Harvesting-Aware Routing for Low-Latency Reliable Transmission in Interference-Powered Networks

1

Harvesting-Aware Routing for Low-Latency Reliable Transmission in Interference-Powered Networks

arXiv:2610.14250v1 [cs.NI] 9 Oct 2026

Azadeh Sheikholeslami, Allan Nguyen Math and Computer Science Department, Suffolk University, Boston, MA

Abstract—We consider a multi-hop wireless network where energy-harvesting nodes rely solely on ambient radio-frequency interference as their power source. The goal is to deliver a packet from source to destination with minimum latency while guaranteeing a target end-to-end outage probability under Rayleigh fading. This gives rise to a joint routing and power allocation problem in which interference plays a dual role: it provides harvestable energy, but also impairs link reliability. We first transform the intricate, multiplicative end-to-end outage constraint into an additive form. Then, we propose two optimal algorithms. The first, HOPE, minimizes the bottleneck harvesting latency, i.e., the longest energy-accumulation time along the path, by reducing the problem to a least-cost path computation on a graph with interference-aware link weights. The second, THOPE, extends the objective to total end-to-end latency, adding the cumulative transmission delay, and likewise admits a closed-form least-cost path solution. Both algorithms are solved in polynomial time via Dijkstra’s algorithm and yield explicit expressions for the optimal transmit powers. Extensive simulations reveal that HOPE and THOPE reduce the average total latency by up to an order of magnitude compared with shortest-path and minimum-energy benchmarks, while exhibiting sharply concentrated delay distributions that are largely insensitive to interferer density.

I. Introduction Next-generation wireless networks, spanning industrial IoT, smart cities, disaster-response systems, and autonomous sensor swarms, must simultaneously satisfy three stringent requirements: reliability (successful data delivery under strict outage limits), low latency (timely delivery from sensing to actuation), and energy selfsufficiency (perpetual operation without battery replacements). Although each of these demands has received substantial attention in isolation, their joint optimization remains an open challenge, especially when energy is harvested from the same radio-frequency (RF) environment that also causes interference. Traditional routing protocols for multi-hop wireless networks assume a continuous power supply [3], [4] and treat interference as a purely detrimental factor [5]–[8]. In energy-harvesting (EH) networks, however, ambient RF interference can be converted into usable energy [9]– [11]. This paradigm shift—from interference avoidance to interference exploitation—enables sustainable operation Parts of this work were presented at the 61st Allerton Conference on Communication, Control, and Computing, 2025 [1], and at the 2026 IEEE MIT Undergraduate Research Technology Conference (URTC) [2].

in scenarios where dedicated power infrastructure is impractical, such as structural health monitoring, precision agriculture, or post-disaster emergency networks. Yet, harvesting from interference also introduces a fundamental tension: the same interferers that provide energy also degrade the signal-to-interference-plus-noise ratio (SINR), threatening link reliability. Most existing ambient-RF harvesting designs either focus on single-hop links [12], [13] or adopt energy-aware routing heuristics without probabilistic outage guarantees [14], [15]. Few works simultaneously consider multihop routing, dynamic power allocation, and end-to-end outage constraints. For instance, Thai [16] proposes trustdriven routing that harvests from jammers, Ghosh [17] analyses fixed multi-hop EH topologies, Wang [18] introduces a clustering protocol for cognitive radio sensors, Shan [19] develops a harvest-then-transmit schedule, and Ren [20] presents opportunistic routing with dynamic power. While these works advance the state of the art, none of them jointly determine both the route and the transmit powers so as to minimize the end-to-end latency that arises from the harvesting bottleneck: the fact that multi-hop forwarding cannot begin until every node on the path has accumulated sufficient energy. Moreover, they rarely consider the additional cumulative transmission delay across hops, which can become significant in dense or long paths. In this paper, we propose two complementary algorithms that close this gap by jointly optimizing routing and power allocation under both energy-neutrality and end-to-end outage constraints, while explicitly treating ambient interference as a harvestable resource. • HOPE (Harvesting-Optimized Path selection with End-to-end outage guarantees): minimizes the bottleneck harvesting latency, i.e., the time required for the slowest node on the path to harvest enough energy. We show that the resulting min-max problem can be transformed into an equivalent shortest-path problem on a weighted graph, enabling polynomialtime solution via Dijkstra’s algorithm (Section III). • THOPE (Total-latency HOPE): extends the objective to minimize the total end-to-end latency, defined as the sum of the bottleneck harvesting time and the cumulative transmission time over all hops. Remarkably, this more complex objective also admits a closed-form shortest-path formulation, with a modified link weight that accounts for both the harvesting

2

time and the constant per-hop transmission duration (Section IV). Both algorithms build on a unified system model in which nodes operate in an energy-neutral manner, using only the RF energy harvested from ambient interferers during a harvest-then-transmit cycle. We derive the perlink transmission time from the SINR threshold and packet size, showing that it becomes network-wide constant under a common rate policy, which allows the entire problem to be cast as a pure graph-theoretic optimization. Extensive numerical comparisons against Shortest-Path routing with uniform outage allocation and Minimum-Energy Routing with outage constraints demonstrate that HOPE and THOPE dramatically reduce total latency (up to a 10fold improvement), exhibit sharply concentrated delay distributions, and adapt gracefully to varying interferer densities, network sizes, and outage requirements (Section VI). The key contributions of this work are as follows. We provide a systematic transformation of the intricate, multiplicative end-to-end outage constraint into an additive sufficient condition under Rayleigh fading in Lemma 1, which decouples link-level variables and enables efficient optimization. Then, we develop the HOPE framework, which formulates the bottleneck harvesting-latency minimization as a convex program for a fixed path, yields a closed-form solution via KKT conditions, and reduces to a shortest-path problem with interference-aware link weights. We extend the framework to total end-to-end latency through THOPE, which additionally accounts for the cumulative transmission delay, and we derive the corresponding shortest-path weights. The link weights for algorithms described above only depend on the aggregate interference power at each node. This property comes from the conservative upper bound used in Lemma 1. We show that this upper bound is very tight, and using the exact outage probability for routing will not improve the harvesting latency beyond a negligible fraction. Finally, we present a comprehensive performance evaluation that reveals the gains of joint routing and power allocation over conventional benchmarks and provides insights into the trade-offs between harvesting time, hop count, and reliability. The remainder of the paper is organized as follows. Section II describes the system model, including the channel, energy-harvesting, and outage characterization. Section III formulates the HOPE problem and presents its solution. Section IV extends the framework to THOPE. Section V analyzes the tightness of the additive outage constraint, and Section VI provides numerical results and comparisons. Finally, Section VII concludes the paper and outlines future research directions. II. System Model and Preliminaries A. Network Model We consider a multi-hop wireless network of energyharvesting (EH) nodes deployed at arbitrary positions in

