 Research
 Open Access
Impact of the environment and the topology on the performance of hierarchical body area networks
 JeanMichel Dricot^{1}Email author,
 Stéphane Van Roy^{1},
 Gianluigi Ferrari^{2},
 François Horlin^{1} and
 Philippe De Doncker^{1}
https://doi.org/10.1186/168714992011122
© Dricot et al; licensee Springer. 2011
 Received: 29 October 2010
 Accepted: 7 October 2011
 Published: 7 October 2011
Abstract
Personal area networks and, more specifically, body area networks (BANs) are key building blocks of future generation networks and of the Internet of Things as well. In this article, we present a novel analytical framework for network performance analysis of body sensor networks with hierarchical (tree) topologies. This framework takes into account the specificities of the onbody channel modeling and the impact of the surrounding environment. The obtained results clearly highlight the differences between indoor and outdoor scenarios, and provide several insights on BAN design and analysis. In particular, it will be shown that the BAN topology should be selected according to the foreseen medical application and the deployment environment.
Keywords
 Sensor Node
 Medium Access Control
 Relay Node
 Medium Access Control Protocol
 Time Division Multiple Access
1. Introduction
Recent advances in ultralow power sensors have fostered the research in the field of bodycentric networks, also referred to as body area networks (BANs) [1–4]. In these networks, a set of nodes (called sensors) is deployed on the human body. They aim at monitoring and reporting several physiological values, such as blood pressure, breath rate, skin temperature, or heart beating rate. Most of the time, sensing is performed at low rates but, in emergency situations, the network load may increase in seconds. Therefore, an indepth analysis of the network outage, throughput, and achievable transmission rate can give insights on the maximum supported reporting rate and the corresponding performance.
In [5, 6], we have considered a preliminary linklevel performance analysis of BANs with centralized topologies. In the current study, we extend this approach, integrating the propagation channel characteristics and the impact of the hierarchy in a general networklevel performance analysis framework. All considered networks will have hierarchical topologies, i.e., the sensor nodes will not be directly connected to a central controller. The modeling of the BAN channel has recently been thoroughly investigated [7–11]. The main findings on the body radio propagation channel can be summarized as follows. First, the average value of the power decreases as an exponential function of the distance. However, unlike classical propagation models, where the received power is a decreasing function of the distance of the form d^{α}, the authors of [12, 13] show that a law of the form 10^{ γd } (γ < 0) characterizes more accurately body radio propagation. Second, the propagation channel is subject to distinct propagations mechanisms with respect to the location of the sensors on the body. More precisely, onbody propagation and reflections from the environment act jointly to create a particular propagation mechanism that is specific to BANs.
This article addresses the development of a specific framework for the accurate evaluation of the impact of the bodyspecific propagation and network topology. Our results are derived by means of the link throughput analysis, this metric being a traditional measure of how much traffic can be delivered, per time unit, by the network [14, 15]. Therefore, our analysis is expedient to understand the level of information which could be collected and processed in bodyrelated applications (e.g., health or fitness monitoring). Furthermore, since energy is critical in the design of autonomous BANs in the context of medical applications [16–18], an accurate evaluation of the impacts of the BAN topology and transmission rate on the energy consumption is of fundamental interest.
The slotted ALOHA multiple access scheme [19] was recently proposed by the IEEE 802.15.6 working group as one of the reference medium access control (MAC) schemes for the wireless body networks in the context of the narrowband communications [20]. In particular, in each time slot, the nodes are assumed to transmit independently with a certain fixed probability [21]. This approach is supported by the observations in [[22], p. 278] and [21, 23], where it is shown that the traffic generated by nodes using a slotted random access MAC protocol can be modeled by means of a Bernoulli distribution. In fact, in more sophisticated MAC schemes, the probability of transmission at a node can be modeled as a function of general parameters, such as queuing statistics, the queuedropping rate, the channel outage probability incurred by fading [24], the adaptation of the sampling to rate to patient's condition [25], the MAC strategy used [26], etc. Since the impact of these parameters is not the focus of the this study, the interested reader is referred to the existing literature [27–29] for further details.
