Energy-Efficient Wireless Packet Scheduling with ...

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Energy-Efficient Wireless Packet Scheduling with Quality of Service Control Xiliang Zhong, Student Member, IEEE, and Cheng-Zhong Xu, Senior Member, IEEE Abstract—In this paper, we study the problem of packet scheduling in a wireless environment with the objective of minimizing the average transmission energy expenditure under individual packet delay constraints. Most past studies assumed that the input arrivals followed a Poisson process or were statistically independent. However, traffic from a real source typically has strong time correlation. We model a packet scheduling and queuing system for a general input process in linear time-invariant systems. We propose an energy-efficient packet scheduling policy that takes the correlation into account. Meanwhile, a slower transmission rate implies that packets stay in the transmitter for a longer time, which may result in unexpected transmitter overload and buffer overflow. We derive the upper bounds of the maximum transmission rate under an overload probability and the upper bounds of the required buffer size under a packet drop rate. Simulation results show that the proposed scheduler improves up to 15 percent in energy savings compared with the policies that assume statistically independent input. Evaluation of the bounds in providing QoS control shows that both deadline misses and packet drops can be effectively bounded by a predefined constraint. Index Terms—Packet scheduling, power control, QoS, wireless networks.

Ç 1

INTRODUCTION

W

IRELESS devices are usually powered by limited battery resources. Reliable content delivery over a wireless channel is a major source of energy expenditure. It is essential to reduce the power consumption of the devices without performance degradation. The objective of this study is to strike a good trade-off between transmission power and delay constraints. It is known that power consumption between two points over a wireless channel is exponentially related to the information transmission rate [2]. A linear increase of transmission time can achieve super linear energy savings. However, applications are often delay sensitive. The transmission time cannot be arbitrarily long. Researchers have proposed different approaches to deal with the energy-delay trade-off. In [2], [5], and [19], the average queuing delay was considered as a constraint in energy minimization. In [9], [25], and [29], a single deadline for all packets was set so that all arrivals in time ½0; T Þ have to be transmitted before time T . Both average delay and deadline constraints provide a delay guarantee for a group of packets; the transmission delay of an individual packet can still be quite large. A more desirable constraint for delay-sensitive applications is the individual packet delay under which each packet transmission must finish before its deadline [3], [10], [19], [21], [28]. However, most existing work can only be applied to special types of packet arrival. For example, the input was assumed to be a Poisson process in [28], independent and identically distributed over each slot in [10], [19], a periodic task with constant interarrival

. The authors are with the Department of Electrical and Computer Engineering, Wayne State University, 5050 Anthony Wayne Drive, Detroit, MI 48202. E-mail: {xlzhong, czxu}@wayne.edu. Manuscript received 18 Feb. 2006; revised 2 Sept. 2006; accepted 27 Nov. 2006; published online 7 Feb. 2007. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TMC-0053-0206. Digital Object Identifier no. 10.1109/TMC.2007.1012. 1536-1233/07/$25.00 ß 2007 IEEE

time in [21], and deterministic with all timing information known offline in [3]. However, we cannot always have full knowledge of input arrivals during packet scheduling and a Poisson process is limited in characterizing real-world traffic. For example, it has been shown that, in the wireless LAN environment, packet arrivals are more bursty [25]. The assumption of statistically dependent packet arrivals may not be valid in reality either. Existing studies confirm the presence of long range dependence (LRD) and selfsimilarity for packets in an Ethernet LAN [26], packets from a variable bit rate (VBR) video [1]. In this work, we focus on packet scheduling with an individual deadline constraint for a general input process in an Additive White Gaussian Noise (AWGN) channel. We present an energyefficient scheduling policy to take into account the input autocorrelation. We demonstrate that the input autocorrelation plays an important role in determining the scheduling policy. Furthermore, we prove that a recent approach [10] is essentially a special case of our proposed policy for statistically independent input. We observe that, because of the online nature of the scheduler and a slower transmission rate for energy efficiency, an energy-aware policy may lead to unexpected overload or packet drops. As packet loss and error are not uncommon in wireless systems, the application layer is usually designed to be tolerant to a certain degree of QoS violation [18]. We thus provide a controllable QoS guarantee by investigating the relationship between the maximum transmission rate and an overload probability and the relationship between queue size and a packet drop rate. The relationships can be used offline for capacity configuration subject to a QoS constraint and online for QoS bounds with a fixed capacity. The rest of the paper is organized as follows: In Section 2, we introduce the system model and present the problem formulation. In Section 3, we propose a modeling of the scheduling and present an energy-efficient scheduling Published by the IEEE CS, CASS, ComSoc, IES, & SPS

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Fig. 1. System model.

policy. Relationships of transmitter capacities and QoS constraints are revealed in Section 4. Section 5 verifies the analytical results through simulation. Section 6 reviews related work. Section 7 concludes the article.

2

PROBLEM FORMULATION

We consider a single user time-slotted system shown in Fig. 1. Let t represent the system scheduling epoch index. The number of input packets to the transmitter at time t is given by a stationary random process xðtÞ. The packets are assumed to be uniform in size and the size is long enough for reliable communication close to the mutual information of the channel. Each packet is associated with a QoS level characterized by a delay constraint td . The packets to be transmitted are first stored in a queue with size smax . The queue backlog length at the beginning of time t is sðtÞ. The transmitter schedules packets out of the queue at a rate of rðtÞ at time t and uses power PðrðtÞÞ for transmission. We consider a transmitter in support of multiple transmission rates. In practice, this can be achieved by using dynamic modulation scaling [20], coding scaling [25], or a change of symbol rate [4]. Consider a AWGN channel between a pair of transmitters and receivers. The maximum channel capacity under optimal channel coding is   1 PðrðtÞÞ bits=transmission; C ¼ log2 1 þ 2 N where N is noise power. Let B denote the channel bandwidth. We can represent the relationship between transmission rate rðtÞ and transmission power PðrðtÞÞ as  2rðtÞ  ð1Þ PðrðtÞÞ ¼ N  2 B  1 : The power function is monotonically increasing and strictly convex with respect to rðtÞ. The same characteristics of power function apply even if suboptimal channel coding is deployed [25]. This means that even a small reduction in the transmission rate or increase in transmission delay can lead to a large energy savings. We consider an input arrival process during time period ½0; T Þ in an AWGN channel. The AWGN channel does not account for the phenomena of channel fading, interference, and dispersion. The average energy consumed P by the input process can be expressed as IE½PðrðtÞÞ ¼ T1 Tt¼0 PðrðtÞÞ. A scheduler is optimal if it leads to the minimum amount of average energy consumption. Such an optimal scheduler relies upon complete knowledge of future packet arrivals, which is not available in online scheduling for a general input process. It is known that IE½PðrðtÞÞ  PðIE½rðtÞÞ due to the convexity of the energy function. That is, the maximum energy savings would be obtained if we operated the transmitter at the long-term average service rate.

