reward based wireless network algorithm

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Maximizing Rewards in WirelessNetworks with Energy and Timing

Constraints for Periodic Data StreamsJiayu Gong, Student Member, IEEE, Xiliang Zhong, Member, IEEE, and

Cheng-Zhong Xu, Senior Member, IEEE

AbstractPower efficiency is an important design issue in mobile devices with limited power supplies. In this paper, we study a

reward-based packet scheduling problem in wireless environments. We consider a general scenario in which a transmitter

communicates with multiple receivers periodically. To guarantee timely transmission of data, each packet is associated with a delay

constraint. The periodic data streams have different importance levels, power functions, and levels of data sizes. The more data a

transmitter delivers, the more rewards it obtains. Our objective is to develop schemes that selectively transmit data streams of different

data sizes at different transmission rates so that the system reward can be maximized under given time and energy constraints. We

show that the problem is NP-hard and develop a dynamic programming algorithm for the optimal solution in pseudopolynomial time. A

fast polynomial-time heuristic approach based on clustering of states in state space is presented to achieve close approximation.

Simulation results demonstrate the effectiveness of the optimal solution and show that the proposed polynomial-time approach can

achieve near-optimal results. Both approaches make a significant improvement over other approaches adapted from existing studiesat a marginal runtime overhead.

Index TermsReward maximization, power-aware packet scheduling, wireless networks, embedded systems.

1 INTRODUCTION

ENERGY is a critical resource of wireless devices poweredby battery with limited capacity. Reliable contentdelivery over a wireless channel is a major source of energyexpenditure. The increasing wireless transmission rateresults in a rapid increase of the energy consumption of

wireless devices. Studies show that energy expenditure forthe transmission of a given amount of data can be reduced byreducing the transmission rate with proper wireless channelcoding or modulation schemes [13], [29]. As applications areusually delay-sensitive, packet delivery delays should beallowed only if it is controllable. Different delay constraintswere investigated in energy-efficient packet transmission,such as average delay [3], [6], [24], a common deadline to allpackets [13], [31], [35], and individual deadlines[5], [18], [24],[29], [34], [37].

Most existing work focus on the minimization of the totalenergy consumption under the timing constraints. Mean-while, more and more embedded systems are being built

with renewable energy sources, such as solar power, windpower, and mechanical power, from the environment [19].Wireless nodes powered by these energy sources aresubjected to limited amount of energy which is collected

in each period. Generally, a wireless node may generate asignificant amount of data in a networked environment inperiodic cycle. Due to the limitation in both delay andenergy, it is often impossible for a wireless node to deliverall data in the transmission buffer at a time. Instead, the

node tends to transmit data collected in the bufferselectively under time and energy constraints. The periodicdata streams destined to different receivers may consumedifferent amount of energy. These data streams may alsohave different importance. For example, in a wireless sensornetwork which is monitoring the fire outbreak of an area,the data related to some sensors might be more urgent andshould transmitted with no delays. For applications, such asinformation gathering, multimedia applications, and imagesensing, it is also acceptable to receive partial data in atimely manner [1]. Take image sensing for example, thewireless node may support several formats with differentamount of information, such as in raw data format orcompressed formats in jpeg and jpeg2000 [19]. With a larger

data size, more information can be conveyed. When awireless node cannot send all of its data, it is more desirableto transmit more valuable data first. To quantify the level ofimportance of a packet, we associate a reward to eachpacket transmitted. Generally, the reward increases withmore data transmitted, similar to the Increased Rewardwith Increased Service (IRIS) model [9].

The objective of this study is to maximize system rewardsunder given time and energy constraints. Similar objectiveswere shared by several other studies. Wang and Mandayamtried to maximize system throughput [32] and the prob-ability of successful file transmission [33]. They considered atransmitter that could operate only in two states: either an

active state or an idle power state. Fu et al. sought to

IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 8, AUGUST 2010 1187

. J. Gong and C.-Z. Xu are with the Department of Electrical and ComputerEngineering, 5050 Anthony Wayne Dr., Rm 3100, Wayne StateUniversity, Detroit, MI 48202. E-mail: {jygong, czxu}@wayne.edu.

. X. Zhong is with Microsoft Corporation, One Microsoft Way, Redmond,WA 98052. E-mail: [email protected].

Manuscript received 13 Feb. 2009; revised 25 July 2009; accepted 3 Dec. 2009;published online 22 Apr. 2010.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-2009-02-0052.

Digital Object Identifier no. 10.1109/TMC.2010.82.1536-1233/10/$26.00 2010 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS

Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on June 21,2010 at 11:37:05 UTC from IEEE Xplore. Restrictions apply.


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maximize throughput of a transmitter with multiple trans-mission rates in a fading channel [12]. Zhang and Chansontargeted both throughput and value (reward) maximizationin an Additive White Gaussian Noise (AWGN) channel [35].Although the authors in [12], [35] considered time andenergy constraints, they assumed all packets in the systemshare the same deadline. Individual packet can still suffer

from large transmission delays under the constraint.In thispaper, we studythe reward maximization problems

for packets with individual energy and delay constraints. Weconsider a general scenario in which a wireless nodecommunicates with multiple receivers periodically over anAWGN channel. As the receivers may have differentdistances to the sender, they may require different amountof power under the same data transmission rates. Each datastream has several discrete data size levels and can betransmitted at different transmission rate levels. We associateeach data size with a rewardthat can be obtained by finishingthe data transmission. A wireless transmitter may only workwith a limited numberof rate levels, as suggested by practical

realization of multiple data rates [29], [8]. Practically, theIEEE 802.11a wireless LAN standard recommends MQAMwith four rate levels at 1, 2, 4, 6 bits per symbol.

The contributions of our work are two folds. First, wepropose the optimal solution to the time and energyconstrained reward maximization problem. We show thatthe reward optimization problem for periodic data streamswith discrete data sizes and transmission rate levels isNP-hard. We develop a dynamic programming algorithmto solve the problem optimally in pseudopolynomial runningtime. Second, we propose a heuristic approach, namedClustering, to closely approximate the optimal solution witha polynomial-time complexity. Simulation results show theeffectiveness of the algorithms. Furthermore, the Clusteringapproach can effectively approximate the optimal solution ata small cost of space and execution time.

The rest of this paper is organized as follows: In Section 2,we introduce the system model and formulate the problem.In Section 3, we present the optimal solution. The time-efficient approach is presented in Section 4. Section 5evaluates the proposed algorithms through simulations.We summarize the related work in Section 6. Finally, weconclude this paper in Section 7 with remarks on limitationsand future work.

2 SYSTEM MODEL AND PROBLEM FORMULATION

In this part, we first define data and energy consumptionmodels for the reward maximization problem. Then, wepresent a formulation of the reward maximization problemunder given time and energy constraints.

