Research Article  Open Access
Linbo Zhai, Hua Wang, Xiaole Li, "Optimal Task Partition with Delay Requirement in Mobile Crowdsourcing", Wireless Communications and Mobile Computing, vol. 2019, Article ID 5216495, 12 pages, 2019. https://doi.org/10.1155/2019/5216495
Optimal Task Partition with Delay Requirement in Mobile Crowdsourcing
Abstract
Mobile crowdsourcing takes advantage of mobile devices such as smart phones and tablets to process data for a lot of applications (e.g., geotagging for mobile touring guiding monitoring and spectrum sensing). In this paper, we propose a mobile crowdsourcing paradigm to make a task requester exploit encountered mobile workers for highquality results. Since a task may be too complex for a single worker, it is necessary for a task requester to divide a complex task into several parts so that a mobile worker can finish a part of the task easily. We describe the task crowdsourcing process and propose the worker arrival model and task model. Furthermore, the probability that all parts of the complicated task are executed by mobile workers is introduced to evaluate the result of task crowdsourcing. Based on these models, considering computing capacity and rewards for mobile workers, we formulate a task partition problem to maximize the introduced probability which is used to evaluate the result of task crowdsourcing. Then, using a Markov chain, a task partition policy is designed for the task requester to realize highquality mobile crowdsourcing. With this task partition policy, the task requester is able to divide the complicated task into precise number of parts based on mobile workers’ arrival, and the probability that the total parts are executed by mobile workers is maximized. Also, the invalid number of task assignment attempts is analyzed accurately, which is helpful to evaluate the resource consumption of requesters due to probing potential workers. Simulations show that our task partition policy improves the results of task crowdsourcing.
1. Introduction
In recent years, the proliferation of crowdsourcing has shown significant potentials for many application areas. Crowdsourcing, a novel tasksolving paradigm, means that human workers are recruited to solve complicated tasks. Extensive researchers are attracted to pay attention to crowdsourcing due to its success about human intrinsic applications.
Early successful examples are Wikipedia, Yahoo! Answers, and Yelp. As the great potential of crowdsourcing is realized in recent years, several generalpurpose platforms, including oDesk and Amazon Mechanical Turks (AMT), make crowdsourcing more manageable and powerful. These online systems occur to make requesters define tasks and human workers execute them with rewards. For many online crowdsourcing systems, the common issue is inefficiency. For instance, no more than 15% tasks in Amazon mTurk system could be finished within an hour [1].
On the other hand, recent years witness the remarkable proliferation of intelligent mobile devices (e.g., smart phones and tablets) and the sharp growth of mobilebroadband services including data sharing and synchronization ultrahighresolution video streaming and virtual and augmented reality. All these services continue to drive the demand for higher data rates and lead to crowdsourcing application in InternetofThings (IoT), where pervasive interconnected smart objects cooperates together to reach multiple goals. IoT technologies can effectively promote the interactions between environments and the human and enhance the reliability and efficiency of smart cities [2–7].
With the extensive use of mobile devices such as tablets and smart phones, mobile crowdsourcing systems (MCS) is a feasible solution to complete delaysensitive tasks such as geotagging for mobile touring guiding monitoring and local parking space searching which are inevitable in smart cities. To improve the taskexecuting efficiency, a user tends to assign such delaysensitive tasks to mobile devices to complete them and collect executing results through the wireless communication system, which is called mobile crowdsourcing. We should notice that these taskexecuting results must come back within delay requirement. However, mobile devices, which are portable but with less computing ability, are difficult to complete complex tasks within delay requirement. Hence, this is challenging to mobile devices and motivates us to develop a mobile crowdsourcing policy to divide large tasks into several small parts suitable for mobile devices to execute.
In this paper, we propose a mobile crowdsourcing paradigm, dividing a complicated task into small pieces and assigning small subtasks to mobile devices, to obtain highquality executing results which means mobile devices execute as many subtasks as possible. We design the recruited process and present the worker arrival model and task model. Furthermore, the probability that all parts of the complicated task are executed by mobile workers is introduced to evaluate the result of task crowdsourcing. The higher probability means the result of the task crowdsourcing is better. According to these models, we take into consideration other factors such as the complexity and rewards of tasks which will influence mobile workers’ execution and formulate a task partition problem to maximize the probability that all parts of the complicated task are executed by mobile workers. Then, using the Markov chain, we derive a task division policy to realize highquality results, and the invalid number of task assignment attempts is analyzed accurately based on probability generating function. Simulations show that our task partition policy improves the results of task crowdsourcing.
