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0-optimal absolutely continuous Markov auxiliary functions chapter classes of customers close to zero consider continuous Markov transition continuous-time algorithm continuous-time Markov decision customer arrival customer of type defined on 0,T differential equations discrete approximation f e F f maximizes finite number finite state continuous-time finite-horizon problem g e E(f g e F Hence horizon problem implies induction hypothesis initial condition it(t Lebesgue measurable Lebesgue point Lemma 11 lemma 9 Let f Markov decision problem Markov decision process Markov process Markov transition matrix maximizes x(f measurable function measurable policy nonnegative optimal policy optimal to serve piecewise constant policy policy f policy is optimal proves the lemma proves the theorem queueing model queueing theory satisfies the hypothesis satisfies the selective serve a customer servers are free set F solve stationary policy stochastic process t e 0,T Theorem 12 transition matrix function unique solution v(it vector function Vy(f