Linear programming formulations for queueing control problems with action decomposability
Résumé
We consider a special class of continuous-time Markov decision processes (CTMDP) that are action decomposable. An action-Decomposed CTMDP (D-CTMPD) typically models queueing control problems with several types of events. A sub-action and cost is associated to each type of event. The action space is then the Cartesian product of sub-action spaces. We first propose a new and natural Quadratic Programming (QP) formulation for CTMDPs and relate it to more classic Dynamic Programming (DP) and Linear Programming (LP) formulations. Then we focus on D-CTMDPs and introduce the class of decomposed randomized policies that will be shown to be dominant in the class of deterministic policies by a polyhedral argument. With this new class of policies, we are able formulate decomposed QP and LP with a number of variables linear in the number of types of events whereas in its classic version the number of variables grows exponentially. We then show how the decomposed LP formulation can solve a wider class of CTMDP that are quasi decomposable. Indeed it is possible to forbid any combination of sub- actions by adding (possibly many) constraints in the decomposed LP. We prove that, given a set of linear constraints added to the LP, determining whether there exists a deterministic policy solution is NP-complete. We also exhibit simple constraints that allow to forbid some specific combinations of sub-actions. Finally, a numerical study compares computation times of decomposed and non-decomposed formulations for both LP and DP algorithms.
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