Because of the variation of input arrivals, a constant rate may not guarantee all packets transmitted before their deadlines. Considering the delay constraint, we expect an energy-efficient feasible scheduler to assign the transmission rate to the average service rate IE½rðtÞ as close as possible. We denote IE½rðtÞ as r . Our objective is to find an online scheduler that minimizes the mean quare errors between the transmission rate and the constant rate r , IE½ðrðtÞ  r Þ2 .

3

ENERGY-EFFICIENT SCHEDULING

In this section, we first present a model of packet scheduling in a linear system. The model facilitates the derivation of an energy-efficient packet scheduling policy.

3.1 An Energy-Aware Scheduling Model To characterize the transmission process, we define a transmission function as the amount of packets transmitted during each time slot, T ðt; ta ; xðta ÞÞ. It is a function of system time t, packet arrival time ta , and the number of packets arrived during time ½ta  1; ta Þ, xðta Þ. A general scheduler should also consider relative deadline td of each packet. Due to the convexity of the power function, any packet transmitted before its deadline td can be delayed with reduced transmission rate and energy. We hence consider packet transmitted before its deadline to effectively complete at the deadline to achieve maximum energy savings. We define a scheduling function hr ðt; ta Þ as the portion of packets transmitted during time ½t; t þ 1Þ for packets arrived at time ta . The number of transmitted packets from xðta Þ at the slot is then hr ðt; ta Þ  xðta Þ. The total transmission at time t is a sum of all transmitted packets that arrive during the last td time slots. That is, rðtÞ ¼

t X

T ðt; ta ; xðta ÞÞ ¼

ta ¼ttd þ1

t X

xðta Þhr ðt; ta Þ:

ð2Þ

ta ¼ttd þ1

Equation (2) models the scheduling process as a singleinput single-output linear system and the scheduling function hr ðt; ta Þ is its unit impulse response. A scheduler is normally causal, which means no resource will be reserved before packet arrivals, i.e., for all t < ta , hr ðt; ta Þ ¼ 0. We consider a time-invariant scheduler in which its impulse response does not depend on the time when the impulse is applied. Thus, if an impulse, occurring at t ¼ 0, causes the response hr ðtÞ, then an impulse, occurring at t ¼ ta , must cause the response hr ðt  ta Þ. This means hr ðt; ta Þ ¼ hr ðt  ta Þ. The transmission rate can be expressed as rðtÞ ¼

t X ta ¼ttd þ1

xðta Þhr ðt  ta Þ ¼ xðtÞ  hr ðtÞ;

ð3Þ

ZHONG AND XU: ENERGY-EFFICIENT WIRELESS PACKET SCHEDULING WITH QUALITY OF SERVICE CONTROL

where “ ” is a convolution operator. Equation (3) models the scheduling process as a perfect form in a linear, timeinvariant (LTI) system with a transfer function hr ðtÞ. As 0  hr ðtÞ  1, the system represented by hr ðtÞ is a low pass filter.

3.2 An Energy-Efficient Scheduling Policy Consider packets that arrived at time ta , xðta Þ. We try to transmit all the packets before deadline ta þ td . The number of packets to be transmitted is xðta Þ ¼

ta X þtd 1

T ðt; ta ; xðta ÞÞ ¼

ta X þtd 1

t¼ta

¼

hr ðt; ta Þxðta Þ

t¼ta

ta X þtd 1

That is, hr ðt  ta Þ ¼ 1 or

t¼ta

tX d 1

hr ðtÞ ¼ 1:

ð4Þ

t¼0

Define h ¼ ½ hr ð0Þ; hr ð1Þ; . . . ; hr ð td  1 Þ 0 and x ðtÞ ¼ ½xðtÞ; xðt  1Þ; . . . ; xðt  td þ 1Þ as vector forms of the scheduling function and packet arrivals. Let  ¼ IE½xðtÞx0 ðtÞ, being the covariance matrix of input process xðtÞ in the order of td . Consider a wide-sense stationary (WSS) input process, in which the mean of xðtÞ is constant and its autocorrelation function depends only on the time difference. We note that the assumption of WSS is general in modeling self-similar traffic. For example, traffic traces generated from synthetic Fractional Gaussian Noise processes (FGN) are WSS. We present the solution to the optimization problem for a WSS process in the following theorem: Theorem 3.1. The optimal delay bounded scheduler for a WSS process has a scheduling function in the form of 1 u0 : h ¼ u1 u0

ð5Þ

Proof. Rewrite (3) in a vector format as rðtÞ ¼ h0 xðtÞ. We can get the mean value of the transmission process according to [17] 1 X t¼1

Lðh; Þ ¼ h0 h þ ðu  h  1Þ; where u is a unitary vector with td components and u  h ¼ 1 is the vector form of the constraint (4). The gradient of L is 2h þ u0 with a solution of   1 0   u : h ¼  2

hr ðtÞ ¼ x

This theorem reveals the impact of input autocorrelation structure on scheduling. The autocorrelation can be either measured online according to history packet arrivals or offline with a given input. In both cases, the autocorrelation can be computed by summing products of an input arrival with the same arrival delayed by a certain time lag [17]. A special case is independent packet arrivals in which the covariance matrix is diagonal and the solution in (5) becomes uniform. It means an equal amount of transmission is scheduled before the deadline. In this case, the scheduling policy according to Theorem 3.1 is the same as the optimal time invariant solution in [10]. We state this result in the following corollary: Corollary 3.1. If the input arrivals are independent over time, the optimal delay bounded packet scheduling function is hr ðtÞ ¼ 1=td , 0  t < td . The transmission rate can be represented as rðtÞ ¼



r ¼ x

As we assume x is a constant, the problem is reduced to minimizing h0 h. This is a typical optimization problem in signal processing with a linear constraint. It can be solved using Lagrange multipliers. The Lagrangian is

To find the value of the Lagrange multiplier, the solution must satisfy the constraint (4). By imposing the constraint, we obtain a solution as in (5). It is the optimal time-invariant scheduling policy. u t

hr ðt  ta Þxðta Þ:

t¼ta

ta X þtd 1

3

tX d 1

hr ðtÞ:

t¼0

It indicates that the mean value of rðtÞ is equal to the mean of xðtÞ times the area under the impulse response. As the sum of scheduling function hr ðtÞ is one, the average transmission rate becomes r ¼ x . An intuitive explanation is that the transmitter should transmit all arrived packets within their deadlines because we assume no degradation in the transmission (sum of hðiÞ is one). The objective of the energy optimization problem is to minimize IE½ðrðtÞ  r Þ2  ¼ IE½r2 ðtÞ  2r ¼ IE½h0 xðtÞh0 xðtÞ  2x ¼ IE½h0 xðtÞx0 ðtÞh  2x ¼ h0 h  2x :

