Energy-aware scheduling algorithms for network stability

Abstract

A key problem in the control of packet-switched data networks is to schedule the data so that the queue sizes remain bounded over time. Scheduling algorithms have been developed in a number of different models that ensure network stability as long as no queue is inherently overloaded. However, this literature typically assumes that each server runs at a fixed maximum speed. Although this is optimal for clearing queue backlogs as fast as possible, it may be suboptimal in terms of energy consumption. Indeed, a lightly loaded server could operate at a lower rate, at least temporarily, to save energy. Within an energy-aware framework, a natural question arises: "What is the minimum energy that is required to keep the network stable?" In this paper, we demonstrate the following results towards answering that question. Starting with the simplest case of a single server in isolation, we consider three types of rate adaptation policies: a heuristic policy, which sets server speed depending on queue size only, and two more complex ones that exhibit a tradeoff between queue size and energy usage. We also present a lower bound on the best such tradeoff that can possibly be achieved. Next, we study a general network environment and investigate two scenarios. In a temporary sessions scenario, where connection paths can rapidly change over time, we propose a combination of the above rate adaptation policies with the standard Farthest-to Go scheduling algorithm. This approach provides stability in the network setting, while using an amount of energy that is within a bounded factor of the optimum. In a permanent sessions scenario, where connection paths are fixed, we examine an analogue of the well-known Weighted Fair Queueing scheduling policy and show how delay bounds are affected under rate adaptation.

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Energy-aware Scheduling Algorithms for Network
Stability
Matthew Andrews
Bell Labs, Murray Hill, NJ
andrews@research.bell-labs.com
Spyridon Antonakopoulos
Bell Labs, Murray Hill, NJ
spyros@research.bell-labs.com
Lisa Zhang
Bell Labs, Murray Hill, NJ
ylz@research.bell-labs.com
Abstract—A key problem in the control of packet-switched
data networks is to schedule the data so that the queue sizes
remain bounded over time. Scheduling algorithms have been
developed in a number of different models that ensure network
stability as long as no queue is inherently overloaded. However,
this literature typically assumes that each server runs at a fixed
maximum speed. Although this is optimal for clearing queue
backlogs as fast as possible, it may be suboptimal in terms of
energy consumption. Indeed, a lightly loaded server could operate
at a lower rate, at least temporarily, to save energy.
Within an energy-aware framework, a natural question arises:
“What is the minimum energy that is required to keep the
network stable?” In this paper, we demonstrate the following
results towards answering that question.
Starting with the simplest case of a single server in isolation,
we consider three types of rate adaptation policies: a heuristic
policy, which sets server speed depending on queue size only, and
two more complex ones that exhibit a tradeoff between queue
size and energy usage. We also present a lower bound on the
best such tradeoff that can possibly be achieved.
Next, we study a general network environment and investigate
two scenarios. In a temporary sessions scenario, where connection
paths can rapidly change over time, we propose a combination of
the above rate adaptation policies with the standard Farthest-to-
Go scheduling algorithm. This approach provides stability in the
network setting, while using an amount of energy that is within a
bounded factor of the optimum. In a permanent sessions scenario,
where connection paths are fixed, we examine an analogue of the
well-known Weighted Fair Queueing scheduling policy and show
how delay bounds are affected under rate adaptation.
I. INTRODUCTION
In this paper we revisit a number of classical network
scheduling problems and examine how they are affected
when we introduce energy awareness. Most such scheduling
problems have been formulated under the assumption that the
processing rate of a server is fixed. However, modern servers
often allow for dynamic adjustment of their processing rate,
because operating at a lower rate typically requires less power.
This has inspired a growing body of research on how to best
take advantage of this so-called rate adaptation capability in
the interest of maximizing energy savings.
Herein, our main focus is whether it is possible to maintain
stability, or equivalently bounded queues, in a network of
servers in an energy-aware manner. In particular, we inves-
tigate the tradeoff between energy consumption and perfor-
mance measures such as delay and queue size.
We remark that there exists an extensive literature proposing
various scheduling algorithms that keep the network stable, as
long as the servers are always operating at maximum rate.
