{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/asymptotics-for-the-late-arrivals-problem","title":"Asymptotics for the Late Arrivals Problem","arxiv_id":"1302.1999","date":"2013-02-08","proceeding":null,"authors":["Carlo Lancia","Gianluca Guadagni","Sokol Ndreca","Benedetto Scoppola"],"abstract":"We study a discrete time queueing system where deterministic arrivals have i.i.d. exponential delays $\\xi_{i}$. The standard deviation $\\sigma$ of the delay is finite, but its value is much larger than the deterministic unit service time. We describe the model as a bivariate Markov chain, we prove that it is ergodic and then we focus on the unique joint equilibrium distribution. We write a functional equation for the bivariate generating function, finding the solution of such equation on a subset of its set of definition. This solution allows us to prove that the equilibrium distribution of the Markov chain decays super-exponentially fast in the quarter plane. Finally, exploiting the latter result, we discuss the numerical computation of the stationary distribution, showing the effectiveness of a simple approximation scheme in a wide region of the parameters. The model, motivated by air and railway traffic, was proposed many decades ago by Kendall with the name of \"late arrivals problem\", but no solution has been found so far.","url_abs":"http://arxiv.org/abs/1302.1999v6","url_pdf":"http://arxiv.org/pdf/1302.1999v6.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"asymptotics-for-the-late-arrivals-problem","repo_url":"https://github.com/clancia/EDA","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}