Energy-Harvesting Concurrent LoRa Mesh with Timing Offsets for Underground Mine Emergency Communications

Abstract

Underground mine emergencies destroy communication infrastructure when situational awareness is most critical. Current systems rely on centralized network infrastructure, which fails during emergencies when miners are trapped and require rescue coordination. This paper proposes an energy-harvesting LoRa mesh network that addresses self-powered operation, interference management, and adaptive physical layer optimization under severe underground propagation conditions. A dual-antenna architecture separates RF energy harvesting (860 MHz) from LoRa communication (915 MHz), enabling continuous operation with supercapacitor storage. The core contribution is a decentralized scheduler that derives optimal timing offsets by modeling concurrent transmissions as a Poisson collision process, exploiting LoRa’s capture effect while maintaining network coherence. A SINR-aware physical layer adapts spreading factor, bandwidth, and coding rate with hysteresis, controls recomputing timing parameters after each change. Experimental validation in Missouri S&T’s operational mine demonstrates far-field wireless power transfer (WPT) reaching 35 m. Simulations across 2000 independent trials show a 2.2× throughput improvement over ALOHA (49% vs. 22% delivery ratio at 10 nodes/hop), 64% collision reduction, and 67% energy efficiency gains, demonstrating resilient emergency communications for underground environments.

1. Introduction

The United States places significant emphasis on mine safety, as demonstrated by the Mine Improvement and New Emergency Response (MINER) Act of 2006, which strengthened underground miner protection and emergency preparedness [1]. A central provision encourages integrating radio frequency (RF) communication technologies to maintain connectivity during emergencies when wireless infrastructure may be unavailable due to structural failures such as roof collapses. Roof-fall incidents remain one of the most severe causes of mine-related fatalities, shown in Figure 1 [2], disrupting both communication and power supply for miners’ devices and undermining localization, monitoring, and tracking capabilities essential for situational awareness [3,4]. The resulting connectivity loss hinders command centers, delays search and rescue operations, and obstructs real-time coordination. Two requirements therefore emerge: (i) wireless delivery of electrical energy to trapped miners and (ii) interference-resilient, long-range, low-power wide-area network (LPWAN) links to report situational awareness information.

Figure 1. Mining fatalities by types 2023-2024 [2].

LPWAN technologies have attracted increasing attention for energy-efficient, long-range Internet of Things (IoT) applications, [5]. While ZigBee and Bluetooth Low Energy (BLE) offer cost-effective solutions, their transmission ranges rarely exceed tens of meters [6]. Multi-hop mesh networks extend coverage with no additional cost while increasing link success probability and energy efficiency.

More advanced LPWAN technologies include LoRa, SigFox, LTE-M, and NB-IoT. LoRa and SigFox, both operating in unlicensed bands, prioritize range and device lifetime over throughput. LoRa supports the open-source LoRaWAN protocol, which employs a star topology where end devices communicate directly with a central gateway. However, as device count increases, LoRaWAN performance deteriorates due to scalability and capacity limitations [7]. These limitations are especially critical during underground mine emergencies when collocated miners attempt to transmit situational awareness data simultaneously, generating severe inter-device interference that degrades throughput and network reliability.

Several studies have adopted LoRa technology to design mesh multi-hop networks. Hong et al. [5] propose HBEE, a hierarchical LoRa-mesh with contention-free formation and multi-path/multi-channel routing to curb collisions and energy use. Tran et al. [8] design a multi-hop, real-time LoRa protocol for dynamic networks, extending two-hop RT-LoRa with automatic configuration, topology management, and slot schedule updates to sustain high reliability. Bor et al. [9] provide early LoRa characterization and introduce LoRaBlink, demonstrating non-destructive concurrent transmissions and simple multi-hop operation under controlled timing. Leenders et al. [10] present an asynchronous multi-hop protocol that saves energy via prolonged preamble sampling and payload aggregation, validated in-field and large-scale simulation. Liao et al. [11] demonstrate that concurrent transmission (CT)-based flooding can realize efficient multi-hop LoRa, establishing feasibility and characterizing capture/overlap behavior. Table 1 summarizes the differences between our proposed LoRa algorithm and existing works. Energy harvesting (EH) provides a promising means of powering LPWAN devices in energy-constrained environments [12,13].

Table 1. Baseline algorithms and key differences from the proposed protocol.

Category	Baseline Algorithm/Reference	Key Features	Key Differences vs. Proposed Protocol
Access-only	ALOHA LoRa/LoRaWAN (Polonelli et al., 2019 [14]; Ibrahim et al., 2020 [15]).	Random channel access; no coordination; all nodes transmit independently using fixed PHY.	Suffer high collision rate and poor scalability under concurrent traffic; no timing offsets, no SINR adaptation, and no RF-energy harvesting awareness. The proposed method introduces deterministic offsets, SINR-aware PHY selection, and energy gating to prevent energy-starved transmissions.
Link-aware	Adaptive Data Rate (ADR)/SINR-Only Adaptation. (Kufakunesu et al., 2020 [16]; Serati et al., 2022 [17]).	Adjusts spreading factor (SF), bandwidth (BW), and coding rate (CR) using measured SINR; no coordination between nodes.	Improves link budget but cannot avoid collision in concurrent transmission. Proposed method integrates SINR adaptation into the scheduler and couples it with offset calculation, ensuring both PHY efficiency and collision control.
Scheduling/Offset	Multi-Hop Real-Time LoRa (RT-LoRa) (Tran et al., 2022 [8]).	Slot-based transmission scheduling to reduce collisions; fixed PHY; topology-driven slot updates	Lacks adaptability to dynamic interference and harvested energy conditions. The proposed method derives timing offsets analytically from collision probability ( $p_c$) and interference window ${\eta }_0$ and dynamically updates per-hop PHY and offset based on SINR and relay set.
Flooding/Concurrent	LoRaHop (Tian et al., 2023 [18]), JMAC (Escobar et al., 2020 [19])	Nodes flood packets concurrently, exploiting capture effect; minimal routing overhead.	CT flooding assumes synchronized clocks and battery-powered nodes. The proposed design integrates RF-energy harvesting and safe timing offsets that limit destructive interference, enabling energy-sustainable CT in underground conditions.
Structured Mesh/Tree		Static parent–child (cluster-based) routing; topology fixed or slow to adapt; no concurrent forwarding.	Fails during disasters when cluster heads or gateways are destroyed. Proposed method employs flooding-based forwarding with TTL-bounded propagation and adaptive relay selection, maintaining network connectivity even under partial collapse.

Of the available EH sources, including solar, wind, vibration, and RF, RF-EH is most viable in underground mines. RF-EH systems typically comprise a wireless power transfer (WPT) transmitter and a harvesting module, with the P21XXCSR-EVB board widely adopted [20]. WPT techniques are classified as near-field (non-radiative) or far-field (radiative), with far-field transfer enabling long-range operation [21]. Unlike surface mining environments, underground mines present severe propagation challenges, including high attenuation from rock and moisture, obstructed line-of-sight due to complex geometries, and ohmic losses from poor geological conductivity [22]. Conventional WPT–RF-EH testbeds, optimized for multipath-rich environments, perform poorly in underground mining environments [23,24,25]. Reliable RF-EH operation requires received RF power levels above −15 dBm, which many commercial transmitters fail to deliver.

This work advances underground disaster communications with a multi-hop, flooding-based LoRa mesh that couples RF-WPT, EH, and concurrency-aware scheduling. Our key contributions are the following:

• Disaster-resilient mesh topology with far-field RF-WPT integration: We demonstrate practical long-range wireless power transfer up to 35 m in operational underground mine conditions using dual-antenna separation, external power amplification, and supercapacitor storage. We characterize energy storage trade-offs and self-discharge behavior under harsh tunnel propagation, addressing a critical gap in existing LoRa concurrent transmission literature which rarely validates far-field WPT in underground environments.
• Mathematically derived timing offset scheduler: We formulate optimal per-hop timing offsets through Poisson collision modeling that deliberately staggers transmissions to exploit LoRa’s capture effect and imperfect inter-spreading-factor orthogonality while maintaining sync word coherence. This decentralized approach requires no tight synchronization or centralized coordination, critical when infrastructure is damaged during emergencies.
• Coupled SINR-aware adaptive physical layer: Unlike conventional adaptive data rate (ADR) schemes that adjust only spreading factor, we jointly optimize spreading factor, bandwidth, and coding rate based on measured signal quality with hysteresis-based stability controls. The adaptation recomputes airtime and feeds updated timing parameters back into the scheduler, maintaining protocol coherence across dynamic channel conditions.

The remainder of this article is organized as follows. Section 2 provides the system model. Section 3 presents an overview of LoRa technology. Section 4 discusses the proposed protocol design and its derivations. Section 5 reports the experimental results from the far-field WPT testbed, Section 6 presents both simulated results, and Section 7 concludes the study.

2. System Model

2.1. Network Model

We consider an underground mine environment where a set $V$ of N LoRa-enabled energy harvesting nodes operate in a mesh topology to provide emergency communications coverage (Figure 2). The network topology is represented by an undirected graph $\mathcal{G}=\left(V,E\right)$ with mean degree $d$, where edges represent viable communication links in severe underground propagation conditions. Each LoRa-enabled EH node adopts a dual-antenna design: one antenna harvests RF energy at frequency $f_{WPT}$ 860 MHz while the other maintains LoRa communication at $f_{LoRa}$ set to 915 MHz.

