Fair Energy-Efficient Sensing Task Allocation in Participatory Sensing with Smartphones

Transcription

1 Advance Access publication on 24 February 2017 The British Coputer Society All rights reserved. For Perissions, please eail: doi: /cojnl/bxx015 Fair Energy-Efficient Sensing Task Allocation in Participatory Sensing with Sartphones JIA PENG 1,2,YANMIN ZHU 1,2*,QINGWEN ZHAO 3,HONGZI ZHU 4,JIAN CAO 4, GUANGTAO XUE 4 AND BO LI 5 1 Departent of Coputer Science and Engineering, Shanghai Jiao Tong University, Shanghai, China 2 Shanghai Key Lab of Scalable Coputing and Systes, Shanghai, China 3 ebay, Shanghai, China 4 Shanghai Jiao Tong University, Shanghai, China 5 Hong Kong University of Science and Technology, Hong Kong, China * Corresponding author: yzhu@sjtu.edu.cn With the proliferation of sartphones, participatory sensing using sartphones provides unprecedented opportunities for collecting enorous sensing data. There are two crucial requireents in participatory sensing, fair task allocation and energy efficiency, which are particularly challenging given high cobinatorial coplexity, trade-off between energy efficiency and fairness, and dynaic and unpredictable task arrivals. In this paper, we present a novel fair energy-efficient allocation fraework whose objective is characterized by in ax aggregate sensing tie. We rigorously prove that optiizing the in ax aggregate sensing tie is NP hard even when the tasks are assued as a priori. We consider two allocation odels: offline allocation and online allocation. For the offline allocation odel, we design an efficient approxiation algorith with the approxiation ratio of 2-1, where is the nuber of eber sartphones in the syste. For the online allocation odel, we propose two algoriths: greedy algorith and Robin-Hood algorith, which achieve the copetitive ratio of at ost and + 1, respectively. The results deonstrate that the approxiation algorith reduces over 81% total sensing tie, the online greedy algorith and Robin-Hood algoriths reduce the total sensing tie 73% and 37.5%, respectively. The offline approxiation algorith and online greedy algorith achieve uch better in ax fairness copared to other algoriths. Keywords: participatory sensing; task allocation; fairness; energy efficiency Received 14 July 2016; revised 1 Deceber 2016; editorial decision 30 January 2017 Handling editor: Ing-Ray Chen 1. INTRODUCTION With the proliferation of obile devices, participatory sensing with sartphones becoes a new and iportant paradig for collecting and sharing data with the general public. A lot of collaborative, crowdsourcing-based applications spring up. Exaple applications of participatory sensing include intelligent transportation [2, 4, 11], localization [12, 19, 21], environental onitoring [3, 13] and crowding counting [7]. The typical architecture of a participatory sensing syste is illustrated in Fig. 1. The syste is coprised of a central platfor and a collection of sartphones. The platfor residing on the cloud accepts sensing service requests fro syste users, and allocates sensing tasks to the eber sartphones. After being assigned a sensing task, a sartphone perfors the required sensing service and returns the sensing data to the platfor which forwards the data to the querying user. The sartphones in the participatory sensing syste are assued cooperative (not strategic), which belong to or affiliated to the syste, willing to take sensing tasks and provide sensing services to the syste. We call such sartphones eber sartphones. Such participatory sensing systes are practical and realistic in enterprise or agreeent-based cooperation scenarios. The issue of participation incentive [9, 18] of rational or even strategic sartphone user is out of the scope of the paper.

2 FAIR ENERGY-EFFICIENT SENSING TASK ALLOCATION IN PARTICIPATORY SENSING WITH SMARTPHONES 851 In this paper, we focus on energy efficiency of eber sartphones of a participatory sensing syste. More specifically, we study the task allocation strategy of the platfor for optiizing the energy efficiency of eber sartphones. Processing a sensing task typically requires a sartphone to drive the processor for sapling and processing the data. It consues considerable energy on the sartphone, which is dependent on the required sensing tie length of a sensing task. Recent easureent study [25] has shown that the processor consues up to 25% and an energy hungry sensor like GPS can consue up to 15% of the total energy. As a result, it causes large energy cost as a sartphone contributes to the participatory sensing syste. We ake the iportant observation that it can greatly save energy consuption by reusing the sensing service on the sartphone, which is to allocate overlapping tasks requiring the sae sensing service to the sae sartphone. As illustrated by the exaple in Fig. 2, there are two sensing tasks: Task 1 requests a sensing service fro 10 a to 12 a, and Task 2 requests the sensing service fro 11 a to 13 p. Suppose there are two sartphones A and B. We exaine two strategies. In the first strategy, each sartphone is allocated one task. In this case, the total sensing tie of both A FIGURE 1. The architecture of a participatory sensing syste. FIGURE 2. Two different allocation strategies considered. The left figure considers the case two overlapped tasks are allocated to two different sartphones, and the total sensing tie is 4 hours, while the right figure considers that two overlapped tasks are allocated to one sartphone, and thus the total sensing tie 3 hours. and B is 4 hours. In the second strategy, both tasks are allocated to B. The resulting total sensing tie is 3 hours. In conclusion, overlapping the sensing intervals of different tasks can reuse the sensing service on the sartphone and hence energy consuption can be reduced. Nevertheless, we also find that although the total energy is reduced, the issue of unfairness arises. It is clear that A spends no energy on sensing service while B spends 3 hours. The previous observation otivates us to investigate the crucial proble of allocating sensing tasks for axiizing energy efficiency while aintaining good fairness aong eber sartphones. However, several great challenges reain to be solved. First, there is an intrinsic trade-off between total energy efficiency and fairness aong sartphones. It is highly desirable to strike a good balance between overall energy efficiency and fairness in ters of individual energy consuption. Second, both the nuber of sensing tasks and the nuber of sartphones can be large. The tie coplexity would be high if a straightforward exhaustive search is applied. We rigorously prove that the proble of task allocation for optial in ax energy efficiency is NP hard. Finally, in the real-world sensing tasks ay arrive to the syste at anytie and the arrival process of sensing tasks can be arbitrary and unknown beforehand. Little existing work has jointly considered the energy efficiency issue as well as the fairness issue in allocating sensing tasks in participatory sensing systes. In [6], the authors consider task assignent in a crowdsourcing arket such as Aazon Mechanical Turk. The proble is to atch heterogeneous tasks to workers with different, unknown skill sets. The objective is to axiize the total benefits of the requester who subitted tasks. In [10], the proble of selecting a service provider fro a list of providers is considered, with the objective of axiizing the consuer s satisfaction. There is little existing work for task allocation, which is applicable to the proble of axiizing overall energy efficiency and fairness aong sartphones. The unique characteristics of participatory sensing, such as trade-off and utilization of overlapping intervals, have never been explored. In response to the challenges entioned above, we introduce a fair energy-efficient allocation fraework whose objective is characterized by in ax aggregate sensing tie. Based on the fraework, we propose two sensing task allocation algoriths in participatory sensing systes for different allocation odels: offline allocation and online allocation. For the offline allocation odel, at the tie of scheduling, the platfor has the coplete knowledge of all sensing tasks, including the future tasks to be allocated. We design an efficient approxiation algorith with the approxiation ratio of 2-1,where isthenuberofebersartphones in the syste. For the online allocation odel, sensing tasks arrive to the syste and the allocation decision is ade on the fly. We propose a greedy algorith with a polynoial tie coplexity, which achieves a copetitive ratio of at

