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The Discriminatory Processor Sharing (DPS) model is a multi-class generalization of the egalitarian Processor Sharing model. In the DPS model all jobs present in the system are served simultaneously at rates controlled by a vector of weights {gk > 0; k = 1,..., K }. If there are Nk jobs of class k present in the system, k = 1,..., K, each class-k job is served at rate $$g_k/\sum_{j=1}^K{g_j}{N_j}$$ . The present article provides an overview of the analytical results for the DPS model. In particular, we focus on response times and numbers of jobs in the system.

We analyze a Processor-Sharing queue with Batch arrivals. Our analysis is based on the integral equation derived by Kleinrock, Muntz and Rodemich. Using the contraction mapping principle, we demonstrate the existence and uniqueness of a solution to the integral equation. Then we provide asymptotical analysis as well as tight bounds for the expected response time conditioned on the service time. In particular, the asymptotics for large service times depends only on the first moment of the service time distribution and on the first two moments of the batch size distribution. That is, similarly to the Processor-Sharing queue with single arrivals, in the Processor-Sharing queue with batch arrivals the expected conditional response time is finite even when the service time distribution has infinite second moment. Finally, we show how the present results can be applied to the Multi-Level Processor-Sharing scheduling.