Figure 2. System architecture.

This 55 MHz frequency separation provides sufficient isolation to prevent cross-interference between energy harvesting and communication functions, enabling continuous operation of both subsystems. To bound the spatial and temporal extent of floods and respect airtime and regulatory limits, forwarding is constrained by a time-to-live (TTL) parameter $L\le 4$. Nodes maintain a loose timebase derived from periodic gateway beacons when available, but precise slot synchronization is not required. Instead, transmissions are deliberately staggered using per-hop timing offsets $\Delta t_h$ and a small random jitter $ϵ$. This approach decorrelates transmission start times while preserving controlled concurrency necessary for capture effect exploitation and quasi-orthogonal reception.

2.2. Underground Mine Signal Propagation Model

The underground mine environment exhibits distinct propagation characteristics that differ significantly from surface conditions. The received power at node i from the WPT transmitter at distance $d_{ij}$ is as follows:

$$P_{i,j}^r=P_T-PL\left(d_{ij}\right)+X_{\sigma }$$

(1)

$P_T$ is the total transmitting power including the external power amplifier (PA), $PL\left(.\right)$ is the environment calibrated path loss, and $X_{\sigma }$, captures log-normal shadowing (dB), typically $3–10 \mathrm{d}\mathrm{B}$ in underground environments. For underground mine propagation, we employ a more detailed path loss model accounting for tunnel geometry [22,26]:

$$\begin{array}{c}PL\left(d\right)=PL\left(d_0\right)+n_{Corridor}.10 {\log }_{10}\left({\displaystyle \raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$d_0$}\right.}\right)+N_{Wall}·L_{Wall}\\ +{N_{Junction·}L}_{Junction}+L_{Moist}\hfill \end{array}$$

(2)

• $PL\left(d_0\right)=40$ dB: the reference path loss at $d_0=1\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}$
• $n_{Corridor}=2.2$: the corridor propagation exponent deduced from free space due to waveguide effects.
• $L_{Wall}=25$ dB: additional loss per rock wall or rough surface penetration.
• $L_{Junction}=8$ dB: scattering loss per tunnel intersection or crosscut.
• $L_{Moist}=2–5$ dB: absorption due to water vapor in mine air.
• $N_{Wall}$, $N_{Junction}$: number of walls and junctions traversed along the path.

This model captures underground propagation characteristics: enhanced propagation along tunnel corridors due to waveguide effects, severe losses through geological barriers, and scattering at geometric discontinuities.

2.3. LoRa Wireless Channel Interference Model

For a desired transmitter i using spreading factor $\left({SF}_i\right)$ and a set of $\mathcal{K}$ of concurrent transmitters, the signal-to-interference-plus-noise ratio (SINR) is as follows:

$${SINR}_i={\displaystyle \frac{P_i^r}{{\displaystyle \sum _{k\in \mathcal{K}}}\eta \left({SF}_i,{SF}_k\right)P_k^r+N_0}}$$

(3)

$N_0$ is the noise power, $\eta \left({SF}_i,{SF}_k\right)\in \left[\mathrm{0,1}\right]$ models inter-SF interference, and $P_k^r$ is the received power from interferer $k$. Based on experimental LoRa characterization studies, the following can be defined [12,27]:

• $\eta =1.0$ when ${SF}_i={SF}_k$ (same spreading factor collision).
• $\eta =0.3$ when ${SF}_i\ne {SF}_k$ (imperfect orthogonality between different spreading factors).

A LoRa packet is successfully decoded if ${SINR}_i\ge \mathsf{\Gamma }\left(SF,BW,CR\right)$, where $BW$ and $CR$ are bandwidth and coding rate, respectively. $\mathsf{\Gamma }\left(.\right)$ represents the demodulation threshold for the chosen physical layer parameters. Additionally, capture effects enable successful reception under interference when the desired signal strength exceeds interfering signals by the capture margin $C_{Cap}\ge 3$dB. The distributed scheduling algorithm targets safe partial overlap conditions that preserve successful reception while maximizing network throughput through controlled concurrency. Capture works for both intra-SF collisions (same SF) and sometimes inter-SF scenarios (imperfect orthogonality), so carefully staggered overlaps can remain decodable [11]. However, capture robustness degrades with symbol timing offset (STO) and carrier frequency offset (CFO) as excessive offsets spread energy across frequency bins and reduce effective SINR.

2.4. RF Energy Harvesting Storage Dynamics

We model the energy storage dynamics of each LoRa-enabled device using a discrete-time system. Assuming realistic non-linear harvesting characteristics of the Powercast P21XXCSR-EVB [20], the energy storage evolves according to the following:

$$E_{i,t+1}={clip}_{\left[E_{min},E_{max}\right]}\left(E_{i,t}+ {\eta }_{RF-DC}\left(P_{i,t}\right)\mathsf{\tau }-\left(P_{i,t}^{load}\right)\mathsf{\tau }-\left({\displaystyle \frac{E_{i,t}}{{\rho }_{leak}}}\right)\mathsf{\tau }\right)$$

(4)

where the operator clip $\left[E_{min},E_{max}\right]$constrains the stored energy within the feasible interval $\left[E_{min},E_{max}\right]$. The terms in (4) are defined as follows:

• $E_{i,t}$: residual energy stored in the capacitor or battery.
• ${\eta }_{RF-DC}\left(P_{i,t}\right)\mathsf{\tau }$: harvested RF energy with conversion efficiency ${\eta }_{RF-DC}\in \left(\mathrm{0,1}\right)$
• $\left(P_{i,t}^{load}\right)\mathsf{\tau }$: energy consumed by the microcontroller and LoRa transceiver circuitry.
• $\left({\displaystyle \frac{E_{i,t}}{{\rho }_{leak}}}\right)\tau$: leakage losses due to capacitor self-discharge or battery leakage.
• $\tau$: denotes the energy transmission duration per time slot.

The conversion efficiency ${\eta }_{RF-DC}\left(P_{i,t}\right)$ is a non-linear function of received RF power, as characterized in the Powercast P21XXCSR-EVB datasheet [20], with typical values ranging from 10% at −20 dBm to 50% at −10 dBm.

2.5. Far-Field (Fraunhofer) Region for RF Energy Transfer

The motivation for adopting RF-WPT in post-disaster underground mine scenarios is its ability to deliver far-field energy, critical for sustaining tracking and localization systems when conventional infrastructure is unavailable. The Fraunhofer boundary for an antenna array of maximum aperture size D operating wavelength ${\lambda }_o$ is given by the following [21] $:$

$$d_{far}={\displaystyle \frac{{2D}^2}{{\lambda }_o}}$$

(5)

For a uniform linear array (ULA) with $N_t$ elements and inter-element spacing $d= {\lambda }_o/2$, the effective aperture size is $D=(N_t-1)d$. Substituting this into (5), we obtain the following:

$$d_{far}=2{\left({\displaystyle \frac{\left(N_t-1\right){\lambda }_o}{2}}\right)}^2{\displaystyle \frac{1}{{\lambda }_o}}$$

(6)

As a practical example, consider a UHF-based RF-WPT transmitter operating at $f_{WPT}=860 MHz$, ${\lambda }_o\approx 0.349 m$. For ULA with $N_t=4$ elements, (6) yields a far-field threshold of approximately 1.57 m. Thus, receivers must be positioned at least this distance from the array for far-field assumptions to hold. Our experimental validation extends well beyond this minimum, demonstrating effective energy transfer up to 35 m in underground tunnel conditions deploying a single-antenna system.

3. Concurrent LoRa Optimal Timing-Offset Scheduling Problem

We formalize the design goal of the proposed multi-hop, flooding-based LoRa mesh operating under stringent energy harvesting and duty-cycle constraints typical of underground emergencies. The network must deliver situational awareness traffic over multiple relays while limiting interference via timing offsets and adapting the PHY to local channel conditions. Given energy scarcity and intermittent harvesting, we minimize energy per delivered (EPD) packet subject to reliability and latency constraints. The implementation flowchart of the algorithm is presented in Figure 3.

Figure 3. Implementation flowchart of the proposed algorithm.

3.1. LoRa Preamble Structure

The primary function of the preamble is to help the receiver synchronize with the transmitter. A LoRa packet begins with the preamble, followed by an optional header and the payload. The preamble consists of three distinct components serving different functions in packet detection and synchronization:

(i)

Programmable preamble symbol  $\left(N_{preamble}\right)$: A sequence of upchirps used for coarse timing acquisition and automatic gain control (AGC) settling. The standard configuration uses $N_{preamble}=8$ up-chirps, though this is configurable from 6 to 65,535 symbols.

(ii)

Network synchronization word: A 2.25-symbol sequence encoding the network identifier that distinguishes between different LoRa networks operating on the same frequency. Public LoRaWAN networks use sync word 0 × 34, while private networks typically use 0 × 12, enabling network-level filtering at the physical layer.

(iii)

Detection and synchronization symbols: A 2.25-symbol sequence consisting of two downchirps followed by a quarter-chirp (0.25 symbol) that enables the receiver to perform the following:

• Precisely determine time-of-arrival.
• Estimate and compensate for carrier frequency offset (CFO).
• Lock symbol timing for payload demodulation.
• Execute the capture effect when multiple preambles overlap.

