On optimizing energy consumption for mobile handsets

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 6, NOVEMBER 2004 1927

On Optimizing Energy Consumptionfor Mobile Handsets

Yang Xiao, Senior Member, IEEE, Haizhon Li, Student Member, IEEE, Yi Pan, Senior Member, IEEE,Kui Wu, Member, IEEE, and Jie Li, Member, IEEE

Abstract—To reduce energy consumption (EC), a mobilehandset system can be designed in such a way that while thedata-receiving unit in a mobile handset is receiving and moni-toring data packets, the rest part of the handset (i.e., a processingunit and a user interface) is switched off into a sleep mode. In thispaper, we study the timing when the rest part of the handset shouldwake up. Several schemes (i.e., Always-ON, Always-OFF, Wake-upupon Arrival, Wake-up upon Full, and the fractional thresholdscheme) are studied and compared in terms of energy saving andthe packet dropping probability (PDP). We formulate the totalEC for all theses schemes analytically. Furthermore, we show howto choose optimal thresholds for the fractional threshold schemefor the following two-optimization problems: 1) minimizing theswitch-on rate with a bound on the PDP and 2) minimizing thetotal EC with a bound on the PDP. Our study shows that thefractional threshold scheme is the best scheme. Simulations arecarried out to validate analytic models.

Index Terms—Energy consumption, mobile handset, sleep mode,wake-up mechanism.

I. INTRODUCTION

ONE of the critical limiting operational factors for mobilehandsets is that their operation time is restricted by the

battery capacity, even in stand-alone mode [2]. Therefore,designing the system in an energy consumption (EC)-awaremanner is imperative [3], [4], and it is one of the primaryobjectives in the design of mobile handsets [5]. Significantreduction of battery size and weight can be difficult so thatalternative strategies need to be employed toward the goal ofenergy savings in both communication and networking aspects[5], [6].

Operations of mobile handsets may significantly consumebattery energy. The components consuming energy in a mobilehandset can be designed in such a way that while the data-re-ceiving unit (DRU), a fixed-size memory queue in the handsetfor buffering data packets, is receiving and monitoring datapackets, the rest part of the handset, a user interface and aprocessing unit (PU) that processes received data packets, is

Manuscript received March 21, 2004; revised June 9, 2004.Y. Xiao and H. Li are with the Computer Science Division, The University of

Memphis, Memphis, TN 38152 USA (e-mail: [email protected]; [email protected]).

Y. Pan is with the Department of Computer Science, Georgia State University,Atlanta, GA 30303 USA (e-mail: [email protected]).

K. Wu is with the Department of Computer Science, University of Victoria,Victoria, BC V8W 3P6, Canada (e-mail: [email protected]).

J. Li is with the Institute of Information Sciences and Electronics, Uni-versity of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan (e-mail:[email protected]).

Digital Object Identifier 10.1109/TVT.2004.836966

switched-off into a sleep mode [1]. An example of switching-offthe user interface is a Motorola Star TAC cellular handset underSprint PCS, in which, the user interface is switched-off if thereis no activity for several seconds. For the remainder of thispaper, we refer to the rest part of the handset (the user interfaceand the PU) as the PU in general. Switch-on actions consume agreat amount of energy [7], [8]. Immediately waking up uponreceipt of a packet may cause too many switch-on actions,thus consuming more battery energy and degrading the powerefficiency. Waiting for more packet arrivals before performinga switch-on action avoids many switch-on actions, but causespacket losses due to a limited buffer size. Therefore, there isa tradeoff between energy saving and the number of packetsdropped. In [1], an integer threshold-based scheme was pro-posed in which waking-up happens when the number of packetsin the buffer reaches a fixed threshold, and the switch-on rate(SOR) was studied. However, neither the total EC nor opti-mizing the total EC was studied. In this paper, we considerthe influence of the buffer size of the DRU on the actions ofswitching on/off the PU to reduce EC for mobile handsets. Westudy several wake-up mechanisms, and formulize the total ECanalytically for each mechanism. We study and compare thefollowing schemes:

• Always-ON (AON): the PU is always in the wake-up mode.• Always-OFF (AOFF): the PU is always in the sleep mode.

AOFF is considered for comparison purpose.• Wake-up upon arrival (WA): whenever there is a packet

arrival, the PU wakes up, and whenever there is not anypacket in the buffer, the PU goes to sleep.

• Wake-up upon full (WF): whenever the buffer is full, thePU wakes up, and whenever there is not any packet in thebuffer, the PU goes to sleep.