3 852 J. PENG et al. ost, and another online allocation algorith called Robin-hood, which achieves a copetitive ratio of + 1. The ain technical contributions ade in this paper are suarized as follows: To the best of our knowledge, it is the first work that focuses on both energy efficiency and fairness in allocating sensing tasks to the sartphones of a participatory sensing syste. We introduce a novel fair energy-efficient allocation fraework whose objective is characterized by in ax aggregate sensing tie, and rigorously prove that the proble of optiizing in ax aggregate sensing tie is NP hard even when tasks are known a priori. For the offline allocation odel, we design an efficient approxiation algorith with the approxiation ratio of 2-1, where is the nuber of eber sartphones in the syste. For the online allocation odel, we propose Greedy algorith with a copetitive ratio at ost, and Robin-Hood algorith with a copetitive ratio + 1. We have perfored both theoretical analysis and siulations, and the results deonstrate that copared with the baseline algorith, the approxiation algorith reduces over 81% total sensing tie, the greedy online algorith and Robin-Hood algoriths reduce the total sensing tie 73% and 37.5%, respectively, and our algoriths achieve uch better in ax fairness. The rest of this paper is organized as follows. In Section 2, we present the syste odel and define the proble. Section 3 describes the design of the approxiation algorith for the offline task allocation odel and Section 4 describes the design of the online algorith for the online task allocation odel. Section 5 presents and discusses evaluation results. Section 6 reviews related work. In Section 7, we conclude the paper and present the future work directions. 2. MODEL AND PROBLEM FORMULATION In this section, we first present an epirical study to enhance the otivation of this paper, that is, energy efficiency. Then, we describe syste odel and proble forulation. At last, we give the coplexity analysis of the proble. sae type than running the independently because the sensing service is reused. We obtain the conclusion by intuition. In the following, we perfor experients to deonstrate that a sartphone consues less power when even running two sensing tasks of different types siultaneously than running the independently. Note, however, our work assues that sensing tasks are of the sae type. Allocating sensing tasks of different types is beyond the scope of this paper and is subject to future research Experiental setup We use Power Monitor (i.e. Fig. 3) to evaluate the realtie power consuption of a sartphone and the software PowerTool [1] (i.e. Fig. 4) to show the evaluated data. The sartphone is Sasung galaxy I9100. The voltage of the sartphone is set as 3.8. We can observe the sartphone s current or power to evaluate the energy consuption. Here, we choose the current as the etric. We consider three kinds FIGURE 3. Power onitor Epirical study As entioned in Section 1, we ake an observation that overlapping tasks of the sae type can reduce energy consuptions in participatory sensing systes. Fro another point of view, we can say that a sartphone consues less power when running two overlapped sensing tasks of the FIGURE 4. PowerTool software.

4 FAIR ENERGY-EFFICIENT SENSING TASK ALLOCATION IN PARTICIPATORY SENSING WITH SMARTPHONES 853 of sensing tasks: GPS tasks, caera tasks and audio tasks. A GPS task is to update the location inforation every second, a caera task is to record a video, and a audio task is to record a sound. We evaluate the power consuption of two kinds of tasks when they run siultaneously or independently on a sartphone. One set of experient is GPS tasks and caera tasks, the other is GPS tasks and audio tasks Experient result We first study GPS tasks and caera tasks. The result is given in Fig. 5. During 0-2 inutes, GPS tasks and caera tasks are perfored siultaneously. During 2-4 inutes only caera tasks are perfored. During 5 and 7 inutes, only GPS tasks run on the sartphone. The last lowest segent is the energy consuption when the sartphone does nothing. Note that the transient and very high current between two segents is incurred by huan operations, which can be ignored. We can see that the current is 675 A when GPS tasks and caera tasks are perfored siultaneously, 325 A when only GPS tasks are perfored, and 525 A when only caera tasks are perfored. The basic current is 175 A when the sartphone is idle. Next, we present the results for GPS tasks and audio tasks in Fig. 6. During 1-3 inutes, GPS tasks and audio tasks run on the sartphone siultaneously. During 3-5 and 6-8 inutes, the GPS task and the audio task are perfored, respectively. Siilarly, the transient and very high current can be ignored. Fro the result, we can see that the current is 375 A when two tasks are running siultaneously, 325 A for GPS tasks, and 225 A for the audio tasks. The current is still 175 A when the sartphone is in the idle state. Observation: We can see that a participatory sensing syste reduces energy consuption when overlapping two tasks on one sartphone in coparison with the case that running two tasks independently on two sartphones Syste odel We consider sensing task allocation proble in a participatory sensing syste with cooperative eber sartphones. The platfor is located on the cloud, accepting dynaically arriving tasks. The platfor is responsible for allocating sensing tasks to the sartphones. There are eber sartphones in the syste. A sartphone perfors the task by sapling the required sensing data. On the copletion of a sensing task, the sartphone returns the sensing data back to the platfor which then forwards the data to the user who subitted the sensing task. In this paper, we consider hoogeneous sensing tasks which require the sae sensing service fro sartphones. Discussions on heterogenous tasks will be left for future studies. The sensing tasks are of the sae type. The only FIGURE 5. The GPS task and caera task. During 0 2 inutes, the GPS task and the caera task are perfored siultaneously. During 2 4 inutes only caera task is perfored. Then, during 5 and 7 inutes, only GPS task runs on the sartphone. FIGURE 6. The GPS task and audio tasks. Fro 1 to 3 inutes, the GPS task and audio task run on the sartphone siultaneously. During 3 5 and 6 8 inutes, the GPS task and audio task are perfored, respectively. difference aong tasks is that the tie starting or ending sensing data is different. During a certain tie period, the participatory sensing syste calls for one kind of sensing service to accoplish its functionality. For exaple, it recruits sartphones to collect air quality data in a sall city. One task ay be collecting air quality data fro 10:00 a to 11:00 a. Notations are as follows. Each task r i is associated with a sensing interval Ii =[ si, ei ], indicating that the sensing service starts fro s i and ends at e i. The sensing tie of task r i is ei - si. For ease of presentation, we consider the tie as discrete slots of equal size (1 unit tie). Note that a task can only be allocated to one sartphone and is indivisible.

