Thursday, January 25
Mathematics Colloquium
03:35 PM - 04:30 PM
Location: Science Center Room 323
Cost: Free

Speaker: James Cordeiro, University of Dayton

Title: The Role of the Group Inverse in the Ergodicity of Level-Dependent Quasi-Birth-and-Death Processes (LDQBDs)

Abstract: Quasi-birth-and-death (QBD) processes are a class of structured Markov chains that extend the classical Birth-Death model by permitting a finite number of state transitions that might occur between births and deaths. Over time, it has been found that a very large number of queueing models belong to this class of processes. It has thus been of interest among researchers to determine analytic conditions for the ergodicity (process stability) of QBD processes, and if it should hold, to develop efficient algorithms for the computation of steady state probabilities. Level independent QBD processes, namely those with generator or transition probability matrices with homogeneous block-row structures, were the first to be studied. An analytic stability formula was developed by Neuts in the early 1970s, after which attempts were made to do the same for level dependent (LDQBD) processes. In this presentation, we describe the application of Foster-Lyapunov drift to the determination of necessary and sufficient analytic stability criterion for discrete time LDQBD processes whose transition matrices converge over block rows. It was found that the proof of the criterion has a surprising dependence upon Markov generalized inverse theory, from which the Markov group inverse derives. This is the first known application of the Markov group inverse to an infinite-state process, which is made possible through a divide-and-conquer approach applied to levels of the transition probability matrix. 

Refreshments are available at 3:00 PM in SC 313F.

The department colloquia are held every Thursday (excluding holidays) at 3:35 pm in room SC 323 unless otherwise noted. All are invited to attend. 

Contact Information:
Name: Paul Eloe