In this paper, we study the task partition problem of crowdsourcing process in the wireless system consisting of many mobile devices. The main contributions are summarized as follows:(i)Considering the mobility and capacity of mobile workers, we propose a mobile crowdsourcing paradigm to divide a complicated task into multiple subtasks and assign these subtasks to mobile workers. The mobile worker decides to execute such a subtask or not based on its computing capability and corresponding requirements.(ii)To describe the attributes of each task, the task model is proposed to denote task load, delay requirement, and monetary reward to the mobile workers. Moreover, an arrival model of mobile workers is proposed to describe the encountered process between the requester and the mobile worker. Based on mobile worker features and task attributes, we establish the system model and formulate a task partition problem to maximize the probability that all subtasks are executed by mobile workers and guarantee the result of task crowdsourcing.(iii)To realize the highquality results of task crowdsourcing, using a Markov chain, the state transitions, denoting the number of subtasks executed by mobile workers, can be analyzed. Based on these state transitions within limited period, caused by delay requirement, the optimal task partition can be obtained. Then, according to the optimal task partition, the invalid number of subtask assignment is analyzed accurately with probability generating function in the total crowdsourcing process.(iv)Simulation results show our proposed policy, compared to fix partition policy and the adaptive scheme, approaches the optimal solutions.
The rest of the paper is organized as follows. In Section 2, a review about related works is provided. In Section 3, we describe the task crowdsourcing process and propose the system model. In Section 4, a Markov chain is developed to solve the task partition problem and the number of invalid attempts is analyzed. In Section 5, the proposed algorithm is evaluated with simulation results. Finally, conclusions are shown in Section 6.
2. Related Works
Human computation has been executed for many centuries. Specifically, if a “human” serves to “compute,” there will be a human computation which can be observed. This is the reason why there is a history of Human Computation, which is obviously longer than that of the electronic computer. With the rapid development of Internet web service, especially those facilitating online labor recruiting and managing (e.g., oDesk and Amazon MTurk), human computation begins to experience a new era where the sources of human are not designated exerts or employees but extended to a vast pool of crowds instead. This type of outsourcing to crowds, named crowdsourcing, is receiving countless success in several areas such as logistics, fund raising, monitoring, and so on.
With the extensive use of mobile devices such as tablets and smart phones, the new hybrid architecture occurs to support a massive ad hoc crowd which is composed of distributed mobile nodes and a massive social network around a smart city environment [8]. Therefore, mobile crowdsourcing can be utilized in IoT to complete realtime tasks and improve task efficiency (e.g., localization service, spectrum sensing, and environmental monitoring).
For indoor localization systems, a major bottleneck is the tradeoff between both localization accuracy and site survey costs. In [9], a probabilistic radio map construction method is propose with crowdsourcing collection, taking into account both accuracy and survey costs. In order to obtain the need for location labels, Jung et al. [10] propose the unsupervised learning method to calibrate a localization model based on a globallocal optimization scheme. In this hybrid scheme, an efficient globallocal interaction reduces the task complexity drastically. In [11], a locationaware infrastructure is proposed to combine a broad sensing layer, centralized cloud federation support, and edge computing. With the sensing capability of mobile devices, users can obtain the crowd sensing services in IoT. Credible interaction issues among mobile users, however, are still hard problems. Considering the credible interaction, An et al. focus on assigning the crowdsourcing sensing tasks [12]. In the field of environmental monitoring, a path planning approach is introduced for crowd evacuation in buildings [13]. Besides, there are some related researches on Ehealthcare service for the crowdsourcing IoT and crowdsourcing industrialIoT (IIoT) applications [14, 15].
To improve the executing efficiency, a complicated task is often divided into smaller pieces in crowdsourcing [16]. Then, these small pieces can be assigned to a group of undefined workers [17]. In many existing studies, there is a central service entity which not only collects the information of both workers and requesters but also performs the taskworker assignment to optimize a global utility [18, 19]. If someone hopes to use a crowdsourcing system [20, 21], inevitably, corresponding fee must be paid for using the centralized server.
