Stability and Non-stationary Characteristics of Queues

Palm probability;Optimal stopping;Markov chains;ASTA;Little's law;Continuity results;Drift condition;Approximation of queues

Fralix, Brian Haskel ; Industrial and Systems Engineering

University:Georgia Institute of Technology

Department:Industrial and Systems Engineering

关键词: Palm probability;    Optimal stopping;    Markov chains;    ASTA;    Little's law;    Continuity results;    Drift condition;    Approximation of queues;   

Others  :  https://smartech.gatech.edu/bitstream/1853/14569/1/fralix_brian_h_200705_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

We provide contributions to two classical areas of queueing.The first part of this thesis focuses on finding new conditions for aMarkov chain on a general state space to be Harris recurrent,positive Harris recurrent or geometrically ergodic.Most of ourresults show that establishing each property listed above isequivalent to finding a good enough feasible solution to aparticular optimal stopping problem, and they provide a morecomplete understanding of the role Foster's criterion plays in thetheory of Markov chains.The second and third parts of the thesis involve analyzing queuesfrom a transient, or time-dependent perspective.In part two, weare interested in looking at a queueing system from theperspective of a customer that arrives at a fixed time t.Doingthis requires us to use tools from Palm theory.From an intuitivestandpoint, Palm probabilities provide us with a way of computingprobabilities of events, while conditioning on sets of measurezero.Many studies exist in the literature that deal with Palmprobabilities for stationary systems, but very few treat thenon-stationary case.As an application of our main results, weshow that many classical results from queueing (in particular ASTA and Little's law) can be generalized to a time-dependentsetting.In part three, we establish a continuity result for what we referto as jump processes.From a queueing perspective, we basicallyshow that if the primitives and the initial conditions of asequence of queueing processes converge weakly, then thecorresponding queue-length processes converge weakly as well insome sense. Here the notion of convergence used depends onproperties of the limiting process, therefore our resultsgeneralize classical continuity results that exist in theliterature. The way our results can be used to approximatequeueing systems is analogous to the way phase-type randomvariables can be used to approximate other types of randomvariables.