The principal contributions of this article can be summarized as follows. First, a comprehensive and detailed analytic framework for BAN performance evaluation is developed, obtaining closedform expressions for the link probabilities of outage in the context of multiuser communications. This framework encompasses the effect of the environment, the topology, and the traffic intensity. Next, different topologies, corresponding to various medical applications, are characterized in terms of achievable throughput. Finally, the performance of each topology is discussed, and practical insights are given on how to instantiate a reallife BAN with respect to the application demands and propagation context. Furthermore, throughout this entire article the indoor and the outdoor environments are treated separately and properly compared.
The remainder of this article is organized as follows. In Section 2, the propagation mechanisms are introduced and characterized. In Section 3, the conditional success probability of a link transmission for a node, given the transmitterreceiver and interfererreceiver distances, is derived. In the same section, the minimum required transmit power, over a given link, in the absence of any interfering node is computed in both indoor and outdoor environments. Then, in Section 4, the tree topologies analyzed in this article are presented, and the traffic model is discussed. Finally, in Section 5, an extensive performance analysis, in terms of network throughput and energy consumption, is performed. Section 6 concludes this article.
2. Propagation mechanisms
In order to build an accurate model for the onbody propagation, a Rohde & Schwartz ZVA24 vector network analyzer was employed to capture the complexvalued frequencydomain transfer functions between 3 and 7 GHz, with a frequency step of 50 MHz. Omnidirectional Skycross SMT3T010M ultrawide band antennas were used during the entire measurement campaign. Their smallsize (13.6 mm × 16 mm × 3 mm) and low profile characteristics precisely match the body sensor requirements. These antennas were separated from the body skin by about 5 mm to ensure a return loss value S_{11} ≤ 9 dB. Finally, lowloss and phasestable cables interconnect all components, and the IFbandwidth was set to 100 Hz to enlarge the dynamic range to about 120 dB.
A. Measurements
First, the diffraction mechanism is analyzed by gradually shifting the transmitter around the body. The spatial values of the power are extracted from seven different sites separated by 4 cm each. For each level, the transmitter is also shifted one level below and one level above, and the observed measures are averaged. Second, the impact of the reflections off the surrounding environment was investigated for five positions of the transmitter around the body. Repeated measures are taken by positioning the human body on a rectangular grid of 7 × 7 position, each separated by 4 cm. This procedure is performed for a set of 20 locations in a standard office room with a surface of about 20 m^{2}.
The conclusions of this extensive measurement campaign, also highlighted in [13], can be summarized in three points. Firstly, there is propagation through the body. However, when high transmission frequencies are considered, the attenuation undergone by these waves is relevant and the corresponding contribution can be neglected.
A second mechanism corresponds to guided diffraction around the body. This mechanism is consistent with a surface wave propagation, and its properties depend on the body specific characteristics.
Finally, the last propagation contribution comes from the surrounding environment. More precisely, the third propagation mechanism originates from reflections off the body limbs (arms and legs) and the surrounding objects (walls, floor, and ceiling). Obviously, this mechanism is observed only in an indoor environment.
Based on an extensive measurement campaign, we now present accurate statistical models corresponding to the propagation mechanisms described above.
B. Onbody propagation (guided diffraction)
where P(d) is the instantaneous received power (dimension: [W]) at distance d (dimension: [m]), P is the transmit power (dimension: [W]), d_{ref} is a reference distance (dimension: [m]), L_{ref} is the gain at the reference distance (adimensional, in dB), and γ is a suitable constant (dimension: [m^{1}]). For instance, typical experimental values for these parameters are d_{ref} = 8 cm, L_{ref} = 57.42 dB, and γ = 124 dB/m [31].
C. Reflections off the environment
where ${L}^{\left(\mathsf{\text{env}}\right)}=1{0}^{{L}_{\mathsf{\text{dB}}}^{\left(\mathsf{\text{w}}\right)}\u221510}$.