Due to the selforganized nature, requesters in mobile networks, however, do not obtain worker information in advance. Therefore, requesters have to probe worker ability and make sequential recruitment decision. This problem motivates us to design a task partition policy for mobile worker recruitment.
In [22], Tuncay and Helmy use the location visiting frequency to reflect the worker ability and present a gradient ascend policy which assign the task to a user having higher visiting frequency in the destination area. However, the objective, similar to the traditional routing and data dissemination, is to arrive at the destination as fast as possible with minimum replication overhead. In [23, 24], Chang and Gong analyze the schemes for allocating work segments among participants opportunistically. When determining users’ workloads, the schemes take into account contacting delay, acceptance probability, computing speed, and the existence of resource competition works. The purpose is to minimize the task makespan by partitioning the total task into suitable subtasks corresponding to the encountered worker computation power. In [25], Pu et al. use an online algorithm to maximize the task service quality based on worker’s interest preference. Focusing on devicetodevice networks, the authors study the recruitment problem when the crowdsourcing is delaysensitive [26]. In [27], Tong et al. study a largescale crowdsourcing task which consists of thousands or millions of atomic tasks. To the best of our knowledge, little work has been done on jointly considering the mobile worker features and task partition.
3. System Model
In this section, at first, the task crowdsourcing process is described. Then, the system model, involving task attributes and mobile user features is developed, and the task partition problem is formulated to achieve optimal results.
3.1. Recruitment Description
A mobile user having a task is a requester. The requester can selforganize task crowdsourcing by recruiting several encountered mobile users who are workers in realtime. A mobile user encountering another user means they are close to establish a devicetodevice (D2D) link. The term of “close” means the distance between two mobile users does not exceed WiFidirect distance. The recruitment is described as follows.
When a mobile user (requester) launches a task, the requester is invoked to recruit some encountered mobile users (workers) in realtime. Since a complicated task is difficult for a worker to complete in time, the requester will divide the complex task into several subtasks and recruit corresponding number of workers to complete those subtasks. The requester sends attributes of a subtask including the reward, subtask load, and delay requirement to an arrived worker. The worker decides to execute this subtask or not based on available computing capacity, remaining energy, subtask reward, and delay requirement. If the worker decides to accept this subtask which means it can complete the subtask within delay requirement, its worker ability is set to 1. Otherwise, it is set to zero. Then, the worker ability value is sent to the requester. Finally, the requester decides to recruit the worker or not based on its worker ability.
If the requester decides to recruit the worker, the detailed content of a subtask is sent to the worker. During the subtask execution, it is not necessary for the requester and the worker to connect with each other all the time. After the subtask is finished by the worker, the corresponding subtask execution result will be sent back to the requester by a D2D link or cellular link satisfying the delay requirement. Once the requester receives the result in time, the subtask reward will be given to the worker. Otherwise, if the delay requirement is not satisfied, the requester thinks the worker is fraudulent and does not give the subtask reward to the worker.
The key rationale of the recruiting model is that opportunistic encounters of mobile devices are sufficient and prevalent in modern society [28], which offers lots of opportunities to exploit nearby intelligent devices [29]. If a mobile user has a complex task, it can selforganize its task crowdsourcing by leveraging many encountered mobile users in realtime, and fast response can be realized by interacting with intelligent workers in proximity directly. Furthermore, many mobile crowdsourcing tasks will require locationaware knowledge and information (e.g., mobile touring guiding monitoring and local parking space searching). Hence, nearby intelligent workers are more qualified to execute them than the online workers [22]. In addition, compared to the new emerging paradigm “cyber foraging” about mobile computing, our framework shares the similar spirit so that mobile users can exploit nearby intelligent devices to facilitate their computational task processing [30].
Figure 1 illustrates the procedure of mobile crowdsourcing. A mobile user (requester R), launching a mobile crowdsourcing task, moves around within the network and encounters a mobile user (worker W_{1}). Requester R divides this task and sends subtask attributes to worker W_{1} through a D2D link at position 1. Worker W_{1} decides to execute the subtask and returns the worker ability to requester R. Then, requester R sends the subtask content to worker W_{1}. During the period of subtask execution, worker W_{1} and requester R may continue to move and do not need to connect with each other. When worker W_{1} finishes the subtask, worker W_{1} and requester R are at position 3 and position 2, respectively. The distance between worker W_{1} and requester R is too long to establish D2D link. Therefore, worker W_{1} sends the result to requester R by a cellular link, the wireless transmission through mobile communication system [28]. That means worker W_{1} sends the result to the base station at first. Then, the base station relays the result to requester R. After that, the subtask reward is granted to worker W_{1} by requester R.