• Fractional threshold method: we propose a fractionalthreshold wake-up mechanism in which the switch-onaction is performed with a probability when the numberof packets in the buffer of the DRU reaches a threshold.Whenever there is no packet, the PU goes to sleep. Suchan approach improves the integer threshold approach byallowing the threshold to be a random variable so thatthe threshold is no longer a fixed deterministic integervalue, but an integral random variable with a probabilitydistribution. The mean of this variable can be regardedas the nominal threshold value, above which waking-uphappens. The nominal threshold value can be a non-negative real number rather than an integer. Therefore,the fractional scheme provides a finer grained thresholdvalue.

0018-9545/04$20.00 © 2004 IEEE

1928 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 6, NOVEMBER 2004

The fractional threshold concept has been adopted in wire-less/mobile networks for different problems. For example, thewell-known Fractional Guard Channel scheme is the optimal ap-proach for channel assignments, and it outperforms the GuardChannel approach [12], [13]. Fractional movement-based loca-tion area update scheme [14] outperforms the normal locationarea update scheme. In this paper, the fractional concept hasbeen applied to save EC in mobile handsets, and the followingtwo optimization problems are studied: 1) minimizing the SORwith a bound on the packet dropping probability (PDP) and 2)minimizing the total EC with a bound on the PDP.

The rest of this paper is organized as follows. Wake-upschemes are presented in Section II. Section III provides ana-lytic models and formulation for all these schemes. Section IVprovides optimality analysis and a method to find optimalfractional thresholds. Performance evaluations are reported inSection V. Section VI concludes this paper.

II. WAKE-UP SCHEMES

In this section, we present several wake-up schemes as fol-lows. We present the fractional threshold scheme in Section II-Aand other schemes in Section II-B.

A. Fractional Threshold Scheme

The fractional threshold scheme allows the threshold to berandom. Therefore, the threshold is no longer a fixed determin-istic integer value, but an integral random variable with a proba-bility distribution. The mean of this probability distribution canbe treated as the nominal threshold value, above which the PUwill be switched on, and this nominal threshold can be a non-negative real number instead of an integer.

Specifically, in a mobile handset system with the fractionalthreshold scheme, when the packet-receiving queue is empty,the PU is switched off into the sleep mode; it is switched on tothe wake-up mode with probability ( ) when the number ofpackets in the queue reaches ; otherwise it is switched on whenthe number of packets in the queue reaches . The fractionalthreshold algorithm is shown in Table I.

Definition: The Threshold Random Variable ( ): for anyfixed threshold , the threshold for the fractional thresholdscheme is a random variable , with a probability densityfunction (pdf) defined as follows:

ififotherwise.

(1)

where , , and is the buffer size.Note that in both the description and Table I, we adopt con-

ditional probability, i.e., “otherwise” means the condition thatat the threshold , switch-on action does not happen, whereasin (1), we do not adopt conditional probability. We can provethey are equivalent as follows. From the fractional threshold al-gorithm listed in Table I, we have , and

TABLE IFRACTIONAL THRESHOLD ALGORITHM

This proves that our fractional threshold algorithm in Table Iconforms to our definition of the fractional threshold pdf in (1).The mean of is

(2)

The above mean could be any nonnegative real number in theinterval ( , ) instead of an integer or . The fractionalthreshold takes on real values from [0, ]. We will further dis-cuss the optimality aspect in Section IV.

All mobile handsets run the same algorithm (shown inTable I) in a distributed manner with different optimal thresh-olds calculated based on both the proposed method and themeasured packet arrival rates. How the proposed approach isdynamically adjusted according to different traffic conditionsis discussed in Section IV-C.

B. AON, AOFF, WA, and WF

In the AON scheme, the PU is always in the wake-up mode. Inthe AOFF scheme, the PU is always in the sleep mode. In AON,the energy saving is the worst, but the PDP is the smallest. InAOFF, the energy saving is the best, but the PDP is the worst,i.e., one. AOFF is a nonworking case. AON and AOFF are twoextreme cases in terms of energy saving and the PDP.

In WA scheme, whenever there is a packet arrival, the PUwakes up, and whenever there is not any packet, the PU goes tosleep. In WF scheme, whenever the buffer is full, the PU wakesup; and whenever there is not any packet, the PU goes to sleep.WA and WF are special cases of the fractional threshold scheme.