the plane, modeled as a directed graph G = (N , E), where N is the set of nodes and E is the set of feasible links: a directed link (u, v) belongs to E if and only if du,v ≤ rc , where du,v is the Euclidean distance between u and v and rc is the common communication range. The nodes have neither a long-term battery nor an external power supply: their only energy source is the radio-frequency (RF) interference generated by a set I0 of interferers at arbitrary positions, which may be intentional jammers or uncoordinated co-channel transmitters. Interferer i ∈ I0 transmits continuously at a constant power Pi , and the positions and powers of the interferers remain fixed over the harvest–transmit cycle of a packet. We consider the delivery of a single flow; any concurrent traffic is treated as part of this exogenous interferer set, with a time-averaged power at each node that is likewise constant over the cycle. All nodes and interferers use omnidirectional antennas and share a common frequency band. A packet is to be delivered from a source S to a destination D over a simple path Π, that is, a sequence of directed links (u, v) ∈ E in which no node appears twice. We write u ∈ Π for the transmitting nodes of Π, namely the source and the relays; each transmits on exactly one link of the path, and the destination only receives. B. Channel Model All channels undergo frequency-flat Rayleigh block fading. The fading coefficient hu,v between nodes u and v is constant over the transmission of a packet and independent across links, with |hu,v |2 exponentially distributed and normalized to E[|hu,v |2 ] = 1; the coefficients hi,v from interferer i to node v are defined likewise. When node u transmits, the signal received at node v is s s X Pu Pi hi,v yv = hu,v xu + (1) α xi +nv , dα d u,v i,v i∈I0 {z } | | {z } desired signal

interference

where xu (xi ) is the unit-power symbol transmitted by node u (interferer i), Pu (Pi ) is the corresponding transmit power, du,v (di,v ) is the distance to node v, α is the path-loss exponent, and nv ∼ CN (0, N0 ) is additive white Gaussian noise. So that the received power cannot exceed the transmitted power in the near field, every path-loss distance is floored at a reference distance d0 ; with a slight abuse of notation, du,v denotes max(du,v , d0 ) in all pathloss terms. Receivers treat interference as noise. Under the single-flow assumption of Section II and because the nodes of Π forward the packet sequentially (Section II-E), at most one node transmits at any time, so the only interference in the network is that generated by I0 . C. Transmission Rate and Duration All nodes decode with a common SINR threshold γ (Section II-F) and transmit at the fixed rate ρ = log2 (1 + γ) bits/s/Hz, the rate at which a Gaussian codebook is decodable whenever the instantaneous SINR is at least γ.

3

A packet of B bits sent over a bandwidth W therefore occupies each link for τu,v =

B . W log2 (1 + γ)

(2)

Because γ, B, and W are common to the whole network, the transmission time is the same on every link; we denote this constant value by τ , so that τu,v = τ for all (u, v) ∈ E. D. Energy Harvesting and Energy Neutrality Each node operates in an energy-neutral manner [21], with only short-term buffering and no reliance on longterm battery storage1 . During a harvesting interval of duration Tu , node u collects the energy Euharv = Φ̄u Tu ,

(3)

where Φ̄u > 0 is the time-averaged power delivered by the harvester of node u. It is a non-decreasing function " # X Pi |hi,u |2 X Pi = , (4) Φ̄u = Φ(Iu ), Iu ≜ E α di,u dα i,u i∈I0

i∈I0

of the average interference power Iu received at node u: the stronger the interference reaching a node, the more energy it harvests. This is the precise sense in which interference is a resource here [23], and it is what the routing of Section III exploits. We leave the characteristic Φ otherwise arbitrary. It may be the response of any rectifier, including the limited sensitivity and the saturation exhibited by practical circuits [24]–[26], and every analytical result in this paper holds for any such Φ. For instance, a practical rectifier may be modeled by Φ(I) = ηI between its sensitivity threshold and its saturation knee, and by a bounded function beyond it; the results below are unchanged in either regime, since they depend on Φ only through the per-node constant Φ̄u . Only two properties are used: Φ̄u is constant over the harvesting interval, which holds when that interval spans many fading coherence intervals, and its value is known to node u, which measures it at its own rectifier output. A node with Φ̄u = 0, for instance one below the sensitivity threshold of its rectifier, can never fund a transmission and is simply excluded from the routing graph. For the numerical evaluation of Section VI we adopt the linear model Φ(Iu ) = η Iu , where η ∈ (0, 1) is the energy conversion efficiency; the model is accurate between the sensitivity threshold and the saturation level of a practical rectifier. The parameters of Section VI are normalized rather than drawn from a particular deployment, so the simulations there compare the schemes across regimes rather than predict absolute latencies. The contribution of 1 In practice, we assume each node is equipped with a low-cost electric double-layer supercapacitor (EDLC) that can buffer the harvested energy for short durations (on the order of tens of seconds to a few minutes), sufficient to span the harvest–transmit cycle. Incorporating the supercapacitor buffer preserves energy neutrality while enabling brief scheduling flexibility between harvesting and transmission [22].

Fig. 1. Multi-hop energy-harvesting network and bottleneck latency: The source, R1 , and R2 start harvesting at t = 0; the node with the longest Tu (here R1 ) determines when end-to-end forwarding can begin.

thermal noise to the harvested power is negligible relative to that of the interferers and is omitted. Transmitting at power Pu for the duration τ consumes the energy Eutx = Pu τ . Energy neutrality requires Eutx ≤ Euharv , that is, Pu τ . (5) Tu ≥ Φ̄u Within a harvest–transmit cycle, harvested energy may be buffered as described above; no energy, however, is carried over from one cycle to the next, so each transmission must be funded entirely by the harvesting interval that precedes it. This harvest-use operation is representative of ultralow-power nodes with small energy storage [27] and yields a conservative, worst-case latency. We assume each node v can measure the aggregate interference power Iv received at its own antenna, together with the power Φ̄v its harvester delivers, and that these measurements are available to the source when the route is computed. No knowledge of the number, positions, or powers of the individual interferers is required. E. Harvest-then-Transmit Protocol and Latency Nodes follow a harvest-then-transmit protocol. We assume the route Π is computed before delivery begins and is known to every node on it. At time t = 0, every transmitting node of Π begins harvesting with no stored energy, and node u harvests for Tu . Once all transmitters have completed harvesting, the packet is forwarded in a store-and-forward manner, hop by hop from S to D, each hop occupying τ . Forwarding therefore cannot begin until the slowest transmitter has finished harvesting, as illustrated in Fig. 1: the node with the longest harvesting interval is a bottleneck that dictates the start of forwarding for the entire path. F. Outage Probability and End-to-End Reliability Link (u, v) is in outage when the SINR at its receiver falls below the decoding threshold γ: ) ( Pu |hu,v |2 /dα u,v u,v P <γ , (6) pout = P N0 + i∈I0 Pi |hi,v |2 /dα i,v where the sum in the denominator is the instantaneous interference power at node v. Distinct links involve fading

4

coefficients that are independent by assumption (Section II-B), so their outage events are independent, and the end-to-end outage probability of Π is Y pSD (1 − pu,v (7) out ) . out = 1 − (u,v)∈Π

We impose the end-to-end reliability constraint pSD out ≤ πe , which, by (7), reads 1−

Y

(1 − pu,v out ) ≤ πe .