3.2. System Model
3.2.1. Worker Arrival Model
We use K to denote the set of potential mobile workers who may execute the task. Considering worker mobility, the interencounter time of a requester r and a worker is important. In our system, it is not necessary for the requester r to know the arrival rate of each worker . The total arrival rate , obtained by calculating the number of encountered workers per time unit, is adopted instead. The current value of total arrival rate can be estimated based on the value during the recent time units. When the total arrival rate is obtained, the average interencounter time of a requester r and a worker is .
3.2.2. Task Model
A mobile requester can describe a task i by a set of attributes, <Ta_{i}, D_{i}, R_{i}>, where Ta_{i} denotes the task load of task i; Di represents delay requirement of task i within which a worker must return the result; R_{i} denotes the total reward to workers after the results are returned within D_{i}.
There are three parameters in the task model. Task load and delay requirement are determined by the category of the corresponding task such as content creation or information finding [31]. For the reward, similar to online systems, this model uses a common posted price method where the workers are provided the explicit price offer [32].
3.2.3. Task Partition Formulation
Since a task may be too complex for a mobile worker with less computing ability to complete in time, the task can be divided into many subtasks by the task requester. Then, each subtask of the task is suitable for a worker to finish.
Assume that task i is averagely divided into N subtasks. Then, the load of each subtask is
The reward for a worker finishing a subtask is
The requester sends the subtask attributes including R_{i}(N), STa_{i}(N), and delay requirement D_{i} to a worker . The worker evaluates whether STa_{i}(N) can be completed within D_{i} with satisfying reward R_{i}(N).
Let denote the remaining energy of worker , denote the computing capacity of worker , and denote the probability that worker accepts the subtask of task i. Without loss of generality, it is defined that the probability that a worker accepts a subtask is related to R_{i}(N), , , and STa_{i}(N)/D_{i}. Intuitively, R_{i}(N), , and will have positive impact on the accepting probability, while STa_{i}(N)/D_{i} will have negative impact on the accepting probability. However, the accepting probability is not simply defined as a function which is proportional to R_{i}(N), , and , and inversely proportional to STa_{i}(N)/D_{i} since the real case is more complicated. Let RE denote the expectation reward of a mobile worker. If a mobile worker can gain the reward R_{i}(N) exceeding its expectation RE, it will pay more attention to the ratio between the subtask load STa_{i}(N)/D_{i} and its computing capacity . Otherwise, if the reward it can gain is less than its expectation RE, it will focus on the real reward it can gain. Additionally, a worker’s remaining energy also influences the probability of accepting the subtask. On the one hand, the larger value of will result in the higher accepting probability. On the other hand, the value of will influence the pace of the probability variety. When the remaining energy of a mobile worker is high, it is sensitive and its accepting probability will change drastically as other parameters including R_{i}(N), , and STa_{i}(N)/D_{i} change. When a mobile worker has the low remaining energy, its accepting probability will change slowly as other parameters change.
Based on the aforementioned description, the accepting probability should be the piecewise function. When the reward R_{i}(N) exceeds its expectation RE, has more influence on the probability and the probability will decrease as increases. When the reward R_{i}(N) is lower than its expectation RE, the reward will have greater influence on the probability and the probability will increase as RE increases. To clearly reflect the influence of and R_{i}(N) on the trend of the function variety, we use the square of them rather than their original forms in the piecewise function. Besides, the value of the accepting probability should be in the range (0, 1). Hence, the exponential function is used to guarantee the accepting probability P_{iw} in this range. Then, P_{iw} can be defined aswhere C is an adjustable constant to control the changing speed of accepting probability.
Using (1) and (2), the probability P_{iw} in Equation (3) can be rewritten as
Since accepting probability P_{iw} is the piecewise function based on the relationship between R_{i}(N) and RE and Equation (3) only describes the relationship between R_{i}(N) and RE, using equation (2), we can obtain the relationship between R_{i} and RE. From Equation (4), it can be observed that the accepting probability P_{iw} increases with the growth of subtask number N when N is less than R_{i}/RE. On the contrary, the accepting probability P_{iw} decreases with the growth of the subtask number N when N is higher than R_{i}/RE. This can be explained as follows. When the subtask number N is smaller, the worker is able to gain reward more than its expectation RE. As N increases, a subtask load decreases and the worker is more likely to accept such a subtask. After N exceeds R_{i}/RE, the worker is not able to achieve its expected reward RE and it mainly focuses on the current reward. As N increases, the current reward of a subtask decreases and the worker is less likely to accept such a subtask.