ð6Þ

t 1 X xðta Þ: td ta ¼ttd þ1

ð7Þ

3.3 Example Solutions We next illustrate the solution by examining several traffic patterns with different autocorrelation functions (ACFs). The first is a multimedia trace of the popular Simpsons VBR video from [22]. It is a 20 minute clip consisting of 30,334 frames. The Hurst parameter is used to characterize the long range dependence (LRD) of the traffic. The degree of correlation is high with a large H. The Hurst parameter of the video clip is 0.84 and it has a strong degree of dependence and burstiness. The second is from a Fractional Gaussian Noise process [15] with a Hurst parameter 0.9. The last is a multimedia traffic model with its ACF based on a shifted exponential scene-length distribution [11], i.e., ejj , where  ¼ 1=49 and  is the time lag. Fig. 2 shows the ACFs of the traffic models with different lags. Their impacts on the optimal scheduling functions are shown in Fig. 3 with the delay constraint set to 10 time slots. A Poisson process, which is statistically independent, has been included for comparison. An interesting finding is that, as the degree of input autocorrelation increases (in the order of td ), shown from Figs. 3a, 3b, 3c, and 3d, the

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Fig. 2. Autocorrelation of different input processes.

convexity of the scheduling function also increases. This can be verified from the low-pass nature of the scheduling process. With increased degree of correlation, there are more low frequency components [1]. The low-pass filter needs to have a lower cutoff frequency to effectively smooth the input. As a result, the scheduling function for a Poisson process has the highest bandwidth while the lowest for the multimedia pattern.

4

QoS GUARANTEE

A transmitter is characterized by two factors: maximum reliable transmission rate and queue size. Overload occurs when the required transmission rate exceeds the maximum rate. The energy-efficient scheduling policy represented by (3) tries to postpone packet transmission as late as possible under the delay constraint. As the scheduling is causal without knowledge of future packet arrivals, it is possible to get unexpected overload and packet drops by transmitting at a lower rate. Assume we have two arrivals from an independent process to a transmitter with a maximum transmission rate 100 packets per time slot. One arrives at time 0 with

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100 packets; the other at time 1 with 400 packets. Both have a delay constraint 4. If we do not consider energy consumption and schedule transmission as fast as possible, the first arrival will be scheduled during time [0, 1) and the second during [1, 5), all with 100 packets per slot. This is a feasible schedule. If we use the energy efficient scheduler for an independent process, according to (7), the transmission for each arrival will be scheduled uniformly before its deadline. Thus, for the first arrival, 25 packets per time slot will be transmitted during [0, 4) and 100 packets during [1,5) for the second, as shown in Fig. 4b. We can see that the maximum required transmission rate is 125 packets per time slot. The schedule is no longer feasible in the system. To guarantee error-free packet transmission, we need to apply admission control policies to prevent system from being overloaded. We can either reject the packet arrivals that would otherwise overload the system or admit them but delay their transmissions. By the rejection policy in this example, we decline the second arrival and no deadline miss occurs. In contrast, by the admit-but-delay policy, the second arrival would not be transmitted before its deadline. As can be seen from Fig. 4c, the system cannot meet the deadline of the second arrival because it slows down the transmission of the first to save energy. Deadline miss cannot be totally avoided in energy efficient online packet scheduling without assumed knowledge of future packet arrivals. A similar observation can be made for a transmitter with a limited queue size. When we slow down packet transmission, more packets stay in the transmitter queue. As a result, there could be more queue overflows. Consider a transmitter with a queue size 300 packets. We show in Figs. 5b and 5c that both schedules under power control in Fig. 4 lead to queue overflow, while the power-oblivious schedule does not. The implication of queue overflow is that packets have to be dropped without being transmitted. The number of queue overflow is identical to the number of packet drops.

Fig. 3. Unit impulse response of the transmission process with different ACFs. (a) Response with a Poisson process. (b) Response with a VBR video trace. (c) Response with an FGN process. (d) Response with a multimedia trace.

Fig. 4. Impact of energy-efficient scheduling on delay guarantee. (a) Schedule without power control. (b) Schedule under power control with transmitter overload. (c) Schedule under power control with deadline miss.

ZHONG AND XU: ENERGY-EFFICIENT WIRELESS PACKET SCHEDULING WITH QUALITY OF SERVICE CONTROL

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Fig. 5. Impact of energy-efficient scheduling on queue backlog. (a) Queue backlog without power control. (b) Queue backlog with queue overflow under schedule in Fig. 4b. (c) Queue backlog with queue overflow under schedule in Fig. 4c.

To restrict deadline misses and packet drops in a controllable manner, we investigate the relationship between the maximum transmission rate and an overload probability and the relationship between the queue size and packet drop rate. Note that the four factors are interdependent. For example, an increase of transmission capacity leads to decreases of both overload probability and packet drop rate. The number of packet drops is reduced because more packets can be transmitted and fewer are buffered in the transmitter. It is very difficult to characterize the four factors in one expression. Our approach is to deal with them in two steps. Given a group of packet arrivals, we first determine the transmission capacity or the overload probability according to the analysis in Section 4.1. The basic idea is to obtain the transmission rate distribution that is required to transmit all packets before their deadlines. The transmission rate distribution is then used to provide a bound of the transmission capacity with an overload probability and vice versa. With derived transmission capacity or overload probability, we can then get input statistics to the queuing system and compute either the transmission queue size or the packet drop rate according to the results in Section 4.2.

4.1 Transmission Rate with Overload Probability We provide two bounds on the transmission capacity with a QoS constraint. The first bound makes no assumption about the arrival process while the second assumes independent input with known distribution. The former bound is loose as it holds for any input having the same input mean and variance and the latter is tight. We do not consider the possibility of buffer overflow in this subsection. 4.1.1 General Input Define the overload probability vr as probðrðtÞ > rmax Þ, where rmax is the maximum reliable transmission rate. With known mean r and standard deviation r of the transmission rate rðtÞ, we can estimate the probability distribution tail by Chebyshev’s inequality as follows [7]: Theorem 4.1. The upper bound of the transmission rate that is required to guarantee a predefined overload probability vr for a general input is rffiffiffiffiffiffiffiffiffiffiffiffiffi 1  vr rmax  ð8Þ  r þ r : vr Proof. From Chebyshev’s inequality, it is known that F fIg  a1 IEðuðyÞÞ;

where y is a random variable, I is an interval, F fIg is a distribution function, uðyÞ > a > 0 for all y in I. Substitute y with transmission rate rðtÞ and define uðrðtÞÞ ¼ ðrðtÞ þ xÞ2 with x > 0. It can be verified that uðrðtÞÞ > ðrmax þ xÞ2 > 0 for rðtÞ > rmax > 0. Therefore, probðrðtÞ > rmax Þ 

1 ðrmax þ xÞ2

IE½ðrðtÞ þ xÞ2 :

Since IE½ðrðtÞ þ xÞ2  ¼ 2r þ 2r þ 2xr þ x2 ; we have probðrðtÞ > rmax Þ 

1 ðrmax þ xÞ2

ðx2 þ 2xr þ 2r þ 2r Þ:

It can be proved that the right side of the inequality takes the minimum value at x ¼ r þ 2r =rmax . Replace probðrðtÞ > rmax Þ with vr and compute the minimum value. We get vr 

2r 2r þ ðrmax  r Þ2

:

Solving the inequality for rmax gets the rate bound.