2.1 Data Model

Early studies of energy-efficient problem in wireless net-works were largely targeted at communication channelsover a single-transmitter-single-receiver model; see [6], [3],[31], [12] for examples. A single-transmitter-single-receivermodel is also known as point-to-point communicationwhere there is only one transmitter which will commu-nicate with a single receiver. In recent years, we have seenthe extension of the studies to a more general single-

transmitter-multiple-receiver model [13], [35], [23], [38] in

which a wireless transmitter communicates with multiplereceivers periodically, as shown in Fig. 1. In this model, thetransmitter can only communicate with one receiver at atime and has an energy budget in each transmit cycle. Eachreceiver will receive data from the transmitter periodically.Every transmitter-receiver pair has a maximal amount ofdata to be transmitted in each time period. The receivers arelocated with different distances from the transmitter. Thedata to different receivers can be transmitted at different

transmission rates.We regard the transmission between each transmitter-

receiver pair as a periodic data stream and refer this as atask. It is a sequence of packet transmissions with thesame characteristics that occurs at a regular interval. Weuse Task f1; 2; . . . ; Ng to denote the set of N tasks.The number of tasks is always equal to the number ofreceivers in our model. All the transmission tasks areassumed to be independent and preemptive, scheduled bythe Earliest Deadline First (EDF) policy [20].

EDF is a popular scheduling policy for delay-sensitivewireless packet scheduling [11], [34]. We choose EDF as theunderlying policy because it has been shown to be optimal

for deadline constrained scheduling over optimal linkconditions under various modeling assumptions [14], [21].The preemption can take place between transmissions ofpackets of different tasks. Practically, it can be implementedin a transmitter by rearranging packets in the transmissionbuffer by choosing those with the earliest deadlines. Similarpreemptive model was assumed in [11], [14], [30].

We characterize a task i by a tuple fCi; Ti; Di; Pi; Rig,where Ci denotes the size of data to be transmitted, Ti is thetask period, Pi and Ri are power function and rewardfunction, respectively. The relative deadline Di is assumedto be equal to task period Ti. We assume for each datastream i, there are several discrete levels of sizes of data to

be transmitted, denoted by fc1

i ; c2

i ; . . . ; cK

i g. We haveCi 2 fc1i ; c

2i ; . . . ; c

Ki g, which means the actual amount of

data to be transmitted is not always the maximum andshould be determined due to time and energy constraints.In general, the data stored in the transmit buffer can beeither generated by local host or forwarded from othernodes. Different data may have different priorities andrewards related to the corresponding receiver and data size.For notional brevity, we use ji to represent the task whendata cji is selected for transmission.

The transmitter has a set of discrete levels of transmis-sion rates. We define the set of available transmission ratesof a transmitter as Speed fs1; s2; . . . ; sMg in which the

available rates are indexed in an ascending order.

1188 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 8, AUGUST 2010

Fig. 1. Single-transmitter-multiple-receiver model in a single-hopwireless network.



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2.2 Power Consumption Model

The power consumption of a wireless transmitter can bedivided into two parts: circuit power and transmissionpower. The transmission power usually dominates sincelong-range communications (over 100 m) are common inwireless networks. In order to maintain the same transmis-sion rate, the required transmission power needs to increase

with the distance between the transmitter-receiver pair tooffset the propagation loss. In addition, the circuit power isexpected to decrease as the IC technology advances. Thispart of power only occupies a small portion of the wholepower consumption. So in our work, we assume thetransmission power dominates the negligible circuit power.

In our power model, we assume the channel is slowlytime-varying, which means the channel condition will notchange during transmission. Proper channel coding canreduce the energy consumption effectively during transmis-sion. We take the AWGN channel model as an example,which explains how energy, rate, and data size are related.With optimal channel coding, the maximum transmissionrate is [7]

S B

2log2 1

P0

N0B

; 1

where S is the transmission rate, P0 is the received signalpower, N0 is the spectral density, and B is the channelbandwidth. From this equation, we can descr ibe therelationship between the transmission rate S and thereceived power P0 by the following equation:

P0 N0B

22SB 1

: 2

As we aforementioned, the power will increase withdistance between transmitter and receiver in order to

maintain thesame transmission rate. Considering this powerattenuation, we have

P P0

A

N0B

A

22SB 1

; 3

where P is the transmission power and A is the attenuationfactor for the transmitter-receiver pair. The attenuationfactor A is generally inversely proportional to a function ofthe distance, denoted by l. For example, this function couldbe a square function, A / 1=l2, in [7]. In this paper, we donot assume any specific form of the relationship betweenattenuation factor and distance except that all transmitter-receiver pairs have the same fading functions which areonly affected by distance. It is easy to see that the required

transmission power P is strictly increasing and strictlyconvex in the transmission rate S. This power function PSis continuous in S though we only consider the discretecases for this function in this paper.

Let Pi denote the power consumption function for taski. Let Ci and Si represent the size and rate of datatransmission for i, respectively. The transmission time totransmit data Ci equals to

CiSi

. Therefore, it consumes PiSiCiSi

units of energy. The energy consumed for i for transmis-sion in one period, denoted by Ei, with data size Ci attransmission rate Si becomes

EiCi; Si PiSiCiSi

N0B

Ai

2

2SiB 1

CiSi

; 4

where the coefficient Ai for each transmitter/receiver pairdiffers depending on the distance between them. Similarpower models were defined in [35], [38], as well. As thechannel states and receiving nodes are assumed to be staticduring the transmission period, the power attenuatorfactor Ai is also static. We fix the bandwidth B andassume it is the same for all the streams. To simplify the

problem, we assume the overhead of switching amongdifferent transmission rates is tiny and can be ignored.

2.3 Problem Formulation

We consider the transmission in a hyperperiod T which isdefined as the Least Common Multiple (LCM) of taskperiods T1; T2; . . . ; TN. The consideration of a hyperperiodensures all tasks can finish their periodic transmissions atleast once. Let Emax represent the units of energy budgetallocated to the transmitter during this hyperperiod T. Ourobjective is to maximize the total reward while all tasksmeet their deadlines and the total energy consumption doesnot exceed the budget Emax. In other words, the optimiza-

tion problem in this paper is to find a speed and a data sizefor each task to maximize the overall rewards whilesatisfying delay and energy constraints.

The reward Ri can be a function of multiple variables,such as data size, transmission rate, etc. In this paper, wedefine reward function Ri as a generic function of data size c

ji

to be transmitted, represented by Ricji . It is conceptually

the same as the reward functions in [1], [9] and the utilityfunction in [25]. In this work, however, we do not assumeany specific form of the reward function. It can be reduced todifferent forms in different application contexts. In itssimplest form, the reward function Ri can be interpretedas the amount of data transmitted, with respect to the datasize; the reward maximization problem is then reduced to

throughput maximization, as in [12], [35]. In an imagesensing application, it could be interpreted as the amount ofinformation transmitted using different image formats [19].In a file transferring application, it can be reduced to theprobability of successful file delivery [33].

