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IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. X, NO. X, 2016 1

Maximizing Charging Satisfaction ofSmartphone Users via Wireless Energy TransferWenzheng Xu, Member, IEEE , Weifa Liang, Senior Member,IEEE , Jian Peng, Yiguang Liu, and Yan Wang

Abstract—Smartphones now become an indispensable part of our daily life. However, maintaining a smartphone’s continuingoperation consumes lots of battery energy. For example, a fully-charged smartphone usually cannot support its continuing operation fora whole day. A fundamental issue on a smartphone is its energy issue. That is, how to prolong the lifetime of a smartphone so that itcan run as long as possible to meet its user needs. Wireless energy transfer has been demonstrated as a promising technique toaddress this issue. In this paper, we study a novel smartphone charging problem, through wireless chargers deployed on publiccommuters, e.g., subway trains, to charge energy-critical smartphones when their users take subway trains to work or go home. Sincethe amounts of residual energy of different smartphones are significantly different, the charging satisfactions of different users areessentially different. In this paper, we formulate this charging satisfaction problem as a novel optimization problem that schedules thelimited number of wireless chargers on subway trains to charge energy-critical smartphones such that the overall charging satisfactionof smartphone users is maximized, for a given monitoring period (e.g., one day). For this problem, we first devise a 1

3-approximation

algorithm if the travel trajectory of each smartphone user is given. We then propose an online algorithm to deal with dynamicenergy-critical smartphone charging requests. We also propose a nontrivial distributed scheduling algorithm for a variant of theproblem where the global knowledge of user energy information is unknown. We finally evaluate the performance of the proposedalgorithms through experimental simulations, using a real dataset of subway-taking in San Francisco. The experimental results showthat the proposed algorithms are very promising, and over 90% of energy-critical user smartphones can be satisfactorily charged in aone-day monitoring period.

Index Terms—Smartphone, energy charging, wireless energy transfer, subway trains, charging satisfaction maximization,approximation algorithm, online algorithm, distributed algorithm

F

1 INTRODUCTION

W ITH the advance on micro-electronic technology andwireless communication, more and more people

nowadays rely heavily on portable mobile devices, suchas smartphones, tablets, and Apple watches, for entertain-ment and business purposes. Especially, smartphones nowbecome an indispensable part of our daily life. The eMarketerreported that there were more than 4 billion smartphoneusers globally in 2014 and this number is expected togrow to 5 billion in 2017 [30]. However, smartphones arevery energy-hungry, and a fully-charged smartphone usu-ally cannot support its continuing operation for a wholeday, even if the battery technology for smartphones hasmade substantial progress in the past decades to makesmartphone batteries last much longer and have higherpower densities [26]. For example, the lifetime of iPhone6s is only about 10 hours for Internet usage [9]. Also, it isreported that 62% of smartphones have less than 20% ofresidual power, 33% of smartphones go below 10% power,and 12% of smartphones run out of their power completelyat the end of a day [26]. The limited energy capacities ofsmartphones bring their users many inconveniences. Some

• W. Xu, J. Peng, Y. Liu, and Y. Wang are with College of ComputerScience, Sichuan University, Chengdu, 610065, P.R. China. E-mails:[email protected], [email protected], [email protected], [email protected]

• W. Liang is with the Research School of Computer Science, The Aus-tralian National University, Canberra, ACT 0200, Australia. E-mail:[email protected]

users cannot continue using their smartphones any morelater of the day (e.g., afternoon) due to the energy depletionsof their smartphones. Others get to turn off some valuableyet energy-consuming functionalities such as GPS, 3G/4G,and Wi-Fi installed in their smartphones, to prolong thelifetimes of the smartphones. Consequently, smartphoneusers cannot make use of many features provided by smart-phones such as Twitter, google maps, Email, YouTube, eBay,Pinterest, etc. Alternatively, a smartphone can be chargedby a portable charger if needed. It however is inconvenientfor its user to bring the charging device with him/her alltime. A fundamental problem of smartphone charging thusis: is there any convenient way to charge energy-criticalsmartphones so that their users can enjoy all applicationsprovided by the smartphones at all times without worryingwhether there is enough energy left and turning off someuseful mechanisms (e.g., 4G)? In this paper we tackle thischallenge by using wireless chargers installed on subwaytrains to charge energy-critical smartphones.

The recent breakthrough on wireless energy transfertechnology based on strongly coupled magnetic resonanceshas drawn lots of attentions in the research community [14],[15], [32], [37], [22]. Kurs et al. demonstrated that it is pos-sible to achieve an approximate 40% efficiency of wirelesspower transfer for powering a 60W light bulb within twometers without any wire lines [14]. Engineers at Intel furtherachieved a 75% efficiency of wireless energy transfer fortransferring 60W of power over a distance of up to twoto three feet [12]. This technology has many advantagesin comparison with other wireless charging technologies,




including high wireless energy transfer efficiency over amid-range, immunity to the neighboring environment, norequirement of line-of-sight or any alignment, and chargingmultiple mobile devices simultaneously [15], [28]. Also, it isreported that the technology is safe to human beings sinceit is not radiative [28]. Furthermore, an industry standardgroup A4WP, including Qualcomm Corp., Samsung Corp,etc., has applied the principle of magnetic resonance todevelop a wireless energy transfer system over distance forconsumer electronics [21]. Several commercial products ofwireless energy transfer technology now are available in themarket, e.g., smartphones, electric vehicles, and sensors [10],[20], [21], [28]. It thus is envisioned smartphones supportingwireless energy transfer will be pervasive in the near futureand the wireless energy transfer market is expected to growfrom just $216 million in 2013 to $8.5 billion in 2018 [29].

The wireless energy transfer technique will revolution-ize the way people charge their smartphones. Subway isone of the most popular transportation means in modernmetropolitan cities including New York, London, Tokyo,Hong Kong, Beijing, etc., where people take a subway trainto work or go home. For example, the average daily subwayridership in New York alone is about 5.5 million in a week-day [25], and the number in Beijing even reaches over 10million [24]. Let us consider an application scenario wherethere are multiple chargers deployed on every subway trainand every charger can charge a smartphone via wirelessenergy transfer if the smartphone is within the chargingrange of the charger, e.g., 2 to 3 meters. When a smartphoneuser takes on a subway train, his/her smartphone cansend a charging request automatically if its residual energyfalls below a given threshold, e.g., only 20% energy left,assuming that the user pays the subway company some feefor such a service, e.g., by the amount of energy charged tothe user. Once the request is received, a charger nearby theuser is allocated to charge the smartphone wirelessly.

In this paper we consider a charging satisfaction maxi-mization problem on subway trains as follows. People takeon a subway train at one station and take off at anotherstation, or they may interchange for another train. There-fore, each user has several opportunities to get his/hersmartphone charged when he/she is on a subway train.Furthermore, some people will take more stations thanothers before they take off. Since there are only limitednumbers of chargers installed on subway trains and thenumber of charging requests may be far more than thesum of the service capacities of all chargers, it thus isdesirable to ‘fairly’ replenish energy to the requested smart-phones such that as many smartphone users as possible aresatisfied. Otherwise, some users with sufficiently residualenergy will be fully charged while others with barely leftenergy will miss the charging opportunities and run outof their energy very soon. We thus model the chargingsatisfaction of each requested user as a non-decreasingsubmodular function of the amount of energy charged tothe smartphone of the user, where a submodular functionusually is used to characterize the diminishing return prop-erty. For example, the charging satisfaction of a user vj isfj(Bj) = log2(REj+Bj+1)− log2(REj+1), where REj isthe residual energy of user vj before the user is charged andBj is the amount of energy charged to the user. The charging

satisfaction maximization problem is to allocate the chargersto replenish energy to smartphones of requested users for agiven monitoring period (e.g., one day), such that the sumof charging satisfaction of all users is maximized.

Comparing with the solution that offers wired outlets tocharge smartphones, there are two significant advantagesin deploying wireless energy chargers to replenish energyto smartphones on subway trains. First, it is much moreconvenient to charge smartphones, since users do not needto connect their smartphones to the outlets with wires, andnearby wireless chargers will be automatically allocated tocharge their smartphones wirelessly. Second, smartphoneusers can be fairly charged. In the solution of offering wiredoutlets, a smartphone user may have to wait for an unoccu-pied outlet a long time if all outlets are being used by otherusers, since an occupied user usually will not release hisoutlet until his smartphone has been charged a large amountof energy (compared with his smartphone battery capacity).As a result, the waiting user may miss his charging opportu-nity before he takes off the train, even if the residual energyof his smartphone is very low. In contrast, wireless chargerscan identify energy-emergent smartphones and charge themso that the sum of the charging satisfactions of all users ismaximized.

The charging satisfaction maximization problem is verychallenging, due to the following constraints on chargingsmartphones by wireless chargers. (i) Each user may bewithin the charging ranges of multiple chargers but only onecharger will be allowed to charge the user at any time; (ii)the number of users within the charging range of a charger isusually larger than the charging capacity of the charger andthus only a subset of the users can be chosen for charging;(iii) the energy transfer efficiency between a charger anda user decreases with the physical distance between them;and (iv) some users enjoy many charging opportunitieswhile others may have only a few of them. We thus mustaddress following subproblems: (a) when should each userbe charged? and (b) which charger should be allocated tocharge the user?

Unlike existing solutions to prolonging smartphone life-times by forcing their users to turn off some useful yetenergy-consuming functionalities (e.g., 3G/4G) or offeringinconvenient wired outlets to charge the smartphones onsubway trains, we make use of wireless energy chargersinstalled on subway trains to charge energy-critical smart-phones so that the smartphones can be charged in a conve-nient way (i.e., wireless charging) and their users can enjoyall applications provided by smartphones at all times. Wealso study the problem of ‘fairly’ charging energy-critical s-martphone users, by proposing a novel charging satisfactionmetric and efficient charging scheduling algorithms.

