Abstract
A machine repair model under general operating/repair distributions is considered in the Quality-and-Efficiency Driven asymptotic (QED) regime: both the number of machines and the number of repairmen are large, while the capacity and offered load relate via the square-root staffing rule. Process-level convergence of the number of broken machines is established—the limit is in terms of the corresponding tractable infinite-repairmen process, a stationary centered Gaussian process.
Highlights
The machine repair model is a classic operations research model
Many primary care physicians manage a finite number of patients (Green et al 2007), and our model can be relevant in that setting as well
We focus on a large-scale system operating in the Quality-andEfficiency-Driven (QED) regime
Summary
The machine repair model is a classic operations research model. The system consists of a finite number of machines and repairmen. In Reed (2009), process-level weak convergence for an open system with a general service time distribution in the QED regime was established; steady-state measures were not considered, since no closed form expressions are known to exist. The case of general service times is covered in Bassamboo et al (2009), where a transient regime in a closed system is considered; in this model, each customer receives “service” only once and the analysis reduces to a study of an infinite-server process. The number of broken machines at time t 0 satisfies, as n → ∞, ξn : ξn − kn ⇒ ξ Note that this limit and the assumption on the distribution of residual repair and working times at t 0 (see (2)) specify the state of the system at time t 0. The limiting (QED) steady-state distribution for the machine repair model is not known; even the steady-state distribution for the QED GI/GI/k queue is not known (only partial results exists, e.g., see Gamarnik and Goldberg 2013b)
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