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Join-the-shortest queue (JSQ) is a classical benchmark for the performance of parallel-server queueing systems because of its strong optimality properties. Recently, there has been significant progress in understanding its large-system asymptotic behavior. In this paper, we analyze the JSQ policy in the super-Halfin-Whitt scaling window when load per server [Formula: see text] scales with the system size N as [Formula: see text] for [Formula: see text] and [Formula: see text]. We establish that the centered and scaled total queue length process converges to a certain Bessel process with negative drift, and the associated (centered and scaled) steady-state total queue length, indexed by N, converges to a [Formula: see text] distribution. The limit laws are universal in the sense that they do not depend on the value of [Formula: see text] and exhibit fundamentally different behavior from both the Halfin–Whitt regime ([Formula: see text]) and the nondegenerate slowdown (NDS) regime ([Formula: see text]). Funding: This work was supported by the National Science Foundation to S. Banerjee [Grants CAREER DMS-2141621 and RTG DMS-2134107] and D. Mukherjee and Z. Zhao [Grants CIF-2113027 and CPS-2240982]. 

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]. 

Reactions on distorted graphene membranes show strain dependent reactivity with water as followed by Raman microspectroscopy. 

Abstract We investigate the statistical learning of nodal attribute functionals in homophily networks using random walks. Attributes can be discrete or continuous. A generalization of various existing canonical models, based on preferential attachment is studied (model class $$\mathscr {P}$$ P ), where new nodes form connections dependent on both their attribute values and popularity as measured by degree. An associated model class $$\mathscr {U}$$ U is described, which is amenable to theoretical analysis and gives access to asymptotics of a host of functionals of interest. Settings where asymptotics for model class $$\mathscr {U}$$ U transfer over to model class $$\mathscr {P}$$ P through the phenomenon of resolvability are analyzed. For the statistical learning, we consider several canonical attribute agnostic sampling schemes such as Metropolis-Hasting random walk, versions of node2vec (Grover and Leskovec, 2016) that incorporate both classical random walk and non-backtracking propensities and propose new variants which use attribute information in addition to topological information to explore the network. Estimators for learning the attribute distribution, degree distribution for an attribute type and homophily measures are proposed. The performance of such statistical learning framework is studied on both synthetic networks (model class $$\mathscr {P}$$ P ) and real world systems, and its dependence on the network topology, degree of homophily or absence thereof, (un)balanced attributes, is assessed.