dc.contributor.advisor Goldberg, David dc.contributor.author Li, Yuan dc.date.accessioned 2018-08-20T15:27:48Z dc.date.available 2018-08-20T15:27:48Z dc.date.created 2017-08 dc.date.issued 2017-05-12 dc.date.submitted August 2017 dc.identifier.uri http://hdl.handle.net/1853/60118 dc.description.abstract In the first part of this thesis, we consider the FCFS $GI/GI/n$ queue, and prove the first simple and explicit bounds that scale gracefully and universally as $\frac{1}{1-\rho}$ ($\rho$ being the corresponding traffic intensity), across both the classical and Halfin-Whitt heavy traffic settings. In particular, supposing that the inter-arrival and service times, distributed as random variables $A$ and $S$, have finite $r$th moment for some $r > 2$, and letting $\mu_A (\mu_S)$ denote $\frac{1}{\E[A]} (\frac{1}{\E[S]})$, 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 $\E\big[(A \mu_A)^r\big], \E\big[(S \mu_S)^r\big], r$, and $\frac{1}{1-\rho}$. In the second part of this thesis, we prove that when service times have finite $1 + \epsilon$ moment for some $\epsilon > 0$ and inter-arrival times have finite second moment, the sequence of stationary queue length distributions, normalized by $n^{\frac{1}{2}}$, is tight in the Halfin-Whitt regime. Furthermore, we develop simple and explicit bounds on the stationary queue length in that regime. When the inter-arrival times have a Pareto tail with index $\alpha \in (1,2)$, we prove that in a generalized Halfin-Whitt regime, for general service time distributions, the sequence of stationary queue length distributions, normalized by $n^{\frac{1}{\alpha}}$, is tight. In the third part of this thesis, when service times have a Pareto tail with index $\alpha \in (1,2)$ and inter-arrival times have finite second moment, we bound the large deviation behavior of the weak limits of the $n^{\frac{1}{2}}$ scaled stationary queue lengths in the Halfin-Whitt regime, and derive a matching lower bound when inter-arrival times are Markovian. We find that the tail of the limit has a \emph{sub-exponential} decay, differing fundamentally from the exponential decay in the light-tailed setting. When inter-arrival times have a Pareto tail with index $\alpha \in (1,2)$, we bound the large-deviation behaviors of the weak limits of the $n^{\frac{1}{\alpha}}$ scaled statioanry queue lengths in a generalized Halfin-Whitt regime, and find that our bounds are tight when service times are deterministic. dc.format.mimetype application/pdf dc.language.iso en_US dc.publisher Georgia Institute of Technology dc.subject Many-server queues dc.subject Stochastic comparison dc.subject Kingman’s bound dc.subject Renewal process dc.subject Halfin-Whitt regime dc.subject Heavy tails dc.subject Weak convergence dc.subject Large deviations dc.subject Gaussian process dc.subject Stable law dc.title Stochastic comparison approach to multi-server queues: Bounds, heavy tails and large deviations dc.type Dissertation dc.description.degree Ph.D. dc.contributor.department Industrial and Systems Engineering thesis.degree.level Doctoral dc.contributor.committeeMember Maguluri, Siva Theja dc.contributor.committeeMember Foley, Robert dc.contributor.committeeMember Ayhan, Hayriye dc.contributor.committeeMember Xu, Jun dc.date.updated 2018-08-20T15:27:48Z
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