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1. Load Balancing Under Strict Compatibility Constraints

   https://doi.org/10.1287/moor.2022.1258  

   Rutten, Daan; Mukherjee, Debankur (April 2022, Mathematics of Operations Research)

   Consider a system with N identical single-server queues and a number of task types, where each server is able to process only a small subset of possible task types. Arriving tasks select [Formula: see text] random compatible servers and join the shortest queue among them. The compatibility constraints are captured by a fixed bipartite graph between the servers and the task types. When the graph is complete bipartite, the mean-field approximation is accurate. However, such dense compatibility graphs are infeasible for large-scale implementation. We characterize a class of sparse compatibility graphs for which the mean-field approximation remains valid. For this, we introduce a novel notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs asymptotically match the performance of a fully flexible system. Furthermore, we show that proportionally sparse random compatibility graphs can be constructed, which reduce the server degree almost by a factor [Formula: see text] compared with the complete bipartite compatibility graph. 

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2. A lower bound on critical points of the electric potential of a knot

   https://doi.org/10.1142/S0218216521500267  

   Lipton, Max (April 2021, Journal of Knot Theory and Its Ramifications)

   null (Ed.)

   Consider a knot [Formula: see text] in [Formula: see text] with charge uniformly distributed on it. From the standpoint of both physics and knot theory, it is natural to try to understand the critical points of the potential and their behavior. We show the number of critical points of the potential is at least [Formula: see text], where [Formula: see text] is the tunnel number, defined as the smallest number of arcs one must add to [Formula: see text] such that its complement is a handlebody. The result is proven using Morse theory and stable manifold theory. 

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3. Stability of Parallel Server Systems

   https://doi.org/10.1287/opre.2021.2125  

   Moyal, Pascal; Perry, Ohad (July 2022, Operations Research)

   The fundamental problem in the study of parallel-server systems is that of finding and analyzing routing policies of arriving jobs to the servers that efficiently balance the load on the servers. The most well-studied policies are (in decreasing order of efficiency) join the shortest workload (JSW), which assigns arrivals to the server with the least workload; join the shortest queue (JSQ), which assigns arrivals to the smallest queue; the power-of-[Formula: see text] (PW([Formula: see text])), which assigns arrivals to the shortest among [Formula: see text] queues that are sampled from the total of [Formula: see text] queues uniformly at random; and uniform routing, under which arrivals are routed to one of the [Formula: see text] queues uniformly at random. In this paper we study the stability problem of parallel-server systems, assuming that routing errors may occur, so that arrivals may be routed to the wrong queue (not the smallest among the relevant queues) with a positive probability. We treat this routing mechanism as a probabilistic routing policy, named a [Formula: see text]-allocation policy, that generalizes the PW([Formula: see text]) policy, and thus also the JSQ and uniform routing, where [Formula: see text] is an [Formula: see text]-dimensional vector whose components are the routing probabilities. Our goal is to study the (in)stability problem of the system under this routing mechanism, and under its “nonidling” version, which assigns new arrivals to an idle server, if such a server is available, and otherwise routes according to the [Formula: see text]-allocation rule. We characterize a sufficient condition for stability, and prove that the stability region, as a function of the system’s primitives and [Formula: see text], is in general smaller than the set [Formula: see text]. Our analyses build on representing the queue process as a continuous-time Markov chain in an ordered space of [Formula: see text]-dimensional real-valued vectors, and using a generalized form of the Schur-convex order. 