Additionally, each worker needs some time to complete a subtask. The duration is determined by STa_{i}(N) and the worker computing capacity. Let B = min {} denote the minimum computing capacity of potential workers, where K is the set of total mobile workers. Then, the maximum duration for executing a subtask is derived as
Therefore, the requester should assign subtasks to workers within . Otherwise, the delay requirement cannot be guaranteed. Let T_{i} () denote the duration of subtask assignment. To maximize the service quality, the requester should find as many workers executing subtasks as possible within T_{i}. Thus, denotes the total number of workers which a requester encounters within T_{i}. Since workers may accepting the subtasks following P_{iw}, the number of accepting workers must be less than . Let n denote the number of workers accepting subtasks. If n exceeds the total number of subtasks, which is denoted by N, all subtasks are executed by workers and the service quality is guaranteed.
To evaluate the result of task crowdsourcing, the probability , denoting that all subtasks of a complicated task are executed by mobile workers, is introduced. The higher the probability that n exceeds N is, the better service quality is.
The task partition policy for task i is to divide task i into N subtasks which are suitable for workers to execute. The objective is to obtain the optimal task partition to maximize the probability that all subtasks are executed by workers within T_{i}. Therefore, the optimal task partition problem can be formulated as
4. Partition Policy
In this section, a Markov chain is used to describe the state transitions denoting subtasks assignment and calculate the optimal task partition. Then, using the transition matrix, the unsuccessful number of subtask assignment attempts before the crowdsourcing end can be analyzed. Furthermore, the time complexity of proposed algorithm is analyzed.
4.1. Optimal Task Partition
According to the description in Section 3, for task i, a requester should receive the executing results from encountered workers within delay requirement D_{i}. Additionally, each worker needs a duration SD_{i} to complete a subtask. The duration is determined by STa_{i}(N)/B. Therefore, the requester needs to assign subtasks to workers within to satisfy the delay requirement of task i.
It is defined that each slot duration is where denotes the arrival rate of total mobile workers. Then, a requester encounters one mobile worker in a slot on average. Since each slot duration T_{s} equals , can be divided into the m slots. We can obtain
Figure 2 shows the slot division within .
To realize the optimal task partition, we use a Markov chain to describe state transitions within task assignment duration . For task i which is divided into N subtasks averagely, we use S_{0}, S_{1}, S_{2}, … ,S_{N} to denote N + 1 states, where means j subtasks have been assigned to workers successfully.
The encountered worker will accept a subtask of task i with the probability P_{iw}, and refuse the subtask with the probability 1 − P_{iw}. If the encountered worker accepts the subtask, the system moves to new state S_{j} + 1 (j ≤ N − 1) from state S_{j}. Otherwise, the system still stays at state S_{j} if the worker refuses the subtask. When the system moves to state SN, which means all subtasks of task i have been accepted by workers, the state will not change no matter the requester encounters new workers or not. Therefore, the onestep transition probability matrix , which is a square matrix with order equaling N + 1, can be derived as follows:(i)When the system is currently in the state S_{j} (0 ≤ j ≤ N − 1), after onestep transition, the system will move to state S_{j} + 1 with the probability P_{iw} or stay at S_{j} with the probability 1 − P_{iw}. Thus for 0 ≤ j ≤ N − 1, Q_{j}, j + 1 = P_{iw} and Q_{j}, j = 1 − P_{iw}.(ii)When the system is currently in the state S_{j} (j = N), the system must stay at S_{j} after onestep transition. Thus for j = N, Q_{j}, j = 1.
The onestep transition probability matrix Q is a sparse matrix. We can obtain
From the initial slot to the last slot, the system witnesses m − 1 transition. The system stays at state S_{0} at the initial slot because no subtask is assigned. Hence, the initial distribution is . Let P_{i}(N) denote the probability that a requester has assigned total N subtasks of task i successfully within the duration . Within the delay requirement, there are m slots. We use a Markov chain to describe the state transition from the first slot to the mth slot. Hence, there are m − 1 steps. For each step, the transition probability matrix is Q. Therefore, Q^{m−1} denotes the transition probability matrix of m − 1 steps. Then, we can obtainwhere . P_{i}(N) is related to the task partition number N.