The main contributions of this paper are as follows. Weare the first to consider the use of wireless energy chargersinstalled on subway trains to charge energy-critical smart-phones of users when their users take subway trains. Tofairly charge energy-critical smartphones, we first formulatea novel optimization problem of allocating wireless chargersto charge the smartphones for a given monitoring period,such that the overall charging satisfaction of smartphoneusers is maximized. We then devise a 1

3 -approximationalgorithm for the problem if the travel trajectory of each




user is given. We also propose an online algorithm to dealwith dynamic energy-critical smartphone charging requests.Furthermore, we develop a novel distributed schedulingalgorithm for a variant of the problem when the globalknowledge of user energy information is not known inadvance. We finally evaluate the performance of the pro-posed algorithms, using a real dataset of subway-taking inSan Francisco. Experimental results show that the proposedalgorithms are very promising, and as high as 87.4%, 87.4%and 90% of energy-critical users can be charged through thesolutions delivered by the proposed distributed, online, andapproximation algorithms, respectively.

The rest of the paper is organized as follows. Section 2reviews related work. Section 3 introduces preliminaries anddefines the charging satisfaction maximization problem pre-cisely. Sections 4 and 5 propose approximation and onlinealgorithms for the problem with and without the knowledgeof travel trajectory of each user, respectively. Section 6 de-vises a novel distributed algorithm. Section 7 evaluates thealgorithm performance, and Section 8 concludes the paper.

2 RELATED WORK

The wireless power transfer technology based on stronglymagnetic resonances has drawn a lot of attentions in manyareas, such as smartphones [21], [28], electric vehicles [41],[10], wireless sensor networks [31], [35], [16], [17], [34], [33],etc. For example, Zhu et al. [41] studied the problem ofscheduling n electric vehicles (EVs) to K deployed chargingstations in a road network such that each EV is fully chargedand the average time spent on charging EVs is minimized,where the time for charging an EV includes its travel timeto its assigned charging station, the queuing time, and theactual charging time. Unlike their work, in this paper eachto-be-charged smartphone of a user is not required to befully charged. Instead, the sum of charging satisfactions ofall smartphone users should be maximized.

There are extensive studies in adopting wireless energyreplenishment to prolong the lifetime of wireless sensornetworks (WSNs) [31], [35], [16], [17], [36]. Xie et al. [31]employed a wireless charging vehicle to periodically visiteach sensor in a WSN, where the charging vehicle cancharge each sensor wirelessly when the vehicle travels inthe vicinity of the sensor. On the other hand, Xu et al. [35],[36] and Liang et al. [16], [17] employed multiple chargingvehicles to replenish sensors energy in WSNs, in whichXu et al. studied the problem of finding a series of charg-ing scheduling for the charging vehicles to maintain theperpetual operations of the WSN for a given monitoringperiod such that the total travel distance of the vehicles forthe period is minimized [35], [36], and Liang et al. consid-ered the problem of dispatching the minimum number ofcharging vehicles to charge a set of to-be-charged sensors,assuming that the energy capacity of each charging vehicleis bounded [16], [17]. Unlike these studies, in this paperthe wireless chargers on subway trains are fixed and eachsmartphone may not be fully charged. In fact, the amountof energy received by a smartphone relies on the durationof its user on trains and how much residual energy it has.

There are several closely related studies in partial exe-cutions for interactive services such as web search, where arequest can be partially executed and the response quality to

it improves with increasing resources but diminishing utilitygain margins [8], [40], [38], [39]. He et al. [8] consideredthe problem of allocating processing time to competingservice requests so that the total quality gained by executingall requests is maximized. Zheng et al. [40] investigatedan extended version of the problem in [8], by taking intoaccount both the multi-resource sharing and execution par-allelism. Xu et al. [38] studied the problem of schedulingpartially-executed requests in data centers such that the costof the total energy consumption is minimized, subject tothe Service Level Agreement (SLA) on the response qualityto each executed request. Zheng et al. [39] considered theproblem of scheduling interactive jobs in a data center withmultiple identical machines so that the utility sum of all jobsis maximized. Unlike these studies of request execution ona single data center [8], [40] or a user’s request traffic canbe arbitrarily split among all data centers [38], in this paperwe investigate scheduling multiple wireless chargers, ratherthan a single wireless charger, and each smartphone can becharged by only one charger, instead of being arbitrarilysplit among chargers, at any time. Furthermore, althoughthe problem considered in [39] seems similar to that in thispaper, in terms of there being N machines in [39] and Kchargers in this paper, the N machines are identical witheach having a processing capacity of one and each job isallowed to be allocated to every one of the N machines,the N machines thus can be considered as a single virtualmachine with a more powerful processing capacity of N .Contrarily, the charging efficiencies (i.e., processing capaci-ties) of different chargers for a smartphone user significantlyvary and each smartphone can be allocated to only thewireless chargers nearby, since the energy transfer efficiencydecreases with the increase of the distance between a charg-er and a smartphone. Therefore, the problem considered inthis paper thus is essentially different from those in [8], [40],[38], [39], and the proposed algorithms for partial requestsare inapplicable to the problem considered in this paper.

There are several fairness metrics that are used to mea-sure the fairness in resource allocation, such as Jain’s fair-ness metric [11], the max-min fairness [3], and the propor-tional fairness [13]. In Jain’s fairness metric [11], the valueof J(x1, x2, . . . , xn) (= (

∑ni=1 xi)

2

n·∑n

1 x2i

) is used to measure thefairness, where x1, x2, . . . , xn are the amounts of ener-gy charged to users v1, v2, . . . , vn, respectively. It reachesthe maximum when all users are charged with the sameamount of energy. In the max-min fairness [3], the value ofminni=1{xi} is used to measure fairness and reaches themaximum when all users are replenished with the sameamount of energy, i.e., x1 = x2 = · · · = xn. Finally,in the proportional fairness [13], given the energy demandsd1, d2, . . . , dn of the n users, the value of minni=1{xi

di} is used

to measure fairness, and it achieves the maximum whenx1

d1= x2

d2= · · · = xn

dn. It can be seen that none of these

three mentioned fairness metrics can be used to measurethe charging satisfactions of smartphone users. The rationalebehind is as follows. On one hand, in both Jain’s and themax-min fairness metrics, they achieve their maximumsif each user is charged with the same amount of energy,however, they neglected an important fact, that is, differentusers may have different amounts of residual energy, and




energy-critical users are willing to be charged more energythan the others. On the other hand, the proportional fairnessignores that energy consumption rates of different usersmay be significantly different, and users with low energyconsumption rates will feel satisfied if only small amounts ofenergy are charged to their smartphones, while those userswho consume their energy very quickly will require to becharged with large amounts of energy. Instead, in this paperwe use a general submodular function to characterize thediminishing return property of user charging satisfaction,by incorporating the amounts of residual energy of users,the amounts of energy charged to users, and their energyconsumption rates. We will use this submodular function asthe fairness metric on user charging satisfactions.

3 PRELIMINARIES

In this section, we first present the system model, thenpropose a novel model for characterizing user chargingsatisfaction, and finally define charging satisfaction maxi-mization problems and show the NP-hardness.

3.1 System Model

We assume that the subway system in a metropolis con-sists of multiple subway trains and there are K chargersC1, C2, . . . , CK deployed at K different places on subwaytrains. We consider the charging scheduling of the K charg-ers for a given monitoring period (e.g., one day), and wedivide the period into T equal time slots with each time slotlasting δ units (e.g., one minute). We index the T time slotsby 1, 2, . . . , T . Assume that each charger Ci has a chargingcapacity ci, i.e., it can charge up to ci smartphones at thesame time, where ci ≥ 1 is a positive integer. We furtherassume that the output power of charger Ci is P oi (Watts).Denote by dijt the Euclidean distance between charger Ciand smartphone vj at time slot t, 1 ≤ i ≤ K, 1 ≤ j ≤ n and1 ≤ t ≤ T . Following the seminal work of Kurs et al. [14],the energy transfer efficiency µijt of charger Ci chargingsmartphone vj decreases with the increase of distance dijt.For example, Xie et al. [31] showed that

µijt = −0.0958d2ijt − 0.0377dijt + 1, (1)

where 0 ≤ µijt ≤ 1. The reception power Pijt of smart-phone vj from charger Ci at time slot t thus is

Pijt = µijt · P oi . (2)

To ensure that the reception power Pijt is large enoughto charge smartphone vj , we assume that charger Ci canreplenish energy to smartphone vj if dijt is no more thana maximum charging range D so that the energy transferefficiency µijt is no less than a threshold γ. For example,assume that µijt ≥ γ = 20%. The maximum charging rangethen is D = 2.7 m by Eq. (1).

Assume that there are nc users taking the subway for theperiod of T , in which n (≤ nc) of them are required to becharged at some time, and each user can be charged by onlyone charger at each time slot. Let V = {v1, v2, . . . , vn} bethe set of the n users. Denote by Emaxj the battery capacityof the smartphone of each user vj ∈ V . In the following,we use smartphone vj and user vj interchangeably. Assumeuser vj sends a charging request at time slot tSj in a train andwill take off the train at time slot tFj , clearly tSj < tFj . Then,

user vj can be charged within the time interval from tSj totFj . We further assume that the location of each user does notchange during every time slot, but it is allowed to changeat different time slots. We define the travel trajectory of uservj between time slots tSj and tFj as the set of locations of theuser on the train at every time slot in time interval [tSj , t

Fj ).

We also assume that each wireless charger is capableto measure the distance between the charger and a userwhen the user sends a charging request or his/her locationchanges at some time during his/her journey on the subway,through an indoor positioning technique, such as angle ofarrival (AoA), time of arrival (ToA), received signal strengthindication (RSSI), etc [18], [19]. In case that chargers cannotmeasure the distances or the measured distances are notaccurate enough, every charger can periodically charge s-martphone users nearby in a very short period and measurethe energy transfer efficiencies. Assume that the duration ofthe period is much shorter than the entire monitoring periodT , and thus can be ignored.

