Fig. 1

Sequence of optimal observer placements for increasing transmission variance. We assume the transmission transmission delays {X _uv } _{u v∈E} to be such that \(\mathbf {E}[\!X_{uv}] = w_{uv} \in \mathbb {R}_{+}\) and such that the variance is a growing function of a variance parameter σ, i.e., Var(X _uv )=g(w _uv ,σ) with g(x,0)=0 for all \(x\in \mathbb {R}^{+}\). For σ∈(0,σ ₀) the transmission delays are effectively deterministic (i.e., σ does not affect source localization). For σ∈(σ ₀,σ ₁), σ affects the accuracy of source localization but the optimal observer placement is still \(\mathcal {O}_{0}\). For larger σ, the optimal observer placement might change, possibly multiple times (\(\mathcal {O}_{k}\) denotes the optimal placement for σ∈(σ _k ,σ _k+1)) up to σ=σ _F . For σ>σ _F the optimal placement remains the same (\(\mathcal {O}_{F}\))