The complete preamble duration is, therefore, as follows:

$$T_{Preamble}=\left(N_{Preamble}+4.50\right){\times T}_{sym}$$

(7)

$T_{sym}$ is the LoRa symbol time. The default configuration with $N_{preamble}=8$ yields a total preamble length of 12.50 symbols.

3.2. Implications for Concurrent Transmission

The distinction between preamble components is critical for understanding collision behavior in CT flooding:

(i)

Upchirp phase collision: When timing offsets ${ \Delta }_{th}<N_{Preamble}{\times T}_{sym}$, multiple transmitters’ upchirps partially overlap. However, the receiver’s AGC and coarse timing detection are relatively robust to interference during this phase.

(ii)

Sync word alignment: For concurrent transmissions to be detectable as the same network, their sync words must align within approximately $\pm 0.5$ symbols. Our timing offset policy ensures relay nodes forwarding the same packet maintain sync word coherence by scheduling transmissions relative to a common reference $\left(t_{rx}+G\right)$, where $G$ is the guard time.

(iii)

Capture during detection phase: The capture effect primarily occurs during the 2.25-symbol downchirp detection phase. If one signal’s received power exceeds others by the capture margin $C_{Cap}\ge 3$dB during this phase [27,28], the receiver locks onto the stronger signal and successfully demodulates its payload despite ongoing interference from weaker concurrent transmissions.

(iv)

Vulnerable window definition: Based on this structure, we define the vulnerable window $v_h$ as the time interval during which a collision can prevent successful reception. With capture and quasi-orthogonality, the effective vulnerable window is $v_h\approx {{\eta }_{0}T}_{Pkt}$, where ${\eta }_0\in \left[\mathrm{1,2}\right]$ models the window reduction factor. For pure ALOHA without capture ${\eta }_0=2$, it is vulnerable for the entire packet duration plus the preceding packet duration. With LoRa’s capture effect exploited through controlled timing offsets, we empirically observe ${\eta }_0\approx 1.2–1.5$ in underground mine environments, where

• ${\eta }_0\to 1$ indicates strong capture/isolation only overlaps during critical detection symbols cause collisions.
• ${\eta }_0\to 2$ represents the ALOHA worst-case; all overlaps are destructive.

The LoRa packet time on air (ToA) is calculated as follows [29]:

$$ToA=T_{Preamble}+N_{Payload}{T}_{sym},$$

(8)

with $N_{Payload}$ denoting the payload computed from the payload size, spreading factor SF, coding rate CR, and bandwidth BW according to the LoRa modulation formula [27]:

$$N_{Payload}=8+max\left(⌈ \left({\displaystyle \frac{8PL-4SF+28+16CRC-20IH}{4\left(SF-2DE\right)}}\right)⌉{CR}_{idx}+4.0\right)$$

(9)

$PL$ is payload length in bytes, $H$ is the header enables flag, and $DE$ is the low data rate optimization flag. Our protocol’s timing offset ${ \Delta }_{th}$ derived in Section 4.1 is designed to stagger transmission start times such that the following are performed:

• Capture dominance is maintained: The first relay to typically transmit the one with highest RSSI completes its detection phase before subsequent relays’ signals reach comparable power levels.
• Sync word coherence is preserved: For relays forwarding the same packet, ${ \Delta }_{th}\ll T_{sym}$ensures their sync words remain aligned within the receiver’s tolerance window, preventing network-layer filtering from rejecting legitimate concurrent transmissions.
• Vulnerable window is minimized: By spacing transmissions by (see Equation (18)), we ensure that at most one transmission is in its critical detection phase at any given time with high probability.

3.3. Problem Formulation

Consider a data packet traversing $H$ hops as illustrated in Figure 4. At hop $h$, let $E_{tx}^h$ denotes the transmit energy expenditure for the chosen PHY ${\left(SF,BW,CR\right)}_h$, and let $p_h$ represents the hop success probability including capture effects and energy availability.

Figure 4. Multi-hop mesh network topology.

The end-to-end delivery probability is as follows:

$$P_{\left\{E2E\right\}}^D = \prod _{h=1}^H{p}_h$$

(10)

The optimization problem seeks to minimize expected energy consumption:

$$\underset{\pi }{\min }{\mathbb{E}}_{\pi }\left[\sum _{h=1}^H{E}_h^{tx}\right]$$

(11)

Subject to the following:

$$\prod _{h=1}^H{p}_h\ge \gamma$$

(12)

$$\sum _{h=1}^H{\mathbb{E}}_{\pi }\left[T_{Pkt,h}+{\Delta }_{th}\right]\le D_{max,}$$

(13)

$\pi ={\left\{{\left(SF,BW,CR\right)}_h,{\Delta }_{th},{mode}_h\right\}}_{h=1}^H$ represents the decision variables for each hop. $\gamma \in \left[0.9,0.95\right]$ is the end-to-end reliability target. $D_{max}$ is the maximum allowable end-to-end latency deadline accounting for duty-cycle and energy harvesting constraints.

4. Protocol Design

4.1. Timing Offset Derivation

The key innovation of our protocol is the mathematical derivation of optimal timing offsets that balance collision avoidance with network capacity. We model the problem using collision probability theory and derive closed-form expressions for the timing parameters. For a given hop $h$, we want a local offset ${\Delta }_{th}$ that keeps expected overlaps within a target collision probability risk $p_c$ while allowing safe concurrency and quasi-orthogonality. Let $N_{est,h}$ be the estimated number of relays that may forward the same packet after reception. Each relay transmits one replica with airtime $T_{Pkt,h}$ computed from the LoRa PHY, within a forwarding window of length ${\tau }_h$ and a per-hop latency budget that must also respect the end-to-end deadline $D_{max}$. Accordingly, timing offset ${\Delta }_{th}$ is introduced to stagger the packet starts time and a small jitter $\epsilon ~\mathcal{U}\left(\pm \beta T_{sym,h}\right)$ with $0<\beta \ll 1$ as depicted in Figure 5. The residual alignment respects LoRa’s carrier frequency offset and symbol timing tolerances, while a fixed guard time $G$absorbs processing and timestamping granularity [29]. The transmitting node (source or relay) chooses its offset locally when scheduling a packet. Hence, timing offsets are decentralized and per-hop, computed by each sender using local measurements and globally shared static parameters. This keeps the system robust when infrastructure is impaired.

Figure 5. Concurrent timing offset packet alignment.

4.2. Capture-Aware Vulnerable Window and Collision-Risk Target

Fix a per-hop target collision probability ${ p}_c\in \left(\mathrm{0,1}\right)$ for any tagged relay. Approximate start times ${ \tau }_h$ are measured by a homogeneous Poisson process of rate ${ \lambda }_h=N_{est,h}/{\tau }_h$, a standard conservative proxy for random access. The probability that no other start occurs within $v_h$ of the tagged start is as follows:

$$Pr=\left\{no collision\right\}\approx e^{-{ \lambda }_h{ v}_h}$$

(14)

Enforcing $Pr=\left\{no collision\right\} \ge { p}_c$ gives the spacing condition:

$$e^{-{ \lambda }_h{ v}_h}\ge 1-{ p}_c\iff { \lambda }_h\le {\displaystyle \frac{-ln\left(1-{ p}_c\right)}{{ v}_h}}$$

(15)

Proposition 1. Feasible forwarding window

Given$N_{est,h}$contenders, a forwarding window${ \tau }_h$satisfies the collision-risk target${ p}_c$if

$${\tau }_h\ge {\displaystyle \frac{N_{est,h}{ v}_h}{-ln\left(1-{ p}_c\right)}}={\displaystyle \frac{N_{est,h}{{\eta }_{0}T}_{Pkt}}{-ln\left(1-{ p}_c\right)}}$$

(16)

Interpretation

With more contenders, i.e., larger$N_{est,h}$or a more pessimistic vulnerable window, a larger${\eta }_0$the hop requires a proportionally longer${ \tau }_h$to keep the same risk${ p}_c$.

Corollary 1. Offset under a latency cap

If${ \tau }_h$is fixed by latency, e.g.,${ \tau }_h=k{T}_{Pkt}$for some$k>0$, then equalizing starts in$[0,{ \tau }_h)$yields per-node spacing:

$${\Delta }_{th}={\displaystyle \frac{{ \tau }_h}{max\left(N_{est,h},1\right)}}={\displaystyle \frac{k{T}_{Pkt,h}}{max\left(N_{est,h},1\right)}}$$

(17)

To satisfy (17), choose $k$ at least

$$k\ge {\displaystyle \frac{{\eta }_0}{-ln\left(1-{ p}_c\right)}}$$

(18)

Thus, a risk-linked offset rule is

$${\Delta }_{th}\ge k{\displaystyle \frac{T_{Pkt}}{max\left(N_{est,h},1\right)}};k\triangleq {\displaystyle \frac{{\eta }_0}{-ln\left(1-{ p}_c\right)}}$$

(19)

and transmissions occur at$t_{tx}^{\left(i\right)}=t_{rx}+G+i{\Delta }_{th}+ϵ, i=\mathrm{0,1}$for the$i^{th}$contender.

