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On the study of the running maximum and minimum level of level-dependent quasi-birth–death processes and related models

Part of: Markov processes Special processes

Published online by Cambridge University Press:  19 September 2022

Kayla Javier  and
Brian Fralix Open the ORCID record for Brian Fralix [Opens in a new window]
Kayla Javier*
Affiliation:
Wingate University
Brian Fralix*
Affiliation:
Clemson University
*
*Postal address: 228 Cedar St., Wingate, NC 28174. Email address: k.javier@wingate.edu
**Postal address: O-110 Martin Hall, Box 340975, Clemson, SC 29634. Email address: bfralix@clemson.edu
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Abstract

We present a study of the joint distribution of both the state of a level-dependent quasi-birth–death (QBD) process and its associated running maximum level, at a fixed time t: more specifically, we derive expressions for the Laplace transforms of transition functions that contain this information, and the expressions we derive contain familiar constructs from the classical theory of QBD processes. Indeed, one important takeaway from our results is that the distribution of the running maximum level of a level-