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Utility functions are widely used to model user-perceived service quality. For elastic traffic, the utility function is often concave. An elastic allocation (EA) algorithm has recently been proposed to maximize the total utility obtained by users, assuming resource is infinitesimally divisible and queues are constantly backlogged. We found that the EA algorithm is not always optimal. In fact, the solution it obtains can be infeasible. In this paper, we present a modified EA algorithm that is guaranteed to find the optimal solution under the same assumptions. The result is generalized for a system where queues are generally backlogged. In a real system, there is normally a basic unit for resources. Therefore, we further extend the designs to such a system for both constantly backlogged and generally backlogged queues. To reduce computational complexity, we also propose suboptimal resource allocation algorithms. Simulations are conducted to evaluate the proposed algorithms in terms of utility sum and execution time. Results show that our proposed algorithms perform better than previous works. Moreover, the performances of the proposed suboptimal algorithms are close to those of the optimal algorithms.