III. ANALYTICAL MODELS

Assume that the buffer size of the DRU is in terms ofpackets. More specifically, the buffer can hold up to packetsexactly. We further assume that the packet arrivals to a handsetfollow a Poisson distribution with rate . The service time forprocessing packets follows an exponential distribution withrate . Moreover, the two distributions are independent ofeach other. Therefore, the handset PU can be modeled with aMarkovian process with finite capacity [9], [10]. According

XIAO et al.: ON OPTIMIZING ENERGY CONSUMPTION FOR MOBILE HANDSETS 1929

Fig. 1. State transition diagram for the fractional threshold scheme.

Fig. 2. State transition diagram for the AON scheme.

to these assumptions, we will derive analytic models for allwake-up schemes.

To model the fractional threshold scheme, we must distin-guish two modes: the sleep mode in which packets are beingreceived while the PU is switched-off, and the wake-up mode inwhich packets are being received while the PU is processing thepackets. In the fractional threshold scheme, the PU is switchedoff into the sleep mode when the buffer is empty. It is switchedto the wake-up mode with probability ( ) when the numberof packets in the buffer reaches , and is switched to the wake-upmode with probability when the number of packets in thebuffer reaches . In other words, when the number of packetsaccumulated in the buffer reaches , a switch-on action is per-formed with probability ( ). Otherwise, a switch-on ac-tion is performed when the number of packets accumulated inthe buffer reaches . The state transition diagram for thefractional threshold scheme is shown in Fig. 1, where, state ( ),( ) indicates that the queue has packets, and the PUis in the sleep mode. State ( ), ( ) indicates thatthe queue has packets and the PU is in the wake-up. It is alsoshown in Fig. 1 that when the queue is empty [state (0)], the PUis switched off into the sleep mode. When the queue is in state( ), the next state becomes either state ( ) (the wake-upmode), or state ( ) (the sleep mode). When the queue is in state( ), the next state is switched on into the wake-up mode [state( )].

From [9] and Fig. 1, we can easily derive the instantaneoustransition probabilities from state ( ) to either state ( ) orstate ( ) as follows. Note that and are fixed number here.

(3)

(4)

Therefore, when the queue is in state ( ), the next stateis switched on into the wake-up mode with probability ( )and stays in the sleep mode with probability . In other words,when the number of packets in the queue reaches , the PU isswitched on to the wake-up mode with probability ( ), andstays in the sleep mode with probability .

In AON scheme, the PU is always in the wake-up mode. AONis a special case of the fractional threshold scheme whenand , in which Fig. 1 is changed into Fig. 2. In the AOFF

Fig. 3. State transition diagram for the AOFF scheme.

Fig. 4. State transition diagram for the WA scheme.

Fig. 5. State transition diagram for the WF scheme.

scheme, the PU is always in the sleep mode. AOFF is a specialcase of the fractional threshold scheme when and

, in which, Fig. 1 is changed equivalently into Fig. 3. In WAscheme, whenever there is a packet arrival, the PU wakes up,and whenever there is not any packet, the PU goes to sleep.WA is a special case of the fractional threshold scheme when

and , in which, Fig. 1 is changed into Fig. 4, and theonly state in the sleep mode is the state (0). In WF the scheme,whenever the buffer is full, the PU wakes up; whenever there isnot any packet, the PU goes to sleep. WF is a special case of thefractional threshold scheme when and , in which,Fig. 1 is changed into Fig. 5.

Based on the state transition diagrams in Figs. 1–5, equilib-rium equations can be used to derive stationary probabilitiesfor each scheme. Since all the schemes are special cases of thefractional threshold scheme, next we first study the fractionalthreshold scheme.

For the fractional threshold scheme, based on the state transi-tion diagram in Fig. 1, the following equilibrium equations holdwhen the PU is in a steady state, where denotes the stationaryprobability of state ( ).

States Rate enters states Rate leaves states

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)


Furthermore, we have

(15)

With (5) (15), we can derive the stationary proba-bility for each state. Let ,

,and . With some algebra operations, we can obtain thefollowing equations that are proved in Appendix :

(16)

(17)

Let and denote the SOR and thePDP, respectively. They are functions of ( , ). Intuitively, inthe long run, the portion of time that the PU stays in the state( ) is . During this time period, the number of arrivalsshould be the product of the arrival rate and the time period, i.e.,

. Similarly, the portion of time that the PU stays in state( ) is . During this time period, the number of arrivals shouldbe the product of the arrival rate and the time period, i.e., .Moreover, among the arrivals of , only ( ) portion isswitched on. Therefore, also from (8), the SOR will be

(18)

Intuitively, the SOR will be the same as the switch-off rate inthe long run. This fact can be easily proved by formulae (6)–(7)as follows:

(19)

