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AP-Assisted Online Task Assignment Algorithms for Mobile Crowdsensing

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Abstract

Mobile crowdsensing has become a new way to perceive and collect information due to the widespread of smart devices. In this paper, we study the task assignment problem in mobile crowdsensing systems, which is aimed to reducing the average and largest makespan of all tasks. We consider scenarios where task requester needs the help of mobile users for task completion when they encounter directly or through AP cloud (i.e., several APs connected via wired/wireless links) in an opportunistic manner. We describe the mobile crowdsensing system and formulate the problems under study. We first derive the conditional expected encountering time between requester and different users by jointly considering the opportunities via direct encountering and indirect encountering via AP cloud. Then we propose an AP-assisted average makespan sensitive online task assignment (AP-AOTA) algorithm and an AP-assisted largest makespan sensitive online task assignment (AP-LOTA) algorithm. We present detailed design for both algorithms. We deduce the computational complexities of both algorithms to be O(mn2), where m represents the number of tasks and n represent the number of users. We conduct simulations on a real trace data set and a synthetic trace data set and the results show that our proposed algorithms significantly outperform existing work.

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Notes

  1. For this task re-assignment strategy to work smoothly, the AP cloud should have a task re-assignment module

  2. The AP-AOTA-SIM algorithm here is exactly the same as the algorithm presented in [17] for solving the MAM problem. Similarly, the AP-LOTA-SIM algorithm in the next paragraph is exactly the same as the algorithm presented in [17] for solving the MLM problem.

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Acknowledgements

This work was supported in part by the NSF of China under Grant Nos. 61872331, 61872031, 61471339, the Natural Sciences and Engineering Research Council (NSERC) of Canada (Discovery Grant RGPIN-2018-03792), and the InnovateNL SensorTECH Grant 5404-2061-101.

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Authors and Affiliations

  1. Research Center of Ubiquitous Sensor Networks, University of Chinese Academy of Sciences, Beijing, 100049, China

    Shuo Peng, Wei Gong & Baoxian Zhang

  2. School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing, 100044, China

    Yongxiang Zhao

  3. Faculty of Engineering and Applied Science, Memorial University, St. John’s, NL, A1B 3X5, Canada

    Cheng Li

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  1. Shuo Peng
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  2. Wei Gong
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  3. Baoxian Zhang
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  4. Yongxiang Zhao
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  5. Cheng Li
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Correspondence to Baoxian Zhang.

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Appendix

Appendix

In this appendix, we present the detailed derivation process of Eqs. 6 and 7.

1.1 A. Derivation Process of (6)

Expand Eq. 6, we have

$$ \begin{array}{@{}rcl@{}} EST_{i} &=& {\int}_{0}^{+\infty}f(x_{3})dx_{3}{\int}_{0}^{x_{3}}(x_{1} + E(x_{2}))f(x_{1})dx_{1} \\ &&+ {\int}_{0}^{+\infty}f(x_{1})dx_{1}{\int}_{0}^{x_{1}}x_{3}f(x_{3})dx_{3}. \\ \end{array} $$
(14)

Combine Eqs. 4, 5 and 14, we can get

$$ \begin{array}{@{}rcl@{}} EST_{i} &\!=& {\int}_{0}^{+\infty}\lambda_{3}e^{-\lambda_{3}x_{3}}dx_{3}{\int}_{0}^{x_{3}}(x_{1} + \frac{1}{\lambda_{2}})\lambda_{1}e^{-\lambda_{1}x_{1}}dx_{1} \\ &&+ {\int}_{0}^{+\infty}\lambda_{1}e^{-\lambda_{1}x_{1}}dx_{1}{\int}_{0}^{x_{1}}x_{3}\lambda_{3}e^{-\lambda_{3}x_{3}}dx_{3} \\ &=& {\int}_{0}^{+\infty}\lambda_{3}e^{-\lambda_{3}x_{3}}dx_{3}{\int}_{0}^{x_{3}}x_{1}\lambda_{1}e^{-\lambda_{1}x_{1}}dx_{1} \\ &&+ \frac{1}{\lambda_{2}} {\int}_{0}^{+\infty}\lambda_{3}e^{-\lambda_{3}x_{3}}dx_{3}{\int}_{0}^{x_{3}} \lambda_{1}e^{-\lambda_{1}x_{1}}dx_{1} \\ &&+ {\int}_{0}^{+\infty}\lambda_{1}e^{-\lambda_{1}x_{1}}dx_{1}{\int}_{0}^{x_{1}}x_{3}\lambda_{3}e^{-\lambda_{3}x_{3}}dx_{3} \\ &=& {\int}_{0}^{+\infty}\lambda_{3} e^{-\lambda_{3}x_{3}}dx_{3} (1 - \lambda_{1}x_{3}e^{-\lambda_{1}x_{3}} - e^{-\lambda_{1}x_{3}}) \\ &&+ \frac{1}{\lambda_{2}}{\int}_{0}^{+\infty}\lambda_{3} e^{-\lambda_{3}x_{3}}dx_{3} * (1 - e^{-\lambda_{1}x_{3}}) \\ &&+ {\int}_{0}^{+\infty}\lambda_{1} e^{-\lambda_{1}x_{1}}dx_{1} (1 - \lambda_{3}x_{1}e^{-\lambda_{3}x_{1}} - e^{-\lambda_{3}x_{1}}) \\ &=&\frac{1}{\lambda_{1}}-\frac{\lambda_{3}}{(\lambda_{1}+\lambda_{3})^{2}}-\frac{\lambda_{3}}{\lambda_{1}(\lambda_{1}+\lambda_{3})}+\frac{1}{\lambda_{2}}*\frac{1}{\lambda_{1}+\lambda_{3}}\\ &&+ \frac{1}{\lambda_{3}}-\frac{\lambda_{1}}{(\lambda_{1}+\lambda_{3})^{2}}-\frac{\lambda_{1}}{\lambda_{3}(\lambda_{1}+\lambda_{3})} \\ &=& (1 + \frac{1}{\lambda_{2}}) * \frac{1}{\lambda_{1} + \lambda_{3}}. \end{array} $$
(15)