In general, we formulate the reward maximizationproblem as

maximizeXNi1

T

TiRiCi 5

subject toXNi1

T

Ti

CiSi

T ; 6

XNi1

T

TiEiSi; Ci Emax; 7

Si 2 fs1; s2; . . . ; sMg; 1 i N; 8

Ci 2 fc1i ; c

2i ; . . . ; c

Ki g; 1 i N: 9

Constraint (6) guarantees that all the data streams can becompleted under the EDF scheduling [20]. Whenever there isa schedule that maximizes the reward and can guarantee allstreams transmitted under constraints, we can alwaysconvert that schedule to EDF scheduling with the samereward, according to the proof of EDF optimality in [21].When all the packets in a hyperperiod are ready for

transmission at time 0, (6) reduces to a discrete case as in

GONG ET AL.: MAXIMIZING REWARDS IN WIRELESS NETWORKS WITH ENERGY AND TIMING CONSTRAINTS FOR PERIODIC DATA... 1189



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[35]. If we enable continuous data size and transmission ratein (8) and (9), respectively, this problem is reduced to acontinuous case, similar to that in [35]. In constraint (9), weenable multiple choices of data sizes; the problem is reducedto the one in [27] when we fix available data size of each task iin constraint (9) to be fc1i ; c

2i g, with c

1i 0 and c

2i 6 0, for all

1 i N. Theorem 1 shows the reward maximization

problem for periodical tasks defined by (5)-(9) is NP-hard.Theorem 1. The reward maximization problem for periodical

tasks defined by (5)-(9) is NP-hard.

Proof. Let xij and xil denote two 0-1 decision variables, res-pectively. Let i fs1; s2; . . . ; sMg and i fc1i ; c

2i ; . . . ; c

Ki g

for all 1 i N. If a task i is assigned to transmit data atrate j 2 i, then xij 1; otherwise, xij 0. If a task i isassignedto transmit data with data size l 2 i,then xil 1;otherwise, xil 0. The optimal problem (5)-(9) can berewritten as

maximize XN

i1Xl2i

T

TiRiCilxil 10

subject toXNi1

Xj2i

Xl2i

T

Ti

CilSij

xijxil T ; 11

XNi1

Xj2i

Xl2i

T

TiEiSij; Cilxijxil Emax; 12

Xj2i

xij 1; 1 i N; 13

Xl2i

xil 1; 1 i N: 14

By viewing TTi RiCil as the profit of item l out ofclass i, terms TTi

CilSij

, and TTiEiSij; Cil as the correspond-

ing weights, the optimal problem formulation (10)-(14)becomes to an instance of well-known NP-hard Multi-dimensional Multiple-Choice Knapsack Problem(MMKP) [17]. tu

3 DYNAMIC PROGRAMMING FOR THE OPTIMALSOLUTIONS

A general method of solving the optimal MMKP problem isto search thesolution space until an optimal solution is foundand confirmed [17]. We can use breadth-first search togenerate partial solution along with the sequence ofreceivers. This algorithm enumerates all possible data sizesand transmission rate for each receiver. This process can bevisualized as a state space branch where each nonleaf node inthis tree has M Kchildren if there are Mtransmission ratelevels and K data size levels for each receiver. Therefore, anaive algorithm would generate M Ki nodes at level i.The state space can grow exponentially with the tasknumber. To reach the solution in practical runtime, mostresearchers relied on heuristics to obtain approximatedsolutions [16], [22], [26] or adapted approaches to reducethe computational complexity [10], [4], [36]. These ap-proaches are not readily applicable to our problem as ourproblem involves more decision factors. In the following, wedevelop a dynamic programming algorithm for the rewardmaximization optimization problem with two-dimension

multiple choices of data size and transmission rate.

3.1 Dynamic Programming Algorithm

Consider a tuple "rik; "tik; "eik as a state, where "rik denotes the

reward sum of the first i tasks corresponding to the

accumulated transmission time "uik and energy sum "eik.

We use a list to record all the states of task i

Li h"ri1; "ti1; "ei1; "ri2; "ti2; "ei2; . . . ; "rini ; "tini ; "eini i;

where ni is the number of states after the ith iteration. Due

to the delay and energy constraints, branching from

unpromising states that generate infeasible or nonoptimal

solutions can be avoided. We summarize three circum-

stances, referred to pruning criteria, in which branches from

a certain state will be pruned.

1. Temporal Criterion: The state "rik; "tik; "eik in list Liwill be pruned if it satisfies that

"tik XN

ji1

T

Tj

CminjsM

> T :

The inequality means the partial solution will notmeet the deadline by transmitting the smallest size

of data to the remaining receivers even at the

maximal transmission rate. Therefore, the solutions

generated from this state are infeasible.2. Energy Criterion: The state "rik; "tik; "eik in list Li will

be pruned if it satisfies that

"eik XN

ji1

T

TjEi

s1; C

minj

> Emax:

The inequality means the partial solution will not

meet the energy constraint by transmitting the

smallest size of data to the remaining receivers evenat the minimal transmission rate. Therefore, the

solutions generated from this state are infeasible.3. Dominance Criterion: If two states "rik; "tik; "eik and

"rij; "tij; "eij in a list Li satisfy "rik > "rij and "tik "tij and"eik "eij, or "rik ! "rij and "tik "tij and "eik < "eij, or "rik !"rij and "tik < "tij and "eik "eij, thenthe latter state is saidto be dominated by the former one. Intuitively, if astate uses more energy, requires larger transmissiontime butachievessmaller rewardthan another state,itcan always be replaced by the latter state.

In the following, we use an example to show how these

three pruning criteria work. Suppose there are three datastreams with two transmission rate levels and two data size

levels each, as shown in Table 1. Assume the time and energy

constraint is 7; 7. A rootnode 0 atlevel-0is represented bya

3-tuple 0; 0; 0 which means reward, time, and energy are all

zero since no data stream has been enumerated. Starting

from the root node, we first enumerate data stream 1. Note

that the order of enumerating the data streams will not have

impact on the final solution. Since there are two transmission

rate levels and two data size levels for each data stream, we

have four different nodes generated from root node after

enumerating data stream 1, shown in Fig. 2. We label these

four nodes at level-1 as 1, 2, 3, and 4.




