Stochastic comparison approach to multi-server queues: Bounds, heavy tails and large deviations

Li, Yuan

Title:

Stochastic comparison approach to multi-server queues: Bounds, heavy tails and large deviations

dc.contributor.advisor	Goldberg, David A.
dc.contributor.author	Li, Yuan
dc.contributor.committeeMember	Maguluri, Siva Theja
dc.contributor.committeeMember	Foley, Robert
dc.contributor.committeeMember	Ayhan, Hayriye
dc.contributor.committeeMember	Xu, Jun
dc.contributor.department	Industrial and Systems Engineering
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.date.updated	2018-08-20T15:27:48Z
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.description.degree	Ph.D.
dc.format.mimetype	application/pdf
dc.identifier.uri	http://hdl.handle.net/1853/60118
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	Text
dc.type.genre	Dissertation
dspace.entity.type	Publication
local.contributor.corporatename	H. Milton Stewart School of Industrial and Systems Engineering
local.contributor.corporatename	College of Engineering
relation.isOrgUnitOfPublication	29ad75f0-242d-49a7-9b3d-0ac88893323c
relation.isOrgUnitOfPublication	7c022d60-21d5-497c-b552-95e489a06569
thesis.degree.level	Doctoral