(i); σ(i)%))))))))))<ε
1+ε2.(E.8)
Furthermore, as a consequence of (E.5), for any i∈I,wehave
that |a(s(i); sopt(i); σ(i))|<ε
2, and thus7
))))))
i∈I
a$s(i); sopt(i); σ(i)%))))))<ε
2|I|=ε2(q−Np)<ε
2q
(E.9)
which, in turn, allows us to derive from (E.8) the following
inequality:
)))))
n+p
k=n+1
1
N+1
i∈Ik
a$s(k); sopt(k); σ(i)%)))))<ε1+ε2(1 + q)
N+1 .
(E.10)
Finally, (E.10) and (E.3) lead to the following conclusive
relationship:
)))))
n+p
k=n+1
A$s(k); sopt(k)%)))))<ε1+ε2(1 + q)
N+1 +pε∗Δ
=ε.
(E.11)
We explicitly stress that, to conclude that (E.11) implies the
convergence of (E.2), it is necessary for an index n(ε)to not
depend on the particular pvalue such that (E.11) is met for any
n>nand, for any p, exists. This is indeed true. In fact, by
construction, from (E.11) and (E.4), it follows that ncoincides
with n1, which, in turn, only depends on ε1. The latter can
independently be chosen as a function of εfrom p, e.g., by
posing ε1=ε/3. This is possible, because the value of pjust
influences the choices of ε∗and ε2, which do not impact n.
This means that, for any p, it is possible to choose ε∗, and then
ε2, such that ε2(1 + q(ε∗))/(N(ε∗)+1)+ε∗p<δ, for any δ.
This proves the validity of the following inequality:
∀ε∃n∈Ns.t. ∀n>n∀p∈N,)))))
n+p
k=n+1
A$s(k); sopt(k)
%)))))<ε.
(E.12)
The latter gives a necessary and sufficient condition for the
convergence of (E.2). The proof is now complete.
REFERENCES
[1] M. Van Der Schaar and P. A. Chou, Multimedia Over IP and Wireless
Networks. New York: Academic, 2007.
[2] D. Gross and C. M. Harris, Fundamentals of Queueing Theory,3rded.
New York: Wiley, 1998.
7We explicitly note that, even if I⊂{m+1,...,m+q}, from (E.5), we
cannot argue that |i∈I a(s(i); sopt(i); σ(i))|<ε
2.
BACCARELLI et al.:MINIMUM-DELAY SCHEDULING FOR BURSTY TRAFFIC OVER FADING CHANNELS 3309
[3] K. Park and W. Willinger, Self-Similar Network Traffic and Performance
Evaluation. New York: Wiley, 2000.
[4] R. D. Yates and N. B. Mandayam, “Challenges in low-cost wireless data
transmission,” IEEE Signal Process. Mag., vol. 4, no. 3, pp. 99–102,
May 2000.
[5] M. L. Putterman, Markov Decision Processes: Discrete Stochastic
Dynamic Programming. New York: Wiley, 1994.
[6] S. Asmussen, Applied Probability and Queues, 2nd ed. Berlin,
Germany: Springer-Verlag, 2003.
[7] P. R. Jelenkovic, “Long-tailed loss rates in a single server queue,” in Proc.
IEEE INFOCOM, 1998, pp. 1462–1469.
[8] K. Pahlavan and P. Krishnamurthy, Principles of Wireless Networks.
Englewood Cliffs, NJ: Prentice-Hall, 2002.
[9] R. A. Berry and R. G. Gallager, “Communication over fading channels
with delay constraints,” IEEE Trans. Inf. Theory, vol. 48, no. 5, pp. 1135–
1149, May 2002.
[10] M. Goyal, A. Kumar, and V. Sharma, “Power constrained and delay
optimal policies for scheduling transmission over a fading channel,” in
Proc. IEEE INFOCOM, 2003, pp. 311–320.
[11] B. Prabhakar, E. V. Biyikoglu, and A. E. Gamal, “Energy-efficient trans-
mission over a wireless link via lazy packet scheduling,” in Proc. IEEE
INFOCOM, 2001, pp. 386–394.
[12] D. Rajan, A. Sabharwal, and B. Aazhang, “Delay-bounded packet
scheduling of bursty traffic over wireless channels,” IEEE Trans. Inf.
Theory, vol. 50, no. 1, pp. 125–144, Jan. 2004.
[13] S. Shakkottai and R. Srikant, “Scheduling real-time traffic with dead-
lines over a wireless channel,” Wireless Netw., vol. 8, no. 1, pp. 13–26,
Jan. 2002.
[14] D. V. Djonin and V. Krishnamurthy, “Structural results of the opti-
mal transmission scheduling policies and costs for correlated sources
and channels,” in Proc. 44th Conf. Decision Control, Seville, Spain,
Dec. 2005, pp. 3231–3236.
