A New Stochastic Geometry Model of Coexistence of Wireless Body Sensor Networks

Abstract. Stochastic geometry, in particular Poission point process theory, has been widely used in the last decade to provide models and methods to analyze wireless networks. It is a branch of mathematics which deals with the study of random point processes. There are various models for point processes, typically based on but going beyond the classic homogeneous Poisson point process. Poisson point process cannot be used to model the spatial distribution of the simultaneously active transmitters. A novel framework has been presented for modeling the intensity of simultaneous active transmitters of a random carrier sense multiple access wireless sensor network. This thinning rule uses a second-neighbors distance-dependent method, which controls too many nodes deleted of points close together.


Introduction
WBSNs enable wireless communications between several miniaturized body sensors and a single coordinator worn on the human body. WBSNs offer many promising new applications in the area of remote health monitoring systems to measure specified physiological data and also provide location-based information. Since a WBSNs provide remote and continuous medical monitoring for patients without constraining their movements, it plays a crucial role in next generation healthcare applications [1][2].
WBSNs for health monitoring systems are required to meet stringent performance demands regarding the tradeoff between reliability, latency, and power efficiency. WBSNs feature limited range and bandwidth and they are prone to interference. These modes incur issues regarding the coexistence of multiple WBANs. There is often a group of BSNs together to happen in hospitals for patients and staff or for the elderly at nursing homes [3]. In many cases, several WBSNs coexist in a small area, resulting in very strong inter-BAN interference, which seriously disturbs intra-BAN communications [4].
In this paper, we use stochastic geometry to analyze the coexistence of IEEE802.15.4-based WBSNs Stochastic geometry is an established branch of mathematics that studies uncertainty in geometric structures. The density of effective points in model is a sufficient parameter to describe the network performance. However, The HCPP model still suffers from the underestimation problem. So we proposed a PCMP model to mitigate this problem.
We propose a new thinning method of CSMA network to mitigate the node intensity underestimation problem of HCPP model. We put forward a modified Poisson cluster and marked process (PCMP) to adapt to this network coexistence The PCMP model is generalized for CSMA protocol's network.
Define: Contention domain: node y is in the contention domain of node x if the power received by x from y is above some detection threshold.
Neighbors: the neighbors of a node are the nodes in its contention domain. Second Neighbors: the neighbors of a origin node's neighbors node. The rest of the paper is organized as follows: Section 2 introduces the related work. Section 3 proposes and describes the system model in detail. In Section 4, thinning theory and stochastic geometry are used to investigate the CSMA-based wireless sensor networks coexistence. Section 5 illustrates and analyzes numerical results. We conclude our work in Section 6.
Since the Poisson Point Process (PPP) is highly tractable, it is frequently used to model a variety of networks, such as celluar networks, mobile ad hoc networks, cognitive radio networks and wireless sensor networks [5][6]. In [7], the authors assume the star topology network's transmission nodes to be uniformly distributed according to a PPP. this paper stands in the precise consideration of CSMA algorithms, considering the collision, the success transmit probability and interference in the same network. The matern hard core are often used to model concurrent transmitters in CSMA networks [8][9]. Hesham EISawy et al. [10] adopt the Poisson cluster process to obtain the minimum required number of channels of the coexisting IEEE 802.15.4 networks, but the underestimation problem is not fully resoved.

System model
The carrier sense multiple access with collision avoidance (CSMA-CA) is widely employed in wireless networking due to its simplicity and performance efficiency [11] .
CSMA create exclusion zones to protect scheduled transmissions that nodes which are close by never transmit simultaneously. CSMA-type MACs essentially create a guard zone around the receiver [12]. The traditional homogeneous PPP doesn't suit to this coexisting IEEE 802.15.4 networks, so we proposed PCMP model.
We consider WBSNs, each contains one receiver(CN: the cord node) and N transmitters(SN: sensor node). In our proposed model, the CN is placed at center of the human body. Some SN are deployed on a human body, forming a star topology with the CN such as Figure 1. Each SN has equal power and computation capabilities. : The medium access indicators The lower the carrier-sensing threshold E the higher the sensitivity of each transmitter to other transmissions occurring in the spatial domain, which increases the distances between simultaneously active transmitters and decreases the mutual interference at the expense of decreasing the spatial frequency reuse [13].
To consider a realistic medium-access control (MAC) layer algorithm, successful signal reception is based on the protocol model, and nodes employ carrier-sense multiple access [14]. One of the best choices for such scenarios is to use pure carrier sense multiple access (CSMA), which only incorporates channel sensing. In the CSMA protocol, every communication entity first senses the ongoing transmission in the channel and then determines when to start transmitting [15]. A node which wants to access the shared wireless medium senses its occupation and refrains from transmitting if the channel is already locally occupied.
Throughout this paper, we will say that the transmitter i x is older than the transmitter j x (or equivalently j x is younger than i x ) to denote that i j m m ! .The hard core point process (HCPP) accounts only for the points having the lowest mark in their contention domains [16]. The classical HCPP leads to the following scenario: node 1 x is not retained because of it detects its neighbors 2 x ; moreover node 2 x in turn is not retained because it detects its neighbor 3 The shaded area: According to the distance properties of a PPP: The probability of k x is the lowest mark in the A(d): The probability of the nodes retained:

Simulation results
We have distributed many transmission nodes in the 20m*20m area according to certain density. Set the contain domain R around the origin. If the closet node's distance is less than a certain value, the node with the smallest mark is retained.     We distributed many transmitter nodes in the 100 meters of area according to certain density. Set the contain domain is 2 m. If the closet node's distance less than 2, the node with the smallest mark is retained. The underestimation problem of HCPP model lead to the many of nodes is unselected, so in the Figure 4 the number of remaining nodes in PCMP model is much more than the HCPP mode. Their number of selected node is the same in the low density because the intersect probability of contain domain is very small.