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Then, we proceed to the next level. We examine everynode at level-1 and enumerate data stream 2. Start from

node 1, we find nodes 1 cannot be pruned based on allabove three pruning criteria. So, we can generate four new

nodes at level-2 from node 1 by enumerating data stream 2

and calculate the 3-tuple r;t;e for the new nodes. In thesame way, we can derive another eight new nodes at level-2

starting from nodes 2 and 3. Node 4 will be pruned due to

energy criterion (the current energy consumption is 8 which

exceeds the energy budget 7). So there will be no level-2

nodes derived from node 4. After pruning and enumera-

tion, we obtain 12 nodes at level-2, shown in Fig. 3.When we examine the level-2 nodes, we find node 8, 10,12, and 16 can be pruned according to the energy criterion.Node 15 is pruned due to the temporal criterion. Nodes 6 and14 will be pruned as well because they are dominated bynodes 9 and 11, respectively. Iteratively performing thissequence of steps (nodes generation, 3-tuple calculation, andpruning), we can obtain all nodes at level-3. Thus, wegenerate the complete state space tree for the given task set.The complete state space tree is shown in Fig. 4. Thegenerated nodes at level-3 with solid line are feasiblesolutions while those with dotted line are infeasible solu-tions. Among those feasible solutions, the dark nodenumbered 27 representsthe optimal solution 6; 7; 7becauseit has the largest reward value. The number of final nodes(level-3 nodes) now is 20. It has been reduced dramatically incontrast to that without any pruning, which is 64.

Algorithm 1 shows the pseudocode. Initially, we havelist L0 with zero reward, zero time, and zero energy. ListLi is obtained by the following steps. We first get a numberof lists L0ijk by adding reward, transmission time, andenergy values of task i under each data size j 2 fc1i ;c2i ; . . . ; c

Ki g and each transmission rate k 2 S to the states in

list Li1. The componentwise addition is denoted by L0ijk

Li1 "rijk; "tijk; "eijk. Second, we merge the lists L0ijk into a

list L0i in decreasing order of reward value. Third, weprune all nonfeasible state by the Temporal Criterion and

Energy Criterion. Finally, we use the procedure Prune-Lists() to find the undominated states in L0 and returns itas L00. In this procedure, we check each pair of states toeliminate the dominated states as many as possible inorder to save more space.

Algorithm 1. Reward maximization using dynamicprogramming

1: L0 h0; 0; 0i2: for i 1 to N do3: for all data size j 2 fc1i ; c

2i ; . . . ; c

Ki g do

4: for all transmission rate k 2 fs1; s2; . . . ; sMg do5: L0ijk Li1 "rijk; "tijk; "eijk6: end for7: end for

8: merge L0ijk into a list L0i in a decreasing order of

reward

9: delete all states in L0

i with"t

PNji1

T

Tj

Cminj

sM > T or"e PN

ji1TTj

Eis1; Cmin

j > Emax10: Li Prune-Lists(L

0i)

11: end for

12: return the largest state in LN

13: procedure Prune-Lists(L0)14: L00 ;15: while L0 6 ; do16: choose and delete the largest state "r0; "t0; "e0

from L0

17: flag true18: if L00 ; then

19: add "r0; "t0; "e0 to the end of list L0020: else

21: for all states "r00; "t00; "e00 in L00 do22: if ("t00 < "t0 and "e00 "e0) or ("t00 "t0 and "e00 < "e0)

then

23: flag false24: break25: end if26: end for27: if flag true then28: add "r0; "t0; "e0 to the end of list L00

29: end if

30: end if

31: end while32: return L00

33: end procedure

3.2 Algorithm Analysis

Now consider the running time for the ith iteration when

adding task i. In Algorithm 1, lines 3-7 are linear in the

number of the states in L0i, denoted by jL0ij. jL

0ij equals to

jL0i1j multiplied by the number of data sizes and the number

of transmission rates for task i. The time complexity for

merging sorting is OjL0ij logjL0ij. Line 9 can be completed

in OjL0ij. Line 10 can be completed in OjL0ij

2 because we

check each pair of states to eliminate the dominated ones.


TABLE 1An Example for Four Data Streams with Two Data Size Levels and Two Transmission Rate Levels

Fig. 2. Partial state space tree after one enumeration.



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This part dominates the whole time complexity. As a result,the running time of adding task i is OjL0ij

2. We denote theupper bound of states in L0i by U, i.e., max1iNfjL

0ijg U.

Thus, an upper bound of running time for adding N tasks isONU2. The value ofU may increase significantly with theincrease of the size of task set, the number of data size levels,

and the number of transmission rate levels. So, thisalgorithm has a pseudopolynomial time complexity. Thenecessary memory requirement of this algorithm is boundedby the number of states in a list as well. Since it is notnecessary for us to keep all the lists in the algorithm, thespace requirement would be equal to the size of the last list,which is in the order of OU.

4 TIME-EFFICIENT APPROXIMATION

Although the above three pruning conditions are effective inremoving unpromising states, the state space in Algorithm 1can still expand significantly and it will be computationallyexpensive to get the optimal solution with a large number ofreceivers, data sizes, and transmission rates. In practice, it isnot always necessary to find the optimal solution withlimited time and computation resources. A near-optimalsolution is more desirable if it can be completedin reasonabletime while consuming reasonable computation resources.

In this section, we will first propose a polynomial-timeheuristic approach. Then, we will analyze the complexity ofthis algorithm.

4.1 Polynomial-Time Approximated Approach(Clustering)

We develop a heuristic algorithm, named Clusteringalgorithm, to approximate the optimal solution to the

proposed problem with a polynomial computational time

complexity. This Clustering algorithm is novel for the

proposed problem. The general idea of this algorithm can

be traced back to data clustering in mathematics.This Clustering algorithm is based on a clustering

property of the final states after we enumerate all the

tasks. Fig. 5 shows the result if we plot the all of the final

states for Table 1 after enumerating all possible combina-tions of data sizes and transmission rates into a 3D space

with the coordinates of reward; energy; time. We can find

the nodes, representing the final states, are clustered

instead of randomly scattering. This is attributed to the

discrete levels of data sizes and transmission rates. Those

nodes in the same cluster tend to have close reward values,

energy consumption, and transmission time. We have

similar observations about the intermediate results after

each enumeration.


Fig. 5. The reward-time-energy relationship in a 3D space.

Fig. 3. Partial state space tree after two enumerations.

Fig. 4. The complete state space tree by pruning.



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During each enumeration, if we start from one cluster,the best solution generated from this cluster will have closereward values. If one node in this cluster can generate theoptimal solution, the others in the same cluster canproduce near-optimal solutions as well. Keep only onenode in each cluster and remove the others, we can still getthe near-optimal solution though the optimal solution

might be removed. If none of the nodes in this clustercan lead to the optimal solution, there would be no impacton the final solution if we keep one and remove the others.The benefit of this method is that it can reduce the statespace significantly.

