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We consider a novel queuing problem where the decision-maker must choose toaccept or reject randomly arriving tasks into a no buffer queue which areprocessed by $N$ identical servers. Each task has a price, which is a positivereal number, and a class. Each class of task has a different price distributionand service rate, and arrives according to an inhomogenous Poisson process. Theobjective is to decide which tasks to accept so that the total price of tasksprocessed is maximised over a finite horizon. We formulate the problem as adiscrete time Markov Decision Process (MDP) with a hybrid state space. We showthat the optimal value function has a specific structure, which enables us tosolve the hybrid MDP exactly. Moreover, we prove that as the time step isreduced, the discrete time solution approaches the optimal solution to theoriginal continuous time problem. To improve the scalability of our approach toa greater number of task classes, we present an approximation based on stateabstraction. We validate our approach on synthetic data, as well as a realfinancial fraud data set, which is the motivating application for this work.

Optimal Admission Control for Multiclass Queues with Time-Varying Arrival Rates via State Abstraction