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In this paper, we conduct a study to optimize resource allocation for adaptive real-time and delay-tolerant applications in cellular systems. To represent the user applications via several devices and equipment, sigmoidal-like and logarithm utility functions are used. A fairness proportional utility functions policy approach is used to allocate the resources among the user equipment (UE)s in a utility percentage form and to ensure a minimum level of customer satisfaction for all the subscribers. Meanwhile, the priority of resources allocation is given to the real-time applications. We measure the effect of the stochastic variations of the adaptive real-time applications on the optimal rate allocation process and compare the results of deterministic and stochastic systems. Our work is distinct from the other resource allocation optimization works in that we use bisection method besides convex optimization techniques to optimally allocate rates, and we present the adaptive real- time applications in a stochastic form. We use Microsoft Visual Basic for Applications with Arena Simulation Software interface to simulate the optimization problem. Finally, we present our optimization algorithm results.