To maximize P_{i}(N), by solving the Karush–Kuhn–Tucker (KKT) conditions [33], we obtain
Then, using (4), (5), (9), and (10), we can derive the optimal task partition N′ in theory. Considering the practical limitation of partition number, the optimal task partition is derived aswhere N_{max} denotes the allowable maximum partition number. After is determined, the maximal value of successful assignment probability can be calculated based on (9).
For a complicated task, we use a Markov chain and the state transition matrix to obtain the optimal task partition. When the optimal partition is determined, the probability that all parts of the complicated task are executed by mobile workers can be obtained. Based on the value of probability , we know how often this total assignment occurs. The higher value of means this total assignment will occur more often. The proposed algorithm for task partition is described as Algorithm 1.

Theorem 1. The proposed task partition is optimal.
Proof. Since a complex task is difficult for a mobile device to execute, we divide a complex task into N subtasks and assign each subtask to a mobile device. If each subtask is accepted by a mobile device, the crowdsourcing process is completed. Using a Markov chain, we aim to obtain the optimal value of N to maximize the completed probability of the crowdsourcing process.
At first, the states of the Markov chain are defined. We use S_{0}, S_{1}, S_{2}, … , S_{N} to denote N + 1 states where S_{j} means j subtasks have been accepted by mobile devices. S_{0} means no subtask is assigned while S_{N} means all subtasks have been assigned successfully. Hence, these states describe how many subtasks are accepted.
If the encountered worker accepts the subtask with the probability P_{iw}, the system moves to new state S_{j} + 1 (j ≤ N − 1) from state S_{j}. Otherwise, the system still stays at state S_{j} if the worker refuses the subtask with the probability 1 − P_{iw}. When the system moves to state S_{N}, which means all subtasks of task i have been accepted by workers, the state will not change no matter the requester encounters new workers or not. Therefore, we can obtain onestep transition probability matrix Q, shown in equation (8), to describe state transitions.
Within the delay requirement, the crowdsourcing process is completed if the system moves from S_{0} to S_{N}, which means all subtasks are accepted by workers. We use P_{i}(N), shown in equation (9), to denote this completed probability that the system moves from S_{0} to S_{N}. This probability is determined by the partition value N. Hence, the proper value of N will result in the maximized completed probability. We solve this problem based on equations (10) and (11) and obtain the optimal value to maximize the completed probability, which means optimal partition is obtained.
4.2. Unsuccessful Assigned Attempts
Before a requester assigns all subtasks successfully, there may be some unsuccessful assignment attempts for the requester because some encountered workers do not accept the subtasks. These invalid attempts consume the requester’s resources such as energy and probing time. Based on the optimal task partition , the invalid attempt number can be obtained accurately as follows.
After the optimal task partition is derived, the corresponding onestep transition probability matrix with order equaling is determined. Let U denote the unsuccessful number of subtask assignment attempts before the system reaches state . To derive the average unsuccessful attempt number E[U], the invalid attempts should be distinguished from the state transition. Thus, we introduce the matrix associated to with dummy variables x. The is defined as follows:(i)For 0 ≤ j ≤ , , j +1 = P_{iw}, and , j = x(1 − P_{iw}).(ii)For , , j = 1.
Note that if we take x = 1. Then, to describe invalid attempts among the state transitions, we can define
Let . Then we can obtain
Considering that when x equals 1, based on (13), can also be expressed bywhere the coefficient denotes the probability the system reaches state with q unsuccessful subtask assignment attempts after step transitions from slot 1. So, is the probability generating function of the unsuccessful number of subtask assignment attempts within .
Let E[U] be the average unsuccessful number of subtask assignment attempts within . We can obtain
In this section, “dummy variable ” is introduced to calculate the number of unsuccessful assigned attempts. During the state transition, based on y(x) in (12), all unsuccessful assignments are labeled by the dummy variable . To calculate the unsuccessful assignment number, we use the properties of generation function. Hence, we use in (13) to guarantee the coefficient of each term including dummy variable is in the range (0, 1) based on generation function format. Then, (13) can be rewritten as (14), which is the expression of the probability generating function for the unsuccessful number. Based on the properties of generation function, the average unsuccessful number of subtask assignment attempts is equivalent to the first derivative of at the point . Here, the invalid number of subtask assignment attempts is analyzed accurately, which is helpful to evaluate the resource consumption of requesters due to probing potential workers.
