Scheduling Algorithms for Optimizing the Tradeoffs
between Delay, Queue Size and Energy
Matthew Andrews Bell Labs, Murray Hill, NJ [email protected]
Lisa Zhang Bell Labs, Murray Hill, NJ [email protected]
Abstract—In this paper we propose energy-aware scheduling algorithms that aim to optimize the tradeoff between energy con-sumption and traditional performance measures such as queue size and delay. We use the power-rate functionf(x) =c+xα
for x >0andf(0) = 0to model the energy consumption. Scheduling algorithms have been studied in the past for the special case in which the base power c= 0.
In general the base power c can be significant. We propose a batch-based scheduling algorithm that can keep the energy consumption to asymptotically optimal while paying for a loga-rithmic factor in the queue size. Such a tradeoff is best possible. We also characterize the energy-delay tradeoff for a rate-adaptive version of the Weighted Fair Queuing scheduling algorithm.
I. INTRODUCTION
In this paper we examine a number of problems related to minimizing the energy required to carry data in communi-cation networks. As communicommuni-cation networks achieve higher performance they are accounting for a growing fraction of overall energy consumption. For this reason there is a growing body of literature that looks at the energy performance of networks in addition to more traditional measures such as throughput, latency and reliability.
Our work focuses on the two classical issues of stability, namely bounded queue size, and end-to-end delay. We assume that network elements, e.g. routers or switches, can run at variable speeds. The power required by each element is an increasing function of its current speed. Our first problem examines how to schedule packets so that the queue size at each network element is kept under a target bound with minimal energy consumption. In our second problem we aim to schedule packets from a fixed set of connections in such a way that we achieve a delay bound similar to the classic delay bound for the well-known Weighted Fair Queuing scheduling algorithm. Once again we want to do this while minimizing the energy consumption.
Energy-aware scheduling algorithms have been examined before, e.g. [2], [24], [5]. However, our current work explores a number of extensions. Most significantly, we study a situation where the power consumption is not a purely convex function of server speed. We instead assume that there is a non-zero “startup-cost”, also known as the base power, for running a
server at a non-zero rate. We elaborate on other distinctions later on.
A. Modeling Power Consumption
Consider a server in a network, which operates at a speed re(t) at time t. For convenience we perform our analysis in
continuous time so that if a packet of size`p starts service at
servereat time τ0, it completes service at timeτ1= inf{τ1:
Rτ1
τ0 re(t)dt≥`p}.
Whenever a server is serving packets it incurs an energy cost. In particular we assume that there is a functionf(·)such
that when servereis running at speed re(t) at timetit
con-sumes powerf(re(t)). Therefore, the total energy consumed
during the time interval [τ0, τ1) is given by Rτ0τ1f(re(t))dt.
One common power-rate function has the form
f1(x) =xα, (1)
which is considered in earlier scheduling work such as [2], [24], [5]. The value ofαis typically a small constant no bigger than three [10].
However, in many situations such a smooth convex function may not be entirely appropriate. In particular, we may well be faced with hardware for which although power is an increasing function of rate, there is some fixed power needed to keep the server operational whenever it is operating at a non-zero rate. Hence in these situations it makes more sense to consider a power-rate function of the form:
f2(x) =
c+xα x >0
0 x= 0 . (2)
We refer to the termcas thebase powerand evidence exists
that it is non-negligible [10]. One consequence of this function is that it sometimes makes sense to run a server slightly faster than is strictly necessary to meet network Quality-of-Service constraints if this allows us to turn off the server to rate zero at other times. Another way of saying this is that unlike the case with no base power, whenc >0the power-rate function
is no longer strictly convex. It exhibits aspects of a convex (superadditive) function due to thexαbut also exhibits aspects
of a subadditive function due to thec term.
The distinction between the power-rate functionsf1(·)and
f2(·) has been studied in the context of routing. Here the
goal is to choose routes for a fixed traffic matrix over a given network topology so that the total energy consumption is minimized. Under power-rate functions (1), the routing problem admits a constant approximation, where the constant isα-dependent [4]. In contrast, under (2) the routing problem
admits a polylogarithmic approximation which is essentially best-possible [1]. In the current paper we focus on scheduling alone and study the effect of the base powercon scheduling.
Allowed frequency of rate changes: We remark that one
of the goals of this paper is to analyze the situation where there is an immediate cost to run the server at a non-zero rate. One artificial way to get around this situation would be to rapidly switch between running at non-zero rate and running at zero rate. This would allow us to “convexify” the function such that it has the form min{c+xα, λx} for some parameterλ.
