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Randomized load-balancing algorithms play an important role in improving performance in large-scale networks at relatively low computational cost. A common model of such a system is a network of N parallel queues in which incoming jobs with independent and identically distributed service times are routed on arrival using the join-the-shortest-of-d-queues routing algorithm. Under fairly general conditions, it was shown by Aghajani and Ramanan that as the size of the system goes to infinity, the state dynamics converge to the unique solution of a countable system of coupled deterministic measure-valued equations called the hydrodynamic equations. In this article, a characterization of invariant states of these hydrodynamic equations is obtained and, when d=2, used to construct a numerical algorithm to compute the queue length distribution and mean virtual waiting time in the invariant state. Additionally, it is also shown that under a suitable tail condition on the service distribution, the queue length distribution of the invariant state exhibits a doubly exponential tail decay, thus demonstrating a vast improvement in performance over the case [Formula: see text], which corresponds to random routing, when the tail decay could even be polynomial. Furthermore, numerical evidence is provided to support the conjecture that the invariant state is the limit of the steady-state distributions of the N-server models. The proof methodology, which entails analysis of a coupled system of measure-valued equations, can potentially be applied to other many-server systems with general service distributions, where measure-valued representations are useful. 

ABSTRACT Shear stress imparted by blood flow tends to smoothen endothelial monolayers, a response classically attributed to reduced nuclear height and nuclear reorientation along flow. However, the mechanical basis remains unclear. Here, we tested predictions of the nuclear drop model—which posits that nuclear shape changes occur at constant volume and surface area—in human umbilical vein endothelial cells (HUVECs) under physiological shear stress. HUVEC nuclear morphologies varied from smooth, flat nuclei to wrinkled, tall ones. Applying shear stress reduced the frequency of tall, wrinkled nuclei, explaining the population‐level decrease in nuclear height. Lamin A/C–depleted nuclei are highly irregular and failed to recover shapes postindentation on PDMS microposts, suggesting that lamin A/C confers nuclear surface tension. Nuclear volume and surface area remained constant under shear, consistent with the drop model, and a computational model based on these constraints successfully predicted observed nuclear shapes. Neither lamin A/C nor lamin B1 depletion prevented shear‐induced YAP nuclear localization; instead, shear detached poorly spread cells, increasing spreading, focal adhesion assembly, and cytoskeletal tension in the remaining cells, thereby promoting YAP nuclear localization. These findings revise classical interpretations of flow‐induced endothelial smoothing and show that flow‐induced YAP nuclear localization results from increased cell spreading rather than nuclear deformation.