5 854 J. PENG et al. Before forally defining the proble of the sensing task allocation, we first introduce soe notations siilar to [22]. DEFINITION 2.1 (Cover of sensing tasks). For two tasks r 1, r 2 with sensing intervals I1 =[ s1, e1] and I2 =[ s2, e2] and s1 s2, if e1 ³ e2, we call r 2 is covered by r 1. DEFINITION 2.2 (Union of sensing intervals). Define I I the union of two sensing intervals or interval sets. For exaple, [ 2, 5] [ 3, 6] = [ 2, 6 ], {[ 2, 4 ], [ 5, 7]} [ 3, 6] = {[ 2, 7 ]}. 1 2 as DEFINITION 2.3 (Length of interval). Let l(i) denote the length of interval I or the length of the union of intervals in set I. For exaple, l ([ 1, 3] [ 2, 5]) = l ([ 1, 5]) = 4, l ([ 1, 3] [ 4, 5]) = 3 and l ([ 1, 4]) = 3. DEFINITION 2.4 (Aggregate sensing tie). The aggregate sensing tie l i of a sartphone i is the overall sensing tie that i should spend on copleting the allocated tasks. Given tasks allocated to i is { r1, r2, ¼, rn } with intervals { I1, I2, ¼, In }, then l = l( I) i i n = 1 i Proble forulation We next forally define the task allocation proble whose objective is to optiize energy efficiency and axiize fairness. Since the two objectives are contradictory, we introduce a novel fair energy-efficient allocation fraework whose objective is characterized by in ax aggregate sensing tie. By achieving the in ax aggregate sensing tie objective, we can jointly take fairness axiization and energy efficiency into consideration. DEFINITION 2.5 (Task allocation proble with the objective of in ax aggregate sensing tie). Consider tie period [ 0, T ] where T is a sufficiently large future tie point of interest. During [ 0, T ], the set of sensing tasks that arrive to the syste is denoted by ={ r1, r2, ¼, rn }, with corresponding intervals ={ I1, I2, ¼, In }. The syste has eber sartphones. The proble is to find a sensing task allocation, such that the axiu aggregate sensing tie of sartphones is iniized, i.e. in ax l. ( 1) 1 i i We consider two task allocation odels, i.e. offline allocation and online allocation. DEFINITION 2.6 (Offline task allocation odel). In this odel, at the tie of scheduling the platfor has the coplete knowledge of all sensing tasks, i.e. both and. REMARKS. The offline task allocation odel has liited applications in reality. We consider this odel for studying the proble coplexity and as a baseline for coparison with the online algorith to be proposed. DEFINITION 2.7 (Online task allocation odel). In this odel, the platfor akes the allocation decision once the task arrives to the syste. The platfor has no access to the knowledge of future tasks and their corresponding intervals. REMARKS. This odel is practical and applicable to realworld participatory sensing systes Analysis of NP hardness Optiizing the objective of in ax aggregate sensing tie in a participatory sensing syste is coputationally difficult. In this subsection, we rigorously prove that the task allocation with the objective of optiizing the in ax aggregate sensing tie is NP-hard even under the offline odel. THEOREM 2.1. The task allocation proble with the objective of iniizing the axiu aggregate sensing tie of all sartphones under the offline allocation odel is NP-hard. Proof. We prove the NP hardness by reducing fro a classical NP proble of job scheduling [24] to our task allocation proble. Siilar to our proble, each job in the classical NPC proble has a processing tie. We will show that an instance of the job scheduling proble can be reduced to our task allocation proble. In this case, the processing tie of any two jobs will not be overlapped. For each job, there is a sensing task whose sensing tie is equal to the processing tie of the job. The job scheduling proble can be described as follows: a sequence of jobs need to be scheduled on identical parallel achines. Each job has a processing tie. The goal is to find a optial schedule which iniizes the akespan, which is the total processing tie of all jobs scheduled on the ost loaded achine. This reduction takes an instance of the job scheduling proble as input. Given a set of jobs and identical parallel achines, each job j i has a processing tie d i. Order these jobs arbitrarily, such as ={ j1, j2, ¼, jn }, we construct an instance of the sensing task allocation proble as follows: for each job j i, there is a sensing task with interval Ii =[ ti, ti + di], di > 0. At the sae tie, ti + di < t i + 1, i Î{ 1, 2, ¼, n - 1 }. The reduction can be copleted in polynoial tie. An exaple is shown in Fig. 7. A job schedule with the akespan iniized can be translated into a task allocation with the axiu aggregate sensing tie iniized. As there is no overlap between any two

