Dispatching to Parallel Servers: Solutions of Poisson's Equation for First-Policy Improvement

Policy iteration techniques for multiple-server dispatching rely on the computation of value functions. In this context, we consider the continuous-space M/G/1-FCFS queue endowed with an arbitrarily-designed cost function for the waiting times of the incoming jobs. The associated relative value function is a solution of Poisson's equation for Markov chains, which in this work we solve in the Laplace transform domain by considering an ancillary, underlying stochastic process extended to (imaginary) negative backlog states. This construction enables us to issue closed-form relative value functions for polynomial and exponential cost functions and for piecewise compositions of the latter, in turn permitting the derivation of interval bounds for the relative value function in the form of power series or trigonometric sums. We review various cost approximation schemes and assess the convergence of the interval bounds these induce on the relative value function. Namely: Taylor expansions (divergent, except for a narrow class of entire functions with low orders of growth), and uniform approximation schemes (polynomials, trigonometric), which achieve optimal convergence rates over finite intervals. This study addresses all the steps to implementing dispatching policies for systems of parallel servers, from the specification of general cost functions towards the computation of interval bounds for the relative value functions and the exact implementation of the first-policy improvement step.