However, such rapid switches of speed would not be feasible in practice and so such “convexification” would be impractical. In order to formalize this notion we place an upper bound on how often the rate can be changed. In particular we say that an algorithm for changing a server rate is reactive if it changes
speed at most once in an amortized sense for every packet arrival or departure at the server. Hence we do not allow a server to rapidly change speed in between packet arrival and departure events. All of the algorithms discussed in this paper will be reactive.
B. Traffic Models
We assume a set of data packets that arrive into the network consisting ofmservers. Each packetphas a size denoted by`p
and has a fixed route through the network. We consider packet arrivals that conform to two different models, both of which are standard in the literature. Our first connectionless model,
e.g. [7], is parameterized byσ. We assume that within any time interval of durationτ, for any serverethe total amount of data injected into the network that wishes to pass through servere is at mostσ+τ. Here we implicitly assume that time is scaled so that the maximum long-term injection rate to any server is at most 1. In the second connection-based model, e.g. [23],
we have a set of connectionsF, each of which consists of a route through the network. The injections into each connection i∈F are(σi, ρi)-controlled for some parametersσi, ρiwhich
means that the total size of the injections into connection-i during any interval of length τ is at most σi+ρiτ.
C. Problem Definitions
1) Scheduling for Bounded Queues: We first consider the
problem of scheduling packets so that the queue size is bounded above by a parameter B. For a given sequence of packet arrivals, letOptB(t)be the minimum amount of energy
used up to time tby any algorithm that keeps the queue size bounded by B. Our objective is to relate the queue size of a candidate scheduling algorithm to B and relate its energy consumption toOptB(t)for all timet. We study this problem
in the connectionless model for both a single server in isolation as well as a network of servers.
Our algorithm is based on the Batch algorithm that first presented in [2] for the power-rate function f1(·). The basic
idea of Batch is to wait until a batch of data of a certain size has arrived at the server and then to serve this data using an amount of energy that can be bounded with respect to OptB(t). Underf1, the most energy-beneficial servicing rate
is always as low as possible. However, underf2(·)it can be
more beneficial to turn off the server from time to time but operate it at a higher positive rate at other times.
2) Scheduling with Deadlines: The second problem deals
with scheduling packets each with an arrival time and a dead-line. The goal is to respect all the deadlines while minimizing the energy used. Again, we are interested in a power-rate function of the formf2(·).
We build our algorithm on the Average Rate algorithm (AVR) of Yao et al. [24] that was also studied by Bansal et al.[5]. These papers consider a single server only and a power-rate function of the formf1(·). For a packetpwith arrival time
sp and a deadlinedp, AVR assigns a service rate,
r(t) := X
p:sp≤t≤dp
`p
dp−sp
.
In other words, each packet contributes a rate equal to the minimum average rate required to meet its deadline during the entire interval[sp, dp)for which the packet isactive. The
papers [24], [5] showed that if these service rates are used and packets are scheduled according to the Earliest-Deadline-First algorithm all deadlines are met and the total energy used is within a factor2α−1αα of optimal. Further, for AVR this
analysis is tight.
For a single server, our problem is identical to that in [24], [5], with the exception of the difference in the power-rate function. For multiple servers under the connection-based model, our goal is to come up with an rate-adaptive version of the well-known Weighted-Fair-Queueing (WFQ) algorithm, with a comparable end-to-end delay guarantee and analyzable energy consumption. The traditional WFQ guarantees that every packetpfrom a connectioniwithKi hops and(σi, ρi)
-controlled traffic arrival has a delay bound of σi+Ki`p
ρi
+KiLmax, (3)
whereLmaxis the maximum packet size over all connections
[23], [8]. Indeed, WFQ and its fluid version Generalized Processor Sharing (GPS) bear a resemblance to the processor sharing nature of AVR. We take advantage of AVR and use quantities similar to (3) to set connection-ipacket deadlines.
The paper [2] defined a rate-adaptive WFQ algorithm under the power-rate function f1(·), for which end-to-end delay
bounds were given but no bounds on energy were obtained.
D. Our Results
• In Section II we analyze a single-server queue-energy tradeoff under the power-rate functionf2(·), namely with
non-zero base power. We present a variant of the Batch algorithm for which the queue size is kept at most a log factor timesB and the energy used is kept atO(1)times
OptB(t)where the constant depends on the exponent α
of the power-rate function. The achieved queue-energy tradeoff is asymptotically best possible.