We create clusters of nodes by dividing the reward valueaxis and energy axis evenly to mbins and nbins number ofranges, respectively. Thus, we construct mbins nbinsrectangle prisms (clusters) in a 3D space. In Fig. 5, clusterB represents one of the clusters. Suppose the maximal andminimal sums of energy consumption for all tasks are Emaxand Emin, respectively. We set the length of the bottom ofthe cluster to be e Emax Emin=nbins. Similarly, we set

the width of the bottom of the rectangle prism (cluster) tobe r Rmax Rmin=mbins, where Rmax and Rmin are themaximal and minimal sums of reward values for all tasks,respectively. The scaled reward of task i under data size j,transmission rate k, and reward group size r is rounded tothe next integer, i.e., d

rijkr

e, which represents the scaledreward value in each cluster. In the same way, we can getthe rounded scaled energy value, deijke e, which representsenergy values in each cluster. We introduce drijkr e and d

eijke

einto the 3-tuple representing a state in Algorithm 1 toconstruct a 5-tuple state used in Algorithm 2.

Algorithm 2. A polynomial-time heuristic approach(Clustering).

1: L0 h0; 0; 0; 0; 0i2: calculate Emin, Rmax, and Rmin3: r

RmaxRminmbins

4: e EmaxEmin

nbins

5: for i 1 to N do6: Li ;7: for all data size j 2 fc1i ; c

2i ; . . . ; c

Ki g do

8: for all transmission rate k 2 fs1; s2; . . . ; sMg do9: L0ijk Li1 d

"rijkr

e; "rijk; "tijk; d"eijke

e; "eijk10: end for11: end for12: merge L0ijk into a list L

0i in a decreasing order of

scaled reward13: delete all states in L0i with "t

PNji1

TTj

CminjsM

> T or"e

PNji1

TTj

Eis1; Cmin

j > Emax14: construct mbins nbins buckets

B1; B2; . . . ; Bmbinsnbins15: for all states st d rre;r;t; d

ee

e; ein L0 do16: isNew true17: for all nonempty buckets Bi in

B1; B2; . . . ; Bmbinsnbins do18: select a state st=dr

0

re; r0; t0; de

0

ee; e0 from Bi

19: If (thend rre dr0

re) and (d eee d

e0

ee)

20: put st into Bi

21: isNew false

22: break23: end if

24: end for

25: if isNew true then26: put st into an empty bucket27: end if28: end for

29: for all nonempty bucket n B1; B2; . . . ; Bmbinsnbinsdo

30: select one element with smallest "t and add it tothe end of Li

31: end for

32: end for

33: return the largest state in LN.

When enumerating a task, we generate all states for onetask with pruning conditions. The Clustering approachdivides all the states into buckets, which represent theclusters, and keep one node for each bucket, as shown inAlgorithm 2. In Algorithm 2, the strategy in determining

which node should be kept in each cluster depends onthe factor of transmission time. While the nodes in thesame cluster have close rewards and energy consumption,we always want to keep the one with the smallesttransmission time in order to save time for the remainingtasks. Besides this strategy, we have other criteria to decidewhich node should be reserved in each cluster. Differentmetrics, such as energy, reward, and energy-delay product,can be employed. We conduct simulations to compare theimpact of performance when employing different strategiesfor selecting representative node in the following evalua-tion part. According to the evaluation results, we choosetime as the only metric for this selection since it can lead tothe best approximation.

Clearly, if the intermediate states themselves can clusterwell, it will lead to small number of nonempty clustersafter each enumeration so that the state space can bereduced more, thus the execution time can be shortened.Even in the worst case, as a result of rounding up andkeeping one node per cluster, the number of states can alsobe reduced greatly so are the running time and requiredspace to solve the scaled problem compared with theoptimal approach. The number of nodes at each enumera-tion is bounded by the largest number of clusters, which isequal to mbins nbins.

4.2 Algorithm Analysis

For given numbers of ranges for energy and reward, wehave the upper bound for the number of states in eachiteration by U nbins mbins. In Algorithm 1, the runningtime for adding task i is OjL0ij

2 because we compare eachpair of elements in the list to eliminate the dominated states.Here, we dont have this step. So the running time foradding task i is OjL0ij logjL

0ij, dominated by sorting.

Thus, an upper bound of the running time is ONUlog U,where U denotes the upper bound of states in L0i. Replace Uby nbins mbins, w e g et t he t im e c om pl ex it y o f ONnbins mbins lognbins mbins. Since the valuesofnbins and mbins are initialized according to the problemsize (see evaluation part) which is determined by a

polynomial function with variables of task set size, data




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size levels, and rate levels, the total running time cost ispolynomial in the size of task set, data size set, and rate set.Similar to Algorithm 1, the space cost is in order of OU,which is Onbins mbins, as well.

5 PERFORMANCE EVALUATION

We simulate the following four algorithmsin our experiment:

. Dynamic programming algorithm for optimal solu-tion: We simulate this algorithm to obtain theoptimal solution for reward maximization problem.

. Polynomial-time Clustering algorithm (Clustering):This is the proposed time-efficient Clustering algo-rithm for a near-optimal solution.

. Greedy-pack and greedy-unpack algorithms: Thesetwo algorithms are adapted from [27] as thecompetitors of proposed polynomial-time Clusteringalgorithm. Though the reward maximization pro-blem in [27] is different from that in our work, themethod of designing heuristic approaches is a general

principle and can still be adapted to the problemstudied in this paper.

We compare the solutions of Clustering, greedy-pack,and greedy-unpack with the optimal solution obtained bythe dynamic programming algorithm. We use the metric ofnormalized system reward to show how close thesealgorithms can approximate the optimal solution. We studythe effect of different parameters on the simulation resultsand investigate the execution time cost for differentalgorithms. We conduct simulations to show to whatdegree our proposed dynamic programming algorithmcan restrict the explosion of state space. In addition, weinvestigate the impact on approximation by choosing

different numbers of clusters and strategies of representa-tive node selection for Clustering algorithm.

5.1 Simulation Setup

The wireless channel settings we used are similar to thosedescribed in [2], [35]. The channel bandwidth is 1 MHz. Thebits per transmission is set to be 1, 2, 4, and 6 bits/Hz(BPSK). Each receiver will receive data from the transmitterperiodically at the above transmission rates. These periodicdata streams are generated based on the following para-meters: distance between receiver and transmitter, thenumber of different sizes of data to be transmitted, thesizes of data to be transmitted, period, and reward values.The distances between transmitter and receivers are

uniformly distributed in a range of [20, 200] meters. Underthis setup, a transmission power of 20 mW is required toreliably transmit data at 1 bit/Hz (BPSK) at a distance of100 m. According to power function (3), we can calculatethe power consumption for different receivers with differ-ent transmission rates.

We assumed the number of data size levels to be 5 forall receivers. The data sizes are uniformly distributed inthe range of 1; 16 Mb. The periods for each receiver canbe represented by the number of jobs transmitted to eachreceiver, denoted by fjob1;job2; . . . ;jobNg, within the timeinterval T. So the period Ti can be calculated as Ti

Tjobi

.The value of jobi is a random integer value in the range of

[1, 12].

