# MoN16: Sixteenth Mathematics of Networks meeting

## Aaron Pim (University of Bath) – Stability and rate of convergence of resource allocation within a cellular network

I am investigating the rate of convergence and the stability of known methods as applied to simulated LTE resource block allocation across a cellular radio network. The difficulty lies in the dynamic and nonconvex objective function, the bounded domain of inputs, and the existence of multiple or no solutions.

I constructed algorithms that simulated the network over a large domain and then performed statistical analysis on the rate of convergence, number of iterations it took for the system to reach equilibrium and the final value that it converged to.

I used an autonomous simulation as the benchmark for comparison from which to compare: Newton’s method over a trust region, least squares minimisation and a continuous time approximation.

In the case of no solution, finding the best approximate becomes the objective, I also seek an analytical necessary conditions to determine whether or not a system will have a solution.

I discovered that the the Newton solution was unstable for larger domains due to the increased likelihood that a solution does not exist, however the rate of convergence was an order of magnitude higher than the autonomous simulation. The least squares minimisation underestimated the number resource blocks that were needed to satisfy the data demands. The continuous time approximate converged close to the autonomous solution however its rate of convergence was not much better.

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