D. A unified BAN propagation model
The combination of the two propagation mechanisms presented in Sections 2B and 2C allows to derive a unified propagation model for a generic BAN. It can be observed that the degree of importance of each mechanism depends on the distance between transmitter and receiver. More precisely, in close proximity, the dominant propagation mechanism is the onbody propagation described in Section 2B. Above the crossover distance d_{cross} ≈ 25 cm, the contribution of the environment becomes dominant, and the second propagation mechanism, presented in Section 2C, is the only relevant one.
Therefore, a unified propagation model can be characterized as follows:

in an outdoor environment, the average received power can be computed using (4) (i.e., $E\left[P\left(d\right)\right]\propto P1{0}^{\gamma d}$) and the instantaneous received power is determined by the lognormal fading channel model given by (5);

in an indoor environment,

if d ≤ d_{cross}, the average received power can be computed using (4) (i.e., $E\left[P\left(d\right)\right]\propto P1{0}^{\gamma d}$) and the lognormal fading in (5) is used;

if d > d_{cross}, the average received power is approximately constant (i.e., $E\left[P\left(d\right)\right]=P{L}^{\left(\mathsf{\text{env}}\right)}$) and the instantaneous received power, owing to a Rayleigh faded channel model, has the distribution given by (8).

3. Linklevel performance analysis
In this article, we consider multiuser communicationsin a BAN, all sensors need to transmit to a central controller and, in this sense, the scenario at hand can be interpreted as a multiuser scenario. The transmission over a link of interest is denoted with the subscript "0." Besides the intended transmitter, other nodes may be interfering. Depending on their distance to the receiver, the interfering nodes will be denoted differently. More precisely,

in an indoor scenario, the interferers located at distances shorter than d_{cross} are referred to as "closerange interferers," their number is indicated as N_{close}, and the generic node will be denoted with a subscript $i\in {\mathcal{N}}_{\mathsf{\text{close}}}\triangleq \left\{1,2,\dots ,{N}_{\mathsf{\text{close}}}\right\}$;

in an indoor scenario, the interferers located at distances longer than d_{cross} are referred to as "farrange interferers," their number is indicated as N_{far}, and the generic node will be denoted with a subscript $j\in {\mathcal{N}}_{\mathsf{\text{far}}}\triangleq \left\{1,2,\dots ,{N}_{\mathsf{\text{far}}}\right\}$;

in an outdoor scenario, the number of interferers is indicated as N_{out}, and the generic node will be denoted with a subscript $k\in {\mathcal{N}}_{\mathsf{\text{out}}}\triangleq \left\{1,2,\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{N}_{\mathsf{\text{out}}}\right\}$.
Assuming slotted transmissions (i.e., t can assume multiples of the slot time), a simple random access scheme is such that, at each time slot, a node transmits with probability q [[34], p. 278]. Therefore, ${\left\{{\Lambda}_{i}\left(t\right)\right\}}_{t=1}^{\infty}$, $i\in {\mathcal{N}}_{\mathsf{\text{close}}}$, ${\left\{{\Lambda}_{j}\left(t\right)\right\}}_{t=1}^{\infty}$, $j\in {\mathcal{N}}_{\mathsf{\text{far}}}$, and ${\left\{{\Lambda}_{k}\left(t\right)\right\}}_{t=1}^{\infty}$, $k\in {\mathcal{N}}_{\mathsf{\text{out}}}$ are sequences of Bernoulli RVs with $\mathbb{P}\left\{{\Lambda}_{i}\left(t\right)=1\right\}=\mathbb{P}\left\{{\Lambda}_{j}\left(t\right)=1\right\}=q$, ∀t, i, j, k.
Finally, as typical in the context of BANs, we assume that all nodes use the same transmit power, i.e., P_{ i }(0) = P_{ j }(0) = P_{ k }(0) = P_{0}(0), ∀i, j, k.
A. Link probability of success with shortrange transmission in indoor scenarios
B. Link probability of success with longrange transmission in indoor scenarios
By inserting (21) and (22) into (20), one obtains the final expression (18) for the probability of successful transmission on the link.