• We also extend the Batch algorithm to the case of multiple servers. We state, without further elaboration,
that if we select the speed at each server according to the Batch algorithm and schedule all packets according to the Nearest-to-Source protocol (that gives priority to packets according to the number of hops from the source), then we can bound queue size and energy relative to B and OptB,e respectively. HereOptB,e is the optimal amount
of energy used by servereso that the queue size at server e is never larger thanB.
• In Section III we adapt the AVR algorithm from [24], [5] under the power-rate functionf2(·). In particular we
show that for a single server model with a given set of arrival and departure times, there is an adaptation of AVR that meets all the deadlines and uses energy that is within a constant factor of optimal.
• In Section IV we consider the connection-based multi-hop scenario. To achieve a delay bound that has a similar form to the classical delay bound of (3) for the well-studied Weighted Fair Queuing (WFQ) scheduling algo-rithm, we begin by showing how to convert delay bounds for Generalized Processor Sharing to delay bounds for WFQ in the case of variable server rates. We also show that for a packet-by-packet scheduler such as WFQ we typically need to extend the length of time that a packet contributes to the service rate at a server. Lastly we consider end-to-end delay bounds for each connection and derive a tradeoff between the delay bound that can be obtained versus the energy utilized.
E. Other Related Work
As already mentioned, this paper builds off the earlier work in [2] for understanding the queue-energy tradeoff and builds on [24], [5] for understanding the delay-energy tradeoff. However, only a power-rate function without the base power was considered there. Another common model for energy consumption is the powerdown model, e.g. [3] in which the server operates at either zero rate or a single positive rate. More generally there is a vast literature on adjusting processing rate to traffic which includes the papers [19], [17], [18], [20], [6], [9], [21], [12], [3], [11], [14], [16], [15], [22], [13].
II. QUEUE VS. ENERGY: SINGLE-SERVER
In this section we examine the tradeoff between queue size and energy for a single server in isolation. The paper [2] introduced an algorithm called Batch for the power-rate func-tion f1(·). Batch repeatedly collects a fixed amount of traffic
A in a window and then serves this amount of traffic A in the following window of the same duration. The service rate is A/t if t is the duration of the window that collects the traffic, where the duration changes from window to window depending on the traffic arrival.
Under the new power-rate function f2(·), there is a
min-imum rate ropt under which the server should not operate
unless it is turned off completely.
Lemma 1. To process traffic amount A in time duration T
in a energy efficient manner, the server operates at rate r=
max{ropt, A/T}for the durationT /r, where
ropt = c α−1 1/α (4)
Proof:We observe that, for energy minimization, a server
only needs to operates at one non-zero rate in order to process traffic of amount A in duration T. This is due to the convexity of f2(x) for positivex > 0. Let t be the time
duration for which the server is active. The server operates at rate A/t when active and consumes a total energy of f2(A/t)t= (c+(A/t)α)t.The optimaltis set toA α−c1
1/α
, which sets the derivative of the above expression to zero. Therefore, the minimum energy is achieved at rateroptdefined
in the statement of the lemma.
Algorithm Description: We are now ready to define
the new batch algorithm. Batch partitions time into arrival intervals of length T1, T2, . . . ,. During each arrival interval
traffic of total size 2B arrives. If the ith interval finishes at time ti, then this traffic of size 2B is served during the service interval [ti, ti+Ti0) at rate r = max{ropt,2B/T}
where Ti0 = 2B/r. If r = ropt, such a service interval is a minimum-rate service interval, and the corresponding arrival
interval[ti−Ti, ti)aminimum-rate arrival interval.
Note that multiple service intervals may overlap if the arrival intervals have varying lengths. Let r(I) be the service rate
associated with service intervalI. At timet, the server serves at the rate r(t), which is the total rate P
I:t∈Ir(I) over all
service intervals that contain t.
Analysis: To analyze the energy conumption by Batch,
we need to bound the combined rate r(t) which requires
knowing how service intervals overlap. We first observe that two min-rate service intervals never overlap since such service intervals are shorter than their corresponding arrival intervals which do not overlap one another.
Lemma 2. No two min-rate service intervals overlap. We now bound the combined rate r(t)at time t. Consider all service intervals that contain t. Among those, let I(t) be
the non-min-rate service interval with the latest starting time. Lemma 3. IfI(t)exists,r(t)≤8B/|I(t)|. Otherwise,r(t) =
ropt.
