# Asymptotic analysis of multi-class queues with random order of service .

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The random order of service (ROS) is a natural scheduling policy for systems where no ordering of customers can or should be established. Queueing models under ROS have been used to study molecular interactions of intracellular components in biology. However, these models often assume exponential distributions for processing and patience times, which is not realistic especially when operations such as binding, folding, transcription and translation are involved. We study a multi-class queueing model operating under ROS with reneging and generally distributed processing and patience times. We use measure-valued processes to describe the dynamic evolution of the network, and establish a fluid approximation for this representation. Obtaining a fluid limit for this network requires a multi-scale analysis of its fast and slow components, and to establish an averaging principle in the context of measure-valued process. In addition, under slightly more restrictive assumptions on the patience time distribution, we introduce a reduced, function-valued fluid model that is described by a system of non-linear Partial Differential Equations (PDEs). These PDEs, however, are non-standard and the analysis of their existence, uniqueness and stability properties requires new techniques.