C. Link probability of success in outdoor scenarios
In these scenarios, the links are subject to lognormal fading, and exponential power decreases. The link probability of success can simply be derived using the derivation in Section 3A, setting N_{out} = N_{close} and N_{far} = 0 (this does not mean that there are not far interferers, but that their propagation model is simply the same of close interferers). Therefore, the computation of the link probability of success ${\mathcal{P}}^{\left(\mathsf{\text{outdoor}}\right)}$ is straightforward from (17) and the final expression is given in (19).
D. Minimum transmit powers
where N_{0} has been expressed as Tk_{b}, with T being the room temperature (dimension: [K]) and k_{b} = 1.38 × 10^{23} J/K being the Boltzman's constant, and B being the transmission bandwidth.
On the basis of the results presented in Figures 6 and 7, the following observations can be made. The value of ${\mathcal{P}}_{\mathsf{\text{th}}}$ plays a limited role on the minimum transmit power. If the transmit power is constrained by energy concerns, then only shortrange communications (some tenths of centimeters) will be possible: therefore, a multihop network architecture is to be preferred. Finally, in an indoor environment, as seen from Figure 6, the reflections from the surrounding environment make the minimum transmit power become constant when d ≥ 25 cm.
In the remainder of this study, we will consider only interferencelimited BANs, i.e., scenarios where condition (23) is satisfied. Formally, this is equivalent to assuming that N_{0} B ≪ P_{int}.
4. Tree topologies and multihop communications
A. BAN tree topologies
In [35], a preliminary performance analysis of BANs with star topologies was carried out. Indeed, these topologies are well suited for medical applications since they exhibit lowpower consumption [36] and can perform applicationspecific data aggregation [37–39]. However, in order to limit the transmit power, the use of tree (hierarchical) BAN topologies is appealing.
B. Medical applications of the tree topologies
The three topologies shown in Figure 9 are generic and suitable for a range of medical applications [40, 41]. More precisely, "Configuration A" refers to a multisensor site where highly dense clusters of nodes are deployed. This is representative of medical scenarios where intense monitoring, in a few areas of interest, is needed. Relevant medical applications are mobile EEG (ElectroEncephaloGraphy) or postoperative monitoring of localized critical health conditions.
The second configuration ("Configuration B") is more balanced and corresponds to multiple monitoring sites distributed over the body. Two typical BAN scenarios are encompassed: (i) redundant acquisitions of local physiological signals (for safety reasons) and (ii) multiple independent sensing devices, each having its own relay node (i.e., ECG (ElectroCardioGraphy) combined with limbs monitoring and motion sensors). Relevant medical applications include stroke or Parkinson's disease monitoring (through a combination of EEG, accelerometers, and a gyroscope), and cardiac arrest or ischaemic heart disease monitoring (through a combination of an ECG and a mechanoreceptor).
The third configuration ("Configuration C") is representative of a generic sensing scheme where multiple sensors are networked and distributed all over the body without local clustering. In this sense, it is representative of a star topology, as each intermediate relay is connected to a single sensing unit. A relevant medical application is given by a wearable vest with multiple sensors across it (each node may measure local blood pressure, collect electrical signals for ECG, and measure local accelerations).
C. Multihop traffic model
In this study, we consider a slotted communication model, where T_{slot} (dimension: [s]) denotes the duration of each slot. It is important to distinguish between data generation and data transmissions at the sensors. Data generation, in real applications, depends on the quantity to be measured; data transmission depends on the communication system design. We now show clearly that generation and transmission crossinfluence each other.
In other words, the above inequality shows clearly that the communication/networking technology has an impact on the features of the (medical) sensors. We remark that a careful analysis of the transmission probabilities of the (medical) sensors will more likely lead to different values of λ (and q) for each node, depending on the type of physiological constant and the congestion at the relays, among others. This analysis goes beyond the scope of this article and is the subject of future research. However, whatever the used sensors, it is possible to derive the equivalent value of λ and, therefore, rely on the proposed framework.