6 FAIR ENERGY-EFFICIENT SENSING TASK ALLOCATION IN PARTICIPATORY SENSING WITH SMARTPHONES 855 FIGURE 7. Reduction fro job scheduling to task allocation. sensing intervals in this instance of task allocation proble, the sensing tie of a task r i is exactly equal to the processing tie d i of job j i. Thus, the akespan of an optial scheduling is just the axiu aggregate sensing tie of sartphones. 3. APPROXIMATE OFFLINE TASK ALLOCATION In this section, we consider the offline allocation odel. First, we present the overview of the offline task allocation algorith, and next give the algorith details and finally present soe theoretical analysis Overview As previously proved, the task allocation proble with the objective of in ax aggregate sensing tie is NP hard and thus there is no coputationally efficient algorith for deriving the optial solution. In this section, we design a polynoialtie approxiation algorith for the offline task allocation odel under which the platfor has the coplete knowledge of tasks, including the sensing interval I i of each task ri Î R. There are two steps in the design of the algorith. In the first step, we construct a task precedence graph G ( V È { v0}, E) with V = n. Being a directed graph, a task precedence graph is used to characterize the tiing and intercover relations between sensing tasks. In the second step, we search for paths which start fro v 0. These paths visit all the nodes in the graph collectively. Each node in V in the graph can only be included in a path exactly once. We call the set of such paths -path. With this graph, we are able to convert the original proble of iniizing the axiu aggregate sensing tie to a new proble of finding paths on a directed graph which is easier to handle. CLAIM 1. The optial solution to the task allocation proble with the objective of in ax aggregate sensing tie reains the sae after any task which is covered by another task is reoved fro consideration. REMARKS. Suppose that task r i is covered by r j and r i is reoved. The optial solution with the new set of tasks without r i reains optial after r i is added to the syste because it can be allocated to the sae sartphone which r j is allocated to. It is clear that after r i is added back, the optial aggregate sensing tie does not change. Therefore, we can reove those tasks which are covered by other tasks fro consideration and the platfor only allocate the new set of tasks. After the allocation is done, the reoved tasks then are allocated into the corresponding sartphones. Next, we show how to convert a set of tasks in which no task is covered by another task, into a task precedence graph. Forally, we construct a directed graph G ( VÈ { v 0 }). Each node vi Î V represents a sensing task r i and is attached with an interval [ si, ei ], where s i is the start tie and e i is the end tie. v 0 is an added virtual node with interval [0, 0]. There exists a directed edge ( vi vj) Î E if and only if si < sj. The edge weight w ij is the additional sensing tie needed to coplete task r j after copleting task r i, which can be calculated as w ij = ì í ï l( Ii Ij) - l( Ii), i ¹ j îï 0, i = j ( 2) As task r j ends later than task r i, we have wij > 0. We give a siple exaple to show the constructing process of the task precedence graph (Fig. 8). Three tasks r1, r2, r3 are converted to the nodes v1, v2, v3. v 0 is a virtual node with the interval [ 0, 0 ]. A node with a sall start tie has an edge to a node with large start tie. Thus, there is an edge fro v 0 to v1, v2, v3. The weight of edge ( vi vj) is calculated as l ( Ii Ij) - l( Ii ). For exaple, w = 4-0 = CLAIM 2. An allocation with the axiu aggregate sensing tie of sartphones iniized corresponds to a solution to finding -path with the objective of iniizing the axiu length of those paths Constructing task precedence graph We next explain the construction of the task precedence graph. Before introducing the construction of the graph, we introduce an iportant observation, which is given in the following clai Searching for -path In this section, we propose an approxiation algorith to search for -path. The goal is to iniize the axiu length of the paths. The algorith for searching for paths proceeds in two key steps.

7 856 J. PENG et al. r 1 :[1,5] r 2 :[3,9] r 3 :[10,15] FIGURE 8. The construction of the task precedence graph. Each node in the graph denotes a task with a start tie and an end tie. v 0 is the virtual node and denotes a task with the sensing interval [0,0]. The weight is calculated as equation 2. In the first step, we search for a Hailton path, denoted by P *, fro the constructed task precedence graph. In the second step, we split the obtained Hailton path into sections, each section corresponding to the set of tasks for a sartphone Searching for a Hailton path We refer to a Hailton path in a directed graph as a directed path that goes through each node exactly once, and a path is presented as a sequence of nodes in the reaining part of this paper. A straightforward ethod to find the Hailton path is to enuerate all paths fro node v 0 to other nodes in the graph. Due to the characteristics of the task precedence graph, one can find the Hailton path by the branch and bound algorith which ay have a fast convergent rate. A branch can be pruned if there exists a node unvisited which has a saller start tie than those visited nodes on the branch. In this way, quite a lot of searching branches can be pruned. This is true because there is no possibility that a path revisits a node with a saller start tie. * The length of the Hailton path, denoted by L 1, can be calculated by adding all the weights of the edges of the path Splitting the Hailton path Next, we describe an approxiation algorith which eploys a path splitting heuristic. Given the Hailton path P *, the algorith splits it into sections, ={ P1, P2, ¼, P }. Each section is built by a subcoponent of P * with v 0 added as the source node. For ease of exposition, let d ax denote the axiu distance fro v 0 to other nodes in V. Forally, we have d ax = ax 1 i n w 0 i. The details of the algorith are given in Algorith 1. For each section P i, the distance fro the first node v 0 to the last node v p () i is actually the aggregate sensing tie of corresponding tasks. The nodes in section P1, P2, ¼, P returned by the algorith is the allocation schee for the tasks. To ease the understanding, an exaple of splitting a path into three sections is shown in Fig. 9. v v 0 v v 3 Algorith 1 Approxiation algorith. Require: The Hailtonian path P * and its length L 1* ; d ax ; the nuber of sartphones Ensure: a set of paths { P1, P2, ¼, P }; 1: for each j,1 j < do 2: find the last node v p () j such that the distance fro v 0 to v p () j along P * j( L1* - dax) is not greater than + d ax. 3: Obtain the jth section denoted by a sequence P j. = ì í ï < v0, ¼, vp ( 1) >, j = 1 Pj îï < v0, vpj (- 1)+ 1, ¼, vpj () >, 1 < j < 4: end for 5: P =< v0, vp( - 1)+ 1, ¼, vn> 6: return P1, P2, ¼, P. FIGURE 9. An exaple of path splitting. A Hailtonian path v 0 v 1 v n is split into three sections P 1, P 2, P 3. P 1 consists of v0, v1, ¼, vp ( 1 ). P 2 consists of v0, vp( 1)+ 1, ¼, vp( 2 ), and P 3 consists v, v, ¼, v 0 p( 2)+ 1 n Optiization Constructing the task precedence graph requires O( n 2 ) and searching for a Hailton path in a directed graph is not in polynoial tie [29]. Fortunately, we find that a Hailton path can be constructed siply by sorting all the tasks according to the start tie in an increasing order. Inspired by this observation, we propose an optiization for finding the Hailton path, which has a low coplexity of O( n log n). CLAIM 3. There is only one Hailton path in the task precedence graph, which starts fro v 0 to the node with the largest end tie. As the directed edge ( vi vj) Î E exists when the start tie of v i is saller than that of v j, the Hailton path ust go through all nodes in the order sorted by the start tie, and v 0 is the first node of the Hailton path. Thus, one can sort all the tasks by the start tie in an increasing order. Then,