The proposed algorithm for unsuccessful assignment attempts is described as Algorithm 2.

4.3. Analysis of Time Complexity
The complexity of proposed algorithm is computed as follows.
We first discuss computation complexity of optimal task partition. In line 7 of Algorithm 1, the solution of optimal task partition determines the complexity of Algorithm 1. Considering the allowable maximum partition number N_{max}, the practical partition number cannot exceed this threshold. Meanwhile, the primary computation is matrix multiplication based on (9), and allowable maximum partition number N_{max} limits the matrix size. Thus, the solution complexity is O((N_{max})^{3m}), where m denotes the number of slots within delay requirement. Based on (7), m is determined by the arrival rate of total mobile workers and the delay requirement of task i. Therefore, computation complexity of Algorithm 1 is O((N_{max})^{3m}), which is determined by the allowable maximum partition number, the total arrival rate, and the delay requirement.
Then, we analyze the computation complexity of unsuccessful assignment attempts. In line 2 of Algorithm 2, the calculation of y(x) determines the complexity of Algorithm 2. Based on slot number m and optimal partition obtained in Algorithm 1, the calculation complexity of Algorithm 2 is . It seems that slot number m and optimal partition determine the computation complexity of Algorithm 2. Actually, since slot number m and optimal partition are the results of Algorithm 1, the complexity of Algorithm 2 is also determined by the allowable maximum partition number, the total arrival rate, and the delay requirement.
5. Simulations
In this section, our proposed task partition policy is evaluated by extensive simulations. Compared to the fixed partition scheme and adaptive scheme in [23], our policy can realize high task service quality.
5.1. Comparison Metric
For efficient comparison, in this section, we adopt two metrics: task service quality and invalid number of subtask assignment attempts.
The first metric, task service quality, is denoted by the ratio between the real completed subtasks and the total subtasks. Obviously, the more subtasks have been finished, the better task service quality can be realized. This metric can clearly illustrate how many subtasks are accepted and executed by mobile workers within the duration . By using this metric, we can observe different methods lead to various impacts on real completed subtasks.
The second metric calculates unsuccessful number of subtask assignment attempts within delay requirement. During the process of probing potential workers, some workers do not accept subtasks. Thus, these assignment attempts are unsuccessful. This leads to useless resource consumption of requesters and workers. The larger invalid subtask assignment attempts are, the higher invalid resource consumption is. This metric can clearly illustrate how many subtask assignment attempts are not accepted by mobile workers within the duration . By using this metric, it is helpful for us to understand the resource consumption of requesters and workers in the mobile crowdsourcing system.
The simulation environments are described as follows. As the user mobility model is widely used and validated by research works [34, 35], we use the same parameters of the mobility model as in these studies. The mobile workers come following exponential distribution, and the total arrival rate is denoted by . Note that our task model fits several mobile crowdsourcing tasks including locationbased information finding tasks and content creation tasks. Here, we do the simulation based on the content creation tasks, and all task parameters are the same as in [25]. B, Ta_{i}, D_{i}, and R_{i} are all in the unit of slots. The requester and workers are able to communicate with others by cellular links or D2D links.
5.2. Simulation Results
With the total arrival rate , Figure 3 depicts the result of task crowdsourcing as the task load Ta_{i} increases from 300 to 400. In Figure 3, parameters are set as D_{i} = 100, R_{i} = 10, = 0.2, C = 6, and B = 2. As shown in Figure 3, compared to the fix partition policies, which divide the task into 3 subtasks (N = 3) and 8 subtasks (N = 8), and the adaptive scheme in [23], our partition policy can achieve higher service quality. As the total task load changes, our policy divides the task into various number of small subtasks to improve the accepting probability of mobile workers, while the fix partition method divides the task into fixed number, leading to the low accepting probability of mobile workers. Also, our partition policy is better than the adaptive scheme because the adaptive scheme does not take into consideration the mobile feature of workers. As the task load Ta_{i} increases, the service quality decreases because it is difficult for workers with limited computing capacity to complete complex subtasks.
