One-step dispatching policy improvement in multiple-server queueing systems with Poisson arrivals
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Policy iteration techniques for multiple-server dispatching rely on the computation of value functions. In this context, we consider the M/G/1-FCFS queue endowed with an arbitrarily-designed cost function for the waiting times of the incoming jobs, and we study an undiscounted value function integrating the total cost surplus expected from each state relative to the steady-state costs. When coupled with random initial policies, this value function takes closed-form expressions for polynomial and exponential costs, or for piecewise compositions of the latter, thus hinting in the most general case at the derivation of interval bounds for the value function in the form of power series or trigonometric sums. The value function approximations induced by Taylor polynomial expansions of the cost function prove however to converge only for entire cost functions with low growth orders, and to diverge otherwise. A more suitable approach for assessing convergent interval bounds is found in the uniform approximation framework. Bernstein polynomials constitute straightforward, yet slowly convergent, cost function approximators over intervals. The best convergence rate in the sense of D. Jackson's theorem is achieved by more sophisticated polynomial solutions derived from trigonometric sums. This study is organized as a guide to implementing multiple-server dispatching policies, from the specification of cost functions towards the computation of interval bounds for the value functions and the implementation of the policy improvement step.
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