8 FAIR ENERGY-EFFICIENT SENSING TASK ALLOCATION IN PARTICIPATORY SENSING WITH SMARTPHONES 857 these tasks coprise the nodes of the Hailton path. The weights of edges on the Hailton path are calculated as Definition 2. The total tie coplexity is O( n log n), where n is the nuber of tasks Analysis THEOREM 3.1. Suppose λ is the axiu aggregate sensing tie of sartphones achieved by Algorith 1, and l* is the axiu aggregate sensing tie achieved by an optial allocation. Then we have where is the nuber of sartphones. l l * ( 3 ) Proof. Fro the algorith, we can see that the distance fro v 0 to v p () 1 along P * is no greater than * ( L1 - dax) + dax. For each section j, 1 < j - 1, the distance fro v p(- j 1)+ 1 to v p () j is no greater than * ( L1 - dax ). The distance fro v p( - 1)+ 1 to v n is still * ( L1 - dax ). Thus, for each section j, the axiu * length is no greater than ( L1 - dax) + dax. Therefore, * l ( L1 - dax) + dax * L + ( 1-1 ) d ( 4) 1 ax Due to d ax l* and l * ³ L1 *, then we have l l l* + ( 1-1 ) l* = ( 2-1 ) l* and * 2-1. l The proof is copleted. THEOREM 3.2. The tie coplexity of the approxiation algorith is O(n log n), where n is the total nuber of tasks. Proof. The tie coplexity of finding the Hailton path and calculating the length of it is O( n log n), and the splitting process needs the tie coplexity of O(n). Thus, the total tie coplexity of the algorith is O( n log n). 4. ONLINE TASK ALLOCATION In this section, we consider the online task allocation odel. In this odel, the nuber of eber sartphones is still, and a task can be unknown until it arrives at the platfor, and only can be executed by a sartphone. Since the state of the syste changes only as a consequence of the task allocation, we view tie as being discrete. We say tie t corresponds to the tth coing task. The initial tie is 0, and tie 1 is the tie the first task arrives at the platfor. Note that we do not assue that the task are coing in the order of start tie, but they are valid. Let σ be the task sequence, and s =ár1, r2, ¼ñ.Givenσ, a feasible task allocation is to decide which sartphone a arrived task should be allocated to. OL ( s) is the axiu aggregate sensing tie generated by algorith OL and OPT ( s) is the axiu aggregate sensing tie generated by an optial offline allocation for σ. OL is called r-copetitive if OL ( s) r OPT ( s), for all σ. In the following, we present two online sensing task allocation algoriths, i.e. Greedy algorith and Robin-Hood algorith, and we also provide theoretical analysis on the algoriths Greedy algorith Overview The basic idea of the Greedy algorith is to allocate the incoing task to the least loaded sartphone. We call a sartphone is least loaded if it has the sallest aggregate sensing tie when running an incoing task. The designed Greedy algorith follows three basic rules: Rule 1: allocating a task to the sartphone if it can be covered by tasks which have already been allocated to that sartphone. Rule 2: allocating a task to the sartphone which has the sallest aggregate sensing tie if the task were allocated to the sartphone. Rule 3: if ultiple sartphones eet Rule 2, then allocate the task to the sartphone with the least increased sensing tie. The priority of these three rules decreases fro the first one to the last one Algorith description We present the detailed design of the greedy algorith. Each sartphone aintains a list which stores the ordered tasks that have been allocated to it. A task with a saller start tie is ordered ahead of the one with a larger start tie. The insertion of a new task can be copleted in linear tie. When the tth task r t arrives the platfor, the algorith first calculates the aggregate sensing tie li () t of each sartphone i, Î{ 1, ¼, }, and obtains the increased sensing tie D i () t,andd i()= t li()- t li( t - 1) of sartphone i if r t is allocated to it. Then, the algorith allocates the task to the right sartphone. The pseudocode is given in Algorith 2. The algorith perfors the task allocation following the three rules discussed above. If D i ()= t 0, which eans the incoing task can be covered by tasks allocated to sartphone i, then the algorith allocates r t to sartphone i. Otherwise, if sartphone i is the one with the sallest li () t, the task r t will be allocated to i, and then the algorith returns. In the third case, there exists ultiple sartphones