In order to study the problem we posed earlier, though, our
task is to determine how to adapt these algorithms so as
to minimize energy while preserving stability. The related
objective of guaranteeing bounded end-to-end packet delay is
also considered.
A. Modeling
First of all, let us describe our modeling of the problem. We
consider a network of multiple servers, which process and then
forward incoming data traffic. Every server’s processing rate
may be adjusted independently, taking any value in the interval
[Rmin,Rmax]1, where Rmin and Rmax are given parameters
such that 0 < Rmin < Rmax. Furthermore, the power
consumed by the server e while operating at rate reis given
by a so-called energy function f(re). We adopt the common
assumption that f(s) = sαfor some parameter α > 1, which
is based on properties of CMOS circuits [9], [16].
For every server, a rate-adaptive scheduling algorithm
makes two decisions at any given time, namely setting the
processing rate and determining which data in the server queue
to forward. We concentrate on work-conserving algorithms
only; in other words, the algorithm cannot order a server not
to forward any traffic (for whatever reason) at a time when the
latter’s queue is non-empty. Moreover, with regard to network
traffic we consider two standard models, defined below.
1) Temporary Sessions Model: This model is also known
as the Adversarial Queuing Model (AQM). It aims to capture
situations in which the set of sessions (or connections) carried
by the network is highly volatile. The model stipulates that
packets are injected into the network by some adversarial
process A, henceforth referred to simply as the adversary.
Additionally, A specifies the path along which each packet
must be routed at the time of its injection, and the hop count
of every such path is bounded by a parameter d. Let Ae(t,t?)
be the total size of packets injected into the network during
the time interval [t,t?) that include server e on their paths.
In order for each server not to be inherently overloaded, the
packet injection by A is restricted to be (σ,1−ε)-admissible,
which means that for all e and all intervals [t,t?),
Ae(t,t?) ≤ σ + (1 − ε)Rmax(t?− t) ,
1In practice, only a discrete set of rates would likely be available, due to
hardware constraints. Nevertheless, we extend it to a continuous range for
clarity of exposition. This entails no loss of generality for our results.

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where σ ≥ 0 is the burst size and ε > 0 reflects the load factor.
We then say that A is a bounded adversary of rate (σ,1−ε).
2) Permanent Sessions Model: This model is sometimes
referred to as the connection-oriented model, and it reflects
a setting where all data is transported along pre-defined
connections. Each connection i is specified by a path Pi, a
burst size σi, and an injection rate ρi. Let Hi(t,t?) be the
total injection into connection i during the time interval [t,t?).
Again, to ensure servers are not inherently overloaded, the
injection is restricted to be admissible in the following sense.
Hi(t,t?) ≤ σi+ ρi(t?− t)
?
Remark. It is easy to see that any traffic admissible to the
permanent sessions model is also admissible to the temporary
sessions model. However, the converse statement is not true.
Indeed, consider two paths P1and P2that share a server e. Let
us partition time into intervals of arbitrarily large lengths. The
temporary sessions model allows injections that alternate be-
tween only injecting along P1during odd-numbered intervals
and only injecting along P2during even-numbered intervals,
at rate 0.9Rmax in both cases. This cannot happen in the
permanent sessions model, because the connection rates must
be set at ρ1= ρ2= 0.9Rmax. As a result,?
are arbitrarily large, admissibility in the permanent sessions
model cannot be achieved by setting a large burst size for
each connection.) Therefore, strictly more traffic injections are
admissible in the temporary sessions model.
We say that the network is stable if the aggregate queue
size remains bounded over time. Our goal in this paper is to
maintain stability while also tailoring server rates to traffic
loads in such a way that energy usage is minimized. For
example, if the long-term traffic through server e satisfies
Ae(t,t?) ? Rmax(t?− t) for t?? t, then the latter can
operate at a rate much smaller than Rmaxwithout jeopardizing
stability. However, there is no way to know exactly how much
traffic will arrive at e in the future, and thus it is not clear
a priori how to set its rate for optimal energy efficiency.
This constitutes a scheduling problem which, to the best of
our knowledge, has not been addressed in conjunction with
network stability before.