Packets will be dropped when the buffer is full, i.e., when thePU is in state ( ). Therefore, the is calculated as

(20)

Let and denote ECs of the sleep mode and thewake-up mode, respectively, per unit time, where .Let and denote ECs of the switch-on actionand the switch-off action, respectively, per time. The SOR ismeasured in terms of the times of switch-on per unit time. Let

denote the probability of the PU in the sleep mode. We have

(21)

TABLE IITHE BINARY SEARCH ALGORITHM

The total EC is defined as

(22)

From (18), (20), and (22) for the fractional threshold scheme,we can derive corresponding equations for other special casesof schemes. For AON, AOFF, WA, and WF, we have

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

IV. OPTIMALITY ANALYSIS AND A METHOD

TO FIND OPTIMAL THRESHOLDS

In this section, we study optimality problems for the frac-tional threshold scheme. In Section IV-A, we first provide somefeatures of the performance metrics; in Section IV-B, we showhow to choose the optimal thresholds for the following twoproblems: 1) minimizing the SOR with a bound on the PDP and2) minimizing the total EC with a bound on the PDP. Finally,Section IV-C provides some comments on implementation is-sues for the optimal fractional threshold scheme.


Fig. 6. PDP, SOR, and EC over t + �(� = 0:9).

Fig. 7. PDP over � and t + �.

A. Features of the Performance Metrics

In this section, we study some features of the total EC, theSOR, and the packet-dropping probability for the fractionalthreshold scheme.

Since is a function of and , where andare the integer and fraction parts of , respectively, we

can denote with . Therefore,we use and interchangeably.

In previous section, for fixed values and , we have de-rived the SOR and the PDP. Intuitively, when the mean of thefractional threshold random variable increases, the SOR

will decrease due to fewer switch-on actions,whereas the will increase due to the slower re-


Fig. 8. SOR over � and t + �.

Fig. 9. EC over � and t + �.

sponses for packet processing. Therefore, we have the followingtheorem that is proved in Appendix :

Theorem I: or the fractional threshold scheme, if ,the is a strictly increasing function of and theSOR is a strictly decreasing function. In otherwords, we have

(35)

(36)

Intuitively, when the mean of the fractional threshold randomvariable increases, the total EC decreases since: 1) thePU stays more time on the sleep mode and 2) there are fewerswitch-on actions. Therefore, we can obtain the following the-orem that is proved in Appendix .


Fig. 10. Thresholds over the traffic load � (PDP bound is 3%).

Fig. 11. PDP over the traffic load � for different schemes (PDP bound is 3%).

Theorem II: For the fractional threshold scheme, if we have, the total EC, is a strictly decreasing function

of . In other words, we have

(37)

B. Optimal Thresholds to Minimize the Total EnergyConsumption and the SOR

The objective is to find a good pdf in (1) so that the fractionalthreshold scheme can get better performance. We need to find a2-tuple ( , ). The domain of the 2-tuple ( , ) is

(38)


Fig. 12. EC over the traffic load � for different schemes (PDP bound is 3%).

Fig. 13. SOR over the traffic load � for different schemes (PDP bound is 3%).

Optimization Problem 1 (OP1): For the fractionalthreshold scheme, denote the SOR and the PDP as

and respectively, whenand ( ) is used as the fractional threshold. Giventhat is the predefined PDP bound, find , where

,and .

Optimization Problem 2 (OP2): For the fractionalthreshold scheme, denote the total EC and the PDP as

and respectively, when

and ( ) is used as the fractional threshold. Giventhat is the predefined PDP bound, find , where

and.

Lemma 1: Since is a closed domain, there must exist op-timal solutions of the above two optimization problems.

Since the total EC is a strictly decreasing function of the frac-tional threshold, and the PDP is a strictly increasing function ofthe fractional threshold, the optimal fractional threshold is themaximum fractional threshold under the bound. This is also true


Fig. 14. EC over the traffic load � for different ratios of E : E : E : E (PDP bound is 3%).

for the optimal fractional threshold for the SOR. Furthermore,the optimal value is unique. Therefore, we have the followingLemma 2 and Lemma 3. We also develop the Binary Search Al-gorithm, shown in Table II. Lemma 3 holds since the total EC isa strictly decreasing function of the fractional threshold, and thePDP is a strictly increasing function of the fractional threshold.

Lemma 2: The optimal fractional thresholds for OP1 andOP2 are the same and unique.

Lemma 3: The binary search algorithm can find an optimalfractional threshold for OP1 and OP2.

