We prove a monotonicity result for the problem of optimal service rate control in certain queueing networks. Consider, as an illustrative example, a number of ·/M/1 queues which are arranged in a cycle with some number of customers moving around the cycle. A holding cost hi(xi) is charged for each unit of time that queue i contains xi customers, with hi being convex. As a function of the queue lengths the service rate at each queue i is to be chosen in the interval , where cost ci(μ) is charged for each unit of time that the service rate μis in effect at queue i. It is shown that the policy which minimizes the expected total discounted cost has a monotone structure: namely, that by moving one customer from queue i to the following queue, the optimal service rate in queue i is not increased and the optimal service rates elsewhere are not decreased. We prove a similar result for problems of optimal arrival rate and service rate control in general queueing networks. The results are extended to an average-cost measure, and an example is included to show that in general the assumption of convex holding costs may not be relaxed. A further example shows that the optimal policy may not be monotone unless the choice of possible service rates at each queue includes 0.
Cust-omers arrive at a service area according to a Poisson process. An arriving customer must choose one of K servers without observing present congestioii levels. The only available information about the k-th server is Lhe service time distribution (with expected duration Vk -I ) and the cost per unit time of waiting at the k-th server (hk). Although service distributions may differ from server to server and need not be exponential, it is
We review models for the optimal control of networks of queues, Our main emphasis is on models based on Markov decision theory and the characterization of the structure of optimal control policies.
We introduce the terms dynamic and static, respectively, to identify the prevailing approaches to the single-leg airline yield-management problem: those allowing customers of different fare classes to book concomitantly (dynamic), and those assuming that the demands for the different fare classes arrive separately in a predetermined order (static). We present a coherent frame-work linking these seemingly disparate models through the underlying dynamic program common to both. We develop a discrete-time Markov decision process formulation mirroring that of Janakiram et al. Transp. Sci. 33, 147–167 (1999) to solve the single-leg problem without cancellations, overbooking, or discounting. Borrowing a result from the queueing-control literature, we prove the concavity of the associated optimal value functions and, subsequently, the optimality of a booking limit policy. We then apply this same technique to the more influential papers from the single-leg literature, at once unifying the static and dynamic models and establishing the connection between the yield-management and queueing-control problems. Finally, we propose an omnibus formulation that yields the static and dynamic models as special cases.
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