∀i,∀[t,t?)
∀e
i:e∈Pi
ρi≤ (1 − ε)Rmax
(1)
i:e∈Piρi> Rmax,
which would violate (1). (Note that since the interval lengths
B. Previous Work
1) Network stability: If every server runs at rate Rmax
constantly, then there exist well-known scheduling algorithms
ensuring a time-independent upper bound on the size of
all queues, in the temporary sessions model, for any given
bounded adversary and any network topology. Such algorithms
are called universally stable. More specifically, it was shown
in [1] that several scheduling algorithms – including Farthest-
to-Go (FTG) and Nearest-to-Source (NTS), whose definitions
we provide later – are universally stable, and also guarantee
bounded end-to-end packet delay. By contrast, some very
natural algorithms such as First-in-First-out (FIFO) and Last-
in-First-out (LIFO) are not universally stable.
In the permanent sessions model, the most widely studied
scheduling algorithm is Generalized Processor Sharing (GPS),
in particular its packetized form that is sometimes called
Weighted Fair Queueing (WFQ) [8]. Under GPS and WFQ,
each server operates by splitting service among all backlogged
connections according to some predetermined weights. We
shall focus on the version known as Rate Proportional Pro-
cessor Sharing (RPPS), in which the weight for connection
i is equal to ρi at each server. Parekh and Gallager [13],
[14] proved that if each server always runs at rate Rmaxthen
the packetized version of RPPS is stable and they derived
an end-to-end delay bound for each connection. (See (2) in
Section I-C3.) We note that RPPS highlights the difference
between the two traffic models in question, since it has
been established that RPPS is not necessarily stable in the
temporary sessions model [3].
2) Energy Efficiency: The study of energy minimization via
rate adaptation was initiated by Yao et al. [16], where they con-
sidered energy functions of the form f(x) = xα. Subsequently,
a large number of papers focused on the problem of conserving
energy on a single server. Most of this prior work falls in two
categories. In the first one (e.g. [5], [7]), every job has an
associated deadline, and the goal is to minimize energy while
meeting all the deadlines. In the second category (e.g. [4],
[15]), jobs do not have individual deadlines and the goal is
to minimize the sum of the energy used plus the aggregate
response time of the jobs. Note that even in the single-server
case neither of these objectives directly addresses our goal of
minimizing energy consumption while maintaining stability.
Another body of work focuses on the powerdown model,
in which the servers cannot alter their processing rates but
can toggle between the on and off states at a switching
cost. For example, [10] discusses the energy consequences of
putting router and switch components to sleep. The survey
[11] presents known results for both the powerdown and the
rate adaptation models.
As already mentioned, much less attention was paid to
scheduling for energy minimization in networks of servers.
Nedevschi et al. [12] considered both rate adaptation and
powerdown in the context of multiple servers and concluded
that energy could be saved if we batch together packets
with the same source and destination. Alternative techniques
for batching packets in the powerdown model and thereby
minimizing the number of transmissions between states were
also explored in [2]. To the best of our knowledge, no prior
work has attempted to tackle the specific problem that we
deal with here, namely that of minimizing energy usage while
maintaining network stability.
C. Results
Our paper is divided into three main sections.
1) Single-server results: In Section II we focus on schedul-
ing a single server in isolation. This allows us to study basic
issues such as the tradeoff between queue sizes and energy

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in this simplest setting. For example, suppose optB(t) is the
optimal energy necessary to keep the queue size bounded
by B up to time t. We show that no online algorithm can
simultaneously keep the energy consumption within a certain
factor of optBand the queue size within a corresponding
constant factor of B. This establishes a concrete limit on
what one can hope for in terms of maintaining stability while
minimizing energy.
We then propose three rate-adaptive policies to set server
rates: Batch, SlowStart and queue-based. Roughly speaking,
the Batch policy accumulates data of size 2B before serving
it at a rate equal to the average arrival rate of this data. This
policy ensures a maximum queue size of O?B log2
with rate Rmin at the beginning of each interval where the
queue is non-empty, and after some time it ramps up the
processing rate linearly, in a deterministic fashion. SlowStart
ensures a maximum queue size of O?σ+BR2
rather family of policies) sets the server rate solely as a
function of the current queue size. Although we cannot bound
the energy usage of this approach for all admissible traffic,
when the arrival rate is constant it can keep the queue size
bounded by O(B) while consuming O(optB(t)) energy.