Proof:If only a min-rate service interval containst, then r(t) = ropt. Otherwise, let us concentrate on non-min-rate
intervals only. Let t1 < t2 < · · · < tJ < t be the starting
time of those intervals, wheretJ is the starting time ofI(t).
Note that t−t1 ≥2(t−t2) ≥... ≥ 2J−1(t−tJ) and that
t−tJ−1≥ |I(t)|. We have r(t) ≤ ropt+ 2B |I(t)|+ X 1≤j<J 2B t−tj ≤ ropt+ 2B |I(t)|+ 4B t−tJ−1 ≤ ropt+ 6B |I(t)| ≤ 8B |I(t)|
Lemma 4. The energy consumption under Batch is at most
8αOptB.
Proof: If I(t) does not exist, then t is in a min-rate service interval. The corresponding min-rate arrival interval has traffic arrivals of size2B, for which the optimal algorithm has to serve at least B during the arrival interval. A lower bound on the optimal energy consumption to serve traffic B is obtained by serving it at rateropt. Meanwhile, Batch serves
traffic of size 2B at rateropt. Hence, the energy consumption
due to Batch, f2(ropt)(2B/ropt), at most doubles the optimal
consumption, f2(ropt)(B/ropt).
If I(t) exists, the energy-optimal algorithm could either
serve B bits at ropt or at rate B/|I(t)|. In the former case,
B/I(t)< ropt and2B/I(t)≥ropt since I(t)is not a
min-rate service interval. This shows that the optimal algorithm has to operate at ropt for a time duration that is at leastI(t)/2.
Hence, the ratio of the energy consumption due to Batch to the optimal is at most, f2(2B/|I(t)|)|I(t)|
f2(ropt)|I(t)|/2) = 2.
In the latter case, Batch operates at rate at most 8B/|I(t)| and an energy-optimal algorithm operates at rate at least B/|I(t)|. Hence the ratio is at most f2f2(8(B/B/||II((tt))||))<8α.
With respect to the queue bound, we have the following. Lemma 5. The queue size under Batch is O(Blog2roptσ ).
Proof:As argued in the proof of Lemma 3, the length of
non-min-rate service intervals essentially halves, which means the average arrival rates of the corresponding arrival intervals doubles. The arrival rate in a single time slot cannot be higher than the burst sizeσ. At the same time it cannot be lower than roptfor a non-min-rate interval. Hence, at mostlog2roptσ non-min-rate service intervals can overlap. Further, from Lemma 2 at most one min-rate service interval can overlap. Since each interval has at most2Btraffic, the maximum queue size under Batch is 2Blog2roptσ .
The limit on the queue size and energy consumption tradeoff under the power-rate functionf2 is similar to the limit under
functionf1 [2]. We omit the proof here.
Lemma 6. If a scheduling algorithm keeps the energy
con-sumption at mostν times the minimal consumption that keeps queue size bounded byB, then the resulting queue size has to be Ω(Blogν roptσ ).
Combining Lemmas 4, 5 and 6 we have the following.
Theorem 7. Batch can keep queues bounded by
O(Blog2roptσ ) using energy O(OptB). This tradeoff is asymptotically the best possible.
III. DELAY VS. ENERGY: SINGLESERVER
We now consider the tradeoff between delay and energy consumption. In [24] a simple algorithm called Average Rate (AVR) is proposed for meeting delay deadlines while mini-mizing energy consumption under the power-rate function f1
of (1). Under AVR, each packet p of size `p arrives with an
arrival time sp and a deadline dp. The server allocates rate
`p/(dp−sp)with respect to packetpthroughout the duration
[sp, dp]. At time t, the total service rate is defined to be
r(t) =P
p:t∈[sp,dp]`p/(dp−sp). At any time, AVR serves the packet with the earliest deadlinedp to serve, allowing packet
preemption. Because of the definition ofr(t), it can be shown
easily that AVR respects all packet deadlines. Further, Yao et al. show,
Theorem 8. [24] Under power-rate functionf1, the
preemp-tive AVR algorithm guarantees every packet meets its deadline and the energy consumption is at most gαtimes the optimal, where
gα= 2α−1αα. (5)
Note that without preemption, AVR cannot guarantee all deadlines are met. Consider the following simple example. A packet p1 with a large size arrives at time 0 with a lenient
deadline, and p1 is serviced starting time 0 as it is the only
packet. At time > 0, packet p2 arrives with a stringent
deadline. Without preemption, p2 cannot be served until p1
is done. Sincep1 is a large packet,p2 can miss its deadline
by a lot.