9 858 J. PENG et al. Algorith 2 Greedy online allocation algorith. Require: The set of allocated tasks on each sartphone; Ensure: The sartphone which task r t should be allocated to. 1: inload = INF; // store the iniu sensing tie 2: D in = INF; // store the iniu increent 3: for each sartphone i Î{ 1, ¼, } do 4: store the current aggregate sensing tie li ( t - 1; ) 5: calculate the aggregate sensing tie of sartphone i after allocating r t to it, li () t ; 6: D i()= t li()- t li( t - 1) 7: if D i () t = 0 then 8: choice = i; 9: break; 10: else 11: if li ()< t inload then 12: choice = i; 13: inload = li ( t ); 14: D in =D() i t ; 15: else if li ()= t inload and D i()<d t in then 16: choice = i; 17: inload = li ( t ); 18: D in =D() i t ; 19: end if 20: end if 21: end for 22: allocate task r t to the chosen sartphone 23: update the aggregate sensing tie of each sartphone 24: return choice. that have the sallest aggregate sensing tie. For exaple, li()= t lj ()= t in l() t, the algorith further checks their increased sensing tie D i() t and D j () t. If D i()<d() t j t, allocate r t to sartphone i, otherwise to sartphone j Analysis We present soe theoretical analysis on the Greedy algorith. THEOREM 4.1. The copetitive ratio of the greedy algorith is at ost for the online sensing task allocation proble, where is the nuber of eber sartphones. Proof. Let l i * be the aggregate sensing tie of sartphone i generated by the optial offline algorith, and L * presents the axiu aggregate sensing tie of the optial online allocation. We have, L* ³ l* ³ L *, ( 5) å i i= 1 * where L 1 is the iniu aggregate sensing tie of all the tasks. On the other hand, we can easily conclude that the 1 axiu aggregate sensing tie L resulting fro the greedy algorith will never exceed L 1 *, i.e. L L1 *. By induction, L L1 * L * L and *. L THEOREM 4.2. The tie coplexity of the greedy algorith for per task allocation is O(n), where is the nuber of sartphones and n is the nuber of sensing tasks that have been allocated so far. Proof. The calculation of the aggregate sensing tie l t i on each sartphone i during the allocation of task r t can be copleted in O(n), and the total tie coplexity for calculating the aggregate sensing tie of tasks on sartphones is O( n ). Thus, the total tie coplexity of the greedy algorith for per task allocation is O(n) Robin-Hood algorith Overview The basic idea of Robin-Hood algorith is to allocate tasks to sartphones which are not heavily loaded. We call a sartphone is heavily loaded if its aggregate sensing tie is larger than a threshold, which will be defined later. We aintain an estiate L(t) for OPT(t) at any tie t, i.e. after the tth task arrives, satisfying L () t OPT() t. OPT(t) is the axiu aggregate sensing tie generated by an optial offline allocation at tie t. li () t still denotes the aggregate sensing tie on sartphone i at tie t. A sartphone is heavily loaded at tie t if li ()³ t L() t, and is poor otherwise. A new task is assigned to soe poor sartphone with high priority. The state of a sartphone ay alternate between being heavily loaded and poor over tie Algorith description We present the detailed design of Robin-Hood algorith. First we introduce how to estiate L(t). For the first arrived task, the algorith assigns it to an arbitrary sartphone, and L(1) is the sensing tie of the first task. When a new task r t arrives, we have the following forula ì ()= í ï 1 L t ax L( t - 1 ), l( It ), îï è æ å ç D ( ) + ( - ) ö in t li t 1 1 i ü ý ï ø þï ( 6) where l ( I t ) is the length of task r t and D in()= t in1 i { li()- t li( t - 1)} is the iniu delta incurred by assigning task r t to sartphone i,1 i. The last quantity is the aggregate sensing tie of sartphones at tie t, divided by the nuber of sartphones. Note that the recoputation of L(t) ay cause soe heavily loaded sartphones to be reclassified as poor ones.

10 FAIR ENERGY-EFFICIENT SENSING TASK ALLOCATION IN PARTICIPATORY SENSING WITH SMARTPHONES 859 According to the result derived fro equation (6), the algorith randoly assigns the new task to a poor sartphone i, i.e. the sartphone with aggregate sensing tie li ( t - 1)< L() t. If there are no poor sartphones, the task is assigned to the heavily loaded sartphone with the sallest aggregate sensing tie delta. The pseudocode is given in Algorith 3. Algorith 3 Robin-Hood algorith. Require: The set of allocated tasks on each sartphone, L ( t - 1) an incoing task r t ; Ensure: The sartphone which task r t should be allocated to. 1: for each sartphone i Î{ 1, ¼, } do 2: inload = INF; 3: inindex =-1; 4: choice =-1; 5: D in = INF; 6: totalsensetie = 0; 7: Delta =Æ; 8: calculate the aggregate sensing tie of sartphone i after allocating r t to it, li () t ; 9: totalsensetie = totalsensetie + li ( t - 1; ) 10: D i()= t li()- t li( t - 1; ) 11: Delta. append (D i( t)) ; 12: if D in >D() i t then 13: D in =D() i t ; 14: end if 15: end for 1 16: L ()= t ax{ L ( t - 1 ), It, (D + totalsensetie)} ; in 17: Poor =Æ; 18: for each sartphone i Î{ 1, ¼, } 19: if Delta []== i 0 then 20: choice = i; 21: break; 22: end if 23: if li ( t - 1)< L() t 24: Poor. append () i ; 25: end if 26: if inload > li ( t - 1) + Delta [ i] 27: inload = li ( t - 1 ) + Delta [ i] ; 28: inindex = i; 29: end if 30: end for 31: if Poor is not epty then 32: randoly select choice fro Poor; 33: else 34: choice = inindex; 35: end if 36: allocate task r t to sartphone choice 37: update the aggregate sensing tie sartphone choice 38: return choice. REMARKS. This algorith can also be applied in the restriction allocation, where a task can only be executed by a part of the sartphones in the systes, but not any of the Analysis LEMMA 4.1. For any task r t arrives, Robin-Hood algorith guarantees that L () t OPT() t. Proof. The proof is by induction on the nuber of assigned tasks. For the first task, L () 1 OPT holds. As for the inductive part, it suffices to consider only the case where L(t) is increased. We have l ( Ii ) OPT( t) and 1 (D ()+å t l ( t - 1)) OPT( t) in 1 i i, it is not hard to conclude L () t OPT. LEMMA 4.2. There are at ost sartphones can be heavily loaded at any tie, where is the nuber of eber sartphones in the syste. According to the fact that L(t) is an upper bound on the total aggregate sensing tie of sartphones in the syste at tie t, and the aggregate sensing tie of a heavily loaded sartphone is not less than L( t ), the nuber of heavily loaded sartphones is no ore than. THEOREM 4.3. The copetitive ratio of Robin-Hood algorith is + 1 at ost, where is the nuber of eber sartphones in the syste. Proof. At any point in tie t, we will show that the algorith guarantees that li () t L()+ t OPT for any sartphone i. (i) the forula holds if i is poor on the arrival of task r t.(ii)ifiis heavily loaded, the deduction is a little coplicated. According to Lea 4.2, therealways exists poor sartphones. A task cannot be allocated to a sartphones with aggregate sensing tie larger than L( t ).Letr s bethelasttaskassignedtoi that caused i to becoe heavily loaded. Since the length of the task sensing interval l ( I s ) satisfying l ( Is ) OPT, we can conclude that l () t L()+ t l( I) ( + 1) OPT ( 7) i s 5. PERFORMANCE EVALUATION In this section, we evaluate the perforance of the algoriths.