(8)

(9)

(u,v)∈Π

Under Rayleigh fading each factor 1 − pu,v out admits a closed form (Appendix A) that contains a separate term for every interferer, and the product in (9) couples the transmit powers of all transmitters on Π. Taking logarithms turns the product into a sum over hops, and the resulting constraint is in fact convex in the transmit powers, since each summand of (24) is of the form a/Pu or ln(1 + b/Pu ) with a, b > 0, both convex on Pu > 0; the difficulty lies elsewhere. The smallest latency that (9) permits on a given path is defined only implicitly, as the root of a nonlinear equation, so it does not decompose into per-link costs and cannot be minimized over paths by a shortest-path computation; moreover, evaluating it requires the contribution of each interferer individually. The following lemma removes both obstacles by replacing (9) with a conservative condition that is linear in each transmitter’s inverse power and involves the interferers only through their aggregate power. Lemma 1 (Sufficient Condition for End-to-End Outage Constraint). For a given outage threshold πe , let ϵ ≜ −γ −1 ln(1−πe ). Then the end-to-end reliability constraint pSD out ≤ πe is satisfied if X dα u,v (N0 + Iv ) ≤ ϵ, (10) Pu (u,v)∈Π

where Iv is the average interference power at node v, defined in (4). Proof. The proof is provided in Appendix A; it applies ex ≥ 1 + x to each interferer’s contribution in the exact per-link expression (23) below. We refer to (10) as the additive condition, since it is what makes the optimal latency of a path a sum of per-link costs (Section III). Because it involves the interferers only through the aggregate Iv , the resulting algorithms require no more than a received-power measurement at each node, whereas the exact constraint requires each interferer to be resolved individually. The price of this substitution is quantified in Section V, which shows that it is very nearly tight. In the next section, we formulate the joint routing and power-control problem that minimizes the bottleneck harvesting latency while guaranteeing both energy neutrality and the end-to-end outage constraint.

III. Minimizing Bottleneck Harvesting Latency (HOPE) Recall from Section II-E that multi-hop transmission from S to D can only commence once all nodes on Π have harvested sufficient energy. The node requiring the longest harvesting time thus acts as the bottleneck, determining the overall start time of message forwarding. Our goal is to find and characterize the path that minimizes this bottleneck latency while guaranteeing end-to-end reliability. To this end, we propose HOPE (Harvesting-Optimized Path selection with End-to-end outage guarantees), which jointly optimizes routing and transmit power allocation to minimize the maximum harvesting time along the path, (11)

min max Tu ,

Π,Pu

u∈Π

subject to the energy neutrality constraint (5) and the end-to-end outage constraint (10). For a fixed path Π, let TH (Π) denote the minimum of maxu∈Π Tu over transmit powers satisfying the energyneutrality condition (5) and the outage constraint (10), that is, the value of the program   min max Tu , subject to: X (u,v)∈Π

Pu dα u,v

Pu

u∈Π

(N0 + Iv ) ≤ ϵ,

Tu ≥

Pu τ Φ̄u

∀u ∈ Π,

(12)

where Φ̄u is the power harvested by the transmitter u, defined in Section II. The routing problem is then Π⋆ ∈ arg

min

Π∈P SD

TH (Π),

(13)

where P SD is the set of all paths from source S to destination D. This is a mixed-combinatorial and continuous optimization problem. However, due to its structure, it can be solved exactly. The key is to introduce an auxiliary variable T = maxu∈Π Tu and apply the KKT conditions. This leads to the following closed-form result. Theorem 1 (Minimum Bottleneck Harvesting Latency –HOPE). For the HOPE problem defined in (12) and a given path Π, the optimal (minimum) bottleneck harvesting latency is τ X dα u,v (N0 + Iv ) TH (Π) = , (14) ϵ Φ̄u (u,v)∈Π

where ϵ = − ln(1 − πe )/γ. Consequently, the globally optimal path Π⋆ is obtained by solving a shortest-path probdα u,v (N0 +Iv ) . lem with non-negative link weights wu,v = Φ̄u Proof sketch. Introduce an auxiliary variable T = maxu∈Π Tu . For a fixed path Π, the problem becomes minimizing T subject to T ≥ Tu for all u ∈ Π, along with the energy neutrality and the additive outage constraint in (10). The resulting program is convex on Pu > 0, since each summand dα u,v (N0 + Iv )/Pu of the outage constraint is convex, and Slater’s condition holds; the KKT conditions are therefore necessary and sufficient for the global optimum. Solving the stationarity conditions forces every

5

energy-neutrality constraint tight, so T = Pu τ /Φ̄u for every node, and the additive outage constraint active. Substituting gives τ X dα u,v (N0 + Iv ) T = . ϵ Φ̄u

Algorithm 1 HOPE: Minimum-Latency Reliable Routing

Minimizing this expression over all paths Π from S to D yields the claimed optimal bottleneck latency. □ Theorem 1 defines the path cost functional TH (Π) in closed form. This solution provides a systematic method for joint routing and power allocation, as follows.

τ ∑ ⋆ wu,v . ϵ (u,v)∈Π 6: for each transmitter u on Π⋆ do Φ̄u 7: Pu = TH (Π⋆ ) τ 8: end for 9: return Π⋆ , TH (Π⋆ ), {Pu }

(u,v)∈Π

1: Precompute average interference Iv =

∑

nodes v.