We define a non-preemptive AVR under the power-rate funtionf2(·), which we refer to as NP-AVR2. For packetp,
the server allocates rate
rp= (`p+Lmax)/(dp−sp)
throughout the duration[sp, dp], whereLmaxis the maximum
packet size. At time t, the total service rate is defined to be max{ropt, r(t)}, where r(t) =Pp:t∈[sp,dp]rp andropt is
defined in (4).
NP-AVR2 uses an idealized fractional schedule to determine the order in which packets are scheduled. In particular, letGp
be the finishing time if all packets are served fractionally, i.e. if each packetpreceives service at raterp during[sp, dp]. By
definition,Gp=sp+`p/rp.
NP-AVR2 chooses the next packet to serve only when it finishes serving the current packet and it chooses the one with the earliest value of Gp among those already arrived.
The purpose of NP-AVR2 is to provide bounded energy consumption and ensure deadlines are met.
Theorem 9. Under power-rate function f2, the NP-AVR2
algorithm guarantees every packet meets its deadline. Proof: For packet plet Wp be the finishing time under
NP-AVR2. Our goal is to proveWp ≤Gp+Lmax/rp=dp.
Consider a busy period. Order the packets in this period with respect toWp, their finishing time under NP-AVR2. Forp, let
qbe the latest packet, with respect toWp, such thatGq > Gp
and Wq < Wp. In other words q is the latest packet whose
fractional finishing time is afterpbut whose integral finishing time is beforep. LetQbe the set of packets whose NP-AVR2 finishing time is in the interval (Wq, Wp]. (In particular Q
contains packet p but not q.) By the definition of q all the packets in Q must have a fractional finishing time no later
thanGp. Hence by the definition of NP-AVR2 they must have
arrived at ther server no earlier than q.
We have the following two relationships. The first equation holds since packets in Q are served after q under NP-AVR2 and the interval is busy. The second relationship follows from the fact that all packets in Q must arrive after the time W0
q < Wq when packet q starts being serviced under
NP-AVR2. Moreover all packets inQare served during[W0 q, Gp]. Z Wp Wq r(t)dt = X pp∈Q `pp X pp∈Q `pp ≤ Z Gp W0 q r(t)dt. The above implies RWp
Gp r(t)dt≤
RWq W0
q r(t)dt=`q. Our algo-rithm guarantees that the total rate r(t)during [Gp, Qp] is at
leastrp. Therefore,Wp≤Gp+`q/rp≤Gp+Lmax/rp=dp.
Theorem 10. Let γ = maxp`p+`Lmaxp . The energy consump-tion by NP-AVR2 is at mostγααg
α+ 1 times the optimal. Proof: LetOptbe the optimal energy consumption. The consumption under NP-AVR2 is
Z t max{ropt, r(t)}dt = Z t:r(t)≥ropt f2(r(t))dt+ Z t:r(t)<ropt f2(ropt)dt.
We bound the first term by comparing it againstR
tf1(r0(t))dt wherer0 p=lp/(dp−sp)andr0(t) =Pp:t∈[sp,dp]r 0 p. f2(r(t)) = c+r(t)α= (α−1)roptα +r(t)α ≤ αr(t)α=αf1(r(t))≤αγαf1(r0(t)). Therefore, Z t:r(t)≥ropt f2(r(t))dt≤αγα Z t f1(r0(t))dt≤αγαgαOpt
The last inequality comes from Theorem 8, since the optimal energy consumption with respect tof2(r(t))upper bounds the
optimal energy consumption with respect tof1(r0(t)).
We now bound the second term for whenr(t)< ropt. Let
A=R
t:r(t)<roptroptdtbe the total amount of traffic processed by AVR2 during these times. According to Lemma 1, NP-AVR2 consumes the least amount of energy for processing this traffic of sizeA, since at these times the algorithm only works at rate ropt or zero. Further, A is trivally upper bounded by
the total traffic arrivals. Therefore, the corresponding energy consumed by NP-AVR2 is at mostOpt. Combining two cases, we have
Z
t
max{ropt, r(t)}dt≤(αgαγα+ 1)Opt.
IV. DELAY VS. ENERGY: NETWORK OFSERVERS
In this section we examine a network of servers and define versions of the well-known Generalized-Processor-Sharing (GPS) and Weight-Fair-Queueing (WFQ) algorithms [23] so that they are adaptive to traffic load. The traditional GPS algorithm runs at a fixed speed and partitions service among backlogged connections according to the connection injection rateρi. The traditional WFQ is a discretized packet-by-packet
version of GPS.