1.2 B. Derivation Process of (7)

Expand Eq. 7, we have

$$ \begin{array}{@{}rcl@{}} ERT_{i} &=& {\int}_{0}^{+\infty}f(x_{3})dx_{3}{\int}_{0}^{x_{3}}(E(x_{1})+x_{2})f(x_{2})dx_{2} \\ &&+ {\int}_{0}^{+\infty}f(x_{2})dx_{2}{\int}_{0}^{x_{2}}x_{3}f(x_{3})dx_{3}. \end{array} $$
(16)

Combine Eqs. 4, 5 and 16, we can get

$$ \begin{array}{@{}rcl@{}} ERT_{i} &=& {\int}_{0}^{+\infty}\lambda_{3}e^{-\lambda_{3}x_{3}}dx_{3}{\int}_{0}^{x_{3}}(\frac{1}{\lambda_{1}} + x_{2})\lambda_{2}e^{-\lambda_{2}x_{2}}dx_{2} \\ &&+ {\int}_{0}^{+\infty}\lambda_{1}e^{-\lambda_{2}x_{2}}dx_{2} {\int}_{0}^{x_{2}}x_{3}\lambda_{3}e^{\lambda_{3}x_{3}}dx_{3} \\ &=& {\int}_{0}^{+\infty}\lambda_{3}e^{-\lambda_{3}x_{3}}dx_{3} {\int}_{0}^{x_{3}} x_{2} \lambda_{2}e^{-\lambda_{2}x_{2}}dx_{2} \\ &&+ \frac{1}{\lambda_{1}} {\int}_{0}^{+\infty}\lambda_{3}e^{-\lambda_{3}x_{3}}dx_{3} {\int}_{0}^{x_{3}}\lambda_{2}e^{-\lambda_{2}x_{2}}dx_{2} \\ &&+ {\int}_{0}^{+\infty}\lambda_{2} e^{-\lambda_{2} x_{2}}dx_{2} {\int}_{0}^{x_{2}}x_{3}\lambda_{3}e^{\lambda_{3}x_{3}}dx_{3} \end{array} $$
(17)
$$ \begin{array}{@{}rcl@{}} \qquad&=& {\int}_{0}^{+\infty}\lambda_{3} e^{-\lambda_{3}x_{3}}dx_{3} (1 - \lambda_{2}x_{2}e^{-\lambda_{2}x_{3}} - e^{-\lambda_{2}x_{3}}) \\ &&+ \frac{1}{\lambda_{1}} {\int}_{0}^{+\infty}\lambda_{3}e^{-\lambda_{3}x_{3}}dx_{3} (1 - e^{\lambda_{2}x_{3}}) \\ &&+ {\int}_{0}^{+\infty}\lambda_{2}e^{-\lambda_{2}x_{2}}dx_{2} (1 - \lambda_{3}x_{2}e^{-\lambda_{3}x_{2}} - e^{-\lambda_{3}x_{2}}) \\ \qquad&=& \frac{1}{\lambda_{2}}-\frac{\lambda_{3}}{(\lambda_{2}+\lambda_{3})^{2}}-\frac{\lambda_{3}}{\lambda_{2}(\lambda_{2}+\lambda_{3})}+\frac{1}{\lambda_{1}}*\frac{1}{\lambda_{2}+\lambda_{3}}\\ &&+ \frac{1}{\lambda_{3}}-\frac{\lambda_{2}}{(\lambda_{2}+\lambda_{3})^{2}}-\frac{\lambda_{2}}{\lambda_{3}(\lambda_{2}+\lambda_{3})}\\ \qquad&=& (1 + \frac{1}{\lambda_{1}}) * \frac{1}{\lambda_{2} + \lambda_{3}}. \end{array} $$

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Peng, S., Gong, W., Zhang, B. et al. AP-Assisted Online Task Assignment Algorithms for Mobile Crowdsensing. Mobile Netw Appl 25, 1694–1707 (2020). https://doi.org/10.1007/s11036-020-01579-3

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