2) Results for Temporary Sessions Model: In Section IV
we examine the multiple-server scenario in the temporary
sessions model. Our starting point is the universally stable
[1] algorithm Farthest-to-Go (FTG), which gives priority to
data farthest from its destination (in terms of server hops). We
generalize all three rate-adaptive policies mentioned above to
multiple servers. When combined with FTG, both Batch and
SlowStart yield the desired properties of bounded queue size
and bounded energy consumption. Additionally, we derive a
bound on end-to-end delay in the case of SlowStart, although
we are so far unable to establish a similar claim for Batch.
Note that for “traditional” work-conserving scheduling algo-
rithms, bounded delay is equivalent to bounded queue (hence
stability). However, this is no longer true when rate adaptation
is an option, as demonstrated by the example in Section III.
In the interest of space, our presentation focuses on combin-
ing SlowStart with FTG. All results in this section carry over
if we replace FTG with another universally stable algorithm
Nearest-to-Source (NTS), which gives priority to data closest
to its source.
3) Results for Permanent Sessions Model: For the Perma-
nent Sessions Model, our starting point is the work of Parekh
and Gallager [13], [14], who showed that if every server
always runs at Rmax, Weighted Fair Queueing guarantees the
following end-to-end delay bound for each connection i,
Rmax
Rmin
?and
energy usage of O(6α· optB(t)). The SlowStart policy starts
max
R2
min
?and energy
consumption of O(2α· optB(t)). The queue-based policy (or
σi+ (Ki− 1)Li
ρi
+ KimaxiLi
Rmax
(2)
where σi, ρi, Ki and Li are respectively the burst size,
connection rate, hop count and maximum packet size for
connection i. Our major result in this section is that for any
rate-adaptive policy that uses rates between Rminand Rmax
and always uses rate Rmaxwhen the queue exceeds a threshold
U, if we schedule according to Weighted Fair Queueing then
the end-to-end delay is bounded by
?maxiLi
II. THE SINGLE-SERVER CASE
We begin our analysis by focusing on a single server in
isolation. This will allow us to determine some of the basic
tradeoffs between queue size and energy usage. First of all,
we define a simple lower bound on energy consumption to
keep the queue size bounded by B, which shall be used as a
benchmark henceforth. We then present a bound on the optimal
tradeoff between queue size and energy efficiency that can
be achieved by any rate adaptation policy. Subsequently, in
Sections II-C, II-D, and II-E, we propose three approaches to
set the server rate adaptively and show upper bounds on their
energy usage and queue size.
σi+ (Ki− 1)Li
ρi
+ Ki
Rmin
+Rmax
Rmin
U
ρi
?
.
A. Lower bound on energy usage
Recall that optB(t) is the minimum amount of energy
required by time t if we wish to keep the queue bounded
by B at all times. Let optB(t,t?) be defined similarly on the
interval [t,t?). We derive a simple bound on optB(t,t?).
Lemma 1. For a single server e,
?
Proof: Since data of size Ae(t,t?) arrives during the
time interval [t,t?) and the queue size is to be less than B
at time t?, then the amount of data that must be processed
during [t,t?) is at least max{0,Ae(t,t?) − B}. However, we
are assuming that f(·) is a convex function and the minimum
processing rate is Rmin, thus in order to serve that amount
of data with minimal energy usage the server must operate
at speed max?Rmin,(Ae(t,t?)−B)/(t?−t)?throughout the
We shall assume that B ≥ σ, since one cannot hope to
maintain the queue size smaller than the burst size. Even then,
the energy bound of Lemma 1 may not be achievable, for
instance if most of the data injected during [t,t?) is injected
towards the end of the interval.
optB(t,t?) ≥ fmax
?
Rmin,Ae(t,t?) − B
t?− t
??