In fact, the condition λ > q would be equivalent to assuming that the sensor generates, per time unit, more packets than those it can actually transmit. In this case, there would be an overflow at the sensor, and packets would be lost. On the other hand, assuming λ < q is meaningless as well, it is impossible that the transmission probability of a sensor node is higher than its generation probability (what would it transmit?). Therefore, in the considered simplified model, it follows that λ = q, i.e., the generation and transmission processes coincide. Note also that, according to this model, q is equal to the pernode load (defined as the average number of packets generated during an interval equal to the duration of a packet transmission). Therefore, the network load G (adimensional) is simply equal to q · N_{tot}, where N_{tot} denotes the total number of sensor nodes in the BAN.
where ${\mathcal{P}}_{1\mathsf{\text{eaf}}\to \mathsf{\text{relay}}}$ can be either (17) or (19), in indoor or outdoor scenarios, respectively.
Finally, in multipletier topologies (more complex than the 3tier considered in this article), the same approach can be applied to compute the probability of transmission of any node acting as a relay at a given hierarchical level of the network. In the considered 3tier topologies, this approach can be straightforwardly applied to evaluate q_{sink}, i.e., the probability of transmission from the sink (e.g., through a 3G connection).
5. Networklevel performance analysis
The main simulation parameters are set as follows. With reference to the topologies in Figure 9, the distances between a leaf and its relay and between a relay and the sink are 10 and 30 cm, respectively. The SINR threshold is set to θ = 5 dB. The fading power of the lognormal propagation model is σ_{dB} = 8 dB. These values correspond to typical, multikpbs sensor nodes.
A. Performance metrics
In the following, we will consider two key performance metrics: (i) the linklevel throughput, and (ii) the energy consumption rate.
where $\mathcal{P}$ is either equal to ${\mathcal{P}}_{\mathsf{\text{far}}}^{\left(\mathsf{\text{indoor}}\right)}$, ${\mathcal{P}}_{\mathsf{\text{close}}}^{\left(\mathsf{\text{indoor}}\right)}$, or ${\mathcal{P}}^{\left(\mathsf{\text{outdoor}}\right)}$ depending on the scenario at hand. The probabilistic link throughput [42] (adimensional) of a node is defined as follows:

in the fullduplex communication case, it corresponds to the product of (i) $\mathcal{P}$ and (ii) the probability that the transmitter actually transmits (i.e., q);

in the halfduplex communication case, it corresponds to the product of (i) $\mathcal{P}$, (ii) the probability that the transmitter actually transmits (i.e., q), and (iii) the probability that the receiver actually receives (i.e., 1  q).
since it is supposed that the sink acts as a special device and can communicate with the external equipments with probability equal to one (e.g., it is used to store data on a memory card or to send these data by means of a reliable transmission technology, such as, for example, 3G).
where E_{TX} and E_{RX} are the energies (dimension: [J]) consumed by singlepacket transmission and reception acts, respectively.
where $\mathcal{E}\triangleq E\u2215{E}_{\mathsf{\text{TX}}}$ is denoted as energy depletion rate (adimensional) and corresponds to the ratio of the energy consumed by the network in a slot and the energy consumed to transmit a single packet.
We now provide the reader with a performance analysis of all the three hierarchical topologies deployed in outdoor and indoor scenarios.
B. Outdoor Scenarios
In the Configuration A (Figure 9a) the throughputs of the leaves, relays, and sink are increasing functions of q for small values of q and reach the maximum values at q ≃ 0.15. For q > 0.15, τ_{leaf} starts decreasing. In contrast, τ_{relay} remains approximately constant for q ∈ (0.15, 0.45): in fact, the packets' losses at the leaves' are compensated by data transmitted by the relay node itself, so that the overall value of τ_{relay} tends to remain stable. It can also be observed that the throughput at the sink remains approximately constant for q ∈ (0.15, 0.45), and its value is close to the maximum achievable throughput of any slotted ALOHA system without lossy links, which is e^{1} ≈ 0.37 [44]. Since the throughput at the sink can be interpreted as the overall network throughput, it can be concluded that the network Configuration A yields an excellent channel utilization at the sink node. Regarding the operational region of this configuration, it can be seen from Figure 11a that for q ≥ 0.6 one has τ_{leaf} ≃ 0, i.e., the load is too high, and the leaves tend to be disconnected from the network, i.e., packets from the leaves are no longer relayed and successfully transmitted to the sink. Consequently, for q ≥ 0.6, the throughputs at the relays and at the sink tend to decrease.