4.3. Contender Estimation and Window Sizing

Each relay estimates local contention as the following:

$$N_{est,h}⟵\alpha \left(1+{\widehat{\exists }}_h{T}_{Pkt,h}\right)+\left(1-\alpha \right){\widehat{N}}_{obs,h}, \alpha \in \left[\mathrm{0,1}\right]$$

(20)

• ${\widehat{\exists }}_h$: the arrival rate estimate (packets/s) at hop $h$ from recent receptions. The term ${\widehat{\exists }}_h{T}_{Pkt,h}$ is the expected number of contenders during one airtime. The “+1” guarantees at least one potential contender even when the rate is momentarily low.
• ${\widehat{N}}_{obs,h}$: observation-based contender count, obtained from preamble sightings in a sliding window of length $W_{obs}$. If $C_h$ distinct contenders were overheard in that window, scale to the hop’s forwarding window ${ \tau }_h$ via ${\widehat{N}}_{obs,h}={ \tau }_h \left(C_h/W_{obs}\right)$.
• $\alpha$: blending weight. Higher $\alpha$ favors the model-based, smoother term ${\widehat{\exists }}_h{T}_{Pkt,h}$. Lower $\alpha$ favors the observation-driven, more reactive term ${\widehat{N}}_{obs,h}$

4.4. LoRa-Specific SINR-Aware PHY Selection (SF/BW/CR) with Hysteresis

Before scheduling a forward, each node selects a LoRa PHY tuple ${\left(SF,BW,CR\right)}_h$ based on its measured hop SINR ${\widehat{\gamma }}_h$ and recent packet reception rate (PRR) outcomes. The node adopts a local, hop-by-hop ADR-style policy tailored to LoRa PHY parameters and then recomputes airtime, so the timing offset ${ \Delta }_{th}$ updates consistently. The selected LoRa PHY tuple is based on the measured SINR and a threshold $\left\{{\mathsf{\Gamma }}_l\right\}$ with hysteresis to avoid flapping:

• $\widehat{\gamma }>{\mathsf{\Gamma }}_{high}=5 dB$: use low SF, large BW, and small CR to maximize throughput.
• ${0 dB=\mathsf{\Gamma }}_{mid}<\widehat{\gamma }\le {\mathsf{\Gamma }}_{high}$: balanced profile.
• ${-10 dB=\mathsf{\Gamma }}_{low}<\widehat{\gamma }\le {\mathsf{\Gamma }}_{mid}$: higher SF/CR, smaller BW for robustness.
• $\widehat{\gamma }\le {\mathsf{\Gamma }}_{low}$: max SF, max CR, and min BW for range.

After selection, recompute $T_{sym,h}$  $T_{pkt,h}$ and update the offset.

A. Hysteresis Parameters

The adaptive PHY selection employs hysteresis to prevent rapid oscillation (flapping) between configurations due to transient SINR fluctuations:

• ${\mathit{m}}_{\uparrow }$ (upward margin): Total of 2.0 dB. When upgrading to a faster PHY profile (lower SF, higher BW, lower CR), the measured SINR must exceed the profile’s demodulation threshold ${\mathsf{\Gamma }}_{req}\left(p\right)$ by at least 2 dB.
• ${\mathit{m}}_{\downarrow }$ (downward margin): Total of 2.0 dB. When downgrading to a more robust PHY profile, a trigger occurs if SINR falls below the current profile’s threshold by 2 dB or more.
• ${\mathit{T}}_{\mathit{d}\mathit{w}\mathit{e}\mathit{l}\mathit{l}}$ (dwell time): Total of 5 s. Minimum time a node must remain on a PHY configuration before considering an upgrade. This prevents premature switching during temporary SINR improvements.

B. Frame Quality Assessment

A frame is classified as “good” or “bad” based on the following criteria:

• Good frame—A received packet satisfying all the following:

  ○

  CRC validation passes (integrity check).

  ○

  SINR at reception ${\ge \mathsf{\Gamma }}_{req}\left(SF,BW,CR\right)-$ 1 dB demodulation margin.

  ○

  RSSI $\ge -120$ dBm (above noise floor threshold).

• Bad frame—A frame that fails any of the above criteria, including:

  ○

  CRC failure (corrupted payload).

  ○

  SINR below demodulation threshold minus 1 dB margin.

  ○

  Timeout (no packet received within expected time window).

C. Quality Window Parameters

• ${\mathit{K}}_{\mathit{g}\mathit{o}\mathit{o}\mathit{d}}$ (good frame threshold): Three consecutive good frames required before attempting PHY upgrade to faster configuration.
• ${\mathit{K}}_{\mathit{b}\mathit{a}\mathit{d}}$ (bad frame threshold): Two frames classified as bad within the recent window. Triggers immediate downgrade to more robust PHY.
• ${\mathit{W}}_{\mathit{q}\mathit{u}\mathit{a}\mathit{l}\mathit{i}\mathit{t}\mathit{y}}$ (quality assessment window): Ten most recent frames. A sliding window that tracks frame reception outcomes for computing PRR and triggering PHY adaptation.

D. PRR Calculation

The PRR for neighbor r is computed over the sliding window:

$${PRR}_{\mathit{r}}={\displaystyle \frac{\left(Number of good frames from r in{ W}_{quality} \right)}{\left(Total frames expected from rin{ W}_{quality}\right)}}$$

(21)

where “frames expected” are determined by observing preamble detections even if the payload fails. A missing preamble indicates the neighbor did not transmit; a detected preamble with failed payload counts as a transmission attempt.

E. SINR Measurement

The measured SINR ${\gamma }_h$ at hop $h$ is computed from physical layer measurements:

$${\widehat{\gamma }}_h={SNR}_{masured}+10{\log }_{10}\left({\displaystyle \frac{{BW}_h}{1000}}\right) \left[dB\right]$$

(22)

${SNR}_{masured}$ is the Signal-to-Noise Ratio reported by the SX1262 transceiver register, and ${BW}_h$ is the current bandwidth in kHz (125, 250, or 500). The bandwidth correction term accounts for noise power scaling with BW. SINR is updated after each successful packet reception and exponentially averaged:

$${\widehat{\gamma }}_h\leftarrow \vartheta \times {\widehat{\gamma }}_{h,new}+\left(1-\vartheta \right) \times {\widehat{\gamma }}_{h,old}$$

(23)

where the smoothing parameter $\vartheta =0.3,$ balances responsiveness to channel changes against noise in instantaneous measurements.

4.5. Duty-Cycle and Regulatory Compliance

LoRa’s chirp spread spectrum (CSS) operates under FCC Part 15.247 DSSS equivalent rules. The FCC does not mandate duty cycles for CSS systems in the US915 band (902–928 MHz), only EIRP limits (36 dBm for point-to-multipoint). However, we voluntarily impose a duty cycle; DC = 0.01 (1%) to ensure spectrum fairness and demonstrate operation under stringent constraints.

Let $t_{last}\left(\mathcal{C}\right)$ denote the timestamp when a node finishes its most recent transmission on channel $\mathcal{C}$. For a duty-cycle limit DC on channel $\mathcal{C}$, the required wait time after transmitting a packet of duration $T_{Pkt}$ is

$$T_{off}={\displaystyle \frac{1-DC}{DC}}{T}_{Pkt}$$

(24)

A transmission at time $t_{tx}$ on channel $c$ is permitted only if

$$t_{tx}\left(\mathcal{C}\right)\ge t_{last}\left(c\right)+T_{off}\left(c\right)$$

(25)

Implementation Details:

• Nodes maintain separate $t_{last}\left(\mathcal{C}\right)$ per channel; duty-cycle constraints are enforced independently.
• Only completed transmissions update $t_{last}\left(\mathcal{C}\right)$; backoff, sensing, or queuing do not.
• If both energy insufficiency $E_i\left(t\right)<c$ and duty-cycle violations occur, transmission is deferred for $max\left(T_{energy,wait},T_{off}\right)$.
• Step 7 of the scheduling algorithm (Algorithm 1) checks Equation (26) before transmission.

The protocol supports optional multi-channel extension using the US915 LoRaWAN channel plan (64 × 125 kHz + 8 × 500 kHz channels). Channel selection is deterministic: ${Channel}_{Index}=\left(scrID+seqNum\right)mod N_{channels}$. Timing offset ${\Delta }_{th}$ is computed per packet based on PHY parameters (SF, BW, CR), which are identical for all relays forwarding the same packet, ensuring coordination is preserved across channels.

4.6. Energy Harvesting Coupling

Let the energy inflow to node $i$ be denoted as a renewal process $\left\{A_i\left(t\right)\right\}$ with mean rate $v_i$ joules/s. The node maintains an energy buffer $E_i\left(t\right)\in \left[0,B_i\right]$ capacity of $B_i$ joules. Each transmission on hop $h$ consumes a fixed cost of $c$ joules and occupies the channel for airtime $T_{Pkt,h}.$ Energy conversion losses, if present, are incorporated into $v_i$. A packet can start service only when two conditions hold: (i) the data queue is nonempty, and (ii) the buffer has sufficient energy:

$$E_i\left(t\right)\ge c$$

(26)

When $E_i\left(t\right)<c$, the transmitter is unavailable; we interpret this period as a server vacation due to an energy outage. Under this abstraction, each node behaves as an $M/G/1$ queue with vacations, where the service time distribution has mean $T_{Pkt,h}$ and vacations are governed by the EH process. Let ${\phi }_i\triangleq Pr\left[E_i\left(t\right)\ge c\right]$ denote the steady-state probability that node i has enough energy in buffer to initiate a transmission. The effective service rate on hop $h$ is as follows:

$${\mu }_{eff,h}={\displaystyle \frac{1}{T_{Pkt,h}}}{\phi }_i$$

(27)

which captures the fact that the server can enter service immediately in only a fraction of time. Consequently, a necessary stability condition at each hop is ${\exists }_h^{tot}<{\mu }_{eff,h}$, where ${\exists }_h^{tot}$ is the total exogenous and forwarded arrival rate and is defined in the traffic model. In practice, ${\phi }_i$ can be (i) computed analytically for simple EH laws, e.g., Poisson energy arrivals with i.i.d. packet costs; (ii) estimated from measurements of the harvested-power profile, or (iii) obtained via simulation by discretizing ${ E}_i$ and evolving the coupled data/energy queues.

