A Numerical Approach to Stability of Multiclass Queueing Networks

The Multi-class Queueing Network (McQN) arises as a natural multi-class extension of the traditional (single-class) Jackson network. In a single-class network subcriticality (i.e. subunitary nominal workload at every station) entails stability, but this is no longer sufficient when jobs/customers of different classes (i.e. with different service requirements and/or routing scheme) visit the same server; therefore, analytical conditions for stability of McQNs are lacking, in general. In this note we design a numerical (simulation-based) method for determining the stability region of a McQN, in terms of arrival rate(s). Our method exploits certain (stochastic) monotonicity properties enjoyed by the associated Markovian queue-configuration process. Stochastic monotonicity is a quite common feature of queueing models and can be easily established in the single-class framework (Jackson networks); recently, also for a wide class of McQNs, including first-come-first-serve (FCFS) networks, monotonicity properties have been established. Here, we provide a minimal set of conditions under which the method performs correctly. Eventually, we illustrate the use of our numerical method by presenting a set of numerical experiments, covering both single and multi-class networks

Recent Contributions of Theory to Our Understanding of the Atlantic Meridional Overturning Circulation

Revolutionary observational arrays, together with a new generation of ocean and climate models, have provided new and intriguing insights into the Atlantic Meridional Overturning Circulation (AMOC) over the last two decades. Theoretical models have also changed our view of the AMOC, providing a dynamical framework for understanding the new observations and the results of complex models. In this paper we review recent advances in conceptual understanding of the processes maintaining the AMOC. We discuss recent theoretical models that address issues such as the interplay between surface buoyancy and wind forcing, the extent to which the AMOC is adiabatic, the importance of mesoscale eddies, the interaction between the middepth North Atlantic Deep Water cell and the abyssal Antarctic Bottom Water cell, the role of basin geometry and bathymetry, and the importance of a three‐dimensional multiple‐basin perspective. We review new paradigms for deep water formation in the high‐latitude North Atlantic and the impact of diapycnal mixing on vertical motion in the ocean interior. And we discuss advances in our understanding of the AMOC's stability and its scaling with large‐scale meridional density gradients. Along with reviewing theories for the mean AMOC, we consider models of AMOC variability and discuss what we have learned from theory about the detection and meridional propagation of AMOC anomalies. Simple theoretical models remain a vital and powerful tool for articulating our understanding of the AMOC and identifying the processes that are most critical to represent accurately in the next generation of numerical ocean and climate models

Flow instabilities in circular Couette flow of wormlike micelle solutions with a reentrant flow curve

In this work, we numerically investigate flow instabilities of inertialess circular Couette flow of dilute wormlike micelle solutions. Using the reformulated reactive rod model (RRM-R) [Hommel and Graham, JNNFM 295 (2021) 104606], which treats micelles as rigid Brownian rods undergoing reversible scission and fusion in flow, we study the development and behavior of both vorticity banding and finger-like instabilities. In particular, we focus on solutions that exhibit reentrant constitutive curves, in which there exists some region where the shear stress, $\tau$, has a multivalued relation to shear rate, $\dot{\gamma}$. We find that the radial dependence of the shear stress in circular Couette flow allows for solutions in which parts of the domain lie in the region of the flow curve where $\partial \tau /\partial \dot{\gamma} > 0$, while others lie in the region where $\partial \tau /\partial \dot{\gamma} < 0$; this mixed behavior can lead to complex flow instabilities that manifest as finger-like structures of elongated and anisotropically-oriented micelles. In 3D simulations we find that the initial instability is 2D in origin, and 3D finger-like structures arise through the axial instability of 2D sheets. Finally, we show that the RRM-R can capture vorticity banding in narrow-gap circular Couette flow and that vorticity bands are linearly stable to perturbations.Comment: 42 pages, 27 figures, 5 supplemental movie