The performance with the second topologyreferred to as Configuration B (Figure 9b)is analyzed in Figure 11b. It can be observed that the throughput at the relays is, for small values of q, an increasing function of q and reaches a maximum value at q ≈ 0.1. Beyond this value, the throughput at the relays tend to rapidly decrease. On the other hand, τ_{leaf} is an increasing function till q ≈ 0.3: this is because the number of sensors per cluster (3) is smaller than the number of relays (4) and, therefore, the throughput at the leaves continues to increase even if the throughput at the relays starts decreasing. Unlike Configuration A, in Configuration B, the maximum throughput at the leaves is higher than the maximum throughput at the relays. As the number of relays is relatively large, they tend to interfere with each other and, therefore, the throughput of the sink reaches a maximum at q ≈ 0.1 and, then, decreases. It can be observed that the maximum throughput at the sink with Configuration B is approximately equal to that of Configuration A. However, unlike Configuration A, in Configuration B, there is no interval of q where the throughput at the sink tends to remain constant. In other words, this configuration does not support, at network level, a larger interval of values of q.
The last schemedenoted as Configuration C (Figure 9c)is highly centralized. Its performance is investigated in Figure 11c. As each cluster contains only one leaf node, τ_{leaf} is the highest. On the other hand, the relays interfere with each other while communicating to the sink and, therefore, τ_{relay} remains very low (its maximum value is around 0.05). As a consequence, τ_{sink}, after reaching its maximum value for q ≈ 0.08 (similarly the previous configuration), tends to decrease to zero much faster than in Configuration B. Note that the maximum value of the throughput at the leaves is close to the maximum value of the throughput at the sink. Finally, note that for q ≥ 0.5, even if τ_{leaf} is high, τ_{sink} is basically zero: in other words, no data transmitted by the leaves can be successfully transmitted by the sink to an external controller (e.g., through 3G communications).
C. Throughput in indoor scenarios
Regarding Configuration A, it is seen from Figure 12a that the leaves can support a wide range of values of q (i.e., the throughput is nonzero for any value q ∈ (0, 0.6)). As anticipated in the description of the results in outdoor scenarios, the relays effectively accumulate the leaves' and their own data, guaranteeing the highest throughput almost for all values of qfor very low values of q, τ_{relay} < τ_{sink}. However, the last links (i.e., the relaytosink links) are subject to strong interference because of the reflections off the surrounding environment, and the sink throughput is much lower than in the outdoor scenario. More precisely, the throughput reaches a maximum at q ≈ 0.05 and becomes insignificant for q ≥ 0.3.
The performance of Configuration B is presented in Figure 12b. Since the tree is more balanced than in Configuration A (i.e., there are less leaves and more relays), the performance observed at the leaves is better in terms of throughput. However, the increase of the amount of relay nodes and the fact that these are more subject to environment interference (since these are considered as long links) make the throughput decrease significantly. Finally the throughput at the sink remains limited, compared to the outdoor scenario, for the reasons described previously in analysis of the Configuration A.
The third configurationnamely, Configuration Cis shown in Figure 12c. In this configuration, the throughput at the leaves is significant. This could have been expected by taking into account the facts that (i) the links are shorts and, therefore, nearly not subject to interference; and (ii) the amount of concurrent transmissions remains limited. Since there are numerous relay nodes, the throughput at the relays is very low, because of the presence of multiple access collisions. Furthermore, the reflections off the environment reduce drastically the throughput at the sink when the relay probability of transmission increases. It can be observed that the value of τ_{sink} rapidly reaches a maximum for q ≈ 0.05, before decreasing rapidly for increasing values of the parameter q.