4.7. Forwarding Mode

At each hop $h$, a relay chooses one of three forwarding modes to minimize energy per delivered (EPD) packet while meeting a per-hop reliability budget ${ \gamma }_h:$

A. 

Unicast (Single relay)

Let ${\mathcal{N}}_h$ denote the candidate neighbor set filtered by minimum link quality and TTL and let ${ p}_{h,r}$ be the success probability to neighbor $r\in {\mathcal{N}}_h$. These values are estimated from SINR $\to$ PER curves or from the PRR. Let $E_{h,r}^{tx}$ be the transmit energy for one attempt toward $r$ using the locally selected LoRa PHY tuple:

$${Choose j}^{*}=\underset{j\in {\mathcal{N}}_h}{\mathrm{argmin}}{EPD}_{h,j}^{uni}$$

(28)

where ${EPD}_{h,j}^{uni}=E_{h,j}^{tx}/{ p}_{h,j}$. Accept if ${ p}_{h,j^{*}}\ge {\gamma }_h$.

B. 

Dual Relay

Consider two candidate relays $j, k$ with relay set $\mathcal{R}\left\{j,k\right\}$. Under an independence approximation, the dual-path success probability is as follows:

$$p_h^{dual}=1-\left(1-{ p}_{h,j}\right)\left(1-{ p}_{h,k}\right)$$

(29)

Define the energy per delivered packet ${EPD}_{h,\left\{h,k\right\}}^{uni}=E_{h,j}^{tx}+E_{h,k}^{tx}/p_h^{dual}$. Choose the pair minimizing ${EPD}_{h,\left\{h,k\right\}}^{uni}$, subject to ${ p}_h^{dual}\ge {\gamma }_h$.

C. 

Small-set flood

Select a relay set $\mathcal{R},\left|\mathcal{R}\right|=s\le S_{max}$. With independence from the following:

$$p_h^{flood}=1-\prod _{r\in \mathcal{R}}\left(1-{ p}_{h,r}\right)$$

(30)

${EPD}_{h,\mathcal{R}}^{flood}= {\displaystyle \sum _{r\in \mathcal{R}}}{E}_{h,j}^{tx}/{ p}_{h,j}$. Select $\mathcal{R}$ (typically $s=3–4)$ to minimize ${EPD}_{h,\mathcal{R}}^{flood}$ with $p_h^{flood}\ge {\gamma }_h$. A greedy construction that iteratively adds the relay with the largest marginal gain $\ge {\Delta }_p/\Delta E$ performs well and is computationally light.

4.8. Node Resident Control Plane

In emergencies, the LoRaWAN infrastructure consisting of gateways, backhaul, and network servers may be impaired or unreachable. Hence, the system cannot depend on centralized MAC or ADR downlinks to function. A lightweight control plane carries epoch time, policy parameters, $\left(p_c,{\eta }_0,\beta ,G \right),TTL$, and optional duty-cycle hints as illustrated in Figure 6. Packets include a tuple $\left(srcID,seq,TTL \right)$. Relays decrement $TTL$, drop if $TTL=0$, and suppresses duplicates using a sliding bitmap per $\left(srcID,seq \right)$. This keeps floods bounded and prevents storm amplification.

Figure 6. Decentralized node resident control plane.

4.9. Selection Policy and Ordering Rule

Duty-cycle and energy harvesting gates apply to all modes. Any candidate relay that cannot transmit due to energy insufficiency $E_i\left(t\right)<c$ or off-time $T_{off}$ is excluded for this epoch. Table 2 summarizes the relay path selection algorithm.

Table 2. Relay path selection pseudocode.

Step 1: Compute candidates. Build ${\mathcal{N}}_h$ by retaining neighbors with ${ p}_{h,r}\ge p_{min}$ or $PRR\ge p_{min}$ and within a hop-limit/TTL constraint.

Step 2: Try unicast. Evaluate ${EPD}_{h,j}^{uni}$ for the top -K neighbors by path loss or PRR. If the best $j^{*}$ satisfies ${ p}_{h,j^{*}}\ge {\gamma }_h$, choose unicast.

Step 3: Escalate to dual. If unicast fails the reliability test or yields high EPD, examine the best $\mathcal{O}\left({\mathcal{K}}^2\right)$ pairs. If any pair satisfies ${ p}_h^{dual}\ge {\gamma }_h$, choose the minimum-EPD pair.

Step 4: Escalate to small set-flood. If dual fails, run a greedy subset selection up to $S_{max}$ to reach $p_h^{flood}\ge {\gamma }_h$ with minimal ${EPD}_{h,\mathcal{R}}^{flood}$.

Step 5: Energy-aware tie-break. When two modes meet ${\gamma }_h,$ prefer the one with lower EPD. If EPDs are within a small slack $\partial ,$ prefer the smaller set to reduce channel load.

Ordering Rule:

After choosing the mode and relay set $\mathcal{R}$, assign an intra-set order to exploit LoRa’s capture and quasi-orthogonality:

Primary sort: lower RSSI at the destination (or highest PRR first).

Secondary sort: prefer lower SF (shorter airtime) earlier if RSSIs are similar. Ties by node ID.

Transmissions are then staggered using the timing-offset rule. Consequently, stronger links tend to be transmitted slightly earlier, increasing the probability that at most one packet falls within the vulnerable window of any weaker concurrent signal.

Complexity Analysis:

• Computation. Unicast $\mathcal{O}\left(\mathcal{K}\right)$, dual $\mathcal{O}\left({\mathcal{K}}^2\right)$, small-set flood (greedy) $\mathcal{O}\left(\mathcal{K}{S}_{max}\right)$. With $\mathcal{K}\le 8,S_{max}\le 4$. This is lightweight on SX1262-class nodes.
• Parameter choices. Typical defaults: ${ p}_{min}\in \left[\mathrm{0.5,0.7}\right],\mathcal{K}=5-8,S_{max} 3-4,\alpha =1.05.$
• Fairness/energy spread. Rotate the primary relay across epochs’ round-robin among top candidates to avoid draining a single node.

4.10. Node-Side Scheduler with SINR-Aware LoRa PHY and Timing Offsets Algorithm

We formally present our scheduling Algorithm 1. The algorithm introduces controlled timing offsets and small jitters to preserve capture effect and tone separability while reducing destructive overlap. The algorithm is divided into three sections:

• Mesh Section (Steps 1–2): Maintains awareness of link quality of neighboring nodes and updates connectivity metrics.
• Contention Section (Steps 3–7): Manages medium access control (MAC) by selecting appropriate radio parameters, calculating timing to avoid collisions, and ensuring regulatory/energy compliance.
• Multi-hop Section (Steps 8–10): Handles multi-hop routing decisions by choosing which neighbors to use as relays, determining transmission modes, and coordinating the forwarding process for end-to-end delivery.

The algorithm integrates all three aspects: it uses mesh connectivity information to make contention-aware radio parameter choices, then applies those choices to execute multi-hop forwarding strategies. Appendix A summarizes the features of the underground mine leveraged during the protocol design.

Algorithm 1. SINR-aware LoRa PHY and timing offsets algorithm.

Inputs

Measured SINR ${\widehat{\gamma }}_h$, recent PRR per neighbor, LoRa PHY options $SF\in \left\{7,\dots ,12\right\},BW\in \left\{\mathrm{125,250,500}\right\}kHz,CR\in \left\{4/6,4/7,4/6\right\}$, energy buffer ${ E}_i,$  $TTL,N_{preamble},$ PER $\to$ SINR table, ${\mathsf{\Gamma }}_{req}\left(SF,BW,CR\right)$, last-TX time $t_{last}$, policy $\left(p_c,{\eta }_0,\beta ,G \right)$, per-hop reliability target ${\gamma }_h$, observation window ${ W}_{obs}$, caps K (neighbors), duty-cycle DC per sub-band, per-TX cost $c$, flood size $S_{max}$, hysteresis $\left(m_{\uparrow },m_{\downarrow },K_{good},K_{bad},T_{dwell} \right)$.

Outputs

Selected ${\left(SF,BW,CR\right)}_h$, timing offset ${\Delta }_{th}$, relay set $\mathcal{R}$, transmit time $t_{tx}$, mode $\in \left\{Unicast, Dual, Flood\right\}$.

Procedure.

1:  RX packet ready: Upon reception or source readiness, timestamp $t_{rx}$.

2:  Measure SINR/PRR: Update ${\widehat{\gamma }}_h$ and per-neighbor PRR (sliding window).