α i∈I0 Pi /di,v for all

2: Set ϵ = −γ −1 ln(1 − πe ). 3: Assign weight to each link (u, v): wu,v = 4: Find path Π⋆ = Dijkstra(G, S, D, w). 5: Compute max harvesting time:

dα u,v (N0 + Iv ) . Φ̄u TH (Π⋆ )

=

A. Routing Algorithm and Power Allocation Let us assign the following weight to each feasible directed link (u, v) in the network: wu,v ≜

dα u,v (N0 + Iv ) ≥ 0, Φ̄u

(15)

where nodes with Φ̄u = 0 have been excluded from G as described in Section II. Since N0 > 0 and Φ̄u > 0, every weight is strictly positive, so the shortest path returned by Dijkstra’s algorithm is simple and no node is revisited. Because τ /ϵ is a path-independent positive constant, minimizing TH (Π) is equivalent to minimizing P (u,v)∈Π wu,v . Thus, standard shortest-path algorithms, such as Dijkstra’s algorithm, can find a path Π⋆ that P minimizes (u,v)∈Π wu,v (and therefore TH (Π)) in polynomial time, at a cost of O(|E|+|N | log |N |) with a binary heap. Note that the number of interferers does not enter this expression: the weight wu,v depends on the interferers only through the aggregates Iu and Iv , each of which a node obtains directly by measurement, without resolving the individual interferers or knowing their number. We return to this point in Section V. Once the minimum-weight path Π⋆ is found, the minimized maximum harvesting time is obtained by evaluating (14) on Π⋆ : τ X TH (Π⋆ ) = wu,v . (16) ϵ ⋆ (u,v)∈Π

For a fixed path, the KKT conditions force all harvesting constraints to be active, i.e. Pu fu = T with fu = τ /Φ̄u . Hence, the optimal transmit power for each transmitter u ∈ Π⋆ is TH (Π⋆ ) Φ̄u Pu = = TH (Π⋆ ). (17) fu τ Substituting TH (Π⋆ ) yields X

regions may contain many short hops, so that, depending on the regime, the cumulative transmission delay |Π⋆ |τ need not be negligible relative to the harvesting latency. Applications with hard deadlines, such as industrial monitoring or emergency response, must account for both components, which motivates the total-latency objective of the next section. IV. The Extended Problem: Minimizing Total Latency (THOPE) In this section, we propose THOPE (Total‑latency HOPE), which extends the objective to the total end‑to‑end latency. THOPE minimizes the total time from the start of harvesting until the destination receives the first packet. This total latency of a fixed path Π is the sum of the initial bottleneck harvesting latency and the cumulative transmission delay over all hops of Π, i.e. X Total latency = max Tu + τu,v . u∈Π

We define the total latency cost functional as the smallest achievable total latency along Π:   X Ttot (Π) ≜ min max Tu + τu,v . Pu

u∈Π

(u,v)∈Π

As defined in Section II-C, all links use the same transmission time τ . By incorporating the energy neutrality constraint (5) and the end-to-end outage constraint (10), for a given path Π, the optimization problem is formulated as:  X  min max Tu + τ , subject to: Pu

X (u,v)∈Π

dα u′ ,v ′ (N0 + Iv ′ )

(u,v)∈Π

u∈Π

(u,v)∈Π

dα u,v (N0 + Iv ) ≤ ϵ, Pu

Tu ≥

Pu τ Φ̄u

(19) ∀u ∈ Π

(18)

Following a similar KKT-based approach, we derive the optimal solution for this more complex, non-separable objective.

This procedure yields a computationally efficient joint routing and power-allocation solution for minimum harvesting bottleneck latency, as summarized in Algorithm 1. Discussion. HOPE minimizes only the time at which forwarding can begin. The packet must then traverse every hop of Π⋆ , and a route that exploits strong harvesting

Theorem 2 (Minimum Total Latency –- THOPE). For the THOPE problem defined in (19), the optimal (minimum) total end‑to‑end latency is  α  X du,v (N0 + Iv ) τ Ttot (Π) = +1 , (20) ϵ Φ̄u

Pu =

Φ̄u ϵ

(u′ ,v ′ )∈Π⋆

Φ̄

u′

.

(u,v)∈Π

6

where ϵ = − ln(1 − πe )/γ, and all other symbols are as defined in Section II. Proof sketch. Introduce T = maxu∈Π Tu . For a fixed path Π, the objective becomes T + |Π| × τ , where |Π| is the number of hops in the path. Using Lagrangian duality and KKT conditions, optimality forces Pu = T Φ̄u /τ for every node and the outage constraint to be active; the objective differs from that of Theorem 1 by the constant |Π|τ , so convexity and Slater’s condition carry over and the KKT conditions again identify the global optimum. Substituting yields τ X dα u,v (N0 + Iv ) . T = ϵ Φ̄u

latency equals the sum of that common harvesting latency plus the total transmission time). Moreover, these powers satisfy the additive condition (10), so by Lemma 1 the end-to-end outage probability is at most πe ; since that condition is conservative, the true outage probability lies below the target, and Section V shows how little latency this conservatism costs.

Hence, the total latency is  X  dα u,v (N0 + Iv ) T + |Π| × τ = τ +1 . ϵ Φ̄u

V. Tightness of the Additive Outage Constraint Lemma 1 replaces the exact, multiplicative outage constraint by a conservative additive sufficient condition, and this substitution is what makes the closed-form solutions of Theorems 1 and 2 possible. Because both algorithms operate their nodes at the boundary of that condition, any looseness in Lemma 1 translates directly into transmit power that is larger than necessary, and hence into harvesting latency that is larger than necessary. In this section we quantify that looseness.

Minimizing over all paths gives the claimed optimal total latency. □

A. Exact End-to-End Constraint Throughout this section we write

(u,v)∈Π

(u,v)∈Π

A. Routing Algorithm and Power Allocation The expression in (20) is a sum of non-negative edge weights, so minimizing Ttot (Π) over all paths from S to D is equivalent to finding a shortest path in a graph where each directed link (u, v) carries the weight  α  du,v (N0 + Iv ) wu,v ≜ τ + 1 ≥ 0. ϵ Φ̄u The first term is the harvesting cost of the hop, inherited from HOPE’s weight (15) scaled by τ /ϵ; the second is the per-hop transmission time τ . THOPE’s route therefore minimizes the sum of harvesting and transmission costs, and a route that is short in hops is favored over one that is long, even when the latter offers better harvesting. Because all weights are non‑negative, a standard shortest‑path algorithm such as Dijkstra’s algorithm can compute the optimal path Π⋆ in polynomial time (e.g., O(|E| + |N | log |N |) with a binary heap). THOPE is therefore Algorithm 1 with three changes: line 3 usesPthe weight wu,v above, line 5 computes Ttot (Π⋆ ) = (u,v)∈Π⋆ wu,v , and line 7 uses T = Ttot (Π⋆ ) − |Π⋆ |τ in place of TH (Π⋆ ). Once the optimal path Π⋆ is found, using the same reasoning as in Section III-A, the corresponding optimal transmit power for each transmitter u on Π⋆ is Pu =

Φ̄u T = T. fu τ

Substituting the closed‑form expression for T gives X dα u′ ,v ′ (N0 + Iv ′ ) . Pu = Φ̄u ϵ Φ̄u′ ′ ′ ⋆ (u ,v )∈Π

These powers satisfy P the energy‑neutrality condition T = Pu fu = Ttot (Π⋆ )− (u,v)∈Π⋆ τ (i.e., the harvesting latency is the same for all nodes on the path, and the total

