Stability of routing strategies for the maximum lifetime problem in ad-hoc wireless networks

We solve the maximum lifetime problem for a one-dimensional, regular ad-hoc wireless network with one data collector $L_N$ for any data transmission cost energy matrix which elements $E_{i,j}$ are superadditive functions, i.e., satisfy the inequality $\forall_{i\leq j\leq k}\;E_{i,j}+E_{j,k}\leq E_{i,k}$. We analyze stability of the solution under modification of two sets of parameters, the amount of data $Q_i$, $i\in [1,N]$ generated by each node and location of the nodes $x_i$ in the network. We assume, that the data transmission cost energy matrix $E_{i,j}$ is a function of a distance between network nodes and thus the change of the node location causes change of $E_{i,j}$. We say, that a solution $q(t_0)$ of the maximum network lifetime problem is stable under modification of a given parameter $t_0$ in the stability region $U(t_0)$, if the data flow matrix $q(t)$ is a solution of the problem for any $t\in U(t_0)$. In the paper we estimate the size of the stability region $U(Q^0,d^0)$ for the solution of the maximum network lifetime problem for the $L_N$ network in the neighborhoods of the points $Q^0_i=1$, $d^0_i=0$, where $d_i\in (-1,1)$ describes the shift of the nodes from their initial location $x_i^0=i$, i.e., $x_i=i-d_i$.