3:  Selected$\left(\mathit{S}\mathit{F},\mathit{B}\mathit{W},\mathit{C}\mathit{R}\right)$ ADR-like with hysteresis:

3.1 Let $P$ be LoRa profiles ordered by speed (low SF, high BW, low CR).

3.2 if (time on current PHY $\ge T_{dwell}=5 s$),        and ( $K_{good}=3 consecutive \mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{d}$):        Attempt upgrade: Choose fastest profile $p\in P$ s.t         ${\widehat{\gamma }}_h\ge {\mathsf{\Gamma }}_{req}\left(p\right)+m_{\uparrow }$        where good frame = (CRC pass) and (SINR $\ge {\mathsf{\Gamma }}_{req}$ − 1 dB), and (RSSI $\ge -$120 dBm),

else if $\left({\widehat{\gamma }}_h\le {\mathsf{\Gamma }}_{req}\left(current PHY\right)-m_{\downarrow }\right)$ or $\left(K_{bad}=2 \mathrm{b}\mathrm{a}\mathrm{d}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}{ W}_{quality}=10 frames\right):$ Immediate downgrade: Choose most robust profile $p$ meeting $:{\widehat{\gamma }}_h\ge {\mathsf{\Gamma }}_{req}\left(p\right)-m_{\downarrow }$.

Otherwise        Maintain current PHY profile $p$ (no change).

3.3 Set $\left({SF}_h,{BW}_h,{CR}_h\right)\leftarrow p,$ set $DE$ per LoRa rule (e.g., $DE=1 for very long symbols).$

4:  Compute:  $T_{sym}$, $T_{preamble}$, $T_{pkt}$.

5:  Estimate:  $N_{est,h}$ Equation (20).

6:  Compute timing offset$: {\Delta }_{th}$ Equations (17) and (18).

7:  Enforce duty-cycle and energy gates via Equation (19).

8:  Mode selection (nearest-unicast/dual-relay/small-set flood)

Build candidate set ${\mathcal{N}}_h$, top—K (neighbors) by PRR/RSSI within $TTL$ and currently eligible under Step 7.

For each ${r\in \mathcal{N}}_h$, estimate success $p_{h,r}$ from PRR/SINR tables with capture margin and energy $E_{h,r}^{tx} \approx \left(P_{tx}/{\eta }_{PA}\right)T_{pkt,h}+E_{oh}$

Unicast: Equation (28)

Else dual: Equation (29)

Else flood: Equation (30)

9:  Ordering and schedule:

Order the selected set $\mathcal{R}$ by (i) descending RSSI at next hop, (ii) lower SF first, (iii) node ID.

Assign index $i=0,\dots ,\left|\mathcal{R}\right|-1$ and schedule $t_{tx}^{\left(i\right)}=t_{rx}+G+i{\Delta }_{th}+\epsilon$

10:  Transmit and update state:

Send $t_{tx}^{\left(i\right)}$ with $\left({SF}_h,{BW}_h,{CR}_h,DE\right)$, decrement energy by $c$, set $t_{last}\leftarrow$ end-of-tx, update PRR/hysteresis counters.

5. Far-Field (Fraunhofer) RF-WPT Real-World Experiment

In this section, we provide simulated and experimental validation of the far-field RF-WPT as was mentioned in Section 2.

5.1. Underground Mining Environment

Missouri S&T’s Experimental Mine in Figure 7 is used for this study. The underground mine is characterized by highly irregular sidewall surfaces, with variations in surface protrusion reaching approximately 20 cm between highest and lowest points, along with bends and tilts as illustrated in Figure 7. The mine features multiple drifts and slope accesses with supporting pillars that serve as navigational landmarks for miner localization. Layout dimensions vary, with widths ranging from 2.5 to 3 m and heights between 2.2 and 10 m. The section utilized in this study spans 67 m total, consisting of 37 m with line-of-sight (LoS) conditions and 30 m without line-of-sight (NLoS).

Figure 7. Missouri S&T’s underground mine layout.

5.2. Underground Mining Long-Range Far-Field WPT Testbed

The WPT system consists of a transmitter and receiver, illustrated in Figure 8 and Figure 9, with equipment specifications in Table 3. The transmitter configuration includes a high-gain directional Yagi antenna (11 dBi) operating in the 880–960 MHz frequency band, driven by a 12 V/6.8 A, 26 dB gain, 20 W Class AB power amplifier. The high-power amplifier is essential to overcome the substantial path loss encountered in underground mining environments. For this experiment, a frequency-modulated (FM) square wave signal was generated using VSG60 signal generation software. On the receiver side, the Powercast P21XXCSR-EVB board (Powercast Corporation501 Technology DriveSuite 106, Canonsburg, PA, USA) was utilized. The board is equipped with two onboard capacitors for storing harvested DC energy, rated at 2200 μF and 50 mF. However, these capacitors have limited energy storage capacity. To address this limitation, supercapacitors (electric double-layer capacitors, EDLCs) were introduced. Supercapacitors store significantly higher charge than conventional capacitors due to high capacitance values measured in farads. This study focuses on large-cell supercapacitors. Three variants were evaluated: 5 V/1 F, 2.5 V/25 F, and 2.7 V/100 F. The 2.5 V/25 F supercapacitor was selected due to its faster charging characteristics despite the lower energy storage capacity compared to the 2.7 V/100 F variant. Supercapacitors are particularly relevant in underground mining scenarios where conventional AA batteries are prohibited in coal mines due to the risk of explosion or fire during emergencies.

Figure 8. Far-field long range WPT transmitter.

Figure 9. Far-field RFEH receiver.

Table 3. Equipment description.

Device/Software	Description
Laptop	HP Intel(R) Core (TM) i5-1035G1 CPU @ 1.00GHz 1.19 GHz, Windows 10.
Vector Signal Generator	VSG60A 6GHz—Signal Hound hardware, VSG60 Signal Generation Software (VSG60 Software 64-bit).
Transmit Antenna	880–960 MHz high gain directional 11 dBi Yagi
Receiver Antenna	VERT900 vertical antenna omnidirectional, gain 3 dBi.
WPT Receiver	Powercast P21XXCSR-EVB.
Power Amplifier (PA)	880 MHz- 960 MHz, 12 V/6.8 A, 26 dB Gain, 20 W RF power, Class AB
Oscilloscope	KEYSIGHT DSOX1204A.
Digital Multimeter	Gw INSTEK GDM-8341.
Connectors	Laptop to VSG60A: USB2.0, VSG60A to PA: SMA plug male pin To SMA plug male connector -TNC male connector, WPT receiver to Oscilloscope: jumper cables.
DC power supply	30 V/10A DC.

The accumulated voltage was regulated through a 4.1–4.2 V output rail, selected to optimize charging performance. Higher charging voltages enhance charging speed, a critical factor in EH applications. During experimentation, the transmitter was fixed in position while the Powercast RF-EH receiver was incrementally moved away in 5 m steps. Two measurements were conducted: (i) feasibility of long-range far-field energy harvesting in an underground mine using an FM square wave carrier at 860 MHz, and (ii) efficiency of WPT in replenishing energy in the three supercapacitor variants. The voltage across each capacitor was measured using a digital oscilloscope with probes connected between the VOUT and GND pins. Prior to each measurement, capacitors were fully discharged by briefly connecting an external resistive load.

5.3. Link Budget Analysis on the Impact of the External Power Amplifier

Link budget analysis (Table 4) measures every gain and loss from transmitter to receiver, predicting if a link will be reliable or not. The underground environment presents a harsh RF propagation environment. Without power amplification, RF energy harvesting is not feasible, as demonstrated in Figure 10. Figure 10 is simulated using the underground channel propagation model (Equations (1) and (2)). With the external PA, the effective radiated power increases by 25.7 dB, extending the feasible WPT range from ~8 m to 35 m, as demonstrated in Figure 11.

Table 4. Link budget analysis.

Component	Without PA	With 20 W PA
Base transmitter	10 dBm	10 dBm
PA gain	0 dB	26 dB
PA output power	10 dBm	35.5 dBm
Post-PA cable loss	0.5 dB	0.3 dB
Transmit antenna gain	11 dBi	11 dBi
EIRP	20.5 dBm	46.2 dBm
Power gain from PA	—	25.7 dB

Figure 10. Received power vs. distance in underground mine environment.

Figure 11. The impact of using external PA for underground mine RF-EH.

5.4. Long-Range Far-Field WPT Measurement Campaign

The goal of this measurement campaign is to determine the maximum WPT distance. The RF power from the VSG60A interface was set to 10 dBm. On the P21XXCSR-EVB receiver, the jumper was set to J2 (corresponding to the 860 MHz RF band), JP3 was set to 1.2 V, and S1 was set to 4.1–4.2 V to optimize supercapacitor charging. Higher DC voltages enhance charging speed, a critical factor in energy-harvesting applications. S3 was selected to measure the harvested DC voltage between the VOUT and GND terminals.