Based on the above parameters, we can calculate theenergy consumptions by the energy function (4). Eachreceiver has its own reward function. In this simulation, wedefine a linear reward function in order of the size oftransmitted data. The coefficient hi for RiCi hiCi is themultiplication of hi1 and a random value in the range of1:2; 3:0 for i 2; 3; . . . ; N. The coefficient h1 is a random

value in the range of 0:1; 0:2.The energy constraint Emax and the length of hyperper-

iod T for a task set are defined in ways similar to those in[27]. Emax is a multiplication of a parameter and E,where 0 < < 1 and E is the total energy consumptionfor all data streams to transmit the largest size of data atthe largest transmission rate within the interval, i.e., E PN

i1jobiEiCmaxi ; sM. T is the multiplication of a para-

meter and T, where 0 < < 1 and T is the totaltransmission time for all data streams to transmit thelargest size of data at the smallest transmission rate, i.e.,T

PNi1jobi

CKis1

. A l ower v alu e of means morestringent energy constraint and a lower value of means

tighter deadline. We refer to and as energy constraintfactor and time constraint factor, respectively. In Clusteringalgorithm, we assumed nbins mbins 10 N logMlogK by default.

5.2 Simulation Results

In this part, we first study the impact of time and energyconstraint factors on the performance of the four algo-rithms. Second, we investigate the relationship between theperformance and execution time. Third, compared with ourprevious work in [15], we further study the effectiveness ofour dynamic programming algorithm for optimal solution.Finally, we simulate our Clustering algorithm with differentparameters to see the effect on the extent of the approxima-

tion to the optimal solution.

5.2.1 Impact of Time and Energy Constraint Factors

The first experiment is to study the impact of time andenergy constraint factors on the performance. We simulatedoptimal, Clustering, greedy-pack, and greedy-unpack algo-rithms for this reward maximization problem with differenttime and energy constraint factors. The system rewardnormalized to the optimal value is presented in Fig. 6.

Fig. 6a demonstratesthe impact of energyconstraint factoron the performance. We assumed the number of receivers tobe 10 and the time constraint factor to be 0.2. We increasedthe energy constraintfactor from0.1to0.9withastepsizeof

0.1. The normalized reward of Clustering increases with theincrease of energy constraint factor. It is around 95 percentwhen 0:1 but can reach more than 99 percent when ! 0:5. The normalized rewards of two greedy algorithmsalwayshave a biggap largerthan 15 percent from theoptimalvalue even if the energy constraint is loose. The bestnormalized reward is bounded by 85 percent.

Fig. 6b shows the impact of time constraint factor on theperformance. We increased the time constraint factor from0.1 to 0.9 with each step equal to 0.1 and assumed a constanttask size of 10 receivers and a constant energy constraintfactor 0:2. The results of Clustering are always largerthan 95 percent. With a loose time constraint, it may

increase to more than 99 percent. We find the normalized




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system reward of the two greedy algorithms always has agap more than 10 percent from the optimal value when isless than 0.4. They increase rapidly with the increase of timeconstraint factor and can be more over 99 percent of theoptimal value when time constraint factor goes beyond 0.4.There are two reasons for this result.

On one hand, we observe the optimal solutions arealways the same when is fixed to 0.2 and varies from 0.4to 0.9. This can be explained as follows: Since the energy

consumption grows exponentially with the increase oftransmission rate, we can decrease the energy consumptiongreatly by slowing down the transmission rate if the timeconstraint is loose, which means linear increase of time canlead to exponential decrease of energy consumption. In thisway, the time constraint has insensitive impact on thesolution. That is why we can see the optimal solutions arealways same since is 0.4.

On the other hand, the greedy algorithms can alwayshave good performance with a loose time constraint whenthe only constraint is energy consumption. Since the onlyconstraint is energy consumption after we loose the timeconstraint, the greedy algorithms can have good perfor-

mance in this case.

As comparison, for the case in Fig. 6a, a loose energyconstraint will only lead to small decrease of time. These twoconstraints will always have sensitive impact on the solutionthough energy constraint is very loose. In simulation, wenotice the optimal solutions are not the same for the cases inFig. 6a with different values.

From these results, we can see Clustering algorithmalways obtains near-optimal solution when the constraintsare not very tight. With looser constraints, it can even

achieve the optimal solution.

5.2.2 Performance and Execution Time

The second experiment is to investigate the performance andexecution time for these four algorithms in this simulation.The simulation was run on a machine with Pentium D2.8 GHz CPU and 2 GB RAM.We assumed boththe time andenergyconstraint factorto be 0.2. We increasedthe numberofreceivers from 5 to 30 with each step equal to 5.

Fig. 7a demonstrates the average normalized systemreward. The normalized system reward for Clustering fordifferent task size is typically more than 96 percent at theworst case. With more receivers, the normalized system

reward will increase and can reach more than 99 percent.


Fig. 7. Reward and execution time with different numbers of receivers. (a) Normalized reward. (b) Execution time.

Fig. 6. Reward under different time and energy constraints. (a) Energy constraint. (b) Time constraint.



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This isbecause weset the values ofnbins and mbins related to

thenumber of receivers. The largerthese values are, themore

precise solution we can obtain. For greedy-pack and greedy-

unpack, the normalized system reward is usually around 82to 90 percent. There is alwaysa gapmore than 10 percent with

all task sizes. We notice these two greedy algorithms usually

can reach quite similar solution for the data size and

transmission rate in the end although they search for thenear-optimal solution in an opposite way. That is why theyhave similar normalized system reward.

Fig. 7b shows the execution time of these four algorithms.

It is expected that the execution time for the optimal solution

is the most and increases rapidly with the increase of thenumber of receivers. The reason is that the increase of

number of receivers will lead to tremendous increase of the

state space though we have already employed pruning

techniques. Compared with the dynamic programming

algorithm for optimal solution, Clustering can always havetiny execution time cost while achieving near-optimal

solution. It can obtain more than 99 percent reward whileconsuming less than 0.1 percent of execution time of the

optimal solution. With the increase of task size, the execution

time of Clustering increases but is still tiny compared withthat for the optimal solution. Greedy-pack and greedy-

unpack have very close execution time costs. They have the

smallest cost but much worst performance than Clustering,

shown in Fig. 7a. We notice that the execution time of

Clustering is close to or even smaller than that of the greedy

algorithms when the number of receivers is under 10. It is

because the number of clusters in Clustering depends on the

number of receivers. When the number of receivers is small,the number of clusters is small, which leads to less execution

time. While more receivers can lead to more clusters, moreexecution time will be consumed. In the case of 30 receivers,

the execution time of Clustering is almost 100 times of that of

greedy algorithms.Combining the results from Sections 5.2.1 and 5.2.2, we

can see Clustering algorithm can outperform the greedyalgorithms in terms of normalized reward. It is because thatClustering algorithm always keeps intermediate stateswhich lead to optimal or near-optimal solution when thenumber of the cluster is large enough. In the greedyalgorithm, when enumerating tasks, only the data size andtransmission rate that can satisfy the criteria or lead to localoptimization will be selected. This scheme maybe more

likely to miss the optimal or near-optimal solution thusleads to worse performance. From another point of view,the greedy algorithms will reduce the state space moresignificantly compared with Clustering algorithm but itsuffers from more opportunities to miss the optimal or near-optimal solutions. In addition, the increase of the number ofreceivers will not incur the incredible increase of state spacefor greedy algorithms. This explains why the greedyalgorithms can always have smallest execution cost in mostcases and the execution time for greedy algorithms will notincrease greatly with the number of receivers.

