Designing efficient scheduling algorithms is an important problem in a general class of networks with resource-sharing constraints, such as wireless networks and stochastic processing networks [7]. In [5], we proposed a distributed scheduling algorithm that can achieve the maximal throughput in such networks under certain conditions. This algorithm was inspired by CSMA (Carrier Sense Multiple Access). In this paper, we prove the convergence and stability of the algorithm, with properly-chosen step sizes and update intervals. Convergence of the joint scheduling and congestion control algorithm for utility maximization in [5] can be proved similarly.