· (t?− t) .
interval [t,t?). The bound follows.
B. Bound on tradeoff between queue size and energy
Intuitively, guaranteeing a better bound on queue size should
incur higher energy usage. We establish that such a tradeoff
is inherently unavoidable.
Lemma 2. Let x1 =
satisfies
B
Rmin,x2,x3,... be a sequence that
?
?<j
xjf
?B
2xj
≥ ν
?
x?f
?B
x?
?
,
(3)
for some given ν. Further, suppose that a rate adaptation
policy RA uses energy at most ν · optB(t) by time t, for all
times t. Then, the maximum queue size under RA is at least
(J(ν) + 1)B/2, where J(ν) = argmax?j : xj≥
B
Rmax
?.

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4
Proof: We define a sequence t0= 0,t1,...,tJ(ν), with
tj = tj−1+ xj for 1 ≤ j ≤ J(ν). Assume that data of
size B arrives at each time t0,t1,...,tJ(ν). An optimal rate
adaptation policy would serve data of size B during each
interval [tj−1,tj) for 1 ≤ j ≤ J(ν), so that the queue size
remains at most B.
Nevertheless, the policy RA cannot follow this behavior,
for the simple reason that it cannot predict the future. More
precisely, when data of size B arrives at time tj, j ≥ 1, RA
can deduce what the optimal schedule should have been for
serving all data injected at t0,...,tj−1. Then, by Lemma 1,
?
Since RA does not know whether (or when) additional data
will be injected in the future, it can only afford to use at most
ν · optB(tj) energy in the interval [0,tj+1), hence a fortiori
in the interval [tj,tj+1). From (3),
optB(tj) ≥
?≤j
(t?− t?−1)f
?
B
t?− t?−1
?
=
?
?≤j
x?f
?B
x?
?
.
ν · optB(tj) ≤ (tj+1− tj)f
which implies that the amount of data RA can serve in
[tj,tj+1) is no more than B/2. Finally, during the interval
[t0,t1), RA sets the speed to Rminby default and serves data
of size B. Hence, at time tJ(ν) the queue size is at least
(J(ν) + 1)B − B − (J(ν) − 1)B/2 = (J(ν) + 1)B/2.
Corollary 3. Suppose that the energy function has the form
f(s) = sαand that a rate adaptation policy RA uses energy at
most ν ·optB(t) by time t, for all times t. Then, the maximum
queue size under RA is at least Ω?B logν(Rmax/Rmin)?.
Sketch of proof: For j = 1,2,..., let
B
Rmin
?
Algebraic manipulation shows that the above definition of xi
satisfies (3).
?
B/2
tj+1− tj
?
,
xj=
(2αν + 1)
j−1
1−α,
and
J(ν) =(α − 1)log2αν+1
Rmax
Rmin
+ 1
?
.
C. The Batch policy
We now show how to ensure both stability and near-optimal
energy consumption using a rate adaptation policy that we call
Batch. The basic idea is to wait until just enough data has
arrived at the server so that our lower bound on optB(t) allows
for a transmission rate that can serve all this data. Moreover,
in the interest of simplifying the analysis, let us first assume
that the server may operate even at rates higher than Rmax.
To begin with, define a busy interval as a time interval dur-
ing which the queue is non-empty. Suppose that a busy interval
I starts at time τ0, and let τj= min{t ∈ I | Ae(τ0,t) ≥ 2Bj}.
Therefore, in each interval [τj,τj+1) data of size 2B arrives
at the queue, and the policy makes sure that an equivalent
amount of data is served by time 2τj+1−τj. This is achieved
by setting re(t) = max?Rmin,?2B/(τj+1−τj)?, where the
summation is over all indices j such that t ∈ [τj+1,2τj+1−τj).
Denote these indices (if any such exist) by j1< j2< ··· <
jm, and the interval [τjm,τjm+1) by I(t). We have:
Lemma 4. If re(t) > Rmin, then re(t) ≤ 6B/|I(t)|.