In conclusion, Clustering can balance the trade-off of the

performance and the running time cost better than the

greedy algorithms.

5.2.3 Effectiveness of Dynamic Programming Algorithm

Fig. 8 shows the effectiveness of states pruning when we

searched the optimal solutions for 10 periodic data streams.Simulation results with different time and energy constraintfactors are presented.

In an exhausting search algorithm, the number of statesobserved in each iteration increases rapidly. After weemployed pruning criteria, the state space is reduceddramatically though it is still expanding as more iterationsare processed. Especially, the total number of states isreduced to less than 0.00001 percent of its original valuewhen the last iteration is reached. Though the number ofstates still increases as the iterations go on, the increasingrate is much smaller than the original one. The state spacegenerally expands in the enumeration of first several tasksas few nodes can be pruned by the pruning criteria. More

nonfeasible states will be pruned as more tasks areenumerated. With looser time and energy constraints, morestates will be preserved in the space.

5.2.4 Impact of Number of Clusters for Clustering

In the Clustering approach, the size of state space dependson the number of clusters, which affects the accuracy offinal result. In this part, we will study the impact of numberof clusters on the performance for the Clustering algorithm.

In all thesimulations, we assume mbins nbins, and boththe time and energy constraint factors to be 0.2. First, westudy the impact of different number of clusters on theaccuracy approaching optimal solution and time cost. Wetune the value of nbins to be 50, 100, 200, 400, 800, whichmeans the numbers of clusters can be 2,500, 10,000, 40,000,160,000, 640,000. The number of receivers is increased from 5to 30 with each step equal to 5. Fig. 9 shows the impact ofdifferent number of clusters on both the accuracy andexecution time of Clustering algorithm. We notice that boththe normalized system reward and execution increase withthe increase of cluster number in Figs. 9a and 9b. This isbecause larger number of clusters preserves more states ineach iteration. So the states that can lead to optimal or near-optimal solution will be more likely preserved for futuresearch, which will lead to more execution time cost. Morereceivers lead to larger state space. This requires us to have alarge enough number of clusters to get more precisesolution.

If we set the number of clusters too small for a large task set,


Fig. 8. Effectiveness of states pruning with 10 periodic data steams.



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we cannot get the solution close to the optimal one. For

example, we can get the normalized system reward around99 percent for the case containing five receivers by setting2,500 clusters (nbins 50). But the same setting leads tosmaller system reward with more receivers. When there are30 receivers, this setting can only achieve a normalizedsystem reward less than 75 percent.

So the number of clusters should be chosen carefully withthenumber of receivers in order to getnear-optimal solution.

We conduct simulations to study the impact of ratio ofproblem size over cluster number on the performance of

Clustering algorithm. For a problem with Nreceivers, Mratelevels, and K data size levels, the size of state space is

M KN. We define problem size as logM KN

N logM logK. We represent the number of clusters bynbins. Then, the ratio can be calculated by N logM logK=nbins. We set the value of this ratio to be 2; 1;0:5; 0:1; 0:02. Clearly, the smaller the ratio, the larger

number of clusters. We can see small ratio can always leadto more accurate result. From Fig. 10, we find we can obtain

more than 97 percent normalized reward if we set the ratio to

be 0.1. This can approximate the optimal solution well.

Though a better approximation can be achieved if the ratio is

0.02, the execution time will increase tremendously. Com-promising both the effect of approximation and executiontime cost, in our simulation, we set nbins 10 NlogM logK to achieve a near-optimal solution withaffordable computation cost.

5.2.5 Strategy of Representative Node Selection

As shown in Algorithm 2, we keep the node with the smallesttransmission time in each cluster. We use this strategy tosave time for the remaining tasks. Besides this strategy, thereare other methods to determine which node should be keptin each cluster. In this part, we conduct simulations to showthe results of employing different strategies to select

representative node for each cluster.We compare four different strategies using differentmetrics when selecting a representative node in each cluster.These metrics are time, energy, reward, and energy-delayproduct. If we use the metric of time/energy/energy-delay,we will keep the node with the smallest value of this metricin each cluster. The node with the largest reward will be keptin each cluster if we use the metric of reward. From Fig. 11,we can see using time as the metric can always guarantee the


Fig. 9. Impact of number of clusters. (a) Impact on accuracy. (b) Impact on execution time.

Fig. 10. Impact of ratio of problem size over cluster number. Fig. 11. Impact of different strategies selecting representative node.



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best performance. It can achieve a normalized systemreward more than 97 percent, which can beat the secondbest strategy significantly, no matter how many receivers aretaken into account in the simulation setup. Based on thisresult, we always employ time as the only metric whendetermining which node in each cluster should be kept inour Clustering algorithm.

6 RELATED WORK

Delay-constrained package transmission in wireless com-munications. Power consumption in packet transmissioncan be reduced significantly by transmitting packets at alower bit rate. Most existing studies on energy-efficientpacket transmission focused on minimizing the energyexpenditure subject to a time constraint. The time constraintcan be in terms of average response time [3], [6], [24] and asingle deadline [13], [31], [35] to all packets. Both constraintsdo not put a limit to response time of individual packets,which may lead to unexpected large delay. To guaranteetimely packet deliveries, it is more desirable to put a delayconstraint to each packet [5], [18], [24], [29], [34], [37]. Forexample, Khojastepour and Sabharwal considered strictmaximum delay constraint for each packet [18]. Theyestablished the connection between maximum delay sche-duling and a linear filter for an i.i.d. input. Two optimalscheduling approaches were proposed. One is a time-variant policy which makes scheduling decisions accordingto each new packet arrival and uncompleted arrivals in thequeue backlog. Zhong and Xu [37] studied delay con-strained packet scheduling with a focus on quality ofservice control. They derived relation between maximumpacket transmission rate and packet arrival patterns so as toprovide statistical response time and packet drop control.

In this work, we study packet transmission withindividual delay constraints to guarantee packet transmis-sion time. We consider a general scenario in which awireless transmitter communicates with multiple receivers.The transmitter generates data periodically and send thedata to corresponding receivers. The wireless transmitteris powered by renewable energy sources, such as solarpanel. As a result, the transmitter needs to finish packettransmission subjected to both delay and energy constraints.Due to the constraints, it may not be able to send all the data.To ensure that the most valuable information is transmittedto the receivers, we associate a reward (value) to eachpacket. Our objective is reduced to maximizing the total

reward under the time and energy constraints. Similarproblems have been investigated in existing studies [32],[12], [33], [35].