Since is an integer and is a fraction between 0 and 1, wewill use and ( , ) interchangeably without causing confu-sion to represent the nominal threshold. Note that in Table II, weuse neither the total EC nor the SOR. There are two approachesof calculating in Table II. The first approach is to per-form a simulation, and measure . The second approachis to calculate . We choose the second approach. Lemma 2,Theorem I, and Theorem II indicate that the binary search al-gorithm can find an optimal fractional threshold, as stated inLemma 3. is a predefined value of PDP as the QoS re-quirement.

C. Remarks for Implementation Issues Including BeingDynamically Changed Under Different Traffic Conditions

All the mobile handsets periodically measure traffic condi-tions for some time and obtain measured packet arrival rate ;with the measured mean value of traffic load, the algorithm inTable II is used to obtain optimal fractional threshold; finallythe algorithm shown in Table I can be run. Whenever trafficconditions are changed, the above procedures are performed re-peatedly. Therefore, in case that the traffic load is dynamicallychanging, the system can measure the traffic dynamically and

calculate the optimal fractional threshold dynamically based onthe load. To avoid oscillations, window-averaging techniquescan be adopted [11].

V. PERFORMANCE EVALUATION

In this section, we study wake-up schemes over var-ious parameters, as well as the two-optimization problemsvia analytical results and simulation results. Performancemetrics include the PDP, the total energy consumption(EC), and the SOR. Parameters include the traffic load

, the fractional threshold ( ) and the EC ra-tios ( ). We adopt .Section IV-A studies performance of the fractional thresholdscheme and gives validation with simulations. Section IV-Bcompares the optimal fractional threshold scheme with otherschemes. Section IV-C provides a summary of key results.Note that the optimal fractional threshold scheme is differentfrom the fractional threshold scheme.

A. Performance of the Fractional Threshold Scheme andValidation With Simulations

Fig. 6 shows PDP, SOR, and EC versus the fractionalthreshold ( ) for the fractional threshold scheme when

and .As illustrated in this figure, numerical results and simulationresults match pretty well. As increases, PDP increases,and both SOR and EC decrease. These observations confirmTheorem I and Theorem II.

Fig. 7 shows PDP for the fractional threshold scheme over thetraffic load and the fractional threshold . As illustrated inthe figure, PDP increases when either the traffic load increasesor the threshold increases.


Fig. 15. EC and SOR versus PDP (E : E : E : E = 1 : 10 : 30 : 20 and � = 0:9).

Fig. 8 shows SOR for the fractional threshold scheme over thetraffic load and the fractional threshold when

. As illustrated in thisfigure, SOR increases as the fractional threshold decreases. Fora fixed threshold at a small value, when the traffic load increases,SOR increases since the system need to wake up more often;when the traffic load is high enough, SOR decreases since thesystem spends more time in the wake-up mode and less time inthe sleep mode so that the SOR decreases.

Fig. 9 shows EC for the fractional threshold scheme over thetraffic load and the fractional threshold when

. As illustrated in thefigure, EC increases as the fractional threshold decreases. For afixed threshold at a small value, when the traffic load increases,EC increases since the system spends more time in the wake-upmode; when the traffic load is high enough, EC decreases sincethe system spends more time in the wake-up mode and less timein the sleep mode so that the SOR decreases. This observationis also because we use a large value for the EC for the switch-onaction.

B. Comparisons of the Optimal Fractional Threshold SchemeWith Other Schemes

Fig. 10 shows thresholds for WA, the optimal frac-tional threshold, and WF over the traffic load , when

andthe PDP bound is 3%. As illustrated in this figure, the optimalfractional threshold is in the interval between that of WAand that of WF. When the traffic load increases, the optimalfractional threshold deceases since the system needs to wakeup earlier to handle more traffic. One important observation isthat optimal thresholds are real numbers with nonzero fraction,

instead of integers. Therefore, it is perfect to use the fractionalthreshold scheme to obtain optimal values.

Fig. 11 shows PDP for AOFF, WF, the optimal fractionalthreshold scheme, WA, and AON schemes over the traffic load

, when andthe PDP bound is 3%. As illustrated in this figure, AOFF hasthe highest PDP ( ). WA and AON have the same PDP, i.e.,the lowest PDP. PDP of the optimal fractional threshold schemeis between that of WA and WF. We observe that the optimalfractional threshold scheme is exactly bounded by 3%. Otherschemes whose PDPs are bounded by 3% are AON and WA.However, later figures show that AON and WA have either veryhigh EC or very high SOR.