11 860 J. PENG et al Methodology and siulation setup We copare our algoriths with three algoriths: a rando allocation algorith, a Round-Robin algorith and the optial algorith. With the rando allocation algorith, a task is randoly allocated to one of eber sartphones in the syste. The Round-Robin algorith tries to achieve fairness aong the sartphone users. It is a classical algorith for load balancing. We obtain the optial solution of four sallscale cases with an exhaustive search algorith. Two etrics are used for perforance evaluation, i.e. in ax fairness and total sensing tie. Each sartphone has its aggregate sensing tie, and the in ax fairness is the axiu aggregate sensing tie. A saller in ax fairness is desirable. The total sensing tie is the su of the aggregate sensing tie of all sartphones in the syste, which indicates the energy efficiency of the participatory syste as a whole. A saller total sensing tie indicates better energy efficiency. The default setting is as follows. We set the siulation tie 24 hours for both the offline and the online algoriths. The tie slot is 1 inute. Three ipacting factors are investigated, i.e. nuber of tasks, nuber of sartphones and axiu length of intervals denoted by l ax. The length of the sensing interval of a task is chosen randoly in [ 10, lax] inutes. We assue the iniu length of intervals is 10 inutes, and the arrival of tasks obeys the Poisson process for both the offline and the online odels. The default setting is suarized in Table 1. Each data point is an average over 20 independent runs Ipact of nuber of tasks We first investigate the ipact of the nuber of tasks on the perforance. The nuber of tasks is changed fro 200 to 700. The results are shown in Figs. 10 and 14. In Fig. 10, we can find that as the nuber of tasks increases, the in ax fairness of our algoriths is apparently better than that of rando algorith and Round-Robin algorith. The approxiation algorith achieves the best in ax fairness, which is closely followed by the greedy algorith. Robin-Hood perfors slightly worse. The rando algorith produces alost five ties larger in ax fairness copared with the approxiation algorith when the nuber of tasks is 400. The increent of in ax fairness is not Paraeter TABLE 1. Default settings. Value Nuber of tasks 400 Nuber of sartphones 30 Maxiu length of intervals (in) 60 significant for both greedy algorith and approxiation algorith as the nuber of tasks grows up. This shows our algoriths have good scalability. In Fig. 14, the total sensing tie of all the sartphones increases with the nuber of tasks increasing. Rando algorith perfors largely worse than other algoriths. The total sensing tie of the approxiation algorith is still the sallest. Robin-Hood algorith outperfors Round-Robin algorith in total sensing tie. As the nuber of tasks increases, the in ax fairness of sartphones becoes better. This is easy to understand because the total nuber of tasks to be executed becoes larger, the aggregate sensing tie on each sartphone rises correspondingly Ipact of nuber of sartphones To study the ipact of the nuber of sartphones on the perforance of algoriths, the second set of siulations varies the nuber of sartphones fro 10 to 60. The results are shown in Figs. 11 and 15. Figure 11 shows that the in ax fairness decreases as the nuber of sartphones increases. The approxiation algorith achieves the best perforance, and the greedy algorith follows it. The rando algorith perfors the worst. When the nuber of sartphones is 10, the in ax fairness of rando algorith is around five ties as large as that of the approxiation algorith. In Fig. 15, with the increasing nuber of sartphones, the total sensing tie increases. However, the approxiation algorith has saller increase rate of the total sensing tie while the rando algorith has the largest increase rate. Because tasks are allocated to ore sartphones, the overlap between intervals becoes saller, and hence the total sensing tie increases. Min-ax fairness Rando (offline) Approx (offline) RH (online) Greedy (online) RR (online) Nuber of tasks FIGURE 10. Min ax fairness vs. nuber of tasks.

12 FAIR ENERGY-EFFICIENT SENSING TASK ALLOCATION IN PARTICIPATORY SENSING WITH SMARTPHONES Ipact of axiu length of intervals Finally, we study the ipact of the length of intervals on the perforance of algoriths. An intuition is that the ore tasks with longer intervals, the larger overlap between two tasks. As a consequence, ore sensing tie is saved. In this set of siulations, the axiu length of sensing intervals is varied fro 20 to 120 inutes with the increent of 20. Fro Figs. 12 and 16, we can see that when the axiu length of intervals becoes larger, the in ax fairness of sartphones becoes larger. With the approxiation algorith and the greedy algorith, the in ax fairness increases no ore than 1 when the axiu length of intervals changes fro 20 to 120. With the rando algorith, the increent is as high as 3. The total sensing tie, shown in Fig. 16, also increases with the increasing axiu length of intervals. Min-ax fairness Rando (offline) Approx (offline) RH (online) Greedy (online) RR (online) Nuber of sartphones FIGURE 11. Min ax fairness vs. nuber of sartphones. Min-ax fairness Rando (offline) Approx (offline) RH (online) Greedy (online) RR (online) 5.5. Coparison to the optial solution To show the efficiency of our algoriths, we copare the perforance of our algoriths to the optial solution derived fro an exhaustive search algorith. Four sall-scale cases are designed, with the axiu length of intervals 40 inutes and the iniu length of sensing intervals 5 inutes. The total siulation tie is 100 inutes. We change the nuber of tasks and the nuber of sartphones. The configuration of the paraeters in the four cases is given in Table 2. The results are reported in Figs. 13 and 17. In Fig. 13, we can see that the in ax fairness is generally saller when the nuber of sartphones becoes larger by coparing Cases 3, 4 with Cases 1, 2. As the nuber of tasks increases, the in ax fairness becoes worse. This also confirs the results we have observed in the previous siulations. Besides, the in ax fairness both for the greedy allocation algorith and the approxiation allocation algorith is close to the optiu in ax fairness. The greedy allocation algorith achieves a siilar in ax fairness to the approxiation algorith. This result is uch better than what we expected, as the copetitive ratio analyzed in Theore 4.1 is at ost, where is the nuber of eber sartphones. Although Robin-Hood perfors the worse copared with other algoriths, but the perforance is still no ore than + 1 of the optial value. TABLE 2. Configuration of paraeters. Cases Nuber of tasks Nuber of sartphones Case Case Case Case Min-ax fairness Optial Approx RH Greedy RR Maxiu length of intervals (in) 30 (10,2) (15,2) (10,3) (15,3) Different cases: (n,) FIGURE 12. Min ax fairness vs. axiu length of intervals. FIGURE 13. Min ax fairness in different cases.

