AutoCAT: Automated Product-Form Solution of Stochastic Models

We propose algorithms to automatically generate exact and approximate product-form solutions for large Markov processes that cannot be solved by direct numerical methods. Focusing on models that can be described as cooperating Markov processes, which include queueing networks and stochastic Petri nets as special cases, it is shown that finding a global optimum for a non-convex quadratic program is sufficient for determining a product-form solution. Such problems are notoriously hard to solve due to the inherent difficulty of searching over non-convex sets. Using a potential theory for Markov processes, convexification techniques and a class of linear constraints that follow from stochastic characterization of productform solutions, we obtain a family of linear programming relaxations that can be solved efficiently. A sequence of these linear programs is solved under increasingly tighter constraints to determine the exact product-form solution of a model when one exists. This approach is then extended to obtain approximate solutions for nonproduct-form models. Finally, our new techniques are validated with examples and increasingly complex case studies, which show the effectiveness of the method on both conventional and novel performance models.