Λ ≜ γ ϵ = − ln(1 − πe ),

Ii,v ≜

Pi , dα i,v

(21)

so that Ii,v is the average powerPreceived at node v from the single interferer i and Iv = i∈I0 Ii,v . We define the per-interferer interference-to-signal ratio on link (u, v), γ dα u,v Ii,v . (22) Pu The derivation in Appendix A establishes, before any bounding is applied, the exact per-link success probability under Rayleigh fading,  γ d α N0  1
exp − u,v 1 − pu,v . (23) out = Q Pu i∈I0 1 + xi,v xi,v ≜

Substituting (23) into the end-to-end definition (7) and taking logarithms, the constraint pSD out ≤ πe is equivalent to X h γ dα X
i u,v N0 Ψ ≜ + ln 1 + xi,v ≤ Λ. (24) Pu i∈I0

(u,v)∈Π

The gap between (24) and the additive condition of Lemma 1 comes from the per-interferer relaxation ln(1 + x) ≤ x. Both HOPE and THOPE drive the energy-neutrality constraint to equality, i.e. Pu = Φ̄u T /τ with T = maxu∈Π Tu (Theorems 1 and 2). Substituting this into (22) gives xi,v = cu,v Ii,v /T with the link constant cu,v ≜

γ τ dα u,v . Φ̄u

(25)

Hence, in terms of the reciprocal bottleneck latency z ≜ 1/T , the exact constraint (24) is X h X
i ≤ Λ. Ψ(z) = cu,v N0 z + ln 1 + cu,v Ii,v z (u,v)∈Π

i∈I0

(26)

7

(iii) starting from z0 = 1/TH (Π), the Newton iteration

B(zexact ) − Λ

B(z)

Ψ(z)

Λ

Λ − Ψ(zk ) , Ψ′ (zk ) h X X cu,v N0 +

(29)

zk+1 = zk + Ψ′ (z) =

(u,v)∈Π

i∈I0

i

cu,v Ii,v , (30) 1 + cu,v Ii,v z

satisfies zk ≤ zk+1 ≤ zexact for every k.

gain

Proof. The proof is provided in Appendix B. z0

zexact

z = 1/T

Fig. 2. The exact outage exponent Ψ of (26) (solid) and its linear counterpart B of (27) (dashed), as functions of the reciprocal bottleneck latency z = 1/T . B is the tangent to Ψ at the origin, so it lies above Ψ and meets the budget Λ first, at z0 = 1/TH (Π); the exact constraint is met only at zexact = 1/THexact (Π). The slack B(zexact )−Λ is the amount discarded by the relaxation ln(1+x) ≤ x; Theorem 3 bounds it, and hence the horizontal gain, by the fraction x̄/2 of B(zexact ).

Setting Pu = Φ̄u T /τ and substituting z and cu,v in the right hand side of additive condition of (10) in Lemma 1 gives X
B(z) ≜ z cu,v N0 + Iv = Λ TH (Π) z, (27) (u,v)∈Π

Hence, the comparison in this section is between the linear function B and the exact function Ψ. Let us denote the bottleneck harvesting latency of path Π under the exact constraint (26) by  THexact (Π) ≜ min T > 0 : Ψ(1/T ) ≤ Λ (28) which compared to TH (Π) of Theorem 1, is the same quantity obtained from the exact outage constraint. B. Structural Properties Fig. 2 summarizes the geometry that underlies the remainder of this section. Both formulations B and Ψ spend the same outage budget Λ, but they measure the expenditure differently: B overstates the true exponent Ψ, and thus reaches the budget early, at z0 = 1/TH (Π), and HOPE stops there. On the other hand, Ψ does not reach Λ until zexact = 1/THexact (Π). The horizontal separation between z0 and zexact is the recoverable latency. This value is governed entirely by the vertical difference B − Ψ. The following proposition makes this precise. Proposition 1 (Concavity, tangency, and monotone refinement). For any path Π, the function Ψ in (26) satisfies: (i) Ψ(0) = 0, and Ψ is strictly increasing and, provided I0 ̸= ∅, strictly concave on z ≥ 0. Consequently, Ψ(z) = Λ has a unique root zexact and THexact (Π) = 1/zexact . In the degenerate case I0 = ∅ the logarithmic terms are absent, Ψ coincides with B, and the gap studied in this section vanishes; (ii) B is the tangent to Ψ at the origin. Hence, B(z) ≥ Ψ(z) for all z ≥ 0;

Part (iii) has a practical consequence that matters more than the convergence rate: because every iterate satisfies zk ≤ zexact , every iterate is feasible. Therefore, the refinement may be truncated after any number of steps, without ever violating the end-to-end outage guarantee. C. A Bound on the Achievable Latency Gap The refinement of Proposition 1 part (iii) converges to THexact (Π), which is limited by the target outage constraint. The following theorem quantifies the latency gap between THexact (Π) and TH (Π). Theorem 3 (Latency gap of the additive bound). Let πe < 2/3 be the target end-to-end outage probability of (8), and let πe x̄ ≜ eΛ − 1 = (31) 1 − πe denote the largest interference-to-signal ratio xi,v that any single interferer can present at any receiver on a path without, by itself, exhausting the outage budget. Then, for every path Π,  x̄  THexact (Π) ≥ 1 − TH (Π). (32) 2 Proof sketch. Because all summands of Ψ are nonnegative and sum to Λ at the optimum, no single term may exceed Λ, which bounds every xi,v by x̄. Applying x − ln(1 + x) ≤ x2 /2 term by term and using that bound converts the discarded slack into a fraction x̄/2 of B(zexact ), and (27) turns the resulting inequality into the claimed bound on the latency. The detailed proof is given in Appendix C. □ In words, x̄ is the largest ratio at which a single interferer can be present without causing outage on its own, and Theorem 3 says that the additive condition is loose only to the extent that some interferer approaches this limit. Theorem 3 compares the two latencies on a common path. However, because it holds for every path, it also bounds what could be gained by re-optimizing the path under the exact constraint. Corollary 1 (No gain from re-routing). Under the hypotheses of Theorem 3, let Πexact minimize THexact over all paths from S to D, that is, let it be the best route available to a scheme that uses the exact constraint for both power allocation and routing. Then  x̄  TH (Π⋆ ). THexact (Πexact ) ≥ 1 − (33) 2

8

No scheme within the model of Section II, however it allocates power and however it routes, can therefore reduce HOPE’s bottleneck latency by more than the fraction x̄/2 of its value. Proof. Theorem 3 holds for every path, in particular for Πexact , giving THexact (Πexact ) ≥ (1 − x̄/2) TH (Πexact ). The right-hand side is a latency under the additive condition, and Π⋆ minimizes TH over all paths by Theorem 1, so TH (Πexact ) ≥ TH (Π⋆ ). Combining the two gives (33). Theorem 3 is universal but coarse. The following proposition complements it with a bound that is typically sharper by orders of magnitude (as will be shown in Section VI). Proposition 2 (Computable per-instance bound). Fix a path Π, let z0 = 1/TH (Π), and define G(Π) ≜

1 Λ

X X

2 cu,v Ii,v z0 ,

(34)

(u,v)∈Π i∈I0

that is, the sum of the squared ratios xi,v evaluated at z0 , normalized by the outage budget. If πe ≤ 1/2 and G(Π) ≤ 1/2, then p
1 − 1 − 2G(Π) THexact (Π) G(Π) 1− ≤ = +O G(Π)2 . TH (Π) 2 2 (35) Proof. The proof is provided in Appendix D. The two bounds are of different kinds, and each is bought at a price. The bound of Theorem 3 and Corollary 1 is pessimistic, but it requires no knowledge of the network. They only depend on the outage target πe and hold for every network. The bound of Proposition 2 is much sharper, but it needs the output of HOPE or THOPE and the placement of the interferers.