[15] A. K. Karmokar, D. V. Djonin, and V. K. Bhargava, “Optimal and
suboptimal packet scheduling over correlated time-varying flat fading
channels,” IEEE Trans. Wireless Commun., vol. 5, no. 2, pp. 446–456,
Feb. 2006.
[16] B. Collins and R. Cruz, “Transmission policies for time-varying channels
with average delay constraints,” in Proc. Allerton Conf. Commun., Control
Comput., Monticello, IL, Oct. 1999, pp. 1–9.
[17] A. Fu, E. Modiano, and J. N. Tsitsiklis, “Optimal transmission scheduling
over a fading channel with energy and deadline constraints,” IEEE Trans.
Wireless Commun., vol. 5, no. 3, pp. 630–641, Mar. 2006.
[18] C. Schugers, V. Raghunathan, and M. B. Sriwastava, “Power manage-
ment for energy-aware communication systems,” ACM Trans. Embedded
Comput. Syst., vol. 2, no. 3, pp. 431–447, Aug. 2003.
[19] M. A. Khojastepour and A. Sabharwal, “Delay-constrained scheduling:
Power-efficiency, filter design and bounds,” in Proc. IEEE INFOCOM,
2004, pp. 1938–1949.
[20] R. Mangharam, S. Pollin, B. Bougard, R. Rajkumar, F. Catthoor,
L. V. der Perre, and I. Moemao, “Optimal fixed and scalable energy
management for wireless networks,” in Proc. IEEE INFOCOM, 2005,
pp. 114–125.
[21] M. Zafer and E. Modiano, “A calculus approach to minimum energy
transmission policies with quality of services guarantees,” in Proc. IEEE
INFOCOM, 2005, pp. 548–559.
[22] W. Chen and U. Mitra, “Energy efficient scheduling with individual packet
delay constraints,” in Proc. IEEE INFOCOM, 2006, pp. 1136–1144.
[23] J. Lee and N. Jindal, “Energy-efficient scheduling of delay-constrained
traffic over fading channels,” IEEE Trans. Wireless Commun.,vol.8,
no. 4, pp. 1866–1875, Apr. 2009.
[24] S. Gitzenis and N. Bambos, “Power-controlled data prefetching/
caching in wireless packet networks,” in Proc. IEEE INFOCOM, 2002,
pp. 1405–1414.
[25] S. Stidham, Optimal Design of Queueing Systems. Boca Raton, FL:
CRC, 2009.
[26] H. Fattah and C. Leung, “An overview of scheduling algorithms in wire-
less multimedia networks,” Wireless Commun., vol. 9, no. 5, pp. 76–83,
Oct. 2002.
[27] M. J. Neely, “Energy optimal control for time-varying wireless net-
works,” IEEE Trans. Inf. Theory, vol. 52, no. 7, pp. 2915–2934,
Jul. 2006.
[28] N. Cordeschi, “Adaptive QoS transport of multimedia over wireless
connections—A cross layer approach based on calculus of variations,”
Ph.D. dissertation, Univ. Rome, Rome, Italy, 2008.
[29] E. Leonardi, M. Mellia, F. Neri, and M. Ajmone Marsan, “Bounds on
average delays and queue size average and variance in input-queue cell-
based switches,” in Proc. IEEE INFOCOM, 2001, pp. 1095–1103.
[30] L. Georgidas, M. J. Neely, and L. Tassiulas, “Resource allocation and
cross-layer control in wireless networks,” Found. Trends Netw.,vol.1,
no. 1, pp. 1–148, Apr. 2006.
[31] M. J. Neely, E. Modiano, and C. E. Rohrs, “Power and server allocation
in a multi-beam satellite with time varying channels,” in Proc. IEEE
INFOCOM, 2002, pp. 1451–1460.
Enzo Baccarelli received the Laurea degree (summa cum laude) in electronic
engineering, the Ph.D. degree in communication theory and systems, and
the Ph.D. degree in information theory and applications from the Università
degli Studi di Roma “La Sapienza,” Rome, Italy, in 1989, 1992, and 1995,
respectively.
He is currently with the INFOCOM Department, Università degli Studi di
Roma “La Sapienza,” where he was a Research Scientist from 1996 to 1998,
became an Associate Professor of signal processing and radio communications
in 1998, and has been a Full Professor of data communication and coding since
2003. He is also the Dean of the Telecommunication Board and a Member of
the Educational Board of the Faculty of Engineering, Università degli Studi
di Roma “La Sapienza.” From 1990 to 1995, he was a Project Manager
with SELTI ELETTRONICA Corporation, where he worked on the design of
modems for high-speed data transmissions. From 1996 to 1998, he worked on
the international project AC-104: Mobile Communication Services for High-
Speed Trains (MONSTRAIN), where he worked on the equalization and coding
of fast time-varying radio-mobile links. He is currently the Coordinator of the
National Project Wireless 802.16 Multi-antenna mEsh Networks (WOMEN).