Proof: For a given t, note that re(t) > Rminguarantees
the existence of indices j1,j2,...,jm as defined above, for
some m ≥ 1, and thus also the existence of I(t). Since
j? ≥ j?−1+ 1, observe that τj?≥ τj?−1+1. Furthermore,
t ≤ 2τj?+1− τj?implies t − τj?+1≤ τj?+1− τj?. Combining
the above yields t − τj?+1 ≤ τj?+1− τj?−1+1, and hence
2(t − τj?+1) ≤ t − τj?−1+1. Consequently, re(t) equals
m
?
m−1
?
m−1
?
≤
t − τjm−1+1
≤
τjm+1− τjm
=
τjm+1− τjm
where the last inequality is due to the fact that t−τjm−1+1≥
t − τjm≥ τjm+1− τjm.
Lemma 5. The total energy used by Batch up until time t is
at most sups
6α· optB(t).
Proof: Clearly, we need only consider the energy con-
sumption of Batch during busy intervals – or, even more
restrictively, during intervals where re(t) > Rmin. That
holds because Rmin is the most energy-efficient among the
allowable rates, i.e. it minimizes the ratio f(s)/s, owing to
our assumption about the energy function in Section I-A.
Within a busy period, consider some interval [τj,τj+1) and
let I?(τj) = {t | I(t) = [τj,τj+1)}. Obviously |I?(τj)| ≤
τj+1−τj, by the definition of I(t). Due to Lemmas 4 and 1,
the energy consumption during I?(τj) is at most
?
≤ sup
?=1
2B
τj?+1− τj?
=
m−1
?
?=1
2B
τj?+1− τj?
2B
t − τj?+1
2?−m+12B
t − τjm−1+1
4B
+
2B
τjm+1− τjm
2B
τjm+1− τjm
≤
?=1
+
≤
?=1
+
2B
τjm+1− τjm
2B
τjm+1− τjm
2B
τjm+1− τjm
6B
|I(t)|,
+
4B
+
6B
=
f(6s)
f(s)· optB(t). For f(s) = sα, this is at most
f
6B
τj+1− τj
?
· I?(τj) ≤ f
?
6B
τj+1− τj
f(6s)
f(s)
?
· (τj+1− τj)
s
· optB(τj,τj+1) .
Repeating this for every interval of the form [τj,τj+1), across
all busy periods, accounts for the energy consumption of all
intervals where re(t) > Rmin, without overlaps. The lemma
is thus established.
Subsequently, we derive a bound on the queue size.
Lemma 6. The rate adaptation policy Batch guarantees that
the queue size is never larger than 2B?log2(Rmax/Rmin)+4?.

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Sketch of proof: At time t, the amount of data still in
the queue is given by
m
?
?=1
2B ·2τj?+1− τj?− t
τj?+1− τj?
≤ 2B · m .
We need to bound m. On one hand, it is fairly straightfor-
ward to deduce that t−τj1+1≤ τj1+1−τj1≤ 4B/Rmin, oth-
erwise the busy interval would end before τj1+1. On the other
hand, t − τjm−1+1 ≥ τjm+1− τjm≥ B/Rmax, because we
assumed that B ≥ σ. Together with 2(t−τj?+1) ≤ t−τj?−1+1,
the above yield m ≤ log2(Rmax/Rmin) + 4, as required.
Finally, we explain how Batch works on a server whose
maximum speed is strictly Rmax, which is the case of practical
interest. More specifically, Batch tries to simulate its own
behavior on an unrestricted server, which we described earlier.
This involves two modes of operation, normal and catch-up.
While in normal mode, the policy sets the same rate as it
would set on the simulated unrestricted server. As soon as
the latter rate exceeds Rmax, Batch enters catch-up mode, in
which the rate is held at Rmaxconstantly, and only reverts to
normal mode when the queue sizes of the two servers, actual
and simulated, coincide.
