Optimal Data Transmission Strategy for Healthcare-Based Wireless Sensor Networks: A Stochastic Differential Game Approach

Abstract

As the development of information and communication technology (ICT) and the affirmation of the importance of healthcare, using wireless sensor networks (WSNs) is a promising approach to assist modern medical practices. The transmission optimization of HWSNs is an issue worth studying deeply, which is facing challenges such as data diversity, real-time requirement, reliable transmission, dynamic environment and so on. This paper considers four kinds of transmission cost comprehensively, and adopts the stochastic differential game theory to discuss this issue. With the objective of minimizing the transmission cost, three kinds of game models are constructed, i.e., cooperative model, partial cooperative model and non-cooperative model. The optimal transmission strategies under different game modes are obtained for HWSNs. The numerical simulation compares three strategies and verifies the validity of the method present in this paper.

Appendices

Appendix 1: Solution for the Cooperative Game Model (6)–(10)

Solution Let

$$\pi_{i} = \left( {\eta + \omega } \right)r_{i} - \omega ,$$

(22)

and

$$\lambda_{i} = \left( {1 + \beta + \gamma } \right) - \left( {\beta + \gamma c_{i} } \right)r_{i} .$$

(23)

Differentiating the r.h.s. of (12) with respect to $v_{i} \left( t \right)$ and equating it to zero leads to the following optimal strategy

$$v_{i}^{G} = \left[ {{{\pi_{i} } \mathord{\left/ {\vphantom {{\pi_{i} } {\left( {\alpha \lambda_{i} } \right)}}} \right. \kern-0pt} {\left( {\alpha \lambda_{i} } \right)}}} \right] \cdot F^{\prime}\left( {G,s,t} \right).$$

(24)

Let

$$\theta_{i} = {{\pi_{i} } \mathord{\left/ {\vphantom {{\pi_{i} } {\left( {\alpha \lambda_{i} } \right)}}} \right. \kern-0pt} {\left( {\alpha \lambda_{i} } \right)}}.$$

(25)

Then, (22) can be simplified as

$$v_{i}^{G} = \theta_{i} F^{\prime}\left( {G,s,t} \right) .$$

(26)

Substituting (26) into (12) gives

$$\begin{aligned} \rho F\left( {G,s,t} \right) & = \hbox{min} \left\{ {\sum\limits_{i \in G} {\left( {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left[ {\theta_{i} F^{\prime}\left( {G,s,t} \right)} \right]^{2} \left[ {1 + \beta \left( {1 - r_{i} } \right) + \gamma \left( {1 - c_{i} } \right)r_{i} } \right] + \omega s\left( t \right)} \right)} } \right. \\ & \left. \quad + \,F^{\prime}\left( {G,s,t} \right) \,{ \times \,\left[ {\mu s(t) - \eta \sum\limits_{i \in G} {r_{i} \left[ {\theta_{i} F^{\prime}\left( {G,s,t} \right)} \right]} + \omega \sum\limits_{i \in G} {\left( {1 - r_{i} } \right)\left[ {\theta_{i} F^{\prime}\left( {G,s,t} \right)} \right]} } \right]} \right\}. \\ \end{aligned}$$

(27)

Since $F\left( {G,s,t} \right)$ is a liner function, differentiating it with respect to $s\left( t \right)$ leads to

$$F^{\prime}\left( {G,s,t} \right) = {{n\omega } \mathord{\left/ {\vphantom {{n\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}.$$

(28)

Substituting (28) into (26) gives the optimal transmission rate allocation strategy, that is,

$$v_{i}^{G} = {{n\omega \theta_{i} } \mathord{\left/ {\vphantom {{n\omega \theta_{i} } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}.$$

(15)

Arranging (12), there is

$$\begin{aligned} F\left( {G,s,t} \right)\; & = \rho^{ - 1} \left\{ {\sum\limits_{i \in G} {\left( {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {v_{i}^{G} } \right)^{2} + \beta \left( {1 - r_{i} } \right)\left[ {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {v_{i}^{G} } \right)^{2} } \right]} \right.} } \right. \\ & \quad \left. { +\,\gamma \left( {1 - c_{i} } \right)p_{i} \left[ {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {v_{i}^{G} } \right)^{2} } \right] + \omega s^{G} } \right) + {{n\omega } \mathord{\left/ {\vphantom {{n\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}} \\ & \left. \quad { \times \left[ {\mu s^{G} + \sum\limits_{i \in G} {\left[ {\omega \left( {1 - r_{i} } \right) - \eta r_{i} } \right]v_{i}^{G} } } \right]} \right\} \\ & = \rho^{ - 1} \left\{ {\sum\limits_{i \in G} {\left( {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {v_{i}^{G} } \right)^{2} \left[ {1 + \beta \left( {1 - r_{i} } \right) + \gamma \left( {1 - c_{i} } \right)r_{i} } \right] + \omega s^{G} } \right)} + {{n\omega } \mathord{\left/ {\vphantom {{n\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}} \right. \\ & \left. \quad { \times \left( {\mu s^{G} - \sum\limits_{i \in G} {\left[ {\left( {\omega + \eta } \right)r_{i} - \omega } \right]v_{i}^{G} } } \right)} \right\} \\ & = \rho^{ - 1} \left\{ {\sum\limits_{i \in G} {\left[ {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {v_{i}^{G} } \right)^{2} \lambda_{i} + \omega s^{G} } \right]} + {{n\omega } \mathord{\left/ {\vphantom {{n\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}} \times \left( {\mu s^{G} - \sum\limits_{i \in G} {\pi_{i} v_{i}^{G} } } \right)} \right\}. \\ \end{aligned}$$

