In this paper a distributed stochastic network system with incoming tasks that are classified with priorities is studied. The network system is assumed to have variable topology, and agents are not necessarily always connected to each other. In addition, the observations about neighbors' states are supposed to be obtained with random noise and delays. To ensure efficient operation of this network system, a novel control strategy is proposed. With this strategy, network resources are allocated in a randomized way with probabilities corresponding to each priority class. To maintain the balanced load across the network for different priorities, a so-called "differentiated consensuses" problem is examined. This consensus problem is that, in a system with multiple classes, consensus is targeted for each class, which may be different among classes. In this paper, the ability of the proposed control protocol to maintain almost balanced load, i.e. approximate consensus for every priority class across the network, is proved. In addition, a numerical example that illustrates the proposed control strategy and the results of simulations are provided.