Based on Figure 12, long-distance far-field WPT is achievable in underground mines using an FM square wave carrier. Next, we measured how energy is replenished in the three supercapacitor variants. Using the testbed described above, the achievable RF-WPT range in the underground mine is 35 m, as shown in Figure 13, despite all RF losses. As charging distance increases, instantaneous charging voltage decreases. The charging rates of the various supercapacitors were logged using Keysight PathWave BenchVue oscilloscope software (Version 2024.1). Figure 13 and Table 5 shows charging voltage vs. distance for the three supercapacitor variants. The 5 V/1 F provides the fastest voltage rise over distance, while 2.7 V/100 F charges slowest. The energy stored in each capacitor is as follows:

$$U_C={\displaystyle \frac{1}{2}}{CV}^2$$

(31)

Figure 12. Far-field region WPT in underground mines.

Figure 13. Far-field distance charging of various supercapacitors.

Table 5. Supercapacitors’ charging performance.

Distance (Meters)	2.5 Volts 25 F			2.7 Volts 100 F			5.5 Volts 1 F
	Time (s)	Volt (V)	Energy (J)	Time (s)	Volt (V)	Energy (J)	Time (s)	Volt (V)	Energy (J)
5	120	0.250	1.71125	120	0.37	3.125	120	0.44	0.0968
10	120	0.180	1.12500	120	0.3	1.62	120	0.35	0.06125
15	120	0.100	0.60500	120	0.22	0.5	120	0.24	0.0288
20	120	0.060	0.18000	120	0.12	0.18	120	0.19	0.01805
25	120	0.090	0.08000	120	0.08	0.405	120	0.1	0.005
30	120	0.009	0.00061	120	0.007	0.00405	120	0.01	0.00005
35	120	0.015	0.00005	120	0.002	0.01125	120	0.01	0.00005

$C$ is the capacitance of the supercapacitor, and $V$ is the supercapacitor voltage. As shown in Table 5 and Figure 14, the 5 V/1 F supercapacitor charges fastest but stores minimal energy insufficient to power most microcontroller units (MCUs). Conversely, the 2.7 V/100 F stores the highest energy but requires significantly longer charging time. This trade-off between energy storage capacity and charging rate is fundamental to capacitor physics, as described by [30]:

Figure 14. Supercapacitors’ energy storage over long distance.

$$C={\displaystyle \frac{k{e}_{0}A}{d}}$$

(32)

$d$, $A$, and $k{e}_0$ denote plates separation distance, plate area, and dielectric constant of the insulator. Higher capacitance requires a larger area or smaller separation, both of which limit voltage ratings. To address the low voltage output, we integrated a TPS6109x synchronous boost converter (2-A switch) to step up the supercapacitor voltage to the MCU’s required level. Additionally, supercapacitors exhibit inherent equivalent series resistance (ESR), which prevents complete discharge and limits energy utilization. The boost converter mitigates this by extracting residual energy that would otherwise remain inaccessible.

5.5. Study of Supercapacitor Leakage Current

Supercapacitors suffer from self-discharge, as shown in Figure 15 and Table 6. Self-discharge causes the supercapacitor to lose stored energy even without any external load, due to internal ionic redistribution that requires time to stabilize. To characterize this effect, the 2.7 V/100 F supercapacitor was fully charged and connected to a digital oscilloscope with no external load. The voltage was monitored over time. As shown in Figure 15, the initial voltage drop rate was significant (0.9 V over 83 min), but the discharge rate decreased as the supercapacitor stabilized. After approximately 5.5 h, the voltage had decreased from 2.73 V to 2.57 V, representing a 5.9% energy loss. This self-discharge characteristic has important implications for underground mine deployments: continuous or periodic WPT is necessary to maintain sufficient energy levels in the supercapacitors, as prolonged periods without RF power will deplete the stored energy through self-discharge.

Figure 15. Supercapacitor self-discharge curve.

Table 6. 2.7 V/100 F supercapacitors’ self-discharge.

Seconds	0	5000	10,000	15,000	20,000
Minutes	0	83.33	166.67	250	333.33
Hours	0	1.39	2.78	4.17	5.56
Voltage	2.73	2.64	2.61	2.58	2.57
Voltage drop	0	0.9	0.3	0.3	0.1

6. Results and Discussion

This section presents simulation results of the proposed algorithm. We evaluate performance against three baseline algorithms: (i) ALOHA [14], (ii) no offsets (SINR adaptation without timing offsets) [17], and (iii) SINR-only (timing offsets without PHY adaptation) [8]. Simulation metrics and parameters are defined in Table 7 and Table 8. Simulations were implemented in Python 3.

Table 7. Simulation parameters definitions.

Metric	Symbol/Unit	How Measured	Acceptance Target
End-to-End Delivery	EDR  $\mathbf{0}-\mathbf{1}$	Unique packets accepted at sink/packets generated	$\ge \mathbf{0.90}$ for disaster bursts
Packet Error Rate per hop	${\mathit{P}\mathit{E}\mathit{R}}_{\mathit{h}}\left(\mathbf{0}-\mathbf{1}\right)$	1- PRR at each hop from logs	$\le \mathbf{0.30}$ at median hop
End-to-End Latency (95th)	$\mathit{m}\mathit{s}$	Sink accept timestamp—source tx timestamp (excluding duplicates)	$\le {\mathit{D}}_{\mathit{m}\mathit{a}\mathit{x}}$ e.g., 2–5 s
Energy per Delivered Packet	$\mathit{E}\mathit{P}\mathit{D}$ $\left(\mathit{J}/\mathit{p}\mathit{k}\mathit{t}\right)$	Sum node TX/RX/overhead energy/unique delivered packets	$\le \mathbf{0.60}\times$ Baseline (no offsets)
Outage Probability	$\mathit{P}\mathit{r}\left[\mathit{E}<\mathit{c}\right]$	Fraction of epochs with insufficient energy to TX	$\le \mathbf{5}\mathit{\%}$

Table 8. Simulation parameters values.

Parameter	Value	Unit	Description
Network Topology Parameters
Hops count	4	Hops	Number of hops from source to destination
Nodes per hop	[5, 10, 15, 20, 25, 30]	Nodes	Variable load tested across different densities
Area size	100.0	meters	Network deployment area (100 m × 100 m)
Topology	Linear multi-hop	-	Sequential hop topology
Physical Layer Parameters
Pathloss Expo	2.0	-	Path loss exponent (α)
$P_{max}$	3.0	Watts	Maximum transmission power
$P_{min}$	0.1	Watts	Minimum transmission power
${Noise}_{Power}$	$1\times {10}^{-12}$	Watts	Additive White Gaussian Noise (AWGN)
${Target}_{Sinr\_dB}$	3.0	dB	Target SINR threshold
${Target}_{Sinr}$	1.995	Linear	Target SINR $\left({10}^{\left(3/10\right)}\right)$
Bandwidth	$1\times {10}^6$	Hz	Channel bandwidth (1 MHz)
MAC Layer Parameters
Slot duration	0.01	seconds	Time slot duration (10 ms)
Guard time	0.0005	seconds	Guard interval (0.5 ms)
Max retransmissions	10	attempts	Maximum retransmission attempts before drop
Traffic Generation Parameters
Packet generation probability	0.2	probability	Packet generation probability per slot
Traffic model	Poisson	-	Random packet arrivals
Source nodes	Hop 0 only	-	Only source nodes generate traffic
Packet size	25	Bytes	Fixed-size packets

6.1. Simulation Methodology and Reproducibility

To ensure reproducibility and rigorous statistical validation of our results, we provide comprehensive details of the simulation framework implementation. Table 7 and Table 8 define the simulation parameter definitions and values, respectively.

(i)

Random Number Generation and Seeding: All stochastic processes use independent pseudorandom number generators with fixed seeds:

• Master seed: 42.
• Load-specific seed: Master seed × load × 1000.
• Algorithm seed: Load-specific + algorithm index × 100.
• Run-specific seed: Algorithm seed + run number.

(ii)

Trial Structure: Total of 2000 independent simulation runs, each with different random channel/traffic realizations using the seeds above.

6.2. Simulated Results

6.2.1. End-to-End Delivery Rate

The end-to-end delivery of the proposed algorithm and other baseline algorithms is presented in Figure 16. The proposed protocol achieves 95% delivery by shrinking the vulnerable window factor from ${\mathit{\eta }}_{\mathbf{0}}=\mathbf{2.0}$ to ${\mathit{\eta }}_{\mathbf{0}}=\mathbf{1.3}$ through controlled timing offsets with symbol-scale jitter that decorrelates transmission start times while maintaining sync word coherence within ±0.5 symbol tolerance for network-layer filtering. The offsets-only scheme performs worst overall, showing that timing offsets alone cannot guarantee reliability without PHY adaptation or relay selection logic. In contrast, the proposed protocol delivers significantly higher delivery ratios of roughly double the performance of baselines at moderate load, demonstrating its ability to sustain concurrent transmissions while avoiding collapse under congestion. Notably, the proposed scheme maintains above 40% delivery at 15 contenders per hop, where all baselines drop below ~20%, evidencing the combined benefits of adaptive timing offsets, SINR-aware PHY selection, and energy-sustainable relay participation. Overall, the trend confirms that neither random access nor single-dimension adaptation is sufficient for dense underground emergency networks, and coordinated scheduling with PHY adaptation is essential.

Figure 16. End-to-end delivery rate.

