Scaling analysis of an extended machine-repair model

Abstract

We consider an extension of the classic machine-repair model, where we explicitly model the fact that machines, apart from requiring service from a single repairer, also supply service themselves to products. Due to this dual role of the machines, the system exhibits an intricate relation between the processing rate of products and the performance of the repairer. To characterize this relation, we analyze this model under a Halfin-Whitt inspired scaling regime, where we amplify the arrival rate of products, the repair speed of the repairer and the number of machines appropriately. The resulting limiting stationary distribution is elegant, allows for a closed-form expression and provides intuition on the system's behavior resulting from the machines' dual role. With numerical results we illustrate the convergence, and assess under which conditions the limiting distributions lead to accurate approximations. Next to this valuable insight, the analysis in this paper can be viewed as a first step towards a unifying scaling analysis for general closed queueing networks.