13 862 J. PENG et al. Total sensing tie (in) Total sensing tie (in) x Rando (offline) Approx (offline) RH (online) Greedy (online) RR (online) Nuber of tasks FIGURE 14. Total sensing tie vs. nuber of tasks. Rando (offline) Approx (offline) RH (online) Greedy (online) RR (online) Nuber of sartphones FIGURE 15. Total sensing tie vs. nuber of sartphones. In Fig. 17, the approxiation algorith achieves the best perforance in three cases, even copared with the optial offline algorith which achieves the optial in ax fairness. Although the total sensing tie achieves by Robin- Hood allocation is slightly larger, the difference is no >30% of the optial value. 6. RELATED WORK Sartphones [27] have becoe an iportant part of everyone s life, with good Internet connections with WiFi [26, 28] or 3G and beyond. Participatory sensing has recently attracted extensive research attention fro both industry and acadeic due to its attractive applications [2, 3, 13, 15]. Much existing work has been done to address various kinds of participatory sensing issues, such as the privacy proble [14, 16, 17, 20, Total sensing tie (in) x 104 Rando (offline) Approx (offline) RH (online) Greedy (online) RR (online) Maxiu length of intervals (in) FIGURE 16. Total sensing tie vs. axiu length of intervals. Total sensing tie (in) Optial Approx RH Greedy RR (10,2) (15,2) (10,3) (15,3) Different cases: (n,) FIGURE 17. Total sensing tie in different cases. 38] and the incentive echanis design [8, 9, 18, 37, 39]. This paper focuses on the sensing task allocation in participatory sensing systes. Recently, a few pieces of work on data sharing have been proposed in wireless sensor networks (WSNs) [22, 23]. In [23], the proble of data sharing aong ultiple applications is discussed. This work assues each application deployed on a sensor needs to saple discrete data at soe tie points, and these data can be shared by ultiple applications. The work proposed in [22] further considers a continuous interval of sapling data. The overlapped interval of data can also be shared by ultiple applications deployed on a sensor. The goal is to iniize the total sapling tie for copleting all the sensing tasks. However, they only consider the sapling optiization on a sensor, instead of the whole the WSNs, such as load balancing.

14 FAIR ENERGY-EFFICIENT SENSING TASK ALLOCATION IN PARTICIPATORY SENSING WITH SMARTPHONES 863 Quite a lot of studies on job/task assignent which ai to achieve load balancing have been proposed [30, 31, 33, 34], both in offline and online cases. Most of the assue that the ultiple jobs cannot be perfored on a achine concurrently [33, 34], i.e. there is only a job run at a tie. In addition, a job can be perfored at any tie and no executing tie interval is specified, which is different fro the task odel defined in this paper. Although there also exist studies which discuss the assignent of teporary tasks which have liited duration in tie [31, 35], they also assue no ore than one task can be executing at any tie, and tasks are executed independently. Soe existing work has studied the proble of task allocation in crowdsourcing arkets [5, 6, 10, 36], and uch of the consider how to axiize the benefits obtained by service requesters. In [6], the authors consider how to assign heterogeneous tasks to workers with different, unknown skill sets in the crowdsourcing arkets such as Aazon Mechanical Turk. Given a fixed set of tasks and the ties of each task need to be copleted, and workers arrive online and one at a tie, the goal is to allocate the workers to tasks such that the total benefit that the requester obtained axiized. A two-phase exploration exploitation assignent algorith is presented, which is proved to be copetitive with respect to the optial offline algorith which knows the skill levels of each worker. The proble considered in [10] is to select a service provider fro a list of providers which can provide axiu satisfaction to the service requester. An adaptive task scheduling which based on the custoer satisfaction feedbacks is proposed. In [5], Ho et al. investigate the task assignent and label inference for heterogenous classification tasks. Labels are provided for instances (such as websites ) by workers. By applying online prial-dual techniques, a near-optial adaptive assignent algorith is derived. In [9], Luo and Tha link incentive to users deand for consuing services. The proble is to assign an aount of service quota to users with the objective of axiizing fairness or social welfare. In suary, little existing work has studied the proble of sensing task allocation in participatory sensing systes with the objective of axiizing energy efficiency of sartphones and fairness aong sartphones. Note that the preliinary version of this work has been published in [32]. 7. CONCLUSION AND FUTURE WORK In this paper, we have studied the task allocation proble which is of paraount iportance to both energy efficiency and fairness aong sartphones. We have rigorously proven that the task allocation proble of iniizing the axiu aggregate sensing tie is NP hard even under the offline odel. We consider two task allocation odels including offline odel and online odel. Under the offline allocation odel, we have designed a polynoial-tie approxiation algorith that approxiates the offline optiu within a sall factor of 2 1, where is the nuber of sartphones in the syste. Under the online allocation odel, we have designed two online task allocation algoriths: Greedy algorith and Robin-Hood algorith, which achieve a copetitive ratio of at ost and + 1, respectively. We have presented theoretical analysis for both the offline algorith and the online algoriths. We have also conducted extensive siulations and coparison of different allocation algoriths. The results deonstrate that our algoriths can achieve highenergy efficiency while keeping good fairness aong sartphones. In future work, we would further extend our work. First, we will introduce the location constraint when allocating a sensing task to a sartphone. That is, a sensing task can only be allocated to those sartphone users whose locations atch the location of the sensing task. Note, however, introducing the location constraint will liit the allocation of overlapping sensing tasks to a sartphone. Second, our work currently considers all sensing tasks are of the sae type. In the real world, there can be several different types of sensing tasks, differing in their power consuptions. This further coplicates the proble of allocating overlapping sensing tasks of different types to sartphones. FUNDING 973 Progra (No. 2014CB340303), 863 Progra (No. 2015AA015303), NSFC (Nos , and ); STCSM (Grant nos and 15DZ ); Research Grant for Young Faculty in Shenzhen Polytechnic (No K30015); SZSTI (No. JCYJ ); and Singapore NRF (CREATE E2S2). This work is also supported by the Progra for New Century Excellent Talents in University of China, the Progra for Changjiang Young Scholars in University of China, and the Progra for Shanghai Top Young Talents. REFERENCES [1] Power onitor anual. [2] Ali, K., Al-Yaseen, D., Ejaz, A., Javed, T. and Hassanein, H.S. (2012) CrowdITS: Crowdsourcing in Intelligent Transportation Systes. In Proc. IEEE WCNC, Paris, France, April 1 4, pp IEEE, Piscataway. [3] Ballesteros, J., Rahan, M., Carbunar, B. and Rishe, N. (2012) Safe Cities. A Participatory Sensing Approach. In Proc. IEEE LCN, Clearwater Beach, Florida, USA, October 22 25, pp IEEE, Piscataway. [4] Thiagarajan, A., Biagioni, J., Gerlich, T. and Eriksson, J. (2010) Cooperative transit tracking using sart-phones. In