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 functionapproximations 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 polynomialsconstitute 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 URLanized 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.
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 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 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 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 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 value functions and the exact implementation of the first-policy improvement step.
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continuous-space
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, 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
. The associated 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
R
expressions
value functions
R
costs, or
cost functions and
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thus hinting in the most general case at the
in turn permitting the
R
The value functionapproximations induced by Taylor polynomial expansions of the cost function prove however to converge only for entire cost
We review various cost approximation schemes and assess the convergence of the interval bounds these induce on the value function. Namely: Taylor expansions (divergent, except for a narrow class of entire
R
growth orders
orders of growth), and uniform approximation schemes (polynomials
R
and to diverge otherwise. A more suitable approach for assessing convergent interval bounds is found in the uniform approximation framework. Bernstein polynomialsconstitute 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 URLanized as a guide to implementing multiple-server dispatching policies
trigonometric), which achieve optimal convergence rates over finite intervals. This study addresses all the steps to implementing dispatching policies for systems of parallel servers
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general
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exact
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policy
first-policy