ð9Þ u t

The theorem can be applied to determine the maximum transmission rate in the process of system design. Because of the high variability of input arrivals, the variance can be a dominant factor. For an online scheduler with a fixed maximum rate, we can provide a QoS control by applying (9). Practically no transmitter can transmit packets at a rate higher than its maximum rate. Admission control needs to be implemented when the system becomes overloaded. We can adopt two types of admission control policies: .

.

Arrivals that could lead to overload are rejected. This makes sure that all admitted packets can be finished before their deadlines. The number of rejections would be the same as the number of overloading arrivals if there is no admission control. In this case, the rejection rate is equal to the overload probability. Arrivals that could lead to overload are admitted, but not transmitted until the system has enough resource, similar to the task transformation technique in [24]. As a result, admitted packets may miss their deadlines. The number of misses is the same as the number of overloading arrivals if there is no admission control. In this case, the overload probability is equal to the deadline miss rate of the arrivals.

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Under either policy, the overload probability can be used to bound the rejection rate or the deadline miss rate. We can also calculate the maximum transmission rate or overload probability from input statistics instead of from output. The mean and variance of the transmission process can be readily computed according to input in the LTI system. For a WSS input process, we recall from the proof of Theorem 3.1 that the mean value of the transmission rate equals the average arrival rate, r ¼ x . We only need to determine the variance of the transmission process. It can be computed according to [17] 2r

¼

1 X

1 X

Rxx ðt1  t2 Þhr ðt1 Þhr ðt2 Þ 

¼

tX d 1 tX d 1

¼

tY d 1

IE½ej!xðtiÞhr ðiÞ  ¼

OCTOBER 2007

¼

2x ;

tY d 1

tY d 1

x ðhr ðiÞ  !Þ

i¼0

I¼0

ð10Þ

Rxx ðt1  t2 Þhr ðt1 Þhr ðt2 Þ 

NO. 10,

distribution can be expressed as fðrÞ ¼ ttdd f td ðtd  xÞ, where td means td -fold convolution of fðxÞ with itself. As the convolution operation is computationally expensive, we next apply a transform method to simplify the computation of the output distribution. Denote x ð!Þ as the characteristic function of the input PDF. We first get the characteristic function of the transmission rate distribution for a stationary input process, which is Ptd 1 r ð!Þ ¼ IE½ej!rðtÞ  ¼ IE½ej! i¼0 xðtiÞhr ðiÞ 

2x

t1 ¼1 t2 ¼1

VOL. 6,

ð12Þ

x ð!=td Þ:

i¼0

t1 ¼0 t2 ¼0

where Rxx denotes the autocorrelation function of the input process. As hr ðtÞ has a limited length determined by deadline td , the computation cost is low.

4.1.2 Statistically Independent Input If the input process is WSS and statistically independent, we can determine the transmission rate with respect to input statistics as follows: Corollary 4.1. If the input process is WSS and statistically independent, with x and x as its mean and standard deviation, the relationship of deadline, maximum transmission rate, overload, and input statistics can be characterized by rffiffiffiffiffiffiffiffiffiffiffiffiffi 1  vr rmax  ð11Þ   x þ x : vr  td Proof. For independent input, the scheduling function should be set to a constant value hr ðtÞ ¼ 1=td during time 0  t < td for minimizing transmission energy consumption. We can simplify the computation of the variance as 2r ¼ 2x

tX ¼1 t¼1

h2r ðtÞ ¼ 2x

tX d 1 t¼0

h2r ðtÞ ¼

2x : td

As r ¼ x , substituting r and r into Theorem 4.1 completes the proof. u t The bounds by Theorem 4.1 are loose as they are applicable to a general input. If more information about the input is available, we can get a tightened bound. In our prior work, we provided tightened bounds for a unimodal distribution [30]. In this work, we will show that, for a stationary input process, if the input arrivals at different time slots are independent and identically distributed (i.i.d.) with known distribution, we can get an exact analysis for the overload probability. The input process is assumed to be stationary because we require the probability distribution at a fixed time or position to be the same for all times or positions. We first determine the transmission rate distribution from input distribution and the delay constraints td . The distribution is simply a convolution of td input distribution scaled by 1=td . Denote the probability density function (PDF) of input as fðxÞ. The transmission rate

The PDF of rðtÞ can then be found by taking the inverse Fourier transform of the product, ! tY d 1 ð1Þ x ð!=td Þ : fðrÞ ¼ F i¼0

By considering the tail distribution of the transmission rate, we derive the relation between the maximum rate and an overload probability as Z 1 fðrÞdr: vr ¼ probðrðtÞ > rmax Þ ¼ rmax

We need to estimate PDF of incoming traffic. Generally, there are two ways to estimate distributions. One is a parametric method which assumes the input traffic belongs to a certain type of distribution and estimates parameters of the distribution. The other is a nonparametric method. A popular nonparametric method is kernel density estimation [23], which builds up a distribution by adding up distributions from sample points of the input. More details of estimation for specific distributions such as Normal, Gamma, and Pareto can be found in [13].

4.2

Impact of Energy-Efficient Transmission on Packet Drop We have shown that the scheduling process can be modeled as a linear time-invariant (LTI) system. We next prove that with the scheduler as a LTI system, the queuing system can also be modeled as a linear system with a time-invariant transfer function. The model facilitates the analysis of the impact of energy-efficient transmission on queue backlog distribution and packet drop. 4.2.1 Queueing System Analysis Let sðtÞ be the queue backlog length at the start of time t. The dynamics of the queue can be given by sðt þ 1Þ ¼ sðtÞ þ xðt þ 1Þ  rðtÞ:

ð13Þ

The rðtÞ packets to be transmitted at time t are removed from the queue before the next xðt þ 1Þ packets arrive. Substituting rðtÞ with (3) yields sðt þ 1Þ ¼ sðtÞ þ xðt þ 1Þ  xðtÞ  hr ðtÞ:

ð14Þ

ZHONG AND XU: ENERGY-EFFICIENT WIRELESS PACKET SCHEDULING WITH QUALITY OF SERVICE CONTROL

7

Fig. 6. Unit impulse response of the queuing system with different ACFs. (a) Response with a Poisson process. (b) Response with a VBR video. (c) Response with an FGN process. (d) Response with a multimedia trace.

