Abstract: Many server queues naturally arise in a variety of settings. In modern day applications such as call centers, reneging, where by customers that enter the system sometimes abandon prior to receiving service, dramatically influences system performance. The influence of reneging is understood fairly well in the case of exponential reneging distributions [1,2]. In this work [3,4], we consider the case of general reneging distributions. Motivated by the goal to formulate and analyze a fluid control problem, we consider the case of multiple customer classes and introduce a class of admissible control policies (rules for determining when to serve a given customer class). We then formulate a (general) fluid model, or formal functional law of large numbers approximation, for the multiclass many server queue operating under generic control policy, and characterize the invariant states. This generalizes work in the nonidling single-class case [5,6,7], as well as the policy specific work [2], which considers the multiclass case operating under static priority. We also develop a limit theorem that justifies regarding the fluid model as an approximation, the proof of which draws on methodology developed for the nonidling single class case [5,7].
Finally, we introduce the set of Random Buffer Selection (RBS) control policies. Given a probability vector p on the customer classes, RBS policy p determines at random with distribution p which class to serve next. For any p , we formulate a fluid model for RBS policy p . We prove that under mild conditions, a suitably rescaled state descriptor for the many server queue operating under RBS policy p converges to the fluid model solution for RBS policy p . We also show that any invariant state for the (general) fluid model is an invariant states for some RBS policy p . In particular, we demonstrate that, in contrast to static priority policies, RBS policies capture the entire spectrum of invariant states.

References:
[1] Atar, R., C. Giat, and N. Shimkin. 2010. The cμ/θ rule for many-server queues with abandonment. Operations Research. 58(5). 1427-1439.
[2] Atar, R., H. Kaspi, and N. Shimkin. 2014. Fluid limits for many-server systems with reneging under a priority policy. Mathematics of Operations Research. 39(3). 672-696.
[3] Puha, A. and A. R. Ward. 2018. A Fluid Limit for an Overloaded Multi-class Many-server Queue with General Reneging Distribution. Working Paper.
[4] Puha, A. and A. R. Ward. 2018. Tutorial Paper: Scheduling an Overloaded Multiclass Many-Server Queue with Impatient Customers. Working Paper.
[5] Kang, W. and K. Ramanan. 2010. Fluid limits of many-server queues with reneging. Annals of Applied Probability. 20(6). 2204-2260.
[6] Kang, W. and K. Ramanan. 2012. Asymptotic approximations for stationary distributions of many-serve queues with abandonment. Annals of Applied Probability. 22(2). 477-521.
[7] Kaspi, H. and K. Ramanan. 2011. Law of large numbers limits for many-server queues. Annals of Applied Probability. 21(1). 33-114.