Energy-Aware Wireless Networks via Reachability

Abstract

A wireless sensor network deployed in a hardened facility is modelled as a graph where node states are battery levels and inter-node communications drain energy as a function of link distance. Reachability analysis on the joint battery-state space gives the set of initial battery configurations from which the network can sustain a target communication schedule for a fixed mission horizon. Pairing this analysis with a multi-objective routing policy that minimises link-distance energy cost roughly doubles the operational lifetime versus a baseline shortest-path policy across ten test deployments (5 random, 5 uniform).

Problem Formulation

Let G = (V, E) be the network graph with |V| = n nodes. Each node carries a battery state b_i(t) ∈ [0, B_max]. Sending one packet from i to j drains battery proportional to a path-loss model:

Δb_i  =  −E_tx(d_ij)  =  −(α + β · d_ij^k)
Δb_j  =  −E_rx

E_rx is a fixed reception cost. Joint state x = (b_1, ..., b_n) ∈ [0, B_max]^n. Mission requires a fixed traffic schedule (which pairs talk, how often, for how long). The depletion set (failure region: a node has died) is:

F  =  { x : ∃ i with b_i ≤ b_min }

Routing policy π chooses, for each scheduled message, which path through G to take.

Reachability Computation

The HJR step computes the maximal backward-reachable safe set: states from which there exists a routing policy that keeps the trajectory out of F for the full horizon T:

S  =  { x_0 : ∃ π s.t. x(t; π) ∉ F  ∀ t ∈ [0, T] }

For small n this can be done directly on the joint state grid; for larger networks the formulation factors into per-node sufficient conditions (each node's battery treated as a 1-D reachability problem given the per-node duty cycle implied by the policy), which is what the implementation actually solves.

Energy-Aware Routing Layer

On top of the reachability check, the policy selects routes by a multi-objective cost:

cost(p)  =  λ_E · Σ_{(i,j) ∈ p} E_tx(d_ij)
         +  λ_B · Σ_{i ∈ p} 1/(b_i + ε)
         +  λ_H · hops(p)

Three terms: total transmission energy along the path, an avoid-low-battery term 1/(b_i + ε), and a hop-count term as a latency proxy. The 1/(b_i + ε) term is the key bit: routes that pass through nearly-depleted nodes look very expensive, so traffic naturally migrates onto fresher nodes as the network ages. This is the difference between shortest-path (which keeps re-using the same backbone until it dies) and an energy-aware policy (which load-balances across the whole graph).

Network Layouts & Reachability Visualisation

Ten test deployments: five with uniform-grid placement, five with Poisson-random placement at the same node density. The figure shows a representative random deployment with edge weights coloured by per-link E_tx cost. Nodes near the edge of the facility cluster have fewer neighbours and dominate the depletion timeline.

Representative network configuration · nodes coloured by battery, edges by E_tx

Headline Result

Across the ten deployments, mean operational lifetime (time until first node depletes below b_min) was about 2× longer with the reachability-aware energy routing layer than with a shortest-path baseline. The improvement was larger on Poisson-random deployments (more variance in node connectivity → more room for the energy-aware policy to load-balance) than on uniform-grid deployments (everyone has the same number of neighbours, so shortest-path is already close to optimal).