{"paper":{"title":"The supermarket model with arrival rate tending to one","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Graham Brightwell, Malwina Luczak","submitted_at":"2012-01-26T14:10:51Z","abstract_excerpt":"In the supermarket model, there are $n$ queues, each with a single server. Customers arrive in a Poisson process with arrival rate $\\lambda n$, where $\\lambda = \\lambda (n) \\in (0,1)$. Upon arrival, a customer selects $d=d(n)$ servers uniformly at random, and joins the queue of a least-loaded server amongst those chosen. Service times are independent exponentially distributed random variables with mean~1. In this paper, we analyse the behaviour of the supermarket model in a regime where $\\lambda(n)$ tends to~1, and $d(n)$ tends to infinity, as $n \\to \\infty$. For suitable triples $(n,d,\\lambda"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5523","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}