Fig. 12 shows EC for AON, WA, the optimal fractionalthreshold scheme, WF, and AOFF schemes over the traffic load

, when andthe PDP bound is 3%. As illustrated in this figure, AON has thehighest EC and AOFF has the lowest EC. EC of WF is higherthan that of AOFF. EC of the optimal fractional thresholdscheme is in between that of WA and that of WF. We observethat the optimal fractional threshold scheme has very low EC,and the only working scheme (AOFF is not a working scheme)that is lower than the optimal fractional threshold scheme isWF, which is only a little lower than the optimal fractionalthreshold scheme. From Fig. 11, AOFFs PDP is 1 and WF hasmuch higher PDP than the optimal fractional threshold scheme.

Therefore, the optimal fractional threshold scheme is the bestscheme in terms of optimizing the total EC with a bound on PDP.

Fig. 13 shows SOR for AON, WA, the optimal fractionalthreshold scheme WF and AOFF schemes over the traffic load

, when ,and the PDP bound is 3%. As illustrated in this figure, WA hasthe highest SOR. AON and AOFF have the lowest SOR ( ).


Fig. 16. Comparison of the optimal fractional threshold scheme and the optimal integer threshold scheme (PDP bound is 3%).

SOR of the optimal fractional threshold scheme is in betweenthat of WA and that of WF. We observe that the optimal frac-tional threshold scheme has very low SOR, and the only workingschemes (AOFF is not a working scheme) lower than the optimalfractional threshold scheme are WF and AON. From Fig. 11,AOFFs PDP is 1 and WF has much higher PDP than the op-timal fractional threshold scheme.

Therefore, the optimal fractional threshold scheme is the bestscheme in terms of optimizing the total SOR with a bound onPDP.

Fig. 14 shows EC for the optimal fractional threshold schemeover the traffic load for different ratios of

. As illustrated in the figure, as either orincreases relative to , the total EC increases.

Fig. 15 shows SOR and EC versus PDP bound for the op-timal fractional threshold scheme. As illustrated in the figure, asthe PDP bound increases, both EC and SOR decrease. In otherwords, if we loosen the requirement of PDP, the system savesmore energy.

Finally, Fig. 16 compares the optimal fractional thresholdscheme with the optimal integer threshold scheme. The optimalinteger threshold scheme is to find an optimal integer valuewith a bound on PDP. As illustrated in the figure, the optimalfractional threshold scheme is better than the optimal integerthreshold scheme.

C. Summary and Key Results

Our goal is to minimize both the total EC and the SOR witha bound on the PDP. We summarize the key results as follows.

• As the fractional threshold increases, PDP increases, andboth SOR and EC decrease.

• PDP increases when either the traffic load increases or thethreshold increases.

• EC increases as the fractional threshold decreases. For afixed threshold at a small value, when the traffic load in-creases, EC increases first and then decreases.

• SOR increases as the fractional threshold decreases. Fora fixed threshold at a small value, when the traffic loadincreases, SOR increases first and then decreases.

• When the traffic load increases, the optimal fractionalthreshold decreases. One important observation is thatoptimal thresholds are real numbers with nonzero frac-tion, instead of integers. Therefore, it is perfect to use thefractional threshold scheme to obtain optimal values.

• AOFF has the highest PDP ( ). WA and AON havethe lowest PDP. PDP of the optimal fractional thresholdscheme is between that of WA and WF. Given a bound onPDP, the optimal fractional threshold scheme is exactlybounded by the PDP. Other schemes that can meet a givenPDP bound are AON and WA. However, AON and WAhave either very high EC or very high SOR.

• AON has the highest EC and AOFF has the lowest EC.EC of WF is higher than that of AOFF. EC of the optimalfractional threshold scheme is in between that of WA andthat of WF. The optimal fractional threshold scheme hasvery low EC, and the only working scheme that is lowerthan the optimal fractional threshold scheme is WF, whichis only a little lower than the optimal fractional thresholdscheme. However, WF has much higher PDP than the op-timal fractional threshold scheme. Therefore, the optimalfractional threshold scheme is the best scheme in terms ofoptimizing the total EC with a bound on PDP.

• AON has the highest SOR. AON and AOFF have thelowest SOR ( ). SOR of the optimal fractional


threshold scheme is in between that of WA and that ofWF. The optimal fractional threshold scheme has verylow SOR, and the only working schemes that are lowerthan the optimal fractional threshold scheme are WFand AON. AON and WF have much higher PDP thanthe optimal fractional threshold scheme. Therefore, theoptimal fractional threshold scheme is the best scheme interms of optimizing the total SOR with a bound on PDP.

• As either or increases relative to , the totalEC increases.

