Approximation of bounds on mixed-level orthogonal arrays

Abstract

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao-and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

Approximation of excessive backlog probabilities of two tandem queues

“…Furthermore, by the Perron-Frobenius theorem, this convergence occurs exponentially fast. Therefore, p n k provides an excellent approximation of p n for large k. All the prior works [16], [18], [50], [51], [53], [54], and [56] use this type of approximation to compute the quantity of interest 'exactly' to illustrate the numerical performance of the algorithms under consideration and we will do the same. For our numerical study, we set μ 1 = 0.4, μ 2 = 0.5, λ = 0.1, and n = 60.…”

“…The function h of (56) is a Y -harmonic function. There are four terms in the sum (56) defining h ρ 3 , each of these terms corresponds to a node of the graph in Figure 5. None of them is Y -harmonic individually.…”