{"591132":{"#nid":"591132","#data":{"type":"event","title":"PhD Defense by Yuan Li","body":[{"value":"\u003Cp\u003ETitle:\u0026nbsp;Stochastic Comparison Approach to Multi-server Queues: Bounds, Heavy Tails, and Large Deviations\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EAdvisors: Dr. David Goldberg\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003ECommittee members:\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr.\u0026nbsp;Robert Foley\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr.\u0026nbsp;Hayriye Ayhan\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr.\u0026nbsp;Siva Theja Maguluri\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr.\u0026nbsp;Jun Xu\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDate and Time:\u0026nbsp;\u003Cstrong\u003EMay 9th (Tuesday), 2017, at 11:00 AM\u003C\/strong\u003E\u003C\/p\u003E\r\n\r\n\u003Cp\u003ELocation:\u0026nbsp;ISyE\u0026nbsp;\u003Cstrong\u003EMain\u003C\/strong\u003E\u0026nbsp;Building Room\u0026nbsp;\u003Cstrong\u003E228\u003C\/strong\u003E\u0026nbsp;-\u0026nbsp;Executive Classroom\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003ESummary:\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EFor a FCFS\u0026nbsp;\u003Cstrong\u003EGI\/GI\/n\u003C\/strong\u003E\u0026nbsp;model queue, when the number of servers\u0026nbsp;\u003Cstrong\u003En\u003C\/strong\u003E\u0026nbsp;is large, the direct analysis of various performance metrics is generally analytically intractable. Therefore methods like developing bounds and heavy-traffic approximation are crucial to help us understand the system. A particularly popular heavy-traffic regime is the so-called Halfin-Whitt regime, under which the number of servers\u0026nbsp;\u003Cstrong\u003En\u003C\/strong\u003E\u0026nbsp;grows large and the traffic intensity\u0026nbsp;\u003Cstrong\u003E\u0026rho;\u0026nbsp;\u003C\/strong\u003Econverges to unity simultaneously under some square-root staffing rule, allowing the system to strike a balance between quality and efficiency. However, all known bounds for general multi-server queues in this setting suffer from several fundamental shortcomings, such as holding only asymptotically or involving non-explicit constants. \u0026nbsp;Furthermore, even less is known in the presence of heavy-tailed distributions in the model. This thesis primarily addresses these problems.\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EIn the first part of this thesis, we consider the FCFS \u003Cstrong\u003EGI\/GI\/n\u003C\/strong\u003E queue, and prove the first simple and explicit bounds that scale gracefully and universally as \u003Cstrong\u003E1\/(1-\u003C\/strong\u003E\u003Cstrong\u003E\u0026rho;)\u003C\/strong\u003E, across all notions of heavy traffic, including both the classical and Halfin-Whitt settings. In particular, supposing that the inter-arrival and service times, distributed as random variables \u003Cstrong\u003EA\u003C\/strong\u003E and \u003Cstrong\u003ES\u003C\/strong\u003E, have finite \u003Cstrong\u003Er\u003C\/strong\u003Eth moment for some \u003Cstrong\u003Er \u0026gt; 2\u003C\/strong\u003E, and letting\u0026nbsp;\u003Cstrong\u003E\u0026mu;_A\u003C\/strong\u003E\u0026nbsp;(\u003Cstrong\u003E\u0026mu;_S\u003C\/strong\u003E) denote \u003Cstrong\u003E1\/E[A]\u003C\/strong\u003E (\u003Cstrong\u003E1\/E[S]\u003C\/strong\u003E), our main results are bounds for the tail of the steady-state queue length and the steady-state probability of delay, expressed as simple and explicit functions of only \u003Cstrong\u003EE[(A\u0026mu;_A)^r]\u003C\/strong\u003E, \u003Cstrong\u003E\u0026nbsp;E[(S\u0026mu;_S)^r]\u003C\/strong\u003E, \u003Cstrong\u003Er\u003C\/strong\u003E, and\u0026nbsp;\u003Cstrong\u003E1\/(1-\u003C\/strong\u003E\u003Cstrong\u003E\u0026rho;)\u003C\/strong\u003E.