Denote by REj the residual energy of the smartphoneof user vj when the user sends a charging request with 1 ≤j ≤ n. Let binary variable xijt indicate whether chargerCi charges smartphone vj at time slot t, i.e., xijt = 1 ifsmartphone vj is charged by charger Ci at time slot t; xijt =0, otherwise, where 1 ≤ i ≤ K, 1 ≤ j ≤ n, and 1 ≤ t ≤ T .The amount of energy charged into the smartphone of uservj when he/she takes off the subway is

Bj = min{tFj −1∑t=tSj

K∑i=1

Pijt · δ · xijt, Emaxj −REj}, ∀vj ∈ V, (3)

where∑tFj −1t=tSj

∑Ki=1 Pijt · δ · xijt is the amount of effective

energy consumed by chargers for charging user vj , (Emaxj −REj) is the maximum possible amount of energy that canbe charged to user vj , Pijt is the reception power of uservj from charger Ci at time slot t, δ is the duration of eachtime slot, and Emaxj and REj are the energy capacity andresidual energy before charging of user vj , respectively.

3.2 User charging satisfaction

In this subsection we model the charging satisfaction of ev-ery user vj , by incorporating the amount of residual energyREj of user vj when the user sends a charging request,the amount of energy Bj charged to user vj , and his/heraverage energy consumption rate ρj . Before we proceed,we introduce non-decreasing submodular functions, whichusually are used to characterize the diminishing returnproperty [6].

Definition 1. LetE be a finite set and z be a real-valued func-tion with z : 2E 7→ R≥0, function z is a non-decreasingsubmodular function if and only if it has the followingthree properties [6]. (i) z(∅) = 0; (ii) Non-decreasement:z(S1) ≤ z(S2) for any two sets S1, S2 ⊆ E with S1 ⊆ S2;and (iii) Diminishing return property (submodularity):z(S1 ∪ {e}) − z(S1) ≥ z(S2 ∪ {e}) − z(S2) for any twosets S1 and S2 with S1 ⊆ S2 ⊂ E and e ∈ E \ S2.

Note that the average energy consumption rates of dif-ferent users may significantly vary, since there are varioustypes of smartphones, e.g., smartphones manufactured bydifferent companies, and the user behaviors of using their




smartphones are different, e.g., different frequencies of us-ing smartphones. Recall that Bj is the amount of energycharged to user vj . Then, the smartphone operational timeof user vj can be prolonged from REj

ρjto REj+Bj

ρjafter the

amount of energyBj has been charged into the smartphone.Since smartphone users are sensitive to the residual

operational time of their smartphones, we here use a non-decreasing submodular function g(lj) to characterize thelifetime satisfaction of a user vj for a residual operationaltime lj of his/her smartphone. For example, we haveconducted a questionnaire about the lifetime satisfactionfunction g(lj) and the feedback we received from 123 smart-phone users can be approximated by Eq. (4), where eachuser was asked to give a score (from 0 point to 10 points)for a given residual lifetime l of his/her smartphone and0 ≤ l ≤ 24, see Fig. 1.

g(l) = 3.2874 ∗ ln(l + 1)− 0.0341. (4)

0 2 4 6 8 10 12 14 16 18 20 22 24

residual lifetime l (hours)0

1

2

3

4

5

6

7

8

9

10

11

Lif

etim

e Sa

tisfa

ctio

n g(

l)

Experimental dataFitted curve function g(l) = 3.2874 * ln (l+1) - 0.0341

Fig. 1. The lifetime satisfaction function g(l) of the residual lifetime l of asmartphone user.

Given the residual energy REj of user vj and his/heraverage energy consumption ρj , we model the satisfactionfj(Bj) of user vj for charging an amount of energy Bj as

fj(Bj) = g(REj +Bj

ρj)− g(

REjρj

), (5)

where REj

ρjand REj+Bj

ρjare the residual lifetimes of user

vj before and after charging an amount of energy Bj ,respectively. Note that our model fj(Bj) of characterizingthe charging satisfaction has following important properties.

(i) A user vj is more satisfied if more energy is replen-ished to his/her smartphone.

(ii) If two users v1 and v2 have the same energy con-sumption rate (i.e., ρ1 = ρ2) but different amountsof residual energy RE1 and RE2 (assuming thatRE1 < RE2), then user v1 is more satisfied thanuser v2 if they both are charged with the sameamount of energy.

(iii) If two users v1 and v2 have the same residuallifetime (i.e., RE1

ρ1= RE2

ρ2) but different energy con-

sumption rates ρ1 and ρ2 (assuming that ρ1 < ρ2),then user v1 is more satisfied than user v2 if theyboth are charged with the same amount of energyB. The rationale behind is that the prolonged oper-ational time B

ρ1of user v1 is longer than that B

ρ2of

user v2, i.e., Bρ1 >Bρ2

.

3.3 Problem definitionsSince the number of to-be-charged smartphones usually islarger than the sum of the charging capacities of all chargerson trains, we consider scheduling the chargers to charge

energy-critical smartphones for a given monitoring periodconsisting of T time slots, such that the overall chargingsatisfaction of smartphone users is maximized, where wesay a smartphone is energy critical if its residual energy isbelow its defined energy threshold.

We distinguish our discussions into two different cases:offline charging scheduling and online charging scheduling.In the offline charging scheduling, the travel trajectory ofeach to-be-charged user vj from time slot tSj that user vjsends his/her charging request to time slot tFj that theuser takes off the subway is given, and such knowledgecan be obtained through tracking the train-taking historyof user vj or from the ticket information of user vj whenhe/she bought the ticket at a subway station. In the onlinecharging scheduling, we assume that the user travel trajectoryinformation is not available, due to personal security andprivacy concerns.

Given K chargers C1, C2, . . . , CK deployed on subwaytrains with charging capacities c1, c2, . . . , cK , respectively,n to-be-charged users v1, v2, . . . , vn with user vj sendinghis/her charging request at time slot tSj , and the traveltrajectories of these n users, the offline charging satisfactionmaximization problem is allocating the K chargers to chargethe n to-be-charged smartphones for a given monitoringperiod T , so that the accumulative charging satisfaction ofall smartphone users is maximized, i.e., our objective is to

maximize

n∑j=1

fj(Bj), (6)

subject to constraints (2), (3), (5), and following constraints.n∑j=1

xijt ≤ ci, 1 ≤ i ≤ K, 1 ≤ t ≤ T, (7)

K∑i=1

xijt ≤ 1, 1 ≤ j ≤ n, 1 ≤ t ≤ T, (8)

xijt ∈ {0, 1}, 1 ≤ i ≤ K, 1 ≤ j ≤ n, 1 ≤ t ≤ T, (9)xijt = 0, if t < tSj , t ≥ tFj , or dijt > D, (10)

where constraint (7) ensures that each charger Ci cancharge no more than ci users at each time slot t, constrain-t (8) ensures that each user vj can be charged by no morethan one charger at each time slot t, constraint (9) ensuresthat each user vj is either charged by a charger Ci at timeslot t or not, and constraint (10) ensures that each user vjwill not be charged by any charger Ci when the user havenot sent a charging request (i.e., t < tSj ) or have taken off(i.e., t ≥ tFj ), or the distance dijt between them is longerthan the maximum charging range D (i.e., dijt > D).

The online charging satisfaction maximization problem canbe similarly defined as follows. The problem is to allocatethe K chargers to charge the n to-be-charged smartphonesfor a given period T without the knowledge of the traveltrajectory of each user in future, so that the sum of chargingsatisfaction of all smartphone users is maximized.

The travel trajectory information of each user can beused to significantly improve the user charging satisfaction,as users with plenty of charging opportunities can be distin-guished from those with only a few charging opportunities.For example, assume that there are only two lifetime-criticalusers v1 and v2 within the charging range of a charger Ciand the residual lifetimes of v1 and v2 at some time slot




t are 20 minutes and 10 minutes, respectively. We furtherassume that user v1 will take off the train very soon (e.g.,five minutes later) while user v2 will take a longer trip on thetrain. For this scenario, a charging allocation algorithm Awithout the user trajectory information may allocate chargerCi to charge only user v2 since the residual lifetime of userv2 is less than user v1. As a result, user v1 will miss his/heronly charging opportunity while user v2 will be chargedinto a large amount of energy and the residual lifetime ofuser v2 is prolonged from 10 minutes to, for example, 4hours. Then, the sum of charging satisfactions of users v1and v2 when they take off the trains by algorithm A is0 + 3.2874 ∗ ln(4 + 1) − 3.2874 ∗ ln(10/60 + 1) = 4.8 byEq. (4) and Eq. (5). Contrarily, another algorithm B withthe trajectory information may allocate charger Ci to chargeuser v1 before he/she takes off the train and then assigncharger Ci to charger user v2 after user v1 has taken off thetrain. As a result, the residual lifetimes of users v1 and v2when they take off the trains are prolonged to, for example,20 + 30 = 50 minutes and 4− 30

60 = 3.5 hours, respectively.Then, the sum of charging satisfaction of the two users byalgorithm B is 3.2874 ∗ (ln(50/60 + 1) − ln(20/60 + 1) +ln(3.5 + 1)− ln(10/60 + 1)) = 5.9 > 4.8.

For both the offline and online charging satisfactionmaximization problems, we assume that there is a serveron each subway train and each charger can communicatewith the server. Then the server has the global knowledgeof user energy information and the distances between usersand chargers. The server thus can execute a scheduling algo-rithm to find charging scheduling to chargers, the chargersthen perform the charging. However, sometimes the servermay not be installed at each subway train or chargers cannotcommunicate with the server directly. It thus is desirable todevise a scheduling algorithm that operates distributively.We thus define the distributed charging satisfaction maximiza-tion problem as to assign the K chargers to charge users for aperiod of T so that the accumulative charging satisfactionof users is maximized, under the constraint that everycharger has only the knowledge of energy information anddistances of the users within its maximum charging rangeand chargers cannot communicate with each other.

3.4 NP-hardnessTheorem 1. The offline charging satisfaction maximization

problem is NP-hard.

Proof: We show the NP-hardness of the problem byreducing the decision version of the bin packing problemto a special case of the problem of concern. Given k binswith each having a capacity S and a list of n items withsizes a1, a2, . . . , an, respectively, the decision version of thebin packing problem is to decide whether there is a way topack the n items into k bins such that the total size of itemspacked into each bin is no more than the bin capacity S [27].

Given k bins with each having capacity S and n itemswith sizes a1, a2, . . . , an, we construct an instance of the of-fline charging satisfaction maximization problem as follows.

