A c/μ‐Rule for Job Assignment in Heterogeneous Group‐Server Queues

We study a dynamic job assignment problem in queueing systems with one class of Poisson arrivals and K groups of heterogeneous servers. A scheduling policy prescribes the job assignment among servers in each group at every state n (number of jobs in the system). Our goal is to obtain the optimal policy to minimize the long‐run average cost, which involves the increasingly convex holding cost for jobs and the operating cost for working servers. This problem has wide application scenarios in operations management, such as job scheduling in manufacturing systems, packet routing in communication systems, and staffing in service systems. We prove that the optimal policy has monotone structures and quasi bang–bang control forms. Specifically, we discover that the optimal policy is governed by the marginal cost rate c − μG(n), where c is the operating cost rate, μ is the service rate, and G(n) is called the perturbation realization factor at state n. Under the condition of scale economies which can be guaranteed by any increasingly concave operating cost in μ, we prove that the optimal policy obeys a so‐called c/μ‐rule: Servers with a smaller c/μ should be occupied by jobs with higher priority. Optimality of multi‐threshold type policies is further proved when the c/μ‐rule is applied. Our c/μ‐rule in group‐server queues can be viewed as a counterpart of the famous cμ‐rule in polling queues, which both significantly reduce the complexity of optimization problems. By utilizing these optimality structures, we also develop computational‐efficient algorithms to determine the optimal policy numerically. Simulation experiments demonstrate the good scalability and robustness of the c/μ‐rule, which are important for managerial practice.