Wang and Mandayam [32] tried to maximize throughputin a block fading channel under time/energy constraints.They considered a binary rate control with only an on/offmodel in which the transmitter either transmits with aconstant power or remains silent. Under the same model andconstraints,the authors later considered thetransmission of afixed size file with theobjective to maximizing theprobabilityof successful delivery of the entire file [33]. Fu et al. tried tomaximize throughput in a fading channel by considering theeffect of variable transmission rates [12]. All the three studies

are limited to a single-transmitter-single-receiver scenario. In

contrast, we consider a general case with both multipletransmission rates and multiple receivers. In addition, ourobjective is to maximize systemreward, in which throughputandtransmission probability canbe treated as special types ofreward. We consider a wirelessnodethat collects informationperiodically and transmits the most important data to itsreceivers under the time/energy constraints. Each transmis-

sion must be completed before the start of the next datacollection. This is in contrast to the work by Zhang andChanson, which sets a common deadline to all packets in thesystems [35]. The authors in [35] assumed all packets areready to transmit at beginning time and this assumption isnotapplicable to ourperiodicdata model wheremessagesarereleased in a regular interval. In addition to the optimalsolution, we present a time-efficient approach to obtainapproximated solution in our work, which is not available in[35] either. Finally, a wireless transmitter may only work ata limited number of rate levels. Our work focus on atransmitter with discrete rate levels.

Reward maximization in CPU task scheduling. On the

other line of study, there were extensive studies on rewardmaximization in CPU task scheduling by adapting taskservice time [9], or using dynamic voltage and frequencyscaling [1], [27], [28], [4]. For examples, Aydin et al. [1], andRusu et al. [28] studied CPU reward maximization inprocessors with infinite number of speed levels andcontinuous speed reward functions. Their approach are notapplicable to our problem because there are a limitednumber of transmission rates for choice to maximize reward.

The work in [27], [4] were based on discrete CPU speedlevels. Rusu et al. [27] developed heuristic approaches toreward maximization without performance guarantee,which is a Multidimensional Knapsack Problem (d-KP)fundamentally. They targeted on a special case of task setswhen all jobs share the same release times and deadlines.The problem we addressed is MMKP and we study a moregeneral and challenging case without task timing constraint.Additionally, we propose an optimal solution for MMKPwith pseudopolynomial time complexity while it is notavailable [27]. In [4], Chen and Kuo proposed approxima-tion algorithms for tasks with the same power functionswhen CPU speed change can be done at any time. While itis reasonable for tasks to have the same power functions inCPU tasks scheduling, the power characteristics can varygreatly in wireless communication with different transmis-sion distances of receivers. In addition, we proposedoptimal algorithm for reward maximization in pseudopo-

lynomial time instead of approximation.

7 CONCLUSIONS

We study reward maximization problem under time andenergy constraints in wireless networks in this paper.Transmitting different periodic data streams to differentreceivers will consume different energy and producedifferent reward values. Each data stream has several levelsof data sizes while the transmitter can deliver them atseveral levels of transmission rate. Our objective is tomaximize system reward under time and energy constraintsby selecting a certain data size and a certain transmission

rate for each data stream. The exact optimal solution is




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obtained by the dynamic programming algorithm pre-sented in this paper. Instead of searching the optimalsolution with tremendous costs, we propose time-efficientapproximated approaches, including a polynomial-timeheuristic approach and two greedy algorithms, to approx-imate the optimal solution closely at much lower cost.Simulation results demonstrate the effectiveness of the

dynamic programming algorithm for exact optimal solutionand the performance of the polynomial-time heuristicapproach in approximating the optimal solution.

We note that the algorithms are proposed for periodicpacket transmission over an AWGN channel. Practically, itis more desirable to consider a fading channel in wirelesscommunication rather than an AWGN channel. However,we need to have an initial rate setting for all the delay-sensitive streams in the wireless system to begin with, bothin AWGN and fading channels. This start-up solution isnontrivial to obtain as discussed in Section 2.3. The AWGN-based work provides us a base solution and could beextended by adjusting messages priorities dynamically

online in a fading channel.We also note that Resource allocation for fading multiuser

broadcast channels is a popular topic in information theory.However, the resource being allocated is usually power orbandwidth, and the quantity to be maximized is most oftenShannon capacity.

ACKNOWLEDGMENTS

The authors would like to thank the anonymous reviewersfor their constructive comments and suggestions. Thisresearch was supported in part by the US National ScienceFoundation (NSF) grants CCF-0611750, DMS-0624849, CNS-

0702488, CRI-0708232, and CNS-0914330.

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Jiayu Gong r ec ei ve d t he B S a nd MSdegrees in computer science from Nanjing

University, China, in 2002 and 2005, respec-tively. He is currently working toward the PhDdegree in the Department of Electrical andComputer Engineering, Wayne State Univer-sity, Detroit, Michigan. His current researchinterests include power management in serverenvironments, mobile computing, and resourcemanagement in cloud computing. He is a

student member of the IEEE.

Xiliang Zhong received the BS degree intelecommunication from Southeast University,China, in 1997 and the MS degree in electricalengineering from the Beijing University of Postsand Telecommunications, China, in 2000, andthe PhD degree in computer engineering fromWayne State University, Detroit, Michigan, in

2007. He is a software engineer in MicrosoftCorporation. His research interests were powermanagement in embedded systems, mobile

computing, and resource management in distributed systems. He isa member of the IEEE.

Cheng-Zhong Xu received the BS and MSdegrees from Nanjing University in 1986 and1989, respectively, and the PhD degree incomputer science from the University of HongKong in 1993. He is currently a professor in theDepartment of Electrical and Computer Engi-neering of Wayne State University and thedirector of Suns Center of Excellence in OpenSource Computing and Applications. His recent

research interests are mainly in distributed andparallel systems, particularly in scalable and secure Internet services,autonomic cloud management, energy-aware task scheduling inwireless embedded systems, and high performance cluster and gridcomputing. He has published more than 150 articles in peer-reviewed

journals and conferences in these areas. He is the author of Scalableand Secure Internet Services and Architecture (Chapman & Hall/CRCPress, 2005) and the leading coauthor of Load Balancing in ParallelComputers: Theory and Practice(Kluwer Academic/Springer, 1997). Heserves on five journal editorial boards including IEEE Transactions onParallel and Distributed Systems and the Journal of Parallel andDistributed Computing. He was a program chair or general chair of anumber of conferences, including Infoscale 2008, EUC 2008, and GCC2007. He was a recipient of the Faculty Research Award of Wayne StateUniversity in 2000, the Presidents Award for Excellence in Teaching in2002, and the Career Development Chair Award in 2003. He is a seniormember of the IEEE.

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