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In this paper, we consider the problem of random access in wireless local area networks (WLANs) with each station generating either elastic or inelastic traffic. Elastic traffic is usually non-real-time, while inelastic traffic is usually coming from real-time applications. We formulate a network utility maximization (NUM) problem, where the optimization variables are the persistent probabilities of the stations and the utilities are either concave or sigmoidal functions. Sigmoidal utility functions can better represent inelastic traffic sources compared to concave utility functions commonly used in the existing random access literature. However, they lead to non-convex NUM problems which are not easy to solve in general. By applying the dual decomposition method, we propose a subgradient algorithm to solve the formulated NUM problem. We also develop closed-form solutions for the dual subproblems involving sigmoidal functions that have to be solved in each iteration of the proposed algorithm. Furthermore, we obtain a sufficient condition on the link capacities which guarantees achieving the global optimal solution when our proposed algorithm is being used. If this condition is not satisfied, then we can still guarantee that the optimal value of the objective function is within some lower and upper bounds. We perform various simulations to validate our analytical models when the available link capacities meet or do not meet the sufficient optimality condition.