Before revision:  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.
After revision:  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 . 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 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.