Nonlinear Matrix Equations in Applied Probability—Solution Techniques and Open Problems

Abstract

We draw attention to some important nonlinear equations commonly arising in applied probability, review some known techniques to compute their minimal nonnegative solutions, propose improved iterative techniques, and identify a number of open problems. Such equations arise ubiquitously in the numerical solution of many stochastic models in queueing, inventory, communications, and dam theories. The development of efficient algorithms and error analyses for these problems is, therefore, of substantial importance in enlarging the set of stochastic models that can be solved by algorithmic methods.