He is the author of more than 100 international IEEE publications. He is a
coholder of two international patents on adaptive equalization and turbodecod-
ing for high-speed wireless and wired data-transmission systems licensed by
international corporations.
Dr. Baccarelli is an Associate Editor for the IEEE COMMUNICATION
LETTERS. His biography is listed in Who’s Who and Contemporary Who’s Who.
Nicola Cordeschii received the “Laurea degree”
(summa cum laude) in communication engineering
and the Ph.D. degree in information and communi-
cation engineering from the Università degli Studi
di Roma “La Sapienza,” Rome, Italy, in 2004 and
2005, respectively. His Ph.D. dissertation was enti-
tled “Adaptive QoS Transport of Multimedia Over
Wireless Connections—A Cross Layer Approach
Based on Caculus of Variatons.”
He has been a Teaching Assistant for several
courses, such as information theory and coding, sig-
nals and systems, statistical signal processing, and broadband systems. He is
currently a Research Assistant with the INFOCOM Department, Università
degli Studi di Roma “La Sapienza,” whose activity is focused on dynamic
radio-resource allocation for multiantenna active access networks oriented to
multimedia applications. He is the author or coauthor of 44 publications. His
research interests include the design and the optimization of highly performing
systems for wireless applications in centralized and decentralized environments
based on multiantenna platforms, bandwidth adaptation mechanisms, quality-
of-service management for both variable- and constant-bit-rate streams, space-
time coding for imperfect channel knowledge in flat fading, statistical spatial
shaping for multiuser interference mitigation and interference cancellation via
spatial signal processing to reduce external and internal interference, interfer-
ence reduction in ultrawideband infrared (UWB-IR) systems, spatial processing
for multiple-input–multiple-output UWB-IR, and space-time pulse-position-
modulation codes for UWB-IR for maximizing code and diversity gains.
Prof. Cordeschii has been involved in the scientific and managing activities
of the following national (Italian) projects: NEW-INTERNET (2004–2005)
funded by Trento Province, “Wireless 8O2.16 Multi-antenna mEsh Networks
(WOMEN),” PRIN 2005 (Prot. 2005093248), and “Wireless multiplatfOrm
mimo active access netwoRks for QoS-demanding muLtimedia Delivery
(WORLD, Prot.2007R989S).”
3310 IEEE TRA NSACTI ONS ON VEH ICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010
Mauro Biagi (S’98–M’05) received the “Laurea de-
gree” in communication engineering and the Ph.D.
degree in information and communication engi-
neering from the Università degli Studi di Roma
“La Sapienza,” Rome, Italy, in 2001 and 2005,
respectively.
After qualifying as a National Engineers Associ-
ation member, he has been involved as a Teaching
Assistant for several courses, such as information
theory and coding, signals and systems, statistical
signal processing, and broadband systems, since
2002. He currently teaches courses about mobile communications. In 2004,
he obtained a grant for research entitled “Methodologies for space division
distributed and asynchronous access” at the INFOCOM Department, where,
since 2006, he has been an Assistant Professor. He was a Visiting Professor
with the University of British Columbia, Vancouver, BC, Canada, during the
fall–winter of 2009–2010. He is an Associate Editor of the International
Journal of Ultra Wideband Communications and Systems. He is the author or
coauthor of 70 publications. His research interests include space-time coding
and spatial diversity techniques in general, multiuser detection, information-
theory-based secrecy, ultrawideband infrared transceiver architecture and re-
source management, cognitive algorithms and system for spectrum sensing and
access, Smart Grids, and power line communications.
Dr. Biagi is a member of the workgroup on power line communications
in the framework of the IEEE Communication Society. He was involved in
the following European projects: “Dynamic Evolving Large Scale Information
Systems” (DELIS), “Fast Internet for Fast Train Hosts” (FIFTH), and “Dis-
tribution Line Carrier: Verification, Integration and Test of PLC Technologies
and IP Communication for Utilities” (DLC+VIT4IP). He has been involved
in the scientific and managing activities of the following national (Italian)
projects: NEW-INTERNET (2004–2005) funded by Trento Province, “Wireless
8O2.16 Multi-antenna mEsh Networks (WOMEN),” PRIN 2005, and “Wireless
multiplatfOrm mimo active access netwoRks for QoS-demanding muLtimedia
Delivery (WORLD).”