\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EIn the second part of this thesis, we consider the FCFS \u003Cstrong\u003EGI\/GI\/n\u003C\/strong\u003E queue in the Halfin-Whitt heavy traffic regime, in the presence of heavy-tailed distributions. \u0026nbsp;We prove that when service times have finite \u003Cstrong\u003E1 +\u0026nbsp;\u0026epsilon;\u003C\/strong\u003E\u0026nbsp;moment for some\u0026nbsp;\u003Cstrong\u003E\u0026epsilon;\u0026nbsp;\u0026gt; 0\u003C\/strong\u003E and inter-arrival times have finite second moment, the sequence of stationary queue length distributions, normalized by \u003Cstrong\u003En^(1\/2)\u003C\/strong\u003E, is tight in the Halfin-Whitt regime. Furthermore, we develop simple and explicit bounds on the stationary queue length in that regime. For the setting where instead the inter-arrival times have an Pareto tail with index \u003Cstrong\u003E1\u0026lt;\u003C\/strong\u003E\u003Cstrong\u003E\u0026alpha;\u0026lt;2\u003C\/strong\u003E, we extend recent results of\u0026nbsp;\u003Ca href=\u0022https:\/\/projecteuclid.org\/euclid.aap\/1474296313\u0022\u003E[Hurvich and Reed, 2016]\u003C\/a\u003E, by proving for general service time distribution, the sequence of stationary queue length distributions, normalized by \u003Cstrong\u003En^(1\/\u003C\/strong\u003E\u003Cstrong\u003E\u0026alpha;\u003C\/strong\u003E\u003Cstrong\u003E)\u003C\/strong\u003E, is tight (under their scaling,\u0026nbsp;which we refer to as the Halfin-Whitt-Reed scaling regime).\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EIn the third part of this thesis, we further investigate the large deviation behaviors of the limits of the sequence of scaled stationary queue length in the presence of heavy-tailed inter-arrival and\/or service times. When service times have\u0026nbsp;a\u0026nbsp;Pareto tail with index\u0026nbsp;\u003Cstrong\u003E1\u0026lt;\u003C\/strong\u003E\u003Cstrong\u003E\u0026alpha;\u0026lt;2\u003C\/strong\u003E\u0026nbsp;and inter-arrival times have finite second moment, we bound the large deviations behavior of the limiting process and derive a matching lower bound when inter-arrival times are Markovian. \u0026nbsp;We find that the large deviation behavior of the limit has a sub-exponential decay, differing fundamentally from the exponentially decaying tails known to hold in the light-tailed setting. For the setting where instead the inter-arrival times have a Pareto tail with index\u0026nbsp;\u003Cstrong\u003E1\u0026lt;\u003C\/strong\u003E\u003Cstrong\u003E\u0026alpha;\u0026lt;2,\u003C\/strong\u003E\u0026nbsp;we are again able to bound the large-deviations behavior of the\u0026nbsp;limit, and find that our derived bounds do not depend on the particular service time distribution, and are in fact tight even for the case of deterministic service times.\u003C\/p\u003E\r\n","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"Stochastic Comparison Approach to Multi-server Queues: Bounds, Heavy Tails, and Large Deviations"}],"uid":"27707","created_gmt":"2017-04-28 19:21:36","changed_gmt":"2017-04-28 19:21:36","author":"Tatianna Richardson","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2017-05-09T12:00:00-04:00","event_time_end":"2017-05-09T14:00:00-04:00","event_time_end_last":"2017-05-09T14:00:00-04:00","gmt_time_start":"2017-05-09 16:00:00","gmt_time_end":"2017-05-09 18:00:00","gmt_time_end_last":"2017-05-09 18:00:00","rrule":null,"timezone":"America\/New_York"},"extras":[],"groups":[{"id":"221981","name":"Graduate Studies"}],"categories":[],"keywords":[{"id":"100811","name":"Phd Defense"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[{"id":"1788","name":"Other\/Miscellaneous"}],"invited_audience":[{"id":"78761","name":"Faculty\/Staff"},{"id":"78771","name":"Public"},{"id":"78751","name":"Undergraduate students"}],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}