D. Refinement Procedure Proposition 1 part (iii) yields a refinement that is common to both proposed algorithms. Both HOPE and THOPE terminate with the same two operations: a bottleneck latency T computed from the outage budget on the selected path, and the transmit powers Pu = Φ̄u T /τ derived from it. The refinement acts only on the first of these, leaving the routing decision and the power-allocation rule untouched, and is summarized in Algorithm 2. For HOPE the refined bottleneck latency is THexact (Π⋆ ) itself; for THOPE the refined total latency is THexact (Π⋆ )+ |Π⋆ |τ , since the cumulative transmission time is unaffected. Each iteration costs O(|Π⋆ | |I0 |) operations, and by Proposition 1 the procedure may be stopped at any point; in our experiments at most four iterations were required for a relative tolerance of 10−12 , and a single iteration already recovered essentially the entire gap.

Algorithm 2 Exact refinement of the bottleneck latency (common to HOPE and THOPE) Require: path Π⋆ and bottleneck latency T , from Algorithm 1 (line 5) for HOPE, or from THOPE net of the transmission time |Π⋆ |τ 1: Set Λ = − ln(1 − πe ), z ← 1/T α 2: Precompute cu,v = γτ dα u,v /(Φ̄u ) and Ii,v = Pi /di,v for all ⋆ (u, v) ∈ Π , i ∈ I0 3: repeat ∑ [ ] ∑ 4: Ψ ← (u,v)∈Π⋆ cu,v N0 z + i ln(1 + cu,v Ii,v z) [ ] ∑ ∑ 5: Ψ′ ← (u,v)∈Π⋆ cu,v N0 + i cu,v Ii,v /(1 + cu,v Ii,v z) 6: z ← z + (Λ − Ψ)/Ψ′ 7: until convergence 8: THexact (Π⋆ ) ← 1/z 9: Update Pu ← Φ̄u THexact (Π⋆ )/τ for all u ∈ Π⋆ 10: return THexact (Π⋆ ), {Pu }

VI. Numerical Results To validate the proposed optimal routing framework HOPE and THOPE, we simulate a wireless network with |N | system nodes and |I0 | interferers uniformly distributed in a d × d square area. The source node S is placed at (0, 0) and the destination D at (d, d). Unless stated otherwise, we use the following default parameters: d = 10 m, path-loss exponent α = 2.2, noise power is N0 = 10−6 W, interferer transmit power Pi = 0.1W, transmission duration per hop τ = 0.1 s, SINR threshold γ = −10 dB, reference distance d0 = 1 m, communication range rc = d, energy conversion efficiency η = 0.8 in the linear harvesting model Φ̄u = ηIu of Section II, and target end‑to‑end outage probability πe = 0.2. All results are averaged over 1000 independent random network realizations. These values are normalized and illustrative: the path-loss model carries no reference gain, which rescales the harvesting term relative to τ , so the results show how the schemes compare as the ratio of harvesting to transmission cost varies, rather than absolute latencies for a specific deployment. We first illustrate the trade-off between harvesting and transmission delay on a network instance in which the two objectives select substantially different routes (Section VI-A). A. Network Snapshots Fig. 3 shows an instance in which the two objectives select different routes with substantially different hop counts. In this example, THOPE achieves a total latency of 2.82 s, while HOPE yields 3.11 s, but with a different path. HOPE selects a 14‑hop route because its objective focuses solely on the harvesting bottleneck: additional hops that lie in regions of strong ambient interference can shorten the maximum harvesting time even if they increase the total number of transmissions. THOPE, on the other hand, minimizes the sum of the harvesting bottleneck and the cumulative transmission delay. It therefore prefers the 9‑hop path, where the reduction in transmission time compensates for a modest increase in harvesting time. This

9

70

10

HOPE THOPE MER SP

HOPE THOPE

9

60

Average total delay (s)

8 7 6 5 4

50 40 30 20

3

10

2

0 0

20

40

60

80

100

Number of interferers

1

Fig. 4. Average total delay versus number of interferers for HOPE, THOPE, MER, and SP. The number of system nodes is 50.

0 0

1

2

3

4

5

6

7

8

9

10

Fig. 3. Comparison of selected paths in an instance where the hopcount penalty changes the selected route substantially: 50 system nodes (circles) and 50 interferers (×) uniformly distributed. The green solid line shows the HOPE path (14 hops, total latency 3.11 s) and the blue dashed line the THOPE path (9 hops, total latency 2.82 s). HOPE minimizes the harvesting bottleneck and may select longer paths with better energy‑harvesting opportunities; THOPE trades a slight increase in harvesting time for significantly fewer transmission hops, lowering total end‑to‑end latency.

example highlights the distinct trade‑offs between the two design criteria. B. Benchmark Strategies To quantify the benefits of joint routing and power optimization, we compare HOPE and THOPE against two natural benchmarks: Shortest Path (SP) routing with uniform outage allocation: This scheme selects the path ΠSP that minimizes the Euclidean distance, i.e., edge weights are du,v . To meet the end‑to‑end outage constraint πe , we divide the outage budget equally among the L hops of ΠSP , setting the per‑link outage target to πL = 1−(1−πe )1/L . Applying the same upper bound as in Lemma 1, the minimum required transmit power for node u on this path is PuSP =

γ dα u,v (N0 + Iv ) . − ln(1 − πL )

(36)

The corresponding harvesting latency is the maximum of the individual Tu = PuSP τ /(Φ̄u ): PuSP τ . (37) u∈ΠSP Φ̄u The total latency adds the transmission time over LSP hops, i.e. Ttot (ΠSP ) = TH (ΠSP ) + LSP τ . Minimum Energy Routing (MER) with outage constraints: This benchmark minimizes the total transmit energy subject to the additive outage constraint [6]. Each q link (u, v) is assigned a cost dα (N 0 + Iv ), and the u,v transmit power for node u on the optimal MER path ΠMER is q X 1q α du,v (N0 + Iv ) dα PuMER = u′ ,v ′ (N0 + Iv ′ ). ϵ ′ ′ TH (ΠSP ) = max

