## 2007

# Convergence Results for Ant Routing Algorithms via Stochastic Approximation and Optimization

P. Purkayastha and J. S. Baras

*Proceedings of the 46th IEEE Conference on Decision and Control*, pp. 340-345, New Orleans, Louisiana, December 12-14, 2007.

**Abstract**

“Ant algorithms” have been proposed to solve a variety of problems arising in optimization and distributed control. They form a subset of the larger class of “Swarm Intelligence” algorithms. The central idea is that a ‘swarm’ of relatively simple agents can interact through simple mechanisms and collectively solve complex problems. Instances that exemplify the above idea abound in nature. The abilities of ant colonies to collectively accomplish complex tasks have served as sources of inspiration for the design of “Ant algorithms”. Examples of “Ant algorithms” are “Ant Routing” algorithms that have been proposed for communication networks. We analyze in this paper an Ant-Based Routing Algorithm for packet-switched wireline networks. The algorithm is an attractive multiple path probabilistic routing scheme, that is fully adaptive and distributed. Using methods from adaptive algorithms and stochastic approximation, we show that the evolution of the link delay estimates can be closely tracked by a deterministic ODE system. A study of the equilibrium points of the ODE then gives us the equilibrium behavior of the routing algorithm, in particular, the equilibrium routing probabilities, and mean delays in the links under equilibrium. We also show that the fixed-point equations that the equilibrium probabilities satisfy are actually the necessary and sufficient conditions of an appropriate optimization problem. Simulations supporting the analytical results are provided.