Naturally, in normal mode the bounds of Lemmas 5 and 6
remain valid. Suppose that just before switching to catch-up
mode, the queue size is Q ≤ 2B?log2(Rmax/Rmin)+4?. After
queue size becomes ≤ Q+σ+(1−?)Rmaxλ−Rmaxλ ≤ Q+σ,
which also implies that Batch will not stay in catch-up mode
for ever. Furthermore, during the entire interval spent in catch-
up mode, the actual server processed (at constant rate) exactly
the same amount of data as the simulated one (at variable
rate). Since f(·) is convex, the former server consumed no
more energy than the latter.
remaining in that mode for a time interval of length λ, the
Theorem 7. The rate adaptation policy Batch ensures that
maximum queue size is bounded by Ω?B log(Rmax/Rmin)?
D. The SlowStart policy
In addition to Batch, let us introduce another policy named
SlowStart. It stipulates that re(t) = min{Rmax,g(t−θe(t))},
where g(·) is a simple piecewise linear function
?
and consumes at most constant times the optimal energy.
g(x) =
Rmin
R2
min
2B· x
if 0 ≤ x ≤
if x >
2B
Rmin
2B
Rmin
,
and θe(t) denotes the beginning of the busy interval of e that
contains t, if one such exists, else is equal to t.
Within a busy interval, note that SlowStart initially sets the
speed at Rminand later increases it linearly (hence the name),
in a deterministic fashion. Interestingly, traffic arrivals have
no direct effect on the speed, other than by determining when
the busy interval ends. Below we summarize the properties of
SlowStart in the single-server setting.
Theorem 8. The rate adaptation policy SlowStart ensures that
maximum queue size is bounded by Ω?BR2
max/R2
min
?
and
consumes at most sups
t. For f(s) = sα, this is at most 2α· optB(t).
Sketch of proof:
rems 15 and 16 of Section IV, for n = d = 1.
Remark. Although the above queue bound is worse than that
of Batch, the SlowStart policy is nevertheless valuable, as we
employ a variant of it in Section IV.
f(2s)
f(s)· optB(t) energy up until time
Essentially a special case of Theo-
E. Queue-based policies
A queue-based policy sets the server rate according to
re(t) = r(qe(t)), where qe(t) is the queue size at time t, and
r(·) is a non-negative and non-decreasing continuous function.
In particular, if the queue size exceeds some threshold U then
r(qe(t)) is set to the maximum rate Rmax. Otherwise, r(qe(t))
is at least the minimum rate Rmin. If traffic arrives at a constant
rate, it is easy to provide good bounds for both queue and
energy under such a policy.
Theorem 9. Suppose that traffic arrives at the server at
a constant rate ρ ≤ (1 − ε)Rmax. Under a queue-based
rate adaptation policy for which U = B, the queue size
never exceeds B and the energy consumption up to time t
is O(optB(t)), for large enough t.
Proof: Starting from an empty queue at time 0, both
queue size and server speed begin to increase, until the speed
reaches ρ. At that time, the queue size is at most U, because
of the monotonicity of r(·) and the fact that ρ < Rmax. Both
the server speed and queue size remain constant thereafter.
Additionally, the energy usage up to time t is at most t·f(ρ),
whereas optB(t) ≥ t · f?(ρt − B)/t?.
above queue bound increases to B + σ, however we do not
know whether any energy bound can be guaranteed.
For arbitrary admissible traffic, it is easy to see that the
III. BOUNDED QUEUE VS. BOUNDED DELAY
We now turn to a network of multiple servers. Before diving
into more complex results, let us state a simple but perhaps
somewhat unexpected observation on the relationship between
bounded queues and bounded delays. A folklore result states
the equivalence between bounded queue and bounded delay
when server rates are kept at Rmax.
Theorem 10 (Folklore). If a work-conserving algorithm al-
ways works at Rmax, then bounded delay is equivalent to
bounded queue under any bounded adversary of rate (σ,1−ε).
However, the above theorem no longer holds for some
rate-adaptive algorithms. In particular, a stable rate-adaptive
algorithm may not guarantee a bounded delay for every packet.
Lemma 11. Neither FTG nor NTS guarantees a finite packet
delay even in the permanent sessions model.
Sketch of proof: Consider FTG on the following config-
uration. The network consists of three servers e1, e2, e3 on
a line and carries two connections: connection 1 goes to e3
through e1and e2, with rate (1−2ε)Rmax, and connection 2