• As the PDP bound increases, both EC and SOR decrease.In other words, if we loosen the requirement of PDP, thesystem saves more energy.

• The optimal fractional threshold scheme is better than theoptimal integer threshold scheme.

Furthermore, simulation results and analytical results matchpretty well.

VI. CONCLUSION

In this paper, we study the timing when mobile handsetshould wake up. We propose the fractional threshold schemeand prove that there is a unique optimal threshold to minimizeboth the total EC and the SOR with a bound on the PDP. Wecompare the optimal fractional threshold scheme with five otherschemes including the optimal integer threshold scheme, andshow that the optimal fractional threshold scheme is the best.

APPENDIX

Proof of (16) and (17)

Let denote . From (5) (7), we have

(39)

for (40)

From (8), we have

(41)

From (9) and (10), we have

(42)

(43)

From (42) and (43), we can prove the following equation viamathematical induction:

for (44)


(45)

From (12), we have

for (46)

From (39), (45), and (46), we can prove the following equa-tion via mathematical induction:

for (47)

By putting (40), (44), and (47), into (13), we have

(48)

By putting (41), (44), and (47) into (14), we have

(49)

We observe that (48) and (49) are the same. Putting (40), (41),(44), and (47) into (15), we have


(50)

From (49) and (50), for ,Let ,

, and .

(51)

(52)


Then we can have (16) and (17).

Proof of Theorem I

Let ,where , and

We have

Since , and , we have. Since and are the integral part

and the fractional part, respectively, PDP is continuous withrespective to ( ). Therefore, the PDP is astrictly increasing function of . We can easily have

. For the SOR, we have

Since and , we have

Based on the similar reason as that for the pre-vious , the SOR is astrictly decreasing function. In other words, we have

.

Proof of Theorem II


ifif

Therefore, we have

From (23) and (36), we have (see the equation at the top ofthe page).

Using the similar reasoning for in the proof ofTheorem I and based on the above inequality, we can easilyprove .

ACKNOWLEDGMENT

The authors would like to thank Dr. C. L. P. Chen and Dr. K.Kinateder for their involvements in the early stage of this work.

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Yang Xiao (S’97–M’02–SM’04) received the Ph.D.degree in computer science and engineering fromWright State University, Dayton, OH, in 2001.

He was a software engineer, a senior software engi-neer, and a technical lead working in the computer in-dustry from 1991 to 1996. From 1996 to 2001, he wasawarded the DAGSI Ph.D. Fellowship. From August2001 to August 2002, he was with Micro Linear-SaltLake City Design Center as an MAC architect in-volving the IEEE 802.11 (Wireless LAN) standardenhancement work. Since August 2002, he has been

an assistant professor of computer science at The University of Memphis.Dr. Xiao is a voting member of the IEEE 802.11 Working Group, and a

member of ACM. He currently serves on the editorial boards of InternationalJournal of Wireless and Mobile Computing, and International Journal ofSignal Processing. He serves a lead Guest Editor for the (Wiley) Journalof Wireless Communications and Mobile Computing, Special Issue on“Mobility, Paging and Quality of Service Management for Future WirelessNetworks” in 2004–2005, a lead Guest Editor for the International Journalof Wireless and Mobile Computing, Special Issue on “Medium AccessControl for WLANs, WPANs, Ad Hoc Networks, and Sensor Networks” in2004–2005, and an Associate Guest Editor for the International Journal ofHigh Performance Computing and Networking, Special Issue on “Parallel andDistributed Computing, Applications and Technologies” in 2003. His researchareas include wireless LANs, wireless PANs, and mobile cellular networks.


He has published many papers in major journals and refereed conference pro-ceedings related to these research areas, such as the IEEE TRANSACTIONS ON

MOBILE COMPUTING, IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS,IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, IEEETRANSACTIONS ON VEHICULAR TECHNOLOGY, IEEE COMMUNICATIONS

LETTERS, IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, andACM/Kluwer Mobile Networks and Applications (MONET).

Dr. Xiao serves as a symposium Co-Chair for Symposium on Data BaseManagement in Wireless Network Environments in 2003 IEEE VehicularTechnology Conference. He serves as a TPC member for many conferencessuch as IEEE ICDCS, IEEE ICC, IEEE GLOBECOM, IEEE WCNC, IEEEICCCN, IEEE PIMRC, ACM WMASH, etc. He was a recipient of the 1999Gradate Student Excellence Award in recognition of outstanding academicachievements in the Computer Science and Engineering Ph.D. degree programof Wright State University.