(29)

Substituting (15) into (29) gives

$$\begin{aligned} F\left( {G,s,t} \right)\, & = \rho^{ - 1} \left\{ {\sum\limits_{i \in G} {\left( {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left[ {{{n\omega \theta_{i} } \mathord{\left/ {\vphantom {{n\omega \theta_{i} } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}} \right]^{2} \lambda_{i} + \omega s^{G} } \right)} } \right. \\ & \left. {\quad +\,{{n\omega } \mathord{\left/ {\vphantom {{n\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}} \times \left( {\mu s^{G} - \sum\limits_{i \in G} {\pi_{i} \left[ {{{n\omega \theta_{i} } \mathord{\left/ {\vphantom {{n\omega \theta_{i} } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}} \right]} } \right)} \right\}. \\ \end{aligned}$$

(30)

Then, the minimized cost is got as

$$F\left( {G,s,t} \right) = \left[ {{{n\omega } \mathord{\left/ {\vphantom {{n\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}} \right]^{2} \left\{ { - \left( {2\rho } \right)^{ - 1} \sum\limits_{i \in G} {\pi_{i} \theta_{i} } + \left( {n\omega } \right)^{ - 1} \left( {\rho - \mu } \right)s^{G} } \right\} .$$

(16)

This completes the solution of the cooperative game model (6)–(10).

Appendix 2: Solution for the Non-Cooperative Game Model (7)–(10)

Solution Differentiating the r.h.s. of (13) with respect to $v_{i} \left( t \right)$ and equating it to zero leads to the following optimal strategy

$$v_{i}^{N} = \theta_{i} F^{\prime}\left( {N,s,t} \right).$$

(31)

Substituting (31) into (13) gives

$$\begin{aligned} \rho F\left( {N,s,t} \right)\, & = {\text{min}}\left\{ {\left( {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left[ {\theta_{i} F^{\prime}\left( {N,s,t} \right)} \right]^{2} \left[ {1 + \beta \left( {1 - r_{i} } \right) + \gamma \left( {1 - c_{i} } \right)r_{i} } \right]+ \omega s\left( t \right)} \right)} \right. \\ & \left. \quad {+\,F^{\prime}\left( {N,s,t} \right) \times \left[ {\mu s(t) - \eta \sum\limits_{i \in N} {r_{i} \left[ {\theta_{i} F^{\prime}\left( {N,s,t} \right)} \right]} + \omega \sum\limits_{i \in N} {\left( {1 - r_{i} } \right)\left[ {\theta_{i} F^{\prime}\left( {N,s,t} \right)} \right]} } \right]} \right\}. \\ \end{aligned}$$

(32)

Since $F\left( {N,s,t} \right)$ is a liner function, differentiating it with respect to $s\left( t \right)$ leads to

$$F^{\prime}\left( {N,s,t} \right) = {\omega \mathord{\left/ {\vphantom {\omega {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}.$$

(33)

Substituting (33) into (31) gives the optimal transmission rate allocation strategy, that is,

$$v_{i}^{N} = {{\omega \theta_{i} } \mathord{\left/ {\vphantom {{\omega \theta_{i} } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}.$$

(18)

Arranging (13), there is

$$F\left( {N,s,t} \right) = \rho^{ - 1} \left\{ {\left[ {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {v_{i}^{N} } \right)^{2} \lambda_{i} +\omega s^{N} } \right] + {{n\omega } \mathord{\left/ {\vphantom {{n\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}} \times \left( {\mu s^{N} - \sum\limits_{i \in N} {\pi_{i} v_{i}^{N} } } \right)} \right\}.$$

(34)

Substituting (18) into (34) gives

$$\begin{aligned} F\left( {N,s,t} \right)\, & = \rho^{ - 1} \left\{ {\left( {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left[ {{{\omega \theta_{i} } \mathord{\left/ {\vphantom {{\omega \theta_{i} } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}} \right]^{2} \lambda_{i} +\omega s^{N} } \right)} \right. \\ & \quad \left. { + \,{\omega \mathord{\left/ {\vphantom {\omega {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}} \times \left( {\mu s^{N} - \sum\limits_{i \in N} {\pi_{i} \left[ {{{\omega \theta_{i} } \mathord{\left/ {\vphantom {{\omega \theta_{i} } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}} \right]} } \right)} \right\}. \\ \end{aligned}$$