Substituting it into (13) yields

By taking z-transform of (14), we have zSðzÞ ¼ SðzÞ þ zXðzÞ  XðzÞHr ðzÞ; SðzÞ ¼

1  Hr ðzÞz 1  z1

ð15Þ

1

XðzÞ;

i¼0

where SðzÞ, XðzÞ, and Hr ðzÞ are the z-transforms of sðtÞ, xðtÞ, and hr ðtÞ, respectively. Let Hs ðzÞ ¼ ð1  Hr ðzÞz1 Þ=ð1  z1 Þ. It follows that SðzÞ ¼ Hs ðzÞXðzÞ. Given the packet scheduling process modeled as a LTI system with a transfer function hr ðtÞ, the queuing system can also be represented as a LTI system with a transfer function hs ðtÞ with Hs ðzÞ as its z-transform. We refer to this function as a queuing function. With an input process, we can determine hr ðtÞ according to (5). To determine hs ðtÞ, a straightforward way is to first get the z-transform of hr ðtÞ and then compute Hs ðzÞ according to (15). Once Hs ðzÞ is determined, we can get hs ðtÞ by taking the inverse z-transform of Hs ðzÞ. Alternatively, we can calculate the unit impulse response of the queuing system directly from hr ðtÞ, which is much easier. Theorem 4.2. The queuing system can be modeled as an LTI system given that the scheduling process is an LTI system with a transfer function hr ðtÞ. The transfer function of the queuing system, hs ðtÞ, can be calculated by hs ðtÞ ¼ 1 

hs ðtÞ ¼ hs ðt  1Þ þ xðtÞ  rðt  1Þ ! t2 t1 X X ¼ 1 hr ðiÞ þ 0  hr ðt  1Þ ¼ 1  hr ðiÞ:

t1 X

hr ðiÞ:

ð16Þ

i¼0

Proof. According to (15), we know that the queuing system can be modeled as an LTI system with a transfer function Hs ðzÞ (or hs ðtÞ equivalently). We next prove the expression of hs ðtÞ by induction. When t ¼ 0, there is an input arrival with size 1 and the queue is empty. The queue backlog is simply the current arrival. We define hs ð0Þ as 1. Starting from t ¼ 1, we derive the queue backlog according to (13) as hs ð1Þ ¼ hs ð0Þ þ xð1Þ  rð0Þ ¼ 1  hr ð0Þ: Therefore, (16) holds for t ¼ 1, which establishes a basis for the inductive argument. Suppose that the formula holds for t  1: hs ðt  1Þ ¼ 1 

t2 X i¼0

hr ðiÞ:

i¼0

By induction, the result holds for all t  0. Note that, as t  td , since the arrival has been transmitted and there is no new input, the queue backlog hs ðtÞ ¼ 0. We can verify Ptd 1 this from (16) by noting i¼0 hr ðiÞ ¼ 1. u t We have shown that both the queuing system and the transmission process can be modeled as LTI systems. The major difference is that they have different transfer functions, hs ðÞ and hr ðÞ. Fig. 6 plots the queuing functions with respect to different input described in Section 3.3. With the increase of input correlation, the queuing functions turn flat. This is consistent with existing findings that the presence of input correlation makes it less effective to smooth input using a queue. As a result, a higher cutoff frequency is needed for input with higher correlation. For example, in Fig. 6d, the queuing function for the multimedia pattern has the highest bandwidth. In a similar way to the transmission rate and overload analysis, we derive the relation between queue size and an overflow probability. Define the probability as vs ¼ probðsðtÞ > smax Þ, where smax is the queue size. We use the overflow probability to bound packet drop rate. Let s and s denote the mean and standard deviation of the queue backlog. It is not difficult to prove similar results as in Theorem 4.1 and (10) by changing rmax , vr , r , r , hr ðÞ to smax , vs , s , s , hs ðÞ, respectively. Details are omitted for brevity.

4.2.2 Statistically Independent Input When input is WSS and statistically independent, we can characterize the capacity-QoS relation with the following corollary: Corollary 4.2. If the input process is WSS and statistically independent, an upper bound of the queue size with respect to delay constraint and an overflow probability under the delay bounded packet scheduling is rffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  vs ðtd þ 1Þð2td þ 1Þ td þ 1 smax   x þ  x : ð17Þ 6td 2 vs

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Proof. We derive the queue backlog mean as  s ¼ x

tX d 1

hs ðiÞ ¼ x

tX d 1

i¼0

¼ x

tX d 1 i¼0

1

!

! hr ðjÞ

j¼0

i¼0

i1 X 1 1 t j¼0 d

i1 X

ð18Þ

td þ 1 x : ¼ 2

Similarly, the variance of the backlog is ! tX d 1 2 xðt  iÞhs ðiÞ s ¼ V arðsðtÞÞ ¼ V ar i¼0

¼

td 1 X

V arðxðt 

iÞÞh2s ðiÞ

i¼0

¼ 2x

tX d 1 i¼0

¼

1

i1 X

¼

2x

tX d 1 i¼0

!2 hr ðjÞ

Fig. 7. Energy consumption with different delay constraints.

h2s ðiÞ

¼ 2x

j¼0

tX d 1 i¼0

1

i1 X 1

!2

ð19Þ

t j¼0 d

ðtd þ 1Þð2td þ 1Þ 2 x : 6td

By adapting Theorem 4.1 to the queuing system, we rewrite (8) as rffiffiffiffiffiffiffiffiffiffiffiffiffi 1  vs ð20Þ   s þ s : smax  vs Combining (18) and (19) with (20) gets an estimate of queue size in (17). t u If the input process is stationary and i.i.d. with known distribution, we can get a tight queue backlog bound by first deriving the PDF of the backlog. The computation can be simplified by first deriving its characteristic function, similar to the transmission rate analysis. The characteristic function of the queue backlog distribution is ! ! tY tY i1 d 1 d 1 X x ðhs ðiÞ  !Þ ¼ x 1 hr ðjÞ  ! s ð!Þ ¼ i¼0

¼

tY d 1 i¼0

i¼0

j¼0

  td  i x ! : td ð21Þ

We finally note that the choice of admission control policies has impact on the input to the queuing system. If we reject all packets that may lead to overload, the queuing system may have less number of input arrivals; if we admit packets that cannot be transmitted before their deadlines and transmitted them in a best-effort manner, the queuing system may have a larger number of packets. However, the analytical results in this section still hold with an altered input process to the queuing system.