(u ,v )∈ΠMER

MER does not explicitly account for harvesting asymmetry, but by reducing transmit energy it indirectly reduces the required harvested energy. The harvesting latency is PuMER τ , u∈ΠMER Φ̄u

TH (ΠMER ) = max

(38)

and the total latency with LMER hops is Ttot (ΠMER ) = TH (ΠMER ) + LMER τ . C. Performance Comparison to Benchmarks We now compare the average total latency of the four algorithms over 1000 network realizations. The ordering is the same at every operating point we tested: THOPE is fastest, followed by HOPE, with MER next and SP slowest. Only THOPE and HOPE price harvesting and interference in the link weight, so only they can place transmitters where the ambient field is strong; MER, by minimizing transmit energy, lightens the harvesting burden indirectly and so trails them while still beating SP, which ignores interference altogether. The two sweeps below therefore report how each parameter moves the curves rather than re-establishing that ranking. Fig. 4 varies the number of interferers from 5 to 100 with 50 system nodes. The delay of HOPE and THOPE is low and nearly flat, because two opposing effects cancel: a higher interferer density raises the aggregate interference Iv at the receivers, which degrades the SINR and hence the transmit power required, and at the same time raises Iu at the transmitters, which shortens the time needed to accumulate that energy. Beyond a moderate density the two approximately balance. SP, which exploits neither effect, is the most sensitive to interferer density. Fig. 5 varies the number of system nodes from 25 to 100 with 50 interferers. The delays of HOPE, THOPE and MER all decrease as more relays become available, since a denser deployment offers a richer set of candidate paths, from which a route with better harvesting conditions and fewer hops can be assembled. SP’s delay is almost constant, as its route is determined by geometry alone.

10

35

TABLE I

Average total delay (s)

30

HOPE route

HOPE THOPE MER SP

25 20 15

THOPE route

πe

Thm. 3

Prop. 2

measured

Prop. 2

measured

0.05 0.10 0.20 0.30 0.50

2.63% 5.56% 12.50% 21.43% 50.00%

0.04% 0.09% 0.19% 0.31% 0.61%

0.04% 0.09% 0.19% 0.31% 0.59%

0.05% 0.10% 0.25% 0.47% 1.20%

0.05% 0.10% 0.25% 0.46% 1.14%

10 5 0 20

30

40

50

60

70

80

90

100

Number of relay nodes

Fig. 5. Average total delay versus number of system nodes for HOPE, THOPE, MER, and SP. The number of interferers is 50.

Fig. 6. Histograms of total latency over 1000 network realizations for (a) HOPE, (b) THOPE, (c) MER, and (d) SP.

D. Total Latency Histograms To provide insight into the distribution of the delay experienced across different network realizations, Fig. 6 plots histograms of the total latency for each algorithm over 1000 realizations (default parameters). The median total latency values are 2.65 s (THOPE), 2.86 s (HOPE), 6.52 s (MER), and 27.46 s (SP). The distributions for HOPE and THOPE are sharply concentrated around their medians, indicating highly predictable performance. In contrast, SP and, to a lesser extent, MER exhibit heavy tails. This predictability makes HOPE and THOPE particularly attractive for delay‑sensitive applications with strict latency guarantees. E. Tightness of the Additive Outage Constraint We now quantify empirically the latency gap analyzed in Section V. For each network realization we run HOPE and THOPE, apply the refinement of Algorithm 2 to the

selected path to find THexact , and record the relative latency reduction 1 − THexact /TH . Table I reports the mean reduction of the bottleneck harvesting latency as a function of πe , together with the analytical cap of Theorem 3 and the per-instance bound of Proposition 2, evaluated on the route each algorithm returns. The results are averaged over 1000 network realizations with the default parameters. Since Proposition 2 is a per-path statement, the bound is evaluated separately on the route each algorithm returns. It held in every realization, and its hypothesis G(Π) ≤ 1/2 was satisfied throughout (the largest value of G(Π) observed was 0.102, at πe = 0.5) Multiple conclusions stand out from Table I. First, from the measured (exact) latency column, it is apparent that the additive constraint of Lemma 1 is very tight for all values of πe . For instance, at πe = 0.2 the mean recoverable latency is 0.19%. Also, the spread about that mean is narrow with a 95th percentile of 0.40%. Second, although both Theorem 3 and Proposition 2 give bounds on the reduction in latency, Proposition 2 gives much tighter bounds. In most cases, these bounds are identical to the average measured reduction, and are about two orders of magnitude below the pessimistic caps given by Theorem 3. Finally, the reduction in latency grows approximately linearly in πe , mirroring the cap, but remains roughly two orders of magnitude below it throughout. VII. Conclusion and Future Work We have studied joint routing and power allocation for multi-hop networks powered entirely by ambient radiofrequency interference. Replacing the multiplicative endto-end outage constraint by an additive sufficient condition makes the resulting min–max problem exactly solvable: HOPE minimizes the bottleneck harvesting latency and THOPE the total end-to-end latency, each through a single shortest-path computation with interference-aware link weights and closed-form transmit powers. Because the construction rests on a relaxation, we also quantified its cost. In Theorem 3, we proved that the
achievable latency reduction is capped by πe / 2(1 − πe ) on every path. We also proved that a tight per-instance bound is computable from the algorithms’ own output and the geometry of the interferers’ locations (Proposition 2). Numerically, we showed that HOPE and THOPE reduce average total latency by up to an order of magnitude relative to both benchmarks, and their delay distributions

