We consider a queueing network with two single-server stations and two types of customers. Customers of type A require service only at station 1 and customers of type B require service first at station 1 and then at station 2. Each server has a different general service time distribution, and each customer type has a different general interarrival time distribution. The problem is to find a dynamic sequencing policy at station 1 that minimizes the long-run average expected number of customers in the system.
The scheduling problem is approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved, and the solution is interpreted in terms of the queueing system in order to obtain an effective sequencing policy. Also, a pathwise lower bound (for any sequencing policy) is obtained for the total number of customers in the network. We show via simulation that the relative difference between the performance of the proposed policy and the pathwise lower bound becomes small as the load on the network is increased toward the heavy traffic limit.
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Harrison, J.M., Wein, L.M. Scheduling networks of queues: Heavy traffic analysis of a simple open network. Queueing Syst 5, 265–279 (1989). https://doi.org/10.1007/BF01225319
- Brownian approximations
- networks of queues
- scheduling theory
- heavy traffic analysis
- dynamic priority
- pathwise solution