5

PERFORMANCE EVALUATION

We conducted simulations to verify the analytical results. The simulations were designed in three aspects: 1) Investigate the effectiveness of the proposed scheduling policy for energy saving, 2) demonstrate the use of the capacity bounds for QoS control both in offline configuration and online scheduling, and 3) how the effectiveness of the

capacity-QoS relationships in dealing with input variation. We assumed a packet length of 8 KBits and a channel capacity of 8 bits/transmission with a channel bandwidth B ¼ 5  105 transmissions/s. Under this setting, the minimum transmission duration for a packet is 2 ms, which is the time granularity of the transmitter. The power function 5 is according to 1), PðrðtÞÞ ¼ 22rðtÞ=510  1, with the noise power set to 1.

5.1 Effectiveness in Energy Savings The first experiment was conducted with an input of video traces from [22]. An important characteristic of the traces is their time-correlation. To show the performance of the schedulers under traffic with different degrees of autocorrelation, we chose two VBR video traces, JurassicPark I and Simpsons, with Hurst parameters 0.92 and 0.84, respectively. A baseline packet scheduling policy [10] is implemented for comparison. It is a special case of our proposed policy for statistically independent input process with a uniform scheduling function. To compare energy consumption using the proposed scheduling policy with the uniform allocation, we simulated transmission of the video traces. Fig. 7 plots the energy consumptions with different settings of delay constraints with the baseline policy referred as Uniform. The energy savings by both schedulers increases with larger delay constraints. This is because the transmission rate is reduced with relaxed delay constraints. Due to the convexity of the power function, more energy is saved. It is clear that the proposed scheduler determined by input correlation consistently outperforms the uniform allocation. The improvement is up to 15 percent for JurassicPark and 8 percent for Simpsons. The results of JurassicPark show more benefits by consideration of input correlation. This is expected because high input correlation makes a uniform allocation less effective in energy savings. 5.2 Capacity Configuration We next investigated the effectiveness of the bounds in offline capacity configuration subject to a QoS constraint for a general input (8), (17), the tight bounds when input arrivals are statistically independent with a known distribution (12), (21). We generated the number of packets to be transmitted at each time slot by a Gaussian distribution nð20; 4Þ. The length of each time slot is assumed to be 40 ms. We first set the delay constraint to 10 time slots and plot transmission rates with different overload probabilities in Fig. 8. Expectedly, the capacity requirement increases with a

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9

Fig. 8. Maximum transmission rate with different overload probabilities.

Fig. 11. Empirical CDFs of queue size from theoretic analysis and simulation.

Fig. 9. Empirical CDFs of transmission rate from theoretic analysis and simulation.

Fig. 12. Maximum transmission rate with different delay constraints.

Fig. 10. Queue size with different overflow probabilities.

smaller chance of failure. The bounds based on general input are loose as they hold for all input distributions with the same mean and variance. The bounds by (12) are tight with known input distribution. We also estimated the CDF of transmission rate by simulation using a histogram technique. We compare the obtained CDF with that from the analytical results (12) in Fig. 9; the resultant capacity bounds from the CDFs according to an overload rate 1 percent are shown in Fig. 8. We can observe that both the bounds and the CDF from the analysis agree quite well with the simulation results. Similar observations can be made with the relationship between queue size and an overflow probability in Fig. 10 and the CDFs of queue size in Fig. 11. The bounds of transmission rate with different delay constraints are shown in Fig. 12 with overload probabilities of 1 percent and 10 percent. The transmission rate requirement is reduced greatly with increased delay when the delay is smaller than 10 time units. With larger delay, the transmission bound reduces at a slower speed. Note that the transmission rate requirement is an upper bound of the transmission rate, which is determined by the mean and

Fig. 13. Queue size with different delay constraints.

variance of the rate distribution. A theoretical lower bound of the transmission rate is the average arrival rate (4 Mbps), below which, the system is not stable. If the delay is large enough, variance of the output rate approaches zero and the upper bound converges to the lower bound. Fig. 12 also reveals that the upper bounds with a large overload probability of 10 percent tend to flatten out faster than those with a small probability of 1 percent. This is because a smaller overload probability has a high capacity demand and is more dependent on the transmission rate variance. As the delay increases, packets stay in the queue for a longer time and a larger queue size is required, as can be observed from Fig. 13. It is interesting to find that the bounds can be well approximated by a linear relationship with delay constraint. A close examination of (17) with different parameters indicates that the higher order coefficients of td are much smaller than the linear coefficient; for example, the coefficient of t2d is only 0.16 percent of td in most cases. The linear approximation can be used to simplify queue size configuration with different delay constraints.

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Fig. 14. Normal plots of input, transmission rate, and queue backlog.

We point out that when the marginal distribution of the input process is Gaussian, the transmission rate rðtÞ and queue backlog sðtÞ also follow a Gaussian distribution. This can be verified by the normal plots of input, output rate, and queue backlog in Fig. 14. Note that the normal plot for data from a Gaussian distribution is a straight line. The figure shows that both rðtÞ and sðtÞ are Gaussian with a Gaussian arrival. This greatly simplifies the capacity-QoS analysis because the Gaussian distribution is totally characterized by its mean and variance, which can be readily computed according to Section 4. We also observe that the slope of the transmission rate plot is steeper than that of the input. This is consistent with the low pass nature of the scheduler, which smooths the fluctuating traffic with a smaller output variance and leads to better schedulability and QoS assurance.

5.3 Qos Control In the next experiments, we show the effectiveness of the capacity-QoS relationships in providing statistical QoS control during online packet scheduling. We used the same set of parameters as the last experiment, with number of input packets per time slot following a Gaussian distribution nð20; 4Þ, each with a delay constraint of 10. We limited the chance of failure to 1 percent for both overload and overflow. As we do not allow packet transmission at a speed higher than the maximum reliable rate and the queue backlog cannot exceed the queue size, we enhanced the scheduler in handling overload and overflow. In case of transmitter overload, the system transmits under the maximum rate and the unfinished packets are scheduled for transmission in a best-effort mode. As a result, packet transmission may not finish before its deadline. We characterized the QoS degradation with deadline miss rate and response time in the experiments. In case of queue overflow, we applied admission control to drop the packets that cannot be put into the queue. We used drop rate to characterize the failure due to queue overflow. We first experimented with the loose capacity bounds, i.e., 7 Mbps and 188 KBytes for maximum transmission rate

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and queue size, as can be observed from Figs. 8 and 10. We found no deadline misses or packet drops. This shows the conservativeness of the estimate because of the target 1 percent chance of failure. To show the effectiveness of the tight capacity bounds, we experimented with the tight bound for the transmission rate, 4.58 Mbps. Both loose (188 KBytes) and tight (128 KBytes) queue size bounds are experimented with. The simulation results are shown in Table 1. We can observe that there are deadline misses under the tight transmission rate even when the queue size is large enough, under which no packet drop occurs. The observed 0.88 percent miss rate is close to the 1 percent simulation setting. When we further set a tight queue size, both deadline misses and packet drops occur. The occasional deadline misses cause slightly larger response times than the target 10 units. However, the degradation is not much since more than 99 percent of the requests meet their deadlines, the average response time is close to 10, and the variance of the response time is small. The mean and variance of the transmission rate and queue size are smaller with the tight queue bound due to the 0.71 percent packet drop.