(35)

Then, the minimized cost is got as

$$\begin{aligned} F\left( {N,s,t} \right) & = \left[ {{\omega \mathord{\left/ {\vphantom {\omega {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}} \right]^{2} \\ & \quad \times \left\{ { - \left( {2\rho } \right)^{ - 1} \pi_{i} \psi_{i} - \left( \rho \right)^{ - 1} \sum\limits_{j \in N,j \ne i} {\pi_{j} \psi_{j} } + \omega^{ - 1} \left( {\rho - \mu } \right)s^{N} } \right\}.\end{aligned}$$

(19)

This completes the solution of the non-cooperative game model (7)–(10).

Appendix 3: Solution for the Partial Cooperative Game Model (8)–(10)

Differentiating the r.h.s. of (14) with respect to $v_{i} \left( t \right)$ and equating it to zero leads to the following optimal strategy

$$v_{i}^{K} = \theta_{i} F^{\prime}\left( {K,s,t} \right) .$$

(36)

Substituting (36) into (14) gives

$$\begin{aligned} \rho F\left( {K,s,t} \right)\, & = {\text{min}}\left\{ {\sum\limits_{i \in K} \left( {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left[ {\theta_{i} F^{\prime}\left( {K,s,t} \right)} \right]^{2} \left[ {1 + \beta \left( {1 - r_{i} } \right) + \gamma \left( {1 - c_{i} } \right)r_{i} } \right]+\omega s\left( t \right)} \right)} \right. \\ & \quad \left. +\,{F^{\prime}\left( {K,s,t} \right) \times \left[ {\mu s(t) - \eta \sum\limits_{i \in K} {r_{i} \left[ {\theta_{i} F^{\prime}\left( {K,s,t} \right)} \right]} + \omega \sum\limits_{i \in K} {\left( {1 - r_{i} } \right)\left[ {\theta_{i} F^{\prime}\left( {K,s,t} \right)} \right]} } \right]} \right\}. \\ \end{aligned}$$

(37)

Since $F\left( {K,s,t} \right)$ is a liner function, differentiating it with respect to $s\left( t \right)$ leads to

$$F^{\prime}\left( {K,s,t} \right) = {{k\omega } \mathord{\left/ {\vphantom {{k\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}.$$

(38)

Substituting (38) into (36) gives the optimal transmission rate allocation strategy, that is,

$$v_{j}^{K} = {{k\omega \theta_{j} } \mathord{\left/ {\vphantom {{k\omega \theta_{j} } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}.$$

(20)

Arranging (14), there is

$$F\left( {K,s,t} \right) = \rho^{ - 1} \left\{ {\sum\limits_{j \in K} {\left[ {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {v_{j}^{K} } \right)^{2} \lambda_{j} +\omega s^{K} } \right]} + {{k\omega } \mathord{\left/ {\vphantom {{k\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}} \times \left( {\mu s^{K} - \sum\limits_{j \in K} {\pi_{j} v_{j}^{K} } } \right)} \right\}.$$

(39)

Substituting (20) into (39) gives

$$\begin{aligned} F\left( {K,s,t} \right)\, & = \rho^{ - 1} \left\{ {\sum\limits_{i \in K} {\left( {\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left[ {{{k\omega \theta_{i} } \mathord{\left/ {\vphantom {{k\omega \theta_{i} } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}} \right]^{2} \lambda_{i} +\omega s^{K} } \right)} } \right. \\ & \left. \quad +\,{ {{k\omega } \mathord{\left/ {\vphantom {{k\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}} \times \left( {\mu s^{K} - \sum\limits_{i \in G} {\pi_{i} \left[ {{{k\omega \theta_{i} } \mathord{\left/ {\vphantom {{k\omega \theta_{i} } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}} \right]} } \right)} \right\}. \\ \end{aligned}$$

(40)

Then, the minimized cost is got as

$$\begin{aligned} F\left( {K,s,t} \right)\, & = \left[ {{{k\omega } \mathord{\left/ {\vphantom {{k\omega } {\left( {\rho - \mu } \right)}}} \right. \kern-0pt} {\left( {\rho - \mu } \right)}}} \right]^{2} \\ & \quad\times \left\{ { - \left( {2\rho } \right)^{ - 1} \sum\limits_{i \in K} {\pi_{i} \theta_{i} } - \left( \rho \right)^{ - 1} \sum\limits_{i \in G\backslash K} {\pi_{i} \theta_{i} } + \left( {k\omega } \right)^{ - 1} \left( {\rho - \mu } \right)s^{K} } \right\}. \\ \end{aligned}$$

(21)

This completes the solution of the partial cooperative game model (8), (9), (10).