## 4 Our Work
### System Model  
A network is built with \(n\) bio sensor nodes, and single sink node which acts as network coordinator and accumulates the data from the sensor nodes to communicate to the remote medical server. Two nodes are assumed to be implanted inside human body near heart and right knee and rest (n-2) wearable nodes are assumed to be placed on human body (discussed in Section 5.2). The nodes in the network transmit data with transmission power \(P_{tx}\) governed by the proposed transmission  
Figure 3: Structure of State transition matrix (P) and corresponding reward matrix (R) for m system states subject to an action (\(a_{t}\)). strategy following MDP [15] that is elaborated in the following subsection. Intra-BAN communications are carried out via electromagnetic radio frequency (RF) wave. The radiation absorbed by human tissue is quantified in terms of SAR (Watts/Kg). SAR is evaluated for each node in the network due to performing network activities. SAR assessment is based on the effective distance '\(d\)' from each node location to the reference point in the human tissue. If the reference point is located at the near field region with respect to a node, SAR is evaluated as follows [32]  
\[SAR=\frac{\sigma}{\rho}\frac{\mu\omega}{\sqrt{\sigma^{2}+\epsilon_{1}^{2} \omega^{2}}}(\frac{lIsin\theta}{4\pi}e^{-ad}(\frac{1}{d^{2}}+\frac{|\gamma_{1 }|}{d}))^{2} \tag{4}\]  
Here \(\sigma,\epsilon_{1},\mu\) represent conductivity (S/m), permittivity (F/m) and permeability (H/m) of the medium, respectively. \(\gamma_{1}\) is the complex propagation constant. \(\alpha\) is the attenuation constant and \(\rho\) is the density of the medium. \(l\) is the dipole length and current \(I\) is uniform and varies sinusoidally with time. However, if the location of the reference point is at far field region with respect to a node, SAR is formulated as follows [32]  
\[SAR=\frac{\sigma}{\rho}(|\eta||\gamma_{1}|\frac{Idlsin\theta}{4\pi d}e^{-ad})^ {2} \tag{5}\]  
where \(\eta\) is complex intrinsic impedance defined as \(\eta=\frac{\gamma_{1}}{\pi+\mu\omega_{1}}\). A discrete-time model has been taken into account where time is slotted in intervals of unit length. Each node can generate and transmit a single data packet per time slot. Data transmission between source and sink is described as event occurrence that is defined with correlated, two-state process. The remaining energy of each node is classified into several levels \(L_{t}\) depending on predefined range (to be obtained empirically) such that at any time slot t remaining energy of each node belongs to one of the defined energy levels \(L_{t}\epsilon\{0,1,2,3,4...N\}\). If an event is generated in the present slot, probability of generation of another event (respectively, no event) in the next slot is given by \(p_{on}\) (respectively, \(1-p_{on}\)) where \(0.5<p_{on}<1\)[12]. If no event is generated in the present slot, an event is generated (respectively, not generated) in the next slot with probability \(1-p_{off}\) (respectively, \(p_{off}\)) where \(0.5<p_{off}<1\)[12]. However, in both cases the probability of event generation and no event at present time slot are two complementary as well as equally likely outcomes. Hence, the value of \(p_{on}\) and \(p_{off}\) lies between 0.5 to 1. During each time slot Packet Transmission Rate (PR) is compared with a limiting value, i.e., \(PR_{th}\). If the PR reaches beyond threshold \(PR_{th}\) in the current slot, it will be greater than (respectively, less than) \(PR_{th}\) in the subsequent slot with probability \(pr_{on}\) (respectively, \(1-pr_{on}\)) where \(0.5\leq pr_{on}<1\). However, if the PR is less than the threshold \(PR_{th}\) in the present slot, it will remain less than (respectively, greater than) \(PR_{th}\) in the following slot with probability \(pr_{off}\) (respectively, \(1-pr_{off}\)) where \(0.5\leq pr_{off}<1\).