Dispatching to parallel servers

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