5.4 Qos Control under Traffic Variation We have shown that QoS control can be provided by capacity configuration on the transmitter according to input statistics. If the input traffic is not a WSS process, out analysis cannot be applied. In such a case, the service might degrade. Interesting questions are, when there are more arrivals than expected, what capacity is required to keep the same level of QoS or how much degradation there will be with the same capacity. The answer to the first question shows how capacity configuration can be used to keep the same level of QoS. Following the previous experiment with more variable input arrivals, we generate input with a Gaussian distribution nð20; 6Þ. To keep the original deadline of 10 time units and a 1 percent overload probability, we need a higher reliable transmission rate. According to Corollary 4.1, we have 22  ðrmax1  1 Þ2 21  ðrmax2  2 Þ2

¼ 1:

As a result, the new transmission rate is rmax2 ¼ 4:87 Mbps. Similarly, we determined the queue size according to (17) as 227 KBytes. We ran simulations based on the new predicted capacities. The experimental results in the second row in Table 2 show that, with the adjusted rate and queue size, 1.1 percent of the packets miss their deadlines close to the target QoS level. Alternatively, we can adapt the QoS target without changing the transmission rate. The new delay constraint t2 ,

TABLE 1 Scheduling with a Tight Transmission Rate for nð20; 4Þ ðrmax ¼ 4:58 Mbps, td ¼ 10Þ

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TABLE 2 Scheduling with Increased Maximum Transmission Rate or Delay Constraint for nð20; 6Þ

based on the target 1 percent overload probability for nð20; 6Þ, is t2 ¼

22 62  t1 ¼ 2  10 23: 2 4 1

The increase of delay constraint requires a larger queue size to hold more delayed packets. As a result, the queue size has to be increased to 410 KBytes according to (17). In practice, if the queue size is the bottleneck, we can change the target delay constraint online to adapt with the variation of input traffic. We ran simulations with a new deadline of 23 time units. The capacity was kept to 4:58 Mbps, an upper bound as shown in Fig. 8. The results are in the third row of Table 2. We can observe that, in response to the change of incoming traffic, about 99.2 percent of the packets meet their adjusted deadline of 23. This again demonstrates the prediction accuracy of the analysis results.

6

RELATED WORK

Different approaches have been proposed to investigate the trade-off between power and delay. Average queuing delay was considered in [5], [2], [19]. Collins and Cruz proposed an optimal transmission scheme in a fading channel with an average delay constraint and a peak transmitter power [5]. They used a simplistic channel model with a two-state Markov chain and assumed that energy expenditure is linear with transmitted data. Berry and Gallager considered the energy minimization problem with an average queuing delay in a block-fading channel [2]. The energy minimization was turned into a convex optimization problem and dynamic programming was used to find the optimal solution. Rajan et al. studied the energy gain due to source burstiness with mean queuing delay constraints [19]. An optimal scheduler obtained by dynamic programming and a near-optimal approximation was proposed for energy savings over both Gaussian and fading channels. They assumed independent and identically distributed (i.i.d.) packet arrivals at each time slot. They pointed out that i.i.d. source models rarely represent real sources, which usually have a strong time-correlation. However, they focused on the applicability of energy-efficient packet scheduling in a real system and did not study the impact of the correlation. There are also other similar metrics with average delay. For example, Schurgers et al. studied the impact of channel state under an average data rate constraint [21]. A different delay constraint was applied in [25], [9], [8], [29], in which a single deadline was put to all packets. The indirect bound on packet delay requires that all packets arrive before T to be transmitted no later than T . Offline optimal and online near-optimal algorithms were proposed

for a single transmitter-receiver pair by Uysal-Biyikoglu et al. [25]. The authors showed that their proposed scheduler is more energy-efficient than a deterministic constant service policy that results in the same average delay. In their later extension to multiple users, they applied the same delay constraint [9]. Fu et al. proposed a packet transmission policy to send an amount of data within a fixed time period [8]. Their focus was on the impact of fading channel on throughput and energy optimization. Zhang and Chanson targeted maximum system throughput and value in a Gaussian channel under energy and time constraints [29]. Nuggehalli et al. [16] utilized both average delay and deadline constraints, considering the effect of energy recovery during idle periods. They first adapted the scheduling policy in [25] with a battery recovery model and reported more than 50 percent energy savings. They then applied average delay and derived the optimal solution with and without battery recovery. They showed that the battery-aware policy could save substantial energy. A more practical and widely used constraint for delaysensitive applications is an explicit deadline for each packet [21], [19], [10], [14], [28], [3]. Schurgers et al. analyzed energy-efficient real-time packet scheduling in a timeinvariant channel with a detailed uncoded MQAM modulation [21]. They used a sufficient schedulability test to assure no deadline misses. As the condition is only sufficient, not necessary, it is possible to reject more tasks than necessary. They noted that it is too computationally intensive to compute an energy optimized schedule and proposed a heuristic algorithm. However, it is not clear how efficient their heuristics is and, most importantly, the usage of their approach is limited only to a periodic task model with constant packet interarrival times. Rajan et al. [19] used an absolute deadline to each packet for an i.i.d input. They formalized the optimization problem and solved using a value iteration algorithm. However, the computational complexity of their algorithm grows exponentially with the delay constraint and the complexity is prohibitive with delay larger than three units. This limits its use in an environment with limited power. In contrast, the proposed policy in this paper has a low time complexity. Khojastepour and Sabharwal considered a strict maximum delay constraint for each packet [10]. They established the connection between maximum delay scheduling and a linear filter for an i.i.d. input. Two optimal scheduling approaches were proposed. One is a time-variant policy which makes scheduling decisions according to each new packet arrival and uncompleted arrivals in the queue backlog. Each time a packet arrives, all arrivals in the backlog must be iterated in order for the algorithm to derive