Haizhon Li (S’03) received the M.S. degree incomputer science from the University of Memphisin 2003.

He is currently a Ph.D. degree candidate withthe Computer Science Division, The University ofMemphis. His research interests include quality ofservice and MAC enhancement for IEEE 802.11wireless LANs, performance analysis, and inte-gration of wireless LANs, wireless PANs, and 3Gcellular networks.

Yi Pan (M’91–SM’97) received the B.Eng. andM.Eng. degrees in computer engineering fromTsinghua University, China, in 1982 and 1984, re-spectively, and the Ph.D. degree in computer sciencefrom the University of Pittsburgh, PA, in 1991.

Currently, he is a Yamacraw professor with theDepartment of Computer Science, Georgia StateUniversity, Atlanta. His research interests includeparallel and distributed computing, optical networks,wireless networks, and bioinformatics. He has pub-lished more than 80 journal papers with 25 papers

published in various IEEE journals. In addition, he has published more than 90papers in refereed conferences (including IPDPS, ICPP, ICDCS, INFOCOM,and GLOBECOM). He has also coedited 13 books (including proceedings)and contributed several book chapters. His pioneer work on computing usingreconfigurable optical buses has inspired extensive subsequent work by manyresearchers, and his research results have been cited by more than 100 re-searchers worldwide in books, theses, journal, and conference papers. He isa coinventor of three U.S. patents (pending) and five provisional patents. Hisrecent research has been supported by NSF, NIH, AFOSR, AFRL, JSPS, IISFand the states of Georgia and Ohio. He has served as a reviewer/panelist formany research foundations/agencies such as the U.S. National Science Foun-dation, the Natural Sciences and Engineering Research Council of Canada, theAustralian Research Council, and the Hong Kong Research Grants Council.He served as an Editor-in-Chief or editorial board member for eight journalsincluding three IEEE TRANSACTIONS and a Guest Editor for seven specialissues.

Dr. Pan has received many awards from agencies such as NSF, AFOSR, JSPS,IISF, and Mellon Foundation. He has organized several international confer-ences and workshops and has also served as a program committee memberfor several major international conferences such as INFOCOM, GLOBECOM,ICC, IPDPS, and ICPP. He has delivered more than 40 invited talks, includingkeynote speeches and colloquium talks, at conferences and universities world-wide. He is an IEEE Distinguished Speaker (2000–2002), a Yamacraw Distin-guished Speaker (2002), and a Shell Oil Colloquium Speaker (2002). He is listedin Men of Achievement, Who’s Who in Midwest, Who’s Who in America, Who’sWho in American Education, Who’s Who in Computational Science and Engi-neering, and Who’s Who of Asian Americans.

Kui Wu (M’97) received the Ph.D. degree incomputing science from the University of Alberta,Canada, in 2002.

He then joined the Department of Computer Sci-ence, University of Victoria, Canada, where he is cur-rently an Assistant Professor. His research interestsinclude mobile and wireless networks, network per-formance evaluation, and network security.

Jie Li (M’95) received the B.E. degree in computerscience from Zhejiang University, Hangzhou, China,in 1982, and the M.E. degree in electronic engi-neering and communication systems from ChinaAcademy of Posts and Telecommunications, Beijing,China, in 1985. He received the Dr. Eng. degree fromthe University of Electro-Communications, Tokyo,Japan, in 1993.

From 1985 to 1989, he was a Research Engineerwith the China Academy of Posts and Telecommuni-cations, Beijing. Since April 1993, he has been with

the Department of Computer Science, Graduate School of Systems and Infor-mation Engineering, University of Tsukuba, Japan, where he is currently anAssociate Professor. His research interests are in mobile distributed multimediacomputing and networking, transaction processing, OS, Internet technology, se-curity, modeling, and performance evaluation of information systems.

Dr. Li received the Best Paper award from IEEE NAECON’97 and the Ex-cellent Paper awards from SIG on Mobile Computing, Information ProcessingSociety of Japan for 1998 and 1998. He is a member of ACM. He has served as asecretary for Study Group on System Evaluation of the Information ProcessingSociety of Japan (IPSJ) and a committeeman of ACM SIGMOD Japan TechnicalCommittee on Network Transaction Processing. He has also served on the ed-itorial boards for the Information Processing Society of Japan (IPSJ) Journal,Information, and International Journal of High Performance Computing andNetworking. He has organized several international conferences and workshopsand has also served as a program committee member for several major inter-national conferences such as IEEE INFOCOM, IEEE GLOBECOM, and IEEEMASS.

On optimizing energy consumption for mobile handsets

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