Abstract
We consider a parallel server queueing system consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs. Incoming jobs are classified into one of several different classes (or buffers). Jobs within a class are processed on a first-in-first-out basis, where the processing of a given job may be performed by any server from a given (class-dependent) subset of the bank of servers. The random service time of a job may depend on both its class and the server providing the service. Each job departs the system after receiving service from one server. The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs to available servers. We consider a parameter regime in which the system satisfies both a heavy traffic and a complete resource pooling condition. Our cost function is an expected cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. In a prior work, the second author proposed a continuous review threshold control policy for use in such a parallel server system. This policy was advanced as an "interpretation" of the analytic solution to an associated Brownian control problem (formal heavy traffic diffusion approximation). In this paper we show that the policy proposed previously is asymptotically optimal in the heavy traffic limit and that the limiting cost is the same as the optimal cost in the Brownian control problem.
Highlights
We consider a dynamic scheduling problem for a parallel server queueing system
We describe the sequence of parallel server systems to be used in formulating the notion of heavy traffic asymptotic optimality
Assuming the complete resource pooling condition, we describe a simple “continuous review” policy for the sequence of parallel server systems, which allows changes in the control to be made at random times and in particular at times when the system status changes
Summary
We consider a dynamic scheduling problem for a parallel server queueing system. This system might be viewed as a model for a manufacturing or computer system, consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs (see e.g., [23]). Assuming heavy traffic scaling, the current paper provides the only proof of asymptotic optimality of a continuous review policy for the parallel server system with linear holding costs under a complete resource pooling condition. Using a suitable numbering of the buffers, the proof of state space collapse proceeds by induction on the buffer number, highlighting the fact that the queue length for a particular buffer depends (via the threshold policy) on the queue lengths associated with lower numbered buffers Another new aspect of our proof lies, where a certain uniform integrability is used to prove convergence of normalized cost functions associated with the sequence of parallel server systems, operating under the threshold policy, to the optimal cost function for the Brownian control problem. The proof is divided into two separate cases depending on the size of the time index (cf. (9.34)–(9.36)). (In the proof of Theorem 5.3 in [3], the estimates in (173) and (176) should have been divided into two cases corresponding to r2t > 2/ ̃ and r2t ≤ 2/ ̃.)
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