A. Definition of RA-GPS and RA-WFQ
Under a natural definition of rate-adaptive GPS (RA-GPS), a server operates at rate P
iρi where the sum is over all backlogged connections. Each backlogged connection i then receives a service rate of ρi. However, RA-GPS defined as
such when converted to the packetized version rate-adaptive WFQ (RA-WFQ) can lead to a large delay if a packet is stuck behind a large packet and the server is reduced to low service rate. This is the similar to the situation described in Section III and so we do not discuss it further here.
We define RA-GPS as follows. For packet p from con-nection i, we specify a time sequenceapj for 0 ≤ j ≤ Ki,
where [apj−1, apj] is the interval during which our algorithm
will schedulepto go through thejth server. The values ofapj affect the delay and energy consumption bounds. Let
vp,j= (`p+Lmax)/(apj−a p j−1).
Ifsis thejth server, thensallocates ratevp,jduring[apj−1, a
p j]
with respect top. Note that this definition is analogous to NP-AVR2 in Section III. Again, the term Lmax/(apj −a
p j−1) is
to ensure that during the RA-GPS to RA-WFQ conversion a packet will not be stuck behind a large packet for too long, as shown in Theorem 9.
We now emulate RA-GPS to obtain RA-WFQ in the same way that WFQ emulates GPS. RA-WFQ runs RA-GPS in the background. Suppose RA-WFQ finishes serving a packet at time t. RA-WFQ then picks an unserved packet that has the earliest finishing time under RA-GPS, assuming no more packets arrive after t. In addition, RA-WFQ sets the server rate equal to the rate used by RA-GPS.
B. Analysis
Suppose each packetp has an arrival timesp anddp. Let
Opts,d be the optimal energy consumption in order to meet
every packet deadlinedp.
Theorem 11. Letγ= maxp,j(`p`+pLmax(ap )(dp−sp) j−a
p j−1)
. Ifsp≤apj ≤
dpfor allj, then every deadlinedpcan be met and the energy used by RA-WFQ is at mostαγαg
α+ 1 timesOpts,d. Proof: Given specified apj, for 0 ≤ j ≤ Ki, it follows
directly from Theorem 9 that a session-i packet p finishes being served at its jth server by time apj under RA-WFQ. Therefore, every packet meets its deadlinedp.
To bound the energy consumption under RA-WFQ, note that under the energy-optimal schedule the jth server on the
path for packetpserved the packet during the interval[sp, dp].
Therefore the energy used is no less than the energy used if each server is presented with a separate packet with arrival timespand deadlinedp. From here onwards, the proof follows
the line of argument in Theorem 10. For packetp, servereand timetdefinev0
p,e(t) =`p/(dp−
sp)if servereis on the path for packetpandsp≤t≤dpand
v0
p,e(t) = 0otherwise. The following holds sinceOpts,d, the
optimal energy with respect to f2(Ppvp,e) under RA-WFQ,
is an upper bound on the optimal with respect tof1(Ppvp,e0 ),
and the latter is related to the energy consumption of AVR by a factor of gα according to Theorem 8.
Opts,d ≥ 1 gα Z t X e f1( X p vp,e0 (t))dt.
The rest of analysis is similar to that in Theorem 10, and the ratio of αγαg
α+ 1 follows.
As a specific application of the above result, we define a0 = sp+σi/ρi
aj = aj−1+ (`p+Lmax)/ρi forj≥1
then Theorem 11 implies an end-to-end delay bound very much in the spirit of the classic bound of WFQ and at the same time a bound on the energy consumption.
Corollary 12. Let γ = maxp,j Ki(`p+`pLmax). RA-WFQ achieves an end-to-end delay bound of
σi
ρi +
Ki(`p+Lmax)
ρi using energy that is at most αγαg
α+ 1 times the optimal energy required to satisfy this end-to-end delay bound.
V. CONCLUSION
In this paper we develop energy-aware scheduling algo-rithms for the power-rate functions that have a base power component. For a single server, these algorithms are able to achieve essentially the best possible delay-energy and queue-energy tradeoffs. For the network environment, we believe we are the first to provide a rate-adaptive version of Weighted Fair Queueing with provable bounds on delay and energy consumption, for power-rate functions both with and without base power. There is however room for improvement, either by tightening the bounds on energy consumption, or else by providing some limiting results on the tradeoffs that can be achieved.
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