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.
Regarding the linklevel throughput, a transmission will be successful if and only if a transmission link is not in an outage. This corresponds to requiring that the (instantaneous) SINR of the link is above the threshold θ, i.e.,
\mathcal{P}=\mathbb{P}\left\{SINR>\theta \right\}
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).
The probabilistic link throughput can be interpreted as the unconditioned reception probability which can be achieved with a simple automaticrepeatrequest scheme with errorfree feedback [43]. For the slotted ALOHA transmission scheme under consideration in the context of BAN (where transmissions can typically be organized in a fullduplex way), the probabilistic throughput is
\tau \triangleq q\times \mathcal{P}.
More specifically, in our analyses, three different nodes are considered: leaves, relays, and the sink. The corresponding throughput metrics are
{\tau}_{1\mathsf{\text{eaf}}}={q}_{1\mathsf{\text{eaf}}}\times {\mathcal{P}}_{1\mathsf{\text{eaf}}\to \mathsf{\text{relay}}}=q\times {\mathcal{P}}_{1\mathsf{\text{eaf}}\to \mathsf{\text{relay}}},
(28)
{\tau}_{\mathsf{\text{relay}}}={q}_{\mathsf{\text{relay}}}\times {{\mathcal{P}}_{\mathsf{\text{relay}}\to}}_{\mathsf{\text{sink}}}.
(29)
{\tau}_{\mathsf{\text{sink}}}={q}_{\mathsf{\text{sink}}}\times 1={q}_{\mathsf{\text{sink}}},
(30)
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).
The second performance metric of interest is the energy consumption rate. The average energy consumed by the network in a slot, denoted as E (dimension: [J]), can be expressed as
E=q\phantom{\rule{0.3em}{0ex}}{N}_{1\mathsf{\text{eaves}}}{E}_{\mathsf{\text{TX}}}+{q}_{\mathsf{\text{relay}}}\phantom{\rule{2.77695pt}{0ex}}{N}_{\mathsf{\text{relays}}}\left({E}_{\mathsf{\text{TX}}}+{E}_{\mathsf{\text{RX}}}\right)+{q}_{\mathsf{\text{sink}}}\phantom{\rule{2.77695pt}{0ex}}{E}_{\mathsf{\text{RX}}}
where E_{TX} and E_{RX} are the energies (dimension: [J]) consumed by singlepacket transmission and reception acts, respectively.
In most wireless systems, E_{TX} ≈ E_{RX}, and one has
\begin{array}{lll}\hfill E& ={E}_{\mathsf{\text{TX}}}\times \left(q\phantom{\rule{0.3em}{0ex}}{N}_{1\mathsf{\text{eaves}}}+2\phantom{\rule{0.3em}{0ex}}{q}_{\mathsf{\text{relay}}}{N}_{\mathsf{\text{relays}}}+{q}_{\mathsf{\text{sink}}}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ ={E}_{\mathsf{\text{TX}}}\times \mathcal{E},\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}
(31)
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 Figure 11, the pernode throughputs at the three hierarchical levels (i.e., leaf nodes, relaying nodes, and central sink) are shown as functions of the sensors' probability of transmission q. The three subfigures refer to the three topologies in Figure 9. It can be observed that the three considered topologies lead to very different performances (in terms of throughput) for the leaves, the relays, and the sink.
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
In Figure 12, the pernode throughputs at the various hierarchical levels are presented for the three topologies of interest. As a first, general, observation, it is seen that the pernode throughputs are much lower in indoor scenarios than the corresponding ones, as shown in Figure 11, in outdoor scenarios. This can be explained by the presence of a reflections off the limbs and the surrounding objects. Indeed, the initial antenna gain (at d = d_{ref}) is about L_{ref} = 57.42 dB, and this value is not very different from the gain of the environment, i.e., {L}_{\mathsf{\text{dB}}}^{\left(\mathsf{\text{env}}\right)}=78\mathsf{\text{dB}}. Therefore, short links are less affected (since the received signal power is much stronger than the reflected power), while longer links are more likely to suffer significant interference from the reflected waves. This was not the case in outdoor scenarios where distant nodes did not contribute to the interference thanks to the high path loss coefficient (i.e., γ = 124 dB/m).