6.2.2. Packet Error Rate

Figure 17 compares the packet error rate across all the algorithms because of collision packets. The 70% PER reduction validates Poisson collision modeling where arrival rate ${\mathit{\lambda }}_{\mathit{h}}={\mathit{N}}_{\mathit{e}\mathit{s}\mathit{t},\mathit{h}}/{\mathit{\tau }}_{\mathit{h}}$ must satisfy ${\mathit{e}}^{-{\mathit{k}}_{\mathit{h}}{\mathit{v}}_{\mathit{h}}}\ge 1-{\mathit{p}}_{\mathit{c}}$ with target ${\mathit{p}}_{\mathit{c}}=\mathbf{0.2}$, enforcing forwarding window sizing proportional to estimated contenders and vulnerable window duration. The proposed algorithm consistently achieves the lowest error rate (starting at 19% and rising to 89%), demonstrating superior collision avoidance through its hybrid approach of adaptive power control and flexible time-slot coordination. ALOHA and SINR-only exhibit nearly identical performance (converging at ~90% error rate under high load), as neither employs coordinated scheduling to prevent simultaneous transmissions. The offsets-only algorithm performs worst (reaching 99% error rate), paradoxically suffering from its rigid time-slot assignment which creates systematic collision patterns when nodes are tightly scheduled. At low loads (N₀ = 5), the proposed algorithm shows a 3× improvement over baselines (19% vs. 53% error rate), highlighting the significant reliability gains from coordinated medium access in multi-hop wireless networks.

Figure 17. Packet error rate.

6.2.3. End-to-End Latency (95th Percentile)

Despite deliberately introducing timing offsets, the proposed protocol achieves lower latency than ALOHA because collision avoidance eliminates retransmission cascades that dominate delay in random access systems (Figure 18). At low loads (N₀ = 5), all algorithms except offsets-only perform similarly (~10–11 slots), but as congestion grows, their behaviors diverge significantly. The proposed algorithm maintains relatively stable latency (rising from 11 to 20 slots), benefiting from its adaptive mechanisms that reduce collisions and minimize retransmission delays. ALOHA shows the highest latency growth (reaching 21 slots at N₀ = 30) due to frequent collisions forcing packets to wait longer for successful transmission. Surprisingly, offsets-only exhibits the lowest latency at light loads (seven slots) due to deterministic scheduling, but this advantage disappears at moderate loads (N₀ ≥ 20) where its rigid structure causes the high collision rates we saw earlier, and its 95th percentile becomes unreliable due to the tiny fraction of packets that actually succeed.

Figure 18. End-to-end latency (95th percentile).

6.2.4. Energy per Packet Delivered

Figure 19 highlights the energy per packet delivered. The 75% efficiency improvement stems from mode selection between unicast, dual-relay, and small-set flood ( ${\mathit{S}}_{\mathit{m}\mathit{a}\mathit{x}}=4$) based on greedy marginal gain $\Delta \mathit{p}/\Delta \mathit{E}$ calculations that minimize energy expenditure while satisfying per-hop reliability target ${\mathit{\gamma }}_{\mathit{h}}=\mathbf{0.9}$. Energy consumption includes both transmission ${\mathit{E}}_{\mathit{h}}^{\mathit{t}\mathit{x}}\approx \left({\mathit{P}}_{\mathit{t}\mathit{x}}/{\mathit{\eta }}_{\mathit{P}\mathit{A}}\right){\mathit{T}}_{\mathit{p}\mathit{k}\mathit{t}}+{\mathit{E}}_{\mathit{o}\mathit{h}}$ with ${\mathit{\eta }}_{\mathit{P}\mathit{A}}=\mathbf{0.4}$ PA efficiency and overhead, while supercapacitor dynamics model non-linear RF-DC conversion efficiency ${\mathit{\eta }}_{\mathit{R}\mathit{F}-\mathit{D}\mathit{C}}\in \left[\mathbf{10}\mathbf{\%}@-\mathbf{20}\mathbf{d}\mathbf{B}\mathbf{m},\mathbf{50}\mathbf{\%}@-\mathbf{10}\mathbf{d}\mathbf{B}\mathbf{m}\right]$ with leakage constant ${\mathit{\rho }}_{\mathit{l}\mathit{e}\mathit{a}\mathit{k}}=\mathbf{1000} \mathit{s}$ limiting storage effectiveness. The M/G/1 queue with energy-gating vacations accurately predicts system behavior where effective service rate ${\mathit{\mu }}_{\mathit{e}\mathit{f}\mathit{f},\mathit{h}}=\left(\mathbf{1}/{\mathit{T}}_{\mathit{p}\mathit{k}\mathit{t},\mathit{h}}\right){\mathit{\phi }}_{\mathit{i}}$ depends on energy availability probability ${\mathit{\phi }}_{\mathit{i}}\triangleq \mathit{P}\mathit{r}\left[{\mathit{E}}_{\mathit{i}}\left(\mathit{t}\right)\ge \mathit{c}\right]$, requiring stability condition ${\exists }_{\mathit{h}}^{\mathit{t}\mathit{o}\mathit{t}}<{\mathit{\mu }}_{\mathit{e}\mathit{f}\mathit{f},\mathit{h}}.$ Round-robin relay rotation among top candidates prevents draining best-RSSI nodes while maintaining EPD optimality across network-wide operations.

Figure 19. Energy per delivered packet.

6.2.5. Outage Probability

Figure 20 shows outage probability (the fraction of packets that fail to reach their destination across all four hops) as network congestion increases from 5 to 30 contending nodes per hop. The proposed algorithm demonstrates superior reliability, maintaining near-zero outage at low loads (N₀ ≤ 10) and achieving only 81% outage even under severe congestion (N₀ = 30), compared to 95–99% for baselines. ALOHA and SINR-only converge to nearly identical failure rates (~95% outage at N₀ = 30), confirming that power control alone provides minimal benefit without coordinated scheduling in heavily congested multi-hop networks. The offsets-only algorithm catastrophically fails, reaching 99% outage by N₀ = 15, demonstrating that rigid time-slot assignment without adaptation creates systematic collision patterns that prevent packet delivery. At moderate loads (N₀ = 20), the proposed algorithm achieves a 2× improvement in reliability over ALOHA (50% vs. 84% outage), representing the difference between a marginally functional and a severely degraded network. The widening gap between proposed algorithms and baselines at higher loads proves that hybrid adaptive MAC protocols are essential for maintaining reasonable service in congested wireless multi-hop networks, where uncoordinated approaches essentially collapse.

Figure 20. Outage probability.

6.2.6. Network Throughput Analysis

Figure 21 shows network throughput (successfully delivered packets per time slot) as congestion increases, revealing a critical trade-off between offered load and actual delivery capacity in multi-hop wireless networks. The proposed algorithm achieves peak throughput of 0.44 packets/slot at N₀ = 10, demonstrating optimal medium utilization through its adaptive coordination, then gracefully degrades to 0.12 packets/slot at maximum congestion. In stark contrast, ALOHA and SINR-only peak much lower (0.24–0.28 packets/slot at N₀ = 5) and decline monotonically, converging to near-zero throughput (~0.03 packets/slot) at N₀ = 30 where collisions dominate. The proposed algorithm maintains 4× higher throughput than baselines at N₀ = 30 (0.12 vs. 0.03 packets/slot), proving that hybrid MAC protocols prevent the throughput collapse that afflicts uncoordinated approaches. Offsets-only shows catastrophic performance throughout, peaking at only 0.19 packets/slot before plummeting to essentially zero, confirming that rigid scheduling without adaptation is worse than random access. The inverted-U shape of the proposed curve reveals the classic congestion control problem: moderate contention enables high throughput through statistical multiplexing, but excessive contention requires sophisticated coordination—which only the proposed algorithm successfully provides.

Figure 21. Network throughput.

7. Conclusions

This paper presents an energy-harvesting concurrent LoRa mesh network for underground mine emergency communications when conventional infrastructure fails. The system integrates far-field RF wireless power transfer with intelligent timing-based coordination to address energy scarcity and severe interference during disaster scenarios. Experimental validation in Missouri S&T’s operational underground mine demonstrated far-field wireless power transfer (WPT) feasibility at distances up to 35 m using external power amplification and supercapacitor storage. Characterization of three supercapacitor variants revealed fundamental trade-offs between charging speed and energy density, with the 2.5 V/25 F configuration offering the optimal balance. The measured 5.9% self-discharge over 5.5 h confirms the necessity of continuous or periodic WPT for sustained operations.

The proposed timing offset scheduler derives optimal per-hop staggering through Poisson collision modeling, exploiting LoRa’s capture effect and imperfect inter-spreading-factor orthogonality without requiring tight synchronization. The SINR-aware adaptive physical layer jointly optimizes spreading factor, bandwidth, and coding rate with hysteresis controls, feeding updated airtime back into scheduler decisions to maintain protocol coherence under dynamic conditions.

Simulations across 2000 independent trials under emergency burst scenarios demonstrated substantial improvements: a 2.2× throughput improvement over ALOHA (49% vs. 22% packet delivery ratio at 10 nodes per hop), a 64% collision rate reduction, and a 67% energy efficiency improvement. The energy-aware mode selection successfully balanced reliability requirements against energy consumption through dynamic switching among unicast, dual-relay, and flooding strategies. This work establishes that RF-powered concurrent LoRa meshes provide resilient emergency communications in underground environments despite severe propagation challenges and high-contention traffic. Future work should address multi-gateway coordination for larger mine complexes, field trials during simulated emergency exercises, and hybrid energy harvesting combining RF with vibration or thermal sources for enhanced sustainability.