Dynamic routing and jockeying controls in a two-station queueing network

This paper studies the optimal routing and jockeying policies in a two-station queueing network. It is assumed that jobs arrive to the system in a Poisson stream with rate g and are routed to one of the two parallel stations. Each station has a single server and a buffer of infinite capacity. The service times are exponential with server-dependent rates, m_1 and mu_2. Jockeying between stations is permitted. The jockeying cost is c_{ij} when a job in station i jockeys to station j, j=1, 2, and i not= j. The cost for a new job to join any station is zero. The holding cost in station j is h_j, h_1 less than or equal to h_2, per job per unit time. The problem of interest is to characterize the structure of the dynamic routing and jockeying policies that minimize the expected total (holding plus jockeying) cost, for both the discounted and the long-run average cost criteria. We show that the optimal routing and jockeying controls are described by three monotonically nondecreasing functions and study their relationships. We study the properties of the control functions and their asymptotic behaviors. We show that some well-known queueing control models, such as optimal routing to symmetric or heterogeneous servers, preemptive or nonpreemptive scheduling on symmetric or heterogeneous servers, are special cases of our system.
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