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
First, regarding a BAN deployed in an outdoor environment, the energy consumption rate \mathcal{E} is presented in Figure 13 as function of q and for the three configurations of interest. It can be observed that the energy consumption rates of the three configurations present clearly different profiles. More precisely, Configuration A outperforms Configuration B which, in turn, is more energy efficient than Configuration C. Also, it can be observed that this remains true for any value of the node probability of transmission q.
The energy consumption rate in indoor scenarios is shown in Figure 14. It is noticeable that the values of the energy consumption in Figures 13 and 14 are approximately the same. Also, the relative performances of the three configurations in indoor scenarios are the same as in the outdoor case: Configuration A presents the lowest energy consumption rate, whereas Configuration C is the most energy consuming.
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.
In a slotted ALOHA system, the delay is directly related to the amount of (re)transmissions needed to send (or forward) a packet. More precisely, the average number of transmissions can be written as
E\left[r\right]=\sum _{r=1}^{\infty}r{Q}_{r},
where Q_{
r
} is the probability of a successful communication realized after exactly r transmissions (i.e., r  1 unsuccessful transmissions and a successful transmission). It can be expressed as
{Q}_{r}={\left(1{P}_{s}\right)}^{r1}{P}_{s}.
In other words, r is a geometric RV with parameter P_{
s
}, and one obtains
E\left[r\right]=\frac{1}{{P}_{s}},
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}.
In a TDMA system, the delay D_{TDMA} depends on the amount of time to wait before a dedicated slot takes place. In a generic approach, it can be supposed that a relay node will allocate exactly one slot per sensor to receive its data and a slot per sensor for the forwarding uplink. In Figure 15, the slots' allocation (i.e., chronogram) is presented for each relay \left(\mathsf{\text{noted}}{R}_{1},{R}_{2},\dots ,{R}_{{N}_{2}}\right). The time slots have a fixed duration of T_{
s
} (dimension: [s]). As can be seen from Figure 15, each relay needs a frame of N_{1} time slots to collect the (possibly generated) packets from its N_{1} leaves. It then needs another frame (N_{1} time slots) to forward them to the sink. At this point, it needs to remain idle for (N_{2}  2) frames, as the sink is busy collecting the packets from the other relays. This corresponds to assuming the same transmission rates at leaves and relay, and the same TDMAbased approach as at first and second layers. Therefore, the distance between two consecutive slots assigned to a given leaf is equal to N_{1} · N_{2} slots: when a leaf generates a packet, it needs to wait a number of slots between 0 (its slot is the current one) and N_{1} · N_{2}  1 (its slot just passed).^{d} As each number of slot has the same probability, the average delay (expressed in time slots) experienced by a given leaf node is
\begin{array}{lll}\hfill {D}_{\mathsf{\text{TDMA}}}& =\frac{1}{{N}_{2}\cdot {N}_{1}1}\sum _{i=0}^{{N}_{2}\cdot {N}_{1}1}i\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\frac{{N}_{2}\cdot {N}_{1}}{2}.\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}
(32)
The above derivation for D_{TDMA} represents an "average" scenario where a node generates at most a packet in an interval equal to N_{1} · N_{2} time slots: this corresponds to requiring that λ ≤ 1/(N_{1} · N_{2} · T_{
s
}). In this case, expression (32) for the delay is correct. If, on the other hand, λ > 1/(N_{1} · N_{2} · T_{
s
}), then it means that as soon as a leaf has transmitted a packet to its relay, it is likely to generate shortly a new packet, which will wait a period longer than (32). In general, it can be stated that
\frac{{N}_{2}\cdot {N}_{1}}{2}\le {D}_{\mathsf{\text{TDMA}}}\le {N}_{2}\cdot {N}_{1}.
In Figure 16, the average delay (expressed in contention time slots) incurred by a leaf to reach its relay is shown as a function of the transmission probability q. All curves refer to an indoor scenario. In the TDMA scheme, we consider the average expression^{e} (32). As expected, it can observed that, when the node probability of transmission is low, the slotted ALOHA significantly outperforms the TDMA scheme. However, for increasing probability of transmission, i.e., for increasing traffic load, there exists a critical threshold above which the TDMA scheme is to be preferred.
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.