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a new transmission rate for the new arrival and updated rates for those in the backlog. Scalability might be a concern when there are a large number of packet arrivals. In contrast, the time-invariant policy for time independent input in [10] has a constant running time in determining a transmission rate for a packet arrival. In addition, the timevariant policy works more aggressively by setting the packet transmission at a lower rate. It may lead to more deadline misses. Because of its time-variant nature, any analysis of the deadline miss rate according to input is hard, even for time independent input. We therefore focus on the time-invariant policy for time correlated input and provide a QoS control to bound the number of deadline misses or rejections. Mangharam et al. [14] provided QoS support for multiple users with bursty MPEG-4 video transmission over a time-varying channel. They also used deadline miss rate as a QoS requirement and proposed an online greedy algorithm to satisfy the constraint. However, they considered a different scenario in which there is a centralized Access Point (AP). The AP assigns a transmission rate to each node such that the QoS constraints for multiple users are met. In our work, we restrict our focus on communications between a pair of senders and receivers. Zafer and Modiano presented a generalization of energyefficient packet scheduling by a calculus approach [28]. They proposed an offline generalization of the energy minimization problem and an online scheduler for a Poisson arrival. The online solution can be interpreted as the time-invariant scheduler in [10] plus anticipation of future arrivals. The scheduler can achieve less energy expenditure with Poisson input. An offline algorithm was also proposed by Chen and Mitra for a Gaussian channel and proved optimal with individual packet delay [3]. Their algorithm is a generalization of [25] by changing the group deadline to individual deadline. The authors showed that their algorithm can be fully characterized if the input is a Poisson process; for example, the optimal transmission durations of all the packets can be derived. We point out that the knowledge of all packet arrivals is not usually available offline and an online algorithm is more appropriate in dealing with the dynamic nature of input arrivals. In addition, the assumption that the input is a Poisson process is limited as the input is usually time correlated and more bursty. Finally, our work can also be distinguished by providing a statistical QoS guarantee. We studied the impact of slow transmission rate on both deadline misses and queue overflows and provided statistical control of service degradation. The provisioning of a statistical QoS control has been widely investigated in the real-time community; see [6] and [27] for examples of energy-oblivious and energy-aware analysis, respectively. Their approaches are to measure the uncertainty of execution time demand and adjust the scheduling policy based on the distribution information. They are not applicable to packet scheduling because packet size can be determined once a packet is admitted, while, in the CPU counterpart, the exact computation time of a task is not available even after task release. In our approach, instead of statistical analysis of input time demand, we

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analyze the tail distribution of output transmission rate and queue backlog. It is along the line of recent advances in statistical real-time guarantee for CPU using the Earliest Deadline First scheduling [31]. We finally point out that there exists work on the energyaware selection of appropriate computation speed and onchip buffer sizes in System-on-Chip (SoC) design. One recent example is due to Liu et al. [12], in which a calculus approach was used to determine CPU speed and buffer size from a given input. Our work is complementary because it supports transmitter rate and queue size configuration for a wireless interface in SoC design. The proposed approach is general in the sense that it can be applied to a group of inputs having the same statistical characteristics rather than to only a specific input.

7

CONCLUSION

We consider minimal energy transmission over AWGN channels with delay constraints for each packet. In light of the fact that the input process for a wireless transmitter may be time-correlated, we consider packet scheduling for a general input process without a priori knowledge of input distribution. An energy-efficient scheduling policy has been proposed to take into account input autocorrelation, in which independent process is only a special case. As a slow transmission may result in unexpected overload and packet drops, we provide a statistical QoS control by investigating the relationship between transmitter capacity and a QoS constraint. We reveal the inherent relationships between the maximum reliable transmission rate and an overload probability, between the queue size and a packet drop rate. The relationships hold for all input distributions with the same first and second order moments. A tight bound is derived when the input process is time independent with a known distribution. The relationship can be used offline for capacity configuration during system design with a target QoS constraint and online by giving a service degradation bound with a fixed configuration. Simulation results show an improvement up to 15 percent over a recent policy for independent input. We demonstrate the effectiveness of the bounds in offline capacity configuration, online QoS control, and adaptation to input variation. We note that the channel model considered in this paper is an AWGN channel. We will investigate the impact of more realistic wireless fading channels and multiuser environments in our future work.

ACKNOWLEDGMENTS This research was supported in part by US National Science Foundation grants ACI-0203592, CCF-0611750, and DMS0624849 and NASA grant 03-OBPR-01-0049. The authors thank the reviewers for their valuable suggestions.

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[27] W. Yuan and K. Nahrstedt, “Energy-Efficient Soft Real-Time CPU Scheduling for Mobile Multimedia Systems,” Proc. 19th ACM Symp. Operating Systems Principles, 2003. [28] M. Zafer and E. Modiano, “A Calculus Approach to Minimum Energy Transmission Policies with Quality of Service Guarantees,” Proc. INFOCOM, 2005. [29] F. Zhang and S.T. Chanson, “Throughput and Value Maximization in Wireless Packet Scheduling Under Energy and Time Constraints,” Proc. IEEE Real-Time Systems Symp., pp. 324-334, 2003. [30] X. Zhong and C.-Z. Xu, “Delay-Constrained Energy-Efficient Wireless Packet Scheduling with QoS Guarantees,” Proc. IEEE Global Telecomm. Conf., 2005. [31] X. Zhong and C.-Z. Xu, “Energy-Aware Modeling and Scheduling for Dynamic Voltage Scaling with Statistical Real-Time Guarantee,” IEEE Trans. Computers, vol. 56, no. 3, pp. 358-372 2007. Xiliang Zhong received the BS degree in radio engineering from Southeast University, China, in 1997 and the MS degree in electrical engineering from the Beijing University of Posts and Telecommunications, China, in 2000. He worked as a software engineer in the Zhongxing Telecom Corporation from 2000 to 2002. He is a PhD student in the Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan. His current research interests include power management in embedded systems, mobile computing, and resource management in distributed systems. He is a student member of the IEEE. Cheng-Zhong Xu received the BS and MS degrees in computer science from Nanjing University in 1986 and 1989, respectively, and the PhD degree in computer science from the University of Hong Kong in 1993. He is an associate professor in the Department of Electrical and Computer Engineering at Wayne State University. His research interests lie in distributed and parallel systems, particularly in scalable and secure Internet services, adaptive and highly reliable networked computer systems, and resource management in cluster and grid computing. He has published more than 100 peerreviewed articles in journals and conference proceedings in these areas. He is the author of the book Scalable and Secure Internet Services and Architecture (Chapman & Hall/CRC Press, 2005) and a coauthor of the book Load Balancing in Parallel Computers: Theory and Practice (Kluwer Academic/Springer-Verlag, 1997). He serves on the editorial boards of the Journal of Parallel and Distributed Computing, the Journal of Parallel, Emergent, and Distributed Systems, the Journal of High Performance Computing and Networking, and the Journal of Computers and Applications. He was the founding program cochair of the International Workshop on Security in Systems and Networks (SSN), the general cochair of the IFIP 2006 International Conference on Embedded and Ubiquitous Computing (EUC ’06), and a member of the program committees of numerous conferences. His research was supported in part by the US National Science Foundation and NASA. He is a recipient of the Faculty Research Award of Wayne State University in 2000, the President’s Award for Excellence in Teaching in 2002, and the Career Development Chair Award in 2003. He is a senior member of the IEEE.

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