D. Energy depletion rate
E. An illustrative comparison with TDMAbased architectures
In this article, we refer to the IEEE 802.15.6 standard and the slotted ALOHA MAC. Owing to its random access strategy, this MAC protocol could be of less interest if the traffic is high. In that case, a centralized, time division multiple access (TDMA) access might be more interesting and appealing. In this subsection, we provide the reader with an illustrative comparison between the slotted ALOHA and TDMA schemes, focusing on the delay exhibited by both strategiesin fact, the throughput of a TDMA scheme is equal to 1.
where the link probability of success P_{ s } depends on the scenario (i.e., outdoor or indoor, position of the nodes, etc.) and is given by relations (17), (18), and (19). Finally, the delay, expressed in number of time slots, is ${D}_{\mathsf{\text{ALOHA}}}=E\left[r\right]=1\u2215{P}_{s}$.
To summarize, as a TDMAbased scheme has a throughput τ = 1, it becomes very attractive for values of q beyond the maximum of the considered slotted ALOHA system, as the latter becomes unstable, i.e., the value of the delay D_{ALOHA} → ∞ since P_{ s } → 0. In scenarios with low reporting rate, the slotted ALOHA scheme is to be preferred.
6. Conclusions
In this article, we have presented an analytic framework for the evaluation of the link probability of success in interferencelimited BANs subject to fading. This analytic derivation is based on novel experimental measurements which highlight two characteristic propagation mechanisms in BANs deployed in indoor scenarios: onbody propagation, and propagation through reflections from the environment.
Regarding the impact of the topology, three configurations have been analyzed. These analyses showed significantly different performances, in terms of pernode throughput and energy consumption rate. It can be concluded that a decentralized topology in an outdoor environment presents the best tradeoff between the throughputs at the leaves, the relays, and the sink. Moreover, the shape of the throughput curves shows that it is very stable, i.e., any increase or decrease of the node generation rate does not significantly impact the pernode throughput, and it is the most efficient in terms of energy consumption rate. In an indoor environment, the balanced tree seems to be more suitable. Indeed, it presents a higher throughput for the sink, the relays, and the leaves at the same time.
In conclusion, multiuser BANs deployment and operation should take into account the specificities of environment and adapt the routing algorithms and clustering strategies accordingly. In the particular context of an indoor environment or when the traffic load is high, narrowbandshared spectrum techniques are not performant, and other MAC schemes, such as TDMA should be considered instead, even if they are more complex to implement.
Appendix
Coefficients for the approximation of the ζ function
c _{1}  a _{1}  c _{2}  a _{2}  c _{3}  a _{3}  Residual  

σ = 4  0.49  0.75  0.49  0.75  0.03  0.16  4.68 × 10^{5} 
σ = 6  0.38  0.31  0.56  1.21  0.06  0.07  4.23 × 10^{6} 
σ = 8  0.59  1.32  0.34  0.18  0.06  0.02  1.04 × 10^{4} 
σ = 10  0.29  0.09  0.65  1.17  0.05  0.01  7.53 × 10^{4} 
σ = 12  0.04  0  0.24  0.04  0.70  0.93  3.52 × 10^{3} 
σ = 14  0.20  0.01  0.03  0  0.72  0.64  1.03 × 10^{2} 
σ = 16  0.18  0.01  0.70  0.49  0.04  0  1.67 × 10^{2} 
Endnotes
^{a}Note that, even though (3) holds for d ≥ d_{ref}, L_{0} can be intuitively interpreted as the (extrapolated) gain (adimensional, linear scale) at distance d = 0. In other words, L_{0} takes into account the loss due to antenna emission.
^{b}Note that we use the log_{10} variant of the lognormal, since the widely used shadowing model uses an additive Gaussian variation expressed in dB.
^{c}Note also that, with a slight abuse of notation, in (23) and (24), we indicate by ζ^{1}(·) the inverse of ζ(z; σ) with respect to ζ, with the implicit assumption that σ is fixed.
^{d}We assume that packet generation is at the beginning of a slot.
^{e}Note that this expression does not depend on q, as in TDMA systems, a leaf needs to wait for its assigned time slot.
Declarations
Acknowledgements
This study is supported by the Belgian National Fund for Scientific Research (FRSFNRS grants and FRIA grants).
Authors’ Affiliations
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