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4. Load Balancing in Parallel Queues and Rank-Based Diffusions

   https://doi.org/10.1287/moor.2023.0050  

   Banerjee, Sayan; Budhiraja, Amarjit; Estevez, Benjamin (May 2025, Mathematics of Operations Research)

   Consider a queuing system with K parallel queues in which the server for each queue processes jobs at rate n and the total arrival rate to the system is [Formula: see text], where [Formula: see text] and n is large. Interarrival and service times are taken to be independent and exponentially distributed. It is well known that the join-the-shortest-queue (JSQ) policy has many desirable load-balancing properties. In particular, in comparison with uniformly at random routing, the time asymptotic total queue-length of a JSQ system, in the heavy traffic limit, is reduced by a factor of K. However, this decrease in total queue-length comes at the price of a high communication cost of order [Formula: see text] because at each arrival instant, the state of the full K-dimensional system needs to be queried. In view of this, it is of interest to study alternative routing policies that have lower communication costs and yet have similar load-balancing properties as JSQ. In this work, we study a family of such rank-based routing policies, which we will call Marginal Size Bias Load-Balancing policies, in which [Formula: see text] of the incoming jobs are routed to servers with probabilities depending on their ranked queue length and the remaining jobs are routed uniformly at random. A particular case of such routing schemes, referred to as the marginal JSQ (MJSQ) policy, is one in which all the [Formula: see text] jobs are routed using the JSQ policy. Our first result provides a heavy traffic approximation theorem for such queuing systems in terms of reflected diffusions in the positive orthant [Formula: see text]. It turns out that, unlike the JSQ system, where, due to a state space collapse, the heavy traffic limit is characterized by a one-dimensional reflected Brownian motion, in the setting of MJSQ (and for the more general rank-based routing schemes), there is no state space collapse, and one obtains a novel diffusion limit which is the constrained analogue of the well-studied Atlas model (and other rank-based diffusions) that arise from certain problems in mathematical finance. Next, we prove an interchange of limits ([Formula: see text] and [Formula: see text]) result which shows that, under conditions, the steady state of the queuing system is well approximated by that of the limiting diffusion. It turns out that the latter steady state can be given explicitly in terms of product laws of Exponential random variables. Using these explicit formulae, and the interchange of limits result, we compute the time asymptotic total queue-length in the heavy traffic limit for the MJSQ system. We find the striking result that, although in going from JSQ to MJSQ, the communication cost is reduced by a factor of [Formula: see text], the steady-state heavy traffic total queue-length increases by at most a constant factor (independent of n, K) which can be made arbitrarily close to one by increasing a MJSQ parameter. We also study the case where the system is overloaded—namely, [Formula: see text]. For this case, we show that although the K-dimensional MJSQ system is unstable, unlike the setting of random routing, the system has certain desirable and quantifiable load-balancing properties. In particular, by establishing a suitable interchange of limits result, we show that the steady-state difference between the maximum and the minimum queue lengths stays bounded in probability (in the heavy traffic parameter n). Funding: Financial support from the National Science Foundation [RTG Award DMS-2134107] is gratefully acknowledged. S. Banerjee received financial support from the National Science Foundation [NSF-CAREER Award DMS-2141621]. A. Budhiraja received financial support from the National Science Foundation [Grant DMS-2152577]. 

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5. Marchenko–Pastur law with relaxed independence conditions

   https://doi.org/10.1142/s2010326321500404  

   Bryson, Jennifer; Vershynin, Roman; Zhao, Hongkai (January 2021, Random Matrices: Theory and Applications)

   null (Ed.)

   We prove the Marchenko–Pastur law for the eigenvalues of [Formula: see text] sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario — the block-independent model — the [Formula: see text] coordinates of the data are partitioned into blocks in such a way that the entries in different blocks are independent, but the entries from the same block may be dependent. In the second scenario — the random tensor model — the data is the homogeneous random tensor of order [Formula: see text], i.e. the coordinates of the data are all [Formula: see text] different products of [Formula: see text] variables chosen from a set of [Formula: see text] independent random variables. We show that Marchenko–Pastur law holds for the block-independent model as long as the size of the largest block is [Formula: see text], and for the random tensor model as long as [Formula: see text]. Our main technical tools are new concentration inequalities for quadratic forms in random variables with block-independent coordinates, and for random tensors. 

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