11

are sharply concentrated around the median. This matters for applications with hard deadlines. Also, we showed that at the outage target of πe = 0.2, the measured reduction in HOPE’s bottleneck latency with respect to the latency when the exact outage formula is used is 0.19%. The average per-instance bound from Proposition 2 is also 0.19%, while the analytical reduction cap of Theorem 3 is 12.5%. Three extensions suggest themselves. The decoding threshold γ is treated as fixed, but it can be optimized. The energy-neutrality constraint in this paper reflects a worst case in which no energy is carried across intervals; permitting accumulation in the supercapacitor buffer would shift the metric from first-packet latency to steady-state cycle time. Finally, the single-flow assumption could be relaxed by scheduling multiple concurrent flows, whose transmissions would then interfere with and could be harvested from one another. References [1] A. Sheikholeslami, “Minimum-latency reliable routing in interference-powered networks,” in 61st Allerton Conference on Communication, Control, and Computing Proceedings. Allerton Conference on Communication, Control, and Computing, 2025. [2] A. Nguyen and A. Sheikholeslami, “Total-latency optimal routing in RF energy-harvesting networks,” in 2026 IEEE MIT Undergraduate Research Technology Conference (URTC). IEEE, 2026, accepted, to be presented in Oct. 2026. [3] G. Zeng, Y. Zhan, and X. Pan, “Failure-tolerant and low-latency telecommand in mega-constellations: The redundant multi-path routing,” IEEE Access, vol. 9, pp. 34 975–34 985, 2021. [4] M. Dehghan, M. Ghaderi, and D. Goeckel, “Minimum-energy cooperative routing in wireless networks with channel variations,” IEEE Transactions on Wireless Communications, vol. 10, no. 11, pp. 3813–3823, 2011. [5] A. Sheikholeslami, M. Ghaderi, H. Pishro-Nik, and D. Goeckel, “Jamming-aware minimum energy routing in wireless networks,” in 2014 IEEE International Conference on Communications (ICC). IEEE, 2014, pp. 2313–2318. [6] ——, “Energy-efficient routing in wireless networks in the presence of jamming,” IEEE Transactions on Wireless Communications, vol. 15, no. 10, pp. 6828–6842, 2016. [7] P. Bhavathankar, S. Chatterjee, and S. Misra, “Link-quality aware path selection in the presence of proactive jamming in fallible wireless sensor networks,” IEEE Transactions on Communications, vol. 66, no. 4, pp. 1689–1704, 2018. [8] X. Cheng, J. Shi, M. Sha, and L. Guo, “Revealing smart selective jamming attacks in WirelessHART networks,” IEEE/ACM Transactions on Networking, vol. 31, no. 4, pp. 1611–1625, 2023. [9] S. Bi, C. K. Ho, and R. Zhang, “Wireless powered communication: Opportunities and challenges,” IEEE Communications Magazine, vol. 53, no. 4, pp. 117–125, 2015. [10] X. Lu, P. Wang, D. Niyato, D. I. Kim, and Z. Han, “Wireless networks with RF energy harvesting: A contemporary survey,” IEEE Communications Surveys & Tutorials, vol. 17, no. 2, pp. 757–789, 2015. [11] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Transactions on Wireless Communications, vol. 12, no. 5, pp. 1989–2001, 2013. [12] J. Guo, N. Zhao, F. R. Yu, X. Liu, and V. C. M. Leung, “Exploiting adversarial jamming signals for energy harvesting in interference networks,” IEEE Transactions on Wireless Communications, vol. 16, no. 2, pp. 1267–1280, 2017. [13] O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus, and A. Yener, “Transmission with energy harvesting nodes in fading wireless channels: Optimal policies,” IEEE Journal on Selected Areas in Communications, vol. 29, no. 8, pp. 1732–1743, 2011.

[14] L. Lin, N. B. Shroff, and R. Srikant, “Asymptotically optimal energy-aware routing for multihop wireless networks with renewable energy sources,” IEEE/ACM Transactions on Networking, vol. 15, no. 5, pp. 1021–1034, 2007. [15] Z. Mao, C. E. Koksal, and N. B. Shroff, “Near optimal power and rate control of multi-hop sensor networks with energy replenishment: Basic limitations with finite energy and data storage,” IEEE Transactions on Automatic Control, vol. 57, no. 4, pp. 815–829, 2012. [16] C. D. T. Thai, “Opportunistic routing and energy harvesting in multi-tier networks against jamming attacks,” in 2024 International Wireless Communications and Mobile Computing (IWCMC). IEEE, 2024, pp. 214–219. [17] S. Ghosh, A. Al-Dweik, and M.-S. Alouini, “On the performance of end-to-end cooperative NOMA-based IoT networks with wireless energy harvesting,” IEEE Internet of Things Journal, vol. 10, no. 18, pp. 16 253–16 270, 2023. [18] J. Wang, Z. Wang, and L. Zhang, “A simultaneous wireless information and power transfer-based multi-hop uneven clustering routing protocol for EH-cognitive radio sensor networks,” Big Data and Cognitive Computing, vol. 8, no. 2, p. 15, 2024. [19] F. Shan, J. Luo, Q. Jin, L. Cao, W. Wu, Z. Ling, and F. Dong, “Optimal harvest-then-transmit scheduling for throughput maximization in time-varying RF powered systems,” IEEE Journal on Selected Areas in Communications, vol. 42, no. 11, pp. 3140–3156, 2024. [20] Q. Ren and G. Yao, “An opportunistic routing for energyharvesting wireless sensor networks with dynamic transmission power and duty cycle,” IEEE Access, vol. 10, pp. 121 109– 121 119, 2022. [21] S. Ulukus, A. Yener, E. Erkip, O. Simeone, M. Zorzi, P. Grover, and K. Huang, “Energy harvesting wireless communications: A review of recent advances,” IEEE Journal on Selected Areas in Communications, vol. 33, no. 3, pp. 360–381, 2015. [22] K. Subasinghage and K. Gunawardane, “Supercapacitorassisted energy harvesting systems,” Energies, vol. 17, no. 15, p. 3853, 2024. [23] S. Bi, Y. Zeng, and R. Zhang, “Wireless powered communication networks: An overview,” IEEE Wireless Communications, vol. 23, no. 2, pp. 10–18, 2016. [24] E. Boshkovska, D. W. K. Ng, N. Zlatanov, and R. Schober, “Practical non-linear energy harvesting model and resource allocation for SWIPT systems,” IEEE Communications Letters, vol. 19, no. 12, pp. 2082–2085, 2015. [25] P. N. Alevizos and A. Bletsas, “Sensitive and nonlinear farfield RF energy harvesting in wireless communications,” IEEE Transactions on Wireless Communications, vol. 17, no. 6, pp. 3670–3685, 2018. [26] B. Clerckx, R. Zhang, R. Schober, D. W. K. Ng, D. I. Kim, and H. V. Poor, “Fundamentals of wireless information and power transfer: From RF energy harvester models to signal and system designs,” IEEE Journal on Selected Areas in Communications, vol. 37, no. 1, pp. 4–33, 2019. [27] S. Sudevalayam and P. Kulkarni, “Energy harvesting sensor nodes: Survey and implications,” IEEE Communications Surveys & Tutorials, vol. 13, no. 3, pp. 443–461, 2011.

Related documents

• Harvesting-Aware Routing for Low-Latency Reliable Transmission in Interference-Powered Networks#1324851
• Gossiping with bounded size messages in ad hoc radio networks (Extended abstract)#292067
• Adaptive Channel Hopping for IEEE 802.15. 4 TSCH-Based Networks: A Dynamic Bernoulli Bandit Approach#704653
• Wireless power transfer and communications — The Alfred E. Mann Foundation For Scientific Research (US9962085B2)#607638
• To Code or Not to Code: When and How to Use Network Coding in Energy Harvesting Wireless Multi-hop Networks#1064419