Figure 4 depicts the result of task crowdsourcing as the delay requirement D_{i} increases from 80 to 100. In Figure 4, parameters are set as follows: R_{i} = 10, = 0.2, C = 6, B = 2, and Ta_{i} = 350. As shown in Figure 4, the service quality increases as the delay requirement D_{i} increases, because more time is permitted for workers to finish subtasks. Compared to the fix partition policies (N = 3 and 8) and the adaptive scheme, our policy achieves higher quality. Based on the change of delay requirement, our policy adjusts the length of each subtask by task partition to increase the accepting probability of mobile workers, while the fix partition method divides the task into fixed number, which does not consider the accepting probability of mobile workers. Also, our partition policy is better than the adaptive scheme since the adaptive scheme does not take into consideration the mobile feature of workers. Besides, the fix partition policy of N = 8 realizes higher service quality than that of N = 3 when the delay requirement is small. The policy of N = 8 means each subtask is smaller than that of N = 3. Intuitively, a smaller subtask is more likely to be completed within strict delay requirement. Hence, the fix partition policy of N = 8 realizes higher service quality under small delay requirement. As the delay requirement increases, the completed probability of a bigger subtask also grows. When a bigger subtask is completed, it brings more impact on the service quality. Hence, under the large delay requirement, the fix partition policy of N = 3 realizes higher service quality.
Figure 5 depicts the result of task crowdsourcing as the reward R_{i} increases. In Figure 5, parameters are set as follows: Ta_{i} = 350, D_{i} = 80, C = 6, and B = 2. As shown in Figure 5, we can see the service quality increases when the task reward R_{i} increases. The reason is that the probability that workers accept subtask becomes higher with the larger R_{i}. Our partition policy, even under the lower reward condition, can realize higher task service quality because our policy divides the task into small pieces based on the current reward to increase the worker’s accepting probability of each subtask. Furthermore, the fix partition policy of N = 8 grows sharply as the reward increases. Initially, the total reward is low and each subtask is given a small reward. Hence, the accepting probability is low and the service quality is small. When the reward increases, each subtask is corresponding to higher reward. For fix partition policy of N = 8, the accepting probability increases sharply than others, leading to higher service quality.
When total arrival rate of mobile workers changes, the task service quality of our partition policy also changes. In Figure 6, we set task load Ta_{i} = 350 and 400, respectively. Other parameters are set as follows: Ri = 10, Di = 80, C = 6, and B = 2. As shown in Figure 6, we can see the service quality increases when there is higher total arrival rate of mobile workers. The reason is that the requester will encounter more workers, and the chance for task crowdsourcing increases. When the task load Ta_{i} increases, the service quality decreases since it is difficult for workers with limited computing capacity to complete complex subtasks.
Figure 7 depicts the invalid number of subtask assignment attempts when the task load changes. As shown in Figure 7, our partition policy witnesses less invalid number of subtask assignment attempts than the fixed partition policy which divide the task into 5 subtasks (N = 5). Our partition policy is able to adjust the partition number of subtasks based on the task load and increase the accepting probability of mobile workers, while the fixed partition policy only divides task into fixed number no matter how big the task load is. Therefore, our policy realizes less invalid assignment attempts. When the reward R_{i} increases from 15 to 20, both our partition policy and the fixed partition policy experience less invalid number of subtask assignment attempts. The reason is that the workers are willing to accept subtasks with the high reward.
6. Conclusions
In this paper, a mobile crowdsourcing paradigm has been proposed for a task requester to obtain highquality results. We develop a recruitment process and propose system models. Jointly considering the mobile worker features (e.g., the worker computing capacity and mobility) and task partition, a task partition problem is formulated to maximize the service quality. To solve the optimal task partition problem, a Markov chainbased solution is developed to model and perform task partition. With this policy, the requester is able to divide a complex task into optimal number of subtasks and maximize the probability that all subtasks are accepted by workers within limited time. In addition, the invalid number of subtask assignment is precisely analyzed, which is helpful to evaluate the resource consumption of requesters due to probing potential workers. Simulations show that the proposed task partition policy improves the results of task crowdsourcing.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The work was supported in part by Key Research and Development Program of Shandong Province, China (no. 2017GGX10142) and in part by China Scholarship Fund.
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Copyright © 2019 Linbo Zhai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.