There is only K = 1 wireless charger with a chargingcapacity c = 1 and k to-be-charged users v1, v2, . . . , vk onthe subway, and the energy consumption rate ρj of eachuser vj is one, i.e., ρj = 1. Also, the maximum amountEmaxj −REj of energy that can be charged to each user vj is

S, i.e., Emaxj −REj = S, 1 ≤ j ≤ k. Furthermore, there areT = n time slots in the monitoring period and the amount ofenergy that can be charged to each user vj at time slot t is at,1 ≤ t ≤ T . In addition, we assume that the charging satis-faction function fj(Bj) is a linear function of the amount ofcharged energy Bj , which is a special submodular function,i.e., g(lj) = lj and fj(Bj) = g(

REj+Bj

ρj) − g(

REj

ρj) = Bj

as ρj = 1. We can see that the offline charging satisfactionmaximization problem in this special case is to allocate thecharger to charge the k to-be-charged users for a periodof T = n time slots so that the accumulative amount ofenergy charged to the k users is maximized, subject tothat the total amount of energy charged to each user vjis no more than Emaxj − REj = S. Given the maximumamount of energy OPT of the offline charging satisfactionmaximization problem in this special case, it can be seenthat there is a solution to pack the n items to k bins suchthat the total weight of the items packed into each bin isno more than the bin capacity S if OPT =

∑nt=1 at; and

there is not such a solution if OPT <∑nt=1 at. Since the

bin packing problem is NP-hard [27], the offline chargingsatisfaction maximization problem is NP-hard, too.

4 ALGORITHM FOR THE OFFLINE CHARGING SAT-ISFACTION MAXIMIZATION PROBLEM

In this section, we propose a novel 13 -approximation al-

gorithm for the offline charging satisfaction maximizationproblem. We also analyze the approximation ratio and timecomplexity of the proposed algorithm.

4.1 AlgorithmThe basic idea behind the algorithm is as follows. It proceedsthe charging allocation iteratively. Within each iteration, apair (Ct

∗

i∗ , vt∗

j∗) with the maximum increased satisfactionamong all possible pairs is chosen, where charger Ci∗ isallocated to charge user vj∗ at time slot t∗. In the following,we elaborate the approximation algorithm.

Given K chargers C1, C2, . . . , CK with charging capac-ities c1, c2, . . . , cK , respectively, the n to-be-charged usersv1, v2, . . . , vn with user vj sending a charging request attime slot tSj and taking off the subway at time slot tFj . Recallthat the residual energy of user vj is REj and its energyconsumption rate is ρj . Also, the travel trajectory of the useris given. The algorithm proceeds as follows.

Let C = {C1, C2, . . . , CK} and V = {v1, v2, . . . , vn}.We first construct a bipartite graph Gt = (Ct, Vt, Et) foreach time slot t, where Ct is the set of chargers at timeslot t (i.e., Ct = C), Vt is the set of users that havecharging opportunities on the subway at time slot t (i.e.,Vt = {vj | vj ∈ V, tSj ≤ t < tFj }), where t = 1, 2, . . . , T .For each charger Cti ∈ Ct and each user vtj ∈ Vt, there is anedge (Cti , v

tj) in Et if their Euclidean distance dijt at time

slot t is no more than the maximum charging range D ofcharger Cti , i.e., dijt ≤ D. Then, the reception power Pijt ofcharger Cti charging user vtj at time slot t is calculated byEq. (2), and the amount of energy Bijt charged to user vj isBijt = Pijt · δ, where δ is the duration of every time slot.

We then allocate the K chargers to charge the n usersfor the given period T iteratively. Let rej be the amountof residual energy of user vj after allocating some chargersto charge the user. Also, let cti be the maximum residual




number of users that charger Cti can charge at time slott. Initially, rej = REj , where REj is the residual energyof user vj before any charging, 1 ≤ j ≤ n, cti = ci,1 ≤ i ≤ K, and 1 ≤ t ≤ T . At each iteration, for eachedge (Cti , v

tj) ∈ Gt, recall that Bijt is the amount of energy

that can be charged to user vtj if charger Cti charges the userat time slot t, where 1 ≤ t ≤ T . The amount of increasedsatisfaction of user vj then is

∆(Bijt) = g(rej +Bijt

ρj)− g(

rejρj

), (11)

by Eq. (5). We identify an edge (Ct∗

i∗ , vt∗

j∗) from the T graphsG1, G2, . . . , GT such that the increased satisfaction of charg-ing some user vtj by a charger Ctt at time slot t is maximized,i.e., (Ct

∗

i∗ , vt∗

j∗) = arg max(Cti ,v

tj)∈G1∪G2∪···∪GT

{∆(Bijt)}.We then allocate charger Ci∗ to charge user vj∗ at timeslot t∗. We also increase the residual amount of energy rej∗of user vj∗ by Bi∗j∗t∗ and reduce the maximum residualnumber of users ct

∗

i∗ that charger Ct∗

i∗ can charge at time slott∗ by one. Furthermore, we remove the incident edges ofuser node vt

∗

j∗ from graph Gt∗ since user vj∗ can be chargedby no more than one charger at time slot t∗, and remove theincident edges of charger node Ct

∗

i∗ from graphGt∗ if ct∗

i∗ hasbeen decreased to zero as the number of users allocated tocharger Ct

∗

i∗ at time slot t∗ now reaches its charging capacityci. The approximation algorithm continues until no edgesare left in any of the T graphs G1, G2, . . . , GT . We detail thealgorithm in Algorithm 1.

4.2 Algorithm analysisWe now analyze the approximation ratio of the proposedAlgorithm 1. We start by introducing the definition ofmatroids as follows [6].

A matroid M is a pair of (E,F) that meets three prop-erties, where E is a finite set and F is a family of subsetsof E, i.e., F ⊆ 2E . (i) ∅ ∈ F ; (ii) the hereditary property:S1 ⊆ S2 and S2 ∈ F imply that S1 ∈ F for any two subsetsS1, S2 ⊆ E; (iii) the independent set exchange property: forany two sets S1, S2 ∈ F , if |S1| < |S2|, then, there is anelement e ∈ S2 \ S1 such that S1 ∪ {e} ∈ F .

We have the following important lemma, which is thecornerstone of the approximation ratio analysis.Lemma 1. [6] Let E be a finite set and F be a non-empty

collection of subsets of E which has the property thatS1 ⊆ S2 ⊆ E and S2 ∈ F imply that S1 ∈ F . Givena non-decreasing submodular function z : 2E 7→ R+, agreedy heuristic always delivers a 1

k+1 -approximate so-lution to the problem maxS⊆E{z(S) : S ∈ F}, assumingthat (E,F) is described by the intersection of k matroids,where k is a positive integer.

We then show that the objective function∑nj=1 fj(Bj) is

a submodular function by the following lemma.Lemma 2. Function

∑nj=1 fj(Bj) is a non-decreasing sub-

modular function, where fj(Bj) = g(REj+Bj

ρj)−g(

REj

ρj),

g(·) is a given non-decreasing submodular function,REjis the residual energy of user vj before charging, ρj is theaverage energy consumption rate, and Bj is the amountof energy charged to user vj for the period of T .

Proof: We only need to prove that fj(Bj) is a non-decreasing submodular function for each j with 1 ≤ j ≤ n.

Algorithm 1 ApproAlgInput: K deployed chargers C1, C2, . . . , CK with charging

capacities c1, c2, . . . , cK , n to-be-charged smartphoneusers, the residual energy, energy consumption rate, andtravel trajectory of each user, and a given period T .

Output: A charging allocation A that assigns the chargersto the users for the period T such that the sum ofcharging satisfaction of the users is maximized.

1: Construct a bipartite graph Gt = (Ct, Vt, Et) for eachtime slot t, where Ct is the set of the K chargers, Vt isthe set of the users at time slot t, there is an edge (Cti , v

tj)

in Et if the distance between charger Cti and user vtj attime slot t is no more than D with 1 ≤ t ≤ T ;

2: For each edge (Cti , vtj) in the T graphs, compute the

amount of energy Bijt that can be charged to user vtj ifallocating charger Cti to charge the user at time slot t byEq. (2), where 1 ≤ i ≤ K, 1 ≤ j ≤ n, and 1 ≤ t ≤ T ;

3: A ← ∅; /* the charging allocation */4: rej ← REj , 1 ≤ j ≤ n; /* residual energy of user vj */5: cti ← ci, 1 ≤ i ≤ K, 1 ≤ t ≤ T ;6: while there is an edge in any of the T graphsG1, G2, . . . , GT do

7: For each edge (Cti , vtj) ∈ G1∪G2∪ · · · ∪GT , compute

the increased satisfaction ∆(Bijt) of user vj Eq. (11);8: Find an edge (Ct

∗

i∗ , vt∗

j∗) such that (Ct∗

i∗ , vt∗

j∗) =arg max(Ct

i ,vtj)∈G1∪G2∪···∪GT

{∆(Bijt)};9: A ← A∪ {(Ci∗ , vj∗ , t∗)};

10: rej∗ ← rej∗ +Bi∗j∗t∗ ,11: ct

∗

i∗ ← ct∗

i∗ − 1;12: Remove the incident edges of vt

∗

j∗ from graph Gt∗ ;13: Remove the incident edges of charger node Ct

∗

i∗ fromgraph Gt∗ if ct

∗

i∗ decreases to zero;14: end while15: return A.

Then,∑nj=1 fj(Bj) is a non-decreasing submodular func-

tion, as it is a non-negative, linear combination of submod-ular functions.

To this end, we show that function fj(Bj) meets thethree properties of submodular functions (see Definition 1in Subsection 3.2). Let E = E1∪E2∪· · ·∪ET . We first showthat (i) fj(∅) = 0. In this case, we can see that the amountof energy charged to user vj is zero, i.e., Bj = 0. Then,fj(Bj) = fj(0) = 0. We then show that function fj(Bj)meets property (ii) of submodular functions, i.e., fj(S1) ≤fj(S2) for any two charging allocations S1, S2 ⊆ E withS1 ⊆ S2. We can see that the amount of energy charged touser vj by charging allocation S1 is no more than that byallocation S2, i.e., fj(S1) ≤ fj(S2). We finally prove thatfunction fj(Bj) satisfies property (iii) of submodular func-tions, i.e., fj(S1 ∪ {e})− fj(S1) ≥ fj(S2 ∪ {e})− fj(S2) forany two charging allocations S1 and S2 with S1 ⊆ S2 ⊂ E

and e = (Cti , vtj) ∈ E \ S2. Denote by BS1

j , BS1∪{e}j , BS2

j ,and B

S2∪{e}j the amounts of energy charged to user vj by

charging allocations S1, S1 ∪ {e}, S2, S2 ∪ {e}, respectively.Since S1 ⊆ S2, we have BS1

j ≤ BS2j . Then, we know that

g(REj+B

S1j

ρj) ≤ g(

REj+BS2j

ρj) as g(.) is a non-decreasing func-

tion. It can be seen that the increased replenished energyBS1∪{e}j − BS1

j by charging user vj with charger Ci at timeslot t from charging allocations S1 to S1∪{e} is no less thanthat BS2∪{e}

j −BS2j from charging allocations S2 to S2∪{e},




i.e., BS1∪{e}j −BS1

j ≥ BS2∪{e}j −BS2

j , since BS1j ≤ B

S2j and

the amount of energy charged to user vj in any chargingallocation is no more than Emaxj − REj by Eq. (3), wheree = (Cti , v

tj), Emaxj and REj are the energy capacity and

residual energy before charging of user vj , respectively. Insummary, we have

fj(S1 ∪ {e})− fj(S1) = g(REj +B

S1∪{e}j

ρj)− g(

REj +BS1j

ρj)

= g(REj +BS1

j

ρj+BS1∪{e}j −BS1

j

ρj)− g(

REj +BS1j

ρj)

≥ g(REj +BS1

j


j

ρj)− g(

REj +BS1j

ρj) (12)

≥ g(REj +BS2

j


j

ρj)− g(

REj +BS2j

ρj) (13)

= g(REj +B

S2∪{e}j

ρj)− g(

REj +BS2j

ρj) = fj(S2 ∪ {e})− fj(S2),

where Ineq. (12) holds since BS1∪{e}j − BS1

j ≥ BS2∪{e}j −

BS2j and g(.) is a non-decreasing function, and Ineq. (13)

holds asREj+B

S1j

ρj≤ REj+B

S2j

ρjand g(.) is a submodular

function. Therefore,∑nj=1 fj(Bj) is a non-decreasing sub-

modular function. The lemma then follows.We finally analyze the approximation ratio of

Algorithm 1 by the following theorem.Theorem 2. There is a 1

3 -approximation algorithm for the of-fline charging satisfaction maximization problem, whichtakesO(nT 2 log(nT )+KT ) time, where n is the numberof to-be-charged users, K is the number of deployedchargers, and T is the number of time slots in a givenmonitoring period.

Proof: Since the objective function∑nj=1 fj(Bj) of

the problem is a non-decreasing submodular function byLemma 2, in the following, we show that constraints (7)and (8) can be represented by k = 2 matroids. Then, follow-ing Lemma 1, it can be seen that Algorithm 1 delivers a1k+1 = 1

3 -approximate solution.Recall that there is a bipartite graph Gt = (Ct, Vt, Et) for

each time slot twith 1 ≤ t ≤ T . LetX = C1∪C2∪· · ·∪CT andY = V1 ∪ V2 ∪ · · · ∪ VT . Recall that E = E1 ∪E2 ∪ · · · ∪ET .Note that E is the set of the feasible charging allocationsdefined by constraints (9) and (10). We define a set systemMX = (E,FX) on the edge set E, where FX is a family ofsubsets of E such that, for each edge set S ∈ FX (S ⊆ E),the number of edges in S sharing the same endpoint Cti isno more than ci for each charger node Cti ∈ X , and ci isthe maximum number of users that charger Ci can chargeat time slot t. Similarly, we define another set systemMY =(E,FY ), where FY is a family of subsets of E such that, foreach edge set S ∈ FY (S ⊆ E), no two edges in S have thesame endpoint in Y . Following the definitions of set systemsMX andMY , it can be seen that constraints (7) and (8) arerepresented byMX andMY , respectively. In the followingwe only show that MX is a matroid by meeting the threeproperties of matroids. The claim thatMY is a matroid canbe shown similarly, omitted.

(1)MX = (E,FX) meets property (i) of a matroid that∅ ∈ FX . (2) Given any two sets S1, S2 ⊆ E, assume thatS1 ⊆ S2 and S2 ∈ FX . Following the definition of FX and

the fact S2 ∈ FX , it can be seen that the number of edgesin S2 sharing the same endpoint Cti is no more than ci,for each charger node Cti ∈ X . The number of edges in S1

sharing the same endpoint Cti then is no more than ci due toS1 ⊆ S2. Thus, S1 ⊆ S2 and S2 ∈ FX imply that S1 ∈ FX ,meeting property (ii) of a matroid. (3) Given any two setsS1, S2 ∈ FX , assume that |S1| < |S2|. Denote by NS(Cti )the set of edges in S sharing endpoint Cti ∈ X for any setS ∈ FX . We note that {NS(Cti )}Ct

i∈X is a partitioning of setS, since

⋃Ct

i∈XNS(Cti ) = S and NS(Cti ) ∩ NS(Ct

′

j ) = ∅for Cti 6= Ct

′

j , due to that each of the T graphs is abipartite graph and no edge between nodes Cti and Ct

′

j . AsS1, S2 ∈ FX and |S1| < |S2|, there must be a node Cti ∈ Xsuch that the number of edges in S1 sharing endpoint Cti isstrictly less than that in S2, i.e., |NS1

(Cti )| < |NS2(Cti )|. Oth-

erwise, |NS1(Cti )| ≥ |NS2

(Cti )| for each node Cti ∈ X . Then,|S1| =

∑Ct

i∈X|NS1

(Cti )| ≥∑Ct

i∈X|NS2

(Cti )| = |S2|,which contradicts the assumption that |S1| < |S2|. Since|NS1

(Cti )| < |NS2(Cti )| ≤ ci, there is an edge (Cti , v

tj) in

NS2(Cti )\NS1

(Cti ). We then add this edge to S1. It is obviousthat |NS1

(Cti ) ∪ {(Cti , vtj)}| ≤ ci. Also, note that addingedge (Cti , v

tj) in S1 does not increase the number of edges

that share the same endpoint Ct′

j for each node Ct′

j ∈ X

with Ct′

j 6= Cti , as the endpoint vtj of edge (Cti , vtj) does not

belong to set X . Therefore, set S1∪{(Cti , vtj} ∈ FX , meetingproperty (iii) of a matroid. Therefore, MX = (E,FX) is amatroid. It can be seen that the offline charging satisfac-tion maximization problem can be cast as a non-decreasingsubmodular function maximization problem, subject to theconstraints of k = 2 matroids:MX andMy , Algorithm 1thus delivers a 1

k+1 = 13 -approximate solution by Lemma 1.

We finally analyze the time complexity of Algorithm 1.We assume that the number of chargers on a subway thatcan charge each user at each time slot is bounded by aconstant, since chargers usually are not densely deployed.Then, the number of edges in the T graphs G1, G2, . . . , GTis∑Tt=1O(|Vt|) ·O(1) =

∑Tt=1O(|V |) = O(nT ) and we can

construct the T graphs in time∑Tt=1(K + |Vt|) + O(nT ) =

O((K + n)T ). The time complexity of Algorithm 1 de-pends on the data structure we adopt in its implementa-tion. To quickly find the charging allocation A, we hereadopt the priority queue – a max-heap [4] in the whileloop of Algorithm 1. Before the while loop, we associateeach edge (Cti , v

tj) in the T graphs G1, G2, . . . , GT with

a key ∆ijt, which is the increased overall satisfaction byallocating charger Ci to charge user vj at time slot t, i.e.,∆ijt = g(

REj+Bijt

ρj) − g(

REj

ρj) by Eq. (5). Following [4],

we can build a max-heap H in time O(nT ) and theheap includes all edges in the T graphs G1, G2, . . . , GT .For each edge (Cti , v

tj) in heap H , there is an associated

boolean variable bijt for it, which indicates whether theedge has been removed from the heap or not. Initially,bijt = ‘false′. Within each iteration of the while loop,we can find the edge (Ct

∗

i∗ , vt∗

j∗) such that (Ct∗

i∗ , vt∗

j∗) =arg max(Ct

i ,vtj)∈G1∪G2∪···∪GT

{∆(Bijt)} in time O(log(nT )).If variable bi∗j∗t∗ indicates that edge (Ct

∗

i∗ , vt∗

j∗) has alreadybeen removed (i.e., bi∗j∗t∗ = ‘true′) or the number of usersallocated to charger Ct

∗

i∗ in the charging allocation A at timeslot t∗ has already reached to its charging capacity ci∗ , we




simply remove the edge from heap H . Otherwise, we addan allocation (Ci∗ , vj∗ , t

∗) to A. Then, we can remove theincident edges of user node vt

∗

j∗ in heap H in time O(1) byassigning the boolean variables bijts of these edges ‘true′.Note that after we have allocated charger Ci∗ to charge uservj∗ at time slot t∗, the increased overall satisfaction ∆ij∗t

by allocating a charger Ci to charge user vj∗ at time slott ∈ {1, 2, . . . , T} \ {t∗} decreases, since the amount of resid-ual energy of user vj∗ has been increased by Bi∗j∗t∗ due tothe allocation (Ci∗ , vj∗ , t

∗). Assume that rej∗ is the residualenergy of user vj∗ after the allocation. Following Eq. (5), thevalue of ∆ij∗t is updated by ∆ij∗t = g(

rej∗+Bij∗tρj∗

)−g(rej∗

ρj∗)

for each edge (Cti , vtj∗), where 1 ≤ i ≤ K and t ∈

{1, 2, . . . , T} \ {t∗}. Therefore, we have O(T ) such updates.Since the keys ∆ij∗ts of O(T ) edges (Cti , v

tj∗)s in heap

H have been decreased, we can maintain the max-heapproperty of H in time O(T log(nT )). Therefore, the timecomplexity of Algorithm 1 is O((K + n)T ) + O(nT ) +O(nT ) ∗O(T log(nT )) = O(nT 2 log(nT ) +KT ).

5 ALGORITHM FOR THE ONLINE CHARGING SATIS-FACTION MAXIMIZATION PROBLEM

In the previous section, we proposed an approximationalgorithm for the offline charging satisfaction maximizationproblem, assuming that the travel trajectory of each useris given. However, such knowledge sometimes may not beavailable due to personal security and privacy concerns.In this section, we study the online charging satisfactionmaximization problem without the knowledge of user tra-jectories, by developing a heuristic algorithm for it.

5.1 Online algorithm

The basic idea behind the algorithm is that it finds a charg-ing allocation with only the residual energy informationof to-be-charged users provided so that the sum of thecharging satisfaction of users at each time slot is maximized.To this end, we reduce the problem to the maximum weightmatching problem, and an exact solution to the latter in turnreturns a feasible solution to the former.

The online algorithm is invoked at the beginning ofevery time slot for the entire period T and a chargingallocation At will be delivered by the algorithm at everytime slot t with 1 ≤ t ≤ T . As a result, the union of chargingallocations A1,A2, . . . ,AT forms a feasible solution to theproblem. Specifically, assume that the charging allocationsA1,A2, . . . ,At−1 for the previous t−1 time slots have beenobtained, we now find a charging allocation At at time slott to maximize the sum of charging satisfaction of the users.Recall that Vt is the set of users that have charging oppor-tunities at time slot t, i.e., Vt = {vj | vj ∈ V, tSj ≤ t < tFj }.Denote by retj the amount of residual energy of user vjat the beginning of time slot t. We consider the chargingsatisfaction maximization problem at time slot t, that is, howto allocate the K chargers to charge the users in Vt, so thatthe sum of charging satisfaction of the users at this time slotis maximized.

We first construct a bipartite graph Gt = (C, Vt, Et;wt),where C is the set of the K chargers, there is an edge (Ci, vj)in edge set Et if the Euclidean distance between charger Ciand user vj is no more than the maximum charging range

D. For each edge (Ci, vj) ∈ Et, its weight wt(Ci, vj) is thenet satisfaction by allocating charger Ci to charge user vjat time slot t, i.e., wt(Ci, vj) = g(

retj+Bijt

ρj) − g(

retjρj

), whereretjρj

andretj+Bijt

ρjare the residual lifetimes of user vj before

and after charging vj at time slot t, respectively, retj is theresidual energy of the user at the beginning of time slot t,Bijt is the amount of energy that can be charged to the user,and ρj is its energy consumption rate.

We then construct another bipartite graph G′t =(C′, Vt, E′t;w′t) from Gt = (C, Vt, Et;wt) as follows. Foreach charger Ci in C, there are ci ‘virtual charger’ n-odes Ci,1, Ci,2, . . . , Ci,ci in C′, which have the same lo-cation as charger Ci in graph Gt. Thus, the chargingcapacity of each ‘virtual charger’ is exactly one. Also,for each edge (Ci, vj) in graph Gt, there are ci edges(Ci,1, vj), (Ci,2, vj), . . . , (Ci,ci , vj) in edge set E′t, and eachof these ci edges has the same weight as the original edge(Ci, vj), i.e., w′t(Ci,k, vj) = wt(Ci, vj) with 1 ≤ k ≤ ci.

Consider the maximum weight matching problem inG′t = (C′, Vt, E′t;w′t), which aims to find a matching M suchthat the weighted sum of edges in M is maximized. Givena maximum weight matching M in G′t, a charging alloca-tion At for the online charging satisfaction maximizationproblem can then be derived, by adding (Ci, vj , t) to At foreach matched edge (Ci,k, vj) in matching M . The detailedalgorithm is given in Algorithm 2.

Algorithm 2 OnlineAlgInput: K deployed chargers C1, C2, . . . , CK with charging ca-

pacities c1, c2, . . . , cK , nt to-be-charged users v1, v2, . . . , vnt

with their amounts of residual energy retj and energyconsumption rates ρj at time slot t

Output: A charging allocation At assigning the chargers tocharge users at time slot t so that the sum of user chargingsatisfactions is maximized

1: Construct a bipartite graph Gt = (C, Vt, Et;wt), where C ={C1, C2, . . . , CK}, Vt = {v1, v2, . . . , vnt}, there is an edge

(Ci, vj) in Et if dijt ≤ D and wt(Ci, vj) = g(retj+Bijt

ρj) −

g(retjρj

) for each edge (Ci, vj) ∈ Et;2: Construct another graph G′t = (C′, Vt, E′t;w′t) from graphGt = (C, Vt, Et;wt), where there are ci virtual charger n-odesCi,1, Ci,2, . . . , Ci,ci in C′ for chargerCi in C, there are ciedges (Ci,1, vj), (Ci,2, vj), . . . , (Ci,ci , vj) in E′t for each edge(Ci, vj) ∈ Et, w′t(Ci,k, vj) = wt(Ci, vj), and 1 ≤ k ≤ ci;

3: Find a maximum weight matching M in graph G′t;4: At ← ∅; /* the charging allocation for time slot t*/5: For each matched edge (Ci,k, vj) ∈M , add (Ci, vj , t) to At;6: return At.

5.2 Algorithm analysis

We now analyze the time complexity of Algorithm 2through the following theorem.

Theorem 3. There is an algorithm for the online chargingsatisfaction maximization problem, which takes O((n +K)2 log(n + K)) time for charging scheduling at eachtime slot t with 1 ≤ t ≤ T , where n is the number ofto-be-charged users and K is the number of chargers.

Proof: We first show that Algorithm 2 delivers afeasible solution At. Note that there are ci virtual chargernodes Ci,1, Ci,2, . . . , Ci,ci in graph G′t for each charger Ci




in graph Gt and the sets of user nodes in graphs G′t and Gtare the same. Since the charging allocationAt is constructedfrom the maximum weight matching M in graph G′t, weknow that each user in allocation At is assigned to only onecharger and the number of users assigned to each chargerCi in At is no more than its charging capacity ci. Therefore,Algorithm 2 delivers a feasible solution At at each timeslot t with 1 ≤ t ≤ T .

We then analyze the time complexity of Algorithm 2,which is dominated by finding the matching M in graphG′t. Let nt = |Vt| + |C′| and mt = |E′t| be the number ofnodes and edges in graph G′t, respectively. Algorithm 2can find the matching M in time O(mtnt + n2t log nt) =O(n2t log nt) = O((n+Kcmax)2 log(n+Kcmax)) = O((n+K)2 log(n+K)) by applying an algorithm in [7] and notic-ing that O(mt) = O(nt) and nt = O(n + Kcmax), wherecmax = maxKi=1{ci} = O(1).

6 ALGORITHM FOR THE DISTRIBUTED CHARGINGSATISFACTION MAXIMIZATION PROBLEM

So far we assumed that there is a server on each subwaytrain and each charger can communicate with the server.As a result, the server can execute a scheduling algorithmto find a solution of charging allocations to the chargers.However, there may not be such a server on the board orthe chargers cannot communicate with the server. It thusis desirable to devise a scheduling algorithm that operatesin a distributed way. In this section, we propose a noveldistributed algorithm for the problem.

6.1 Distributed algorithm

The distributed algorithm finds a charging allocation At atthe beginning of each time slot t with 1 ≤ t ≤ T . Recallthat Vt is the set of users that have charging opportunitiesat time slot t, i.e., Vt = {vj | vj ∈ V, tSj ≤ t < tFj }. Denoteby retj the amount of residual energy of user vj ∈ Vt at thebeginning of time slot t. Also, let N(Ci) be the set of userswithin the maximum charging range D of each charger Ci,i.e., N(Ci) = {vj | vj ∈ Vt, dijt ≤ D}, and N(vj) be the setof chargers that can charge user vj , i.e., N(vj) = {Ci | Ci ∈C, dijt ≤ D}.

At the beginning of every time slot t, the distributed al-gorithm proceeds the charging allocation iteratively. Withineach iteration, it finds charging allocation for a subset ofusers in Vt. Every charger Ci first calculates the net satis-faction ∆(Bijt) of every user vj ∈ N(Ci) if it is allocatedto charge user vj at time slot t by Eq. (11). Charger Ci thenchooses the top k = min{ci, |N(Ci)|} users v1, v2, . . . , vkin N(Ci) by their net satisfactions, and sends each of thema ‘ChargingPermission’ message, where ci is its chargingcapacity and |N(Ci)| is the number of users within itsmaximum charging range. Every user vj ∈ Vt may or maynot receive ‘ChargingPermission’ messages from chargers,which is distinguished into three cases: (i) user vj doesnot receive any ‘ChargingPermission’ message, no actionis needed; (ii) user vj receives only one message from acharger Ci ∈ N(vj), the user sends a ‘ChargingAcknowl-edgement’ message to charger Ci; and (iii) user vj receivesmultiple messages from the chargers in N(vj), the user thensends a ‘ChargingAcknowledgement’ message to a charger

Ci with the maximum net satisfaction among the chargers,and sends a ‘ChargingRejection’ message to each of thechargers in N(vj) \ {Ci}, as user vj should be chargedby no more than one charger at time slot t. Assume thatevery chargerCi receives kack ‘ChargingAcknowledgement’messages and krej ‘ChargingRejection’ messages from theusers in N(Ci). Charger Ci then decreases its chargingcapacity by kack and removes the users who sent messagesto it from N(Ci), since these users have already been chosento be charged by some chargers in this iteration and willnot be considered in the next iterations. The distributedalgorithm continues until the charging capacity of eachcharger Ci decreases to zero or there are no users left withinits maximum charging range, i.e., N(Ci) = ∅. The detaileddistributed algorithms at chargers and users are given inAlgorithm 3 and Algorithm 4, respectively.

Algorithm 3 DistributedAlg (each charger Ci atevery time slot t)

1: V tc ← ∅; /* the set of to-be-charged users at time slot t*/2: Calculate the net satisfaction ∆(Bijt) of every user vj ∈N(Ci) by Eq. (11);

3: cti ← ci; /*the residual charging capacity of chargerCi*/4: while cti > 0 and N(Ci) 6= ∅ do5: Choose the top k = min{cti, |N(Ci)|} users

v1, v2, . . . , vk in N(Ci) by their net satisfactions;6: Send each of the k users a ‘ChargingPermission’ mes-

sage;7: for each user vj of the k users do8: if charger Ci receives a ‘ChargingAcknowledge-

ment’ message from user vj then9: V tc ← V tc ∪ {vj}; /* user vj will be charged */

10: cti ← cti − 1; /* decrease its charging capacity */11: end if12: end for13: Remove the users who sent their messages to charger

Ci from N(Ci);14: end while15: Perform energy charging to users in V tc .

Algorithm 4 DistributedAlg (each user vj atevery time slot t)

1: Assume that user vj receives p ‘ChargingPermission’messages from p chargers C1, C2, . . . , Cp, respectively,where Ci ∈ N(vj);

2: Select the charger Ci with the maximum net satisfactionamong the p chargers and reply it with a ‘ChargingAc-knowledgement’ message;

3: Reply each of the chargers in N(vj) \ {Ci} with a‘ChargingRejection’ message.

6.2 Algorithm analysis

Theorem 4. There is an algorithm for the distributedcharging satisfaction maximization problem, it takesO(ni logK) time and O(K logK) messages for chargingscheduling at each time slot t with 1 ≤ t ≤ T , where niis the number of users within the maximum chargingrange of charger Ci and K is the number of chargers.

Proof: We first analyze the time complexity of thedistributed algorithm. Notice that the charging capacity ci




of each charger Ci is bounded by a constant in the real life.Then, the execution of each while loop in Algorithm 3takes O(ni + ci) = O(ni) time, assuming that the da-ta structure of the maximum heap is adopted [4], whereni = |N(Ci)|. We calculate how many while loops thatAlgorithm 3 will perform as follows. Denote by nmax themaximum number of chargers that can charge a user, i.e.,nmax = maxvj∈Vt

{|N(vj)|}. Then, a user vj will receiveno more than nmax ‘ChargingPermission’ messages fromchargers and will reply exact one ‘ChargingAcknowledge-ment‘ message to one of the chargers within a while loop ifuser vj receives some ‘ChargingPermission’ messages. Sincethe K chargers on subway will send

∑Ki=1 min{cti, N(Ci)}

‘ChargingPermission’ messages to users, there will be noless than

∑Ki=1 min{cti,N(Ci)}

nmax‘ChargingAcknowledgemen-

t‘ messages received from the chosen users. As a re-sult, there are no more than

∑Ki=1 min{cti, N(Ci)} · (1 −

1nmax

) to-be-allocated users left after every while loop.If nmax = 1, Algorithm 3 then performs only onewhile loop. Otherwise (nmax > 1), Algorithm 3 per-forms the while loop O(log(1− 1

nmax)

1∑Ki=1 min{ci,N(Ci)}

) =

O(log nmaxnmax−1

∑Ki=1 ci) = O(logK) times, where

∑Ki=1 ci =

O(K) and the value of nmax is a constant in the real life,since it is unlikely that there are many chargers denselydeployed at a location. Therefore, the time complexity of thedistributed algorithm is O(ni) ·O(logK) = O(ni logK).

We then analyze the message complexity of thedistributed algorithm. Within every while loop ofAlgorithm 3, the K chargers and the users in Vt willsendO(

∑Ki=1 min{ci, N(Ci)}) = O(K) messages. Thus, the

message complexity of the algorithm is O(K) · O(logK) =O(K logK) at every time slot.

7 PERFORMANCE EVALUATION

In this section, we evaluate the performance of the proposedalgorithms, using a real dataset.

7.1 Experimental settingsWe consider the subway network in San Francisco in theStates, which consists of 6 subway lines and 45 stations [1].The running timetable of the subway trains is obtainedfrom [1], which includes the arrival times of the stations ofeach train. For simplicity, we assume that the length, width,and height of each train are 100 meters, 3.2 meters, and 3.2meters, respectively, and there are 300 seats along the twosides of each train [2]. We divide one day into equal lengthtime slots with each time slot lasting δ = 1 minute. Weassume that the maximum charging range of each wirelesscharger is 2.7 meters [14]. We also assume that the chargingcapacity ci of each charger Ci is 1 and the output power P oiof charger Ci is 10 Watts. We deploy 41 wireless chargersalong the two sides of each train, where Fig. 2 illustratessuch a deployment in a two-dimensional space and theheight of each deployed charger is 0.4 meters (at the positionbelow seats on the train).

We adopt a real subway-taking dataset of San Franciscoin a weekday of November 2014, which specifies the numberof users between each pair of stations [23]. As a result, thetotal number of users nc in a day is 424,763. Also, we gener-ate the boarding time on trains for each user by referring to

5 m

wireless charger

100 m

3.2 m

Fig. 2. The deployment of wireless chargers on each train.

the 2008 station profile study of San Francisco subway [23].In addition, we assume that a user will randomly take avacant seat if there are vacant seats available in the train.Otherwise, the user will randomly stand on the train.

The battery capacity Emaxj of each user vj ’s smartphoneis randomly chosen from 20 kJ (≈ 3.7 V · 1, 500 mAh) to40 kJ (≈ 3.7 V ·3, 000mAh). Also, the energy consumptionrate ρj of user vj is randomly drawn from an interval[0.5 W, 1 W ]. As a result, the lifetime of a fully chargedsmartphone can last from 5.5 (≈ 20 kJ

1 W ·3600 s ) hours to 22(≈ 40 kJ

0.5 W ·3600 s ) hours. Furthermore, a fraction number αof users request to be charged (i.e., the number of to-be-charged users is α · nc), 0 ≤ α ≤ 1. The residual energyREj of each to-be-charged user vj when issuing a chargingrequest is randomly chosen from an interval [0, β · Emaxj ],where 0 ≤ β ≤ 1.

In addition to the three proposed algorithms ApproAlg,OnlineAlg, and DistriAlg, we also implement anoth-er charging allocation algorithm maxThroughput, whichfinds charging allocations such that the sum of amounts ofenergy charged to users is maximized. Similar to algorithmOnlineAlg, algorithm maxThroughput finds a chargingallocation at each time slot t (i.e., in an online way). Unlikealgorithm OnlineAlg that the weight of each edge (Ci, vj)is the net satisfaction by allocating charger Ci to charge uservj at time slot t, the weight of edge (Ci, vj) in algorithmmaxThroughput is the amount of energy charged to user vj .

7.2 Performance evaluation of different algorithms

In the following we evaluate the performance ofthe three proposed algorithms ApproAlg, OnlineAlg,and DistriAlg against the benchmark algorithmmaxThroughput, assuming half users on trains require tobe charged, i.e., α = 0.5. Fig. 3 (a) plots the residual lifetimedistribution of to-be-charged users when they send chargingrequests, from which it can be seen that most users haveshort residual lifetimes. For example, more than 50% of to-be-charged users have residual lifetime less than a half hour.

Fig. 3 (b) demonstrates the residual lifetime distribu-tion of users when they take off trains, from which itcan be seen that the percentages of users with residuallifetime less than a half hour drop from 50% (see Fig. 3(a)) to 16%, 6.3%, 6.3%, and 5% in the charging allocation-s delivered by algorithms maxThroughput, DistriAlg,OnlineAlg, and ApproAlg, respectively. As a result, al-gorithm maxThroughput can identify only a proportionof 68% (= 50−16

50 ) lifetime-critical users, while algorithmsDistriAlg, OnlineAlg, and ApproAlg can identify mostenergy-critical users, which are as high as 87.4% (= 50−6.3

50 ),87.4%, and 90% (= 50−5

50 ), respectively. The rationale behindis that algorithm maxThroughput finds charging alloca-tions only by the amounts of energy charged to users, itthus fails to identify energy-critical users. Unlike algorithmmaxThroughput, algorithms DistriAlg, OnlineAlg and




0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

residual lifetime (hours)0

5

10

15

20

25

30

35

40

45

50

55

perc

ent (

%)

residual lifetime before boarding

(a) residual lifetime distribution of to-be-charged users when boarding the trains.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8residual lifetime (hours)

02468

101214161820222426

perc

ent (

%)

maxThroughputDistriAlgOnlineAlgApproAlg

(b) residual lifetime distribution of to-be-charged users when taking off the trains.

maxThroughput DistriAlg OnlineAlg ApproAlg 1.8

1.85

1.9

1.95

2

satis

fact

ion

per

user

(c) the satisfaction per user delivered bydifferent algorithms

Fig. 3. Performance of algorithms maxThroughput, DistriAlg, OnlineAlg, and ApproAlg when α = 0.5 and β = 0.1.

0.4 0.5 0.6 0.7 0.8 0.9 1fraction α of to-be-charged users

0 0

10000 10000

20000 20000

30000 30000

40000 40000

50000 50000

60000 60000

70000 70000

80000 80000

90000 90000

1e+05 1e+05

num

ber

of u

sers


(a) the number of users with residual lifetimeless than a half hour when taking off the trains.

0.4 0.5 0.6 0.7 0.8 0.9 1fraction α of to-be-charged users

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

satis

fact

ion

per

user


(b) satisfaction per user delivered by the fouralgorithms

Fig. 4. Performance of algorithms maxThroughput, DistriAlg, OnlineAlg, and ApproAlg by varying α from 0.4 to 1 when β is fixed at 0.1.

ApproAlg are able to identify energy-critical users, since thenet satisfaction gain by charging these users are significantlylarger than that by charging those with long residual life-times. On the other hand, the number of users with energycritical residual lifetimes by algorithm ApproAlg is lessthan that by algorithms DistriAlg and OnlineAlg, sincethe latter two algorithms find charging allocations withoutthe knowledge of the travel trajectory of each user as the oneby algorithm ApproAlg. Thus, algorithm ApproAlg candistinguish users with many charging opportunities fromthe users with only a few charging opportunities. Fig. 3 (b)also shows that the number of users with residual lifetimesbetween 1 hour and 3 hours delivered by the three algo-rithms DistriAlg, OnlineAlg, and ApproAlg are muchlarger than that by algorithm maxThroughput. In contrast,the numbers of users with residual lifetimes longer than 3hours by the three algorithms are slightly less than that byalgorithm maxThroughput, since the net satisfaction gainby charging the users with residual lifetime longer than 3hours is marginal in the three algorithms.

Fig. 3 (c) plots the charging satisfaction performancedelivered by the four mentioned algorithms, from which itcan be seen that the charging satisfaction per user deliveredby algorithms DistriAlg, OnlineAlg and ApproAlg arearound 6%, 6%, and 6.7% higher than that by algorithmmaxThroughput, and the satisfaction delivered by algo-rithm ApproAlg is the highest one among them. Also, wecan see from Fig. 3 (c) that the performances of algorithm-s DistriAlg and OnlineAlg are almost identical. Therationale behind is that there are only a limited numberchargers that can charge a smartphone user on a subwaytrain. Then, the solutions found by the centralized algorithm

(i.e., algorithm OnlineAlg) and the distributed algorithm(i.e., algorithm DistriAlg) are close to each other.

7.3 The impact of the number of to-be-charged users

We then study the impact of the number of to-be-chargedusers on algorithm performance, by varying α from 0.4to 1. Fig. 4 (a) plots the number of users with residuallifetimes less than a half hour when they take off trainsby algorithms maxThroughput, DistriAlg, OnlineAlg,and ApproAlg, respectively, from which it can be seen thatthe numbers of users in the charging allocations deliveredby algorithms DistriAlg, OnlineAlg, and ApproAlgare much less than that by algorithm maxThroughput.Furthermore, the number by algorithm ApproAlg alwaysis the smallest one, which is only 88% of numbers of usersby algorithm OnlineAlg and 41.5% of numbers of usersby algorithm maxThroughput when all users request to becharged (i.e., α = 1). The reason why algorithm ApproAlgoutperforms algorithm OnlineAlg is that the former hasthe knowledge of user trajectories and more energy-criticalusers can be charged on time.

Fig. 4 (b) shows that the satisfaction per user by al-gorithms maxThroughput, DistriAlg, OnlineAlg, andApproAlg decreases with the increase of α, since everyto-be-charged user will have less charging opportunities ifthere are more to-be-charged users on subway trains (i.e.,a larger α). Also, the satisfaction by algorithm ApproAlgalways is the highest one. Note that although the satisfactionper user delivered by algorithm ApproAlg is only slightlyhigher than that by algorithm OnlineAlg (about 0.65%),the charging allocations found by algorithm ApproAlg aremuch better than that by algorithm OnlineAlg, since there




0.05 0.1 0.15 0.2max fraction β of residual energy

0 0

10000 10000

20000 20000

30000 30000

40000 40000

50000 50000

60000 60000

70000 70000

num

ber

of u

sers


(a) the number of users with residual lifetimeless than a half hour when taking off the trains.

0.05 0.1 0.15 0.2max fraction β of residual energy

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

satis

fact

ion

per

user


(b) satisfaction per user delivered by the fouralgorithms

Fig. 5. Performance of algorithms maxThroughput, DistriAlg, OnlineAlg, and ApproAlg by varying β from 0.05 to 0.2 when α = 0.5.

are much less numbers of users with energy critical residuallifetime by algorithm ApproAlg when users take off trains,which has already been shown in Fig. 4 (a).

7.4 The impact of residual energy before charging

We finally investigate the impact of residual energy of to-be-charged users when they take on trains, by varying themaximum fraction β of residual energy of users from 0.05to 0.2, where the residual energy REj of user vj beforecharging is randomly chosen from an interval [0, β · Emaxj ]and Emaxj is the battery capacity of the smartphone of theuser. Fig. 5 (a) plots the number of users with residuallifetimes less than a half hour when they take off trains, fromwhich it can be seen that the number of users by each of thefour mentioned algorithms decreases with the increase of β,since there are more amounts of energy in the smartphonesof these users before charging with a larger β. Also, thenumbers of users by algorithms DistriAlg, OnlineAlg,and ApproAlg decrease to less than 5,000 while the numberof users by algorithm maxThroughput is still more than20,000 when β = 0.2. Again, the number of users byalgorithm ApproAlg is the smallest one, which is aroundfrom 73.8% to 82.5% of that by algorithm OnlineAlg andeven is about from 15.8% to 45.8% of that by algorithmmaxThroughput.

Fig. 5 (b) implies that the satisfaction per user by each ofthe four algorithms decreases with the increase of the valueof β. The rationale behind is that a user is less satisfiedfor charging an amount of energy if the user has moreenergy in his/her smartphone before charging. Fig. 5 (b) alsoshows the satisfactions per user by algorithm ApproAlgand OnlineAlg are around 6.4% and 7.1% higher than thatby algorithm maxThroughput, respectively.

8 CONCLUSION

In this paper, we considered the use of wireless energychargers installed on subway trains to charge energy-criticalsmartphones of users, through wireless energy transferwhile users take subway trains to work or go home. We firstformulated a novel optimization problem that scheduleswireless chargers to charge energy-critical smartphones fora given monitoring period, such that the overall chargingsatisfaction of smartphone users is maximized. We thendevised a non-trivial 1

3 -approximation algorithm for theproblem, assuming that the travel trajectory of each to-be-charged smartphone user is given. We also proposed

an online algorithm to deal with dynamic energy-criticalsmartphone charging requests. Furthermore, we developeda distributed algorithm for the problem when the globalknowledge of user energy information is not given. Wefinally evaluated the performance of the proposed algo-rithms, using a real dataset. Experimental results showedthat the proposed algorithms are very promising, and ashigh as 87.4%, 87.4%, and 90% of energy-critical users can becharged on time in the solutions delivered by the proposeddistributed, online, and approximation algorithms.

ACKNOWLEDGMENTS

We really appreciate the three anonymous referees andthe associate editor for their expertise comments and con-structive suggestions, which have helped us improve thequality and presentation of the paper greatly. We also wouldlike to thank the volunteers who participated in the ques-tionnaire. Furthermore, it is acknowledged that the workby Wenzheng Xu was partially supported by 2016 BasicResearch Talent Foundation of Sichuan University (GrantNo. 2082204194050).

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[37] W. Xu, W. Liang, X. Ren, and X. Lin, “On-demand energyreplenishment for sensor networks via wireless energy transfer,”in Proc. IEEE 25th Annu. Int. Symp. Personal Indoor Mobile RadioCommun. (PIMRC), 2014, pp. 1269–1273.

[38] H. Xu and B. Li, “Reducing electricity demand charge for datacenters with partial execution,” in Proc. 5th Int. Conf. Future EnergySyst., 2014, pp. 51–61.

[39] Y. Zheng, B. Ji, N. B. Shroff, and P. Sinha, “Forget the deadline:Scheduling interactive applications in data centers,” in Proc. IEEE8th Int. Conf. Cloud Comput., 2015, pp. 293–300.

[40] Z. Zheng and N. B. Shroff, “Online multi-resource allocation fordeadline sensitive jobs with partial values in the cloud,” in Proc.IEEE Int. Conf. Comput. Commun. (INFOCOM), 2016, pp. 901–909.

[41] M. Zhu, X. Liu, L. Kong, R. Shen, W. Shu, and M. Wu, “Thecharging-scheduling problem for electric vehicle networks,” inProc. IEEE Wireless Commun. Netw. Conf., 2014, pp. 3178–3183.

Wenzheng Xu (M’15) received the BSc, ME,and PhD degrees in computer science from SunYat-Sen University, Guangzhou, P.R. China, in2008, 2010, and 2015, respectively. He current-ly is a Special Associate Professor at SichuanUniversity and was a visitor at the Australian Na-tional University. His research interests includewireless ad hoc and sensor networks, mobilecomputing, approximation algorithms, combina-torial optimization, online social networks, andgraph theory.

Weifa Liang (M’99–SM’01) received the PhDdegree from the Australian National Universityin 1998, the ME degree from the University ofScience and Technology of China in 1989, andthe BSc degree from Wuhan University, Chinain 1984, all in computer science. He is current-ly a full professor in the Research School ofComputer Science at the Australian National U-niversity. His research interests include designand analysis of energy-efficient routing protocolsfor wireless ad hoc and sensor networks, cloud

computing, software-defined networking, online social networks, designand analysis of parallel and distributed algorithms, approximation al-gorithms, combinatorial optimization, and graph theory. He is a seniormember of the IEEE.

Jian Peng is a Professor at College of Comput-er Science, Sichuan University. He received hisB.A. degree and PhD degree from the Universityof Electronic Science and Technology of China(UESTC) in 1992 and 2004, respectively. His re-cent research interests include wireless sensornetworks, big data, and cloud computing.

Yiguang Liu received the M.S. degree fromPeking University in 1998, and the Ph.D. de-gree from Sichuan University in 2004. He was aResearch Fellow, Visiting Professor, and SeniorResearch Scholar with the National University ofSingapore, Imperial College London, and Michi-gan State University, respectively. He was cho-sen into the program for new century excellenttalents of MOE in 2008, and chosen as a Scien-tific and Technical Leader in Sichuan Province in2010. He is currently the Director of the Vision

and Image Processing Laboratory, a Professor at Sichuan University,and a Reviewer for the Mathematical Reviews of American mathematicalSociety. He has co-authored over 100 international journal and con-ference papers, and a chapter of the book entitled it ComputationalIntelligence and Its Applications (ed H. K. Lam). His research interestsinclude computer vision and image processing, pattern recognition, andcomputational intelligence.

Yan Wang received her Bachelor degree, Mas-ter degree and Doctoral degree from SichuanUniversity, Chengdu, China, in 2009, 2012 and2015 respectively. She now is an Assistant Pro-fessor at Sichuan University and was a visitingscholar at University of North Carolina at ChapelHill, NC, USA, from 2014 to 2015. Her mainresearch interests focus on the inverse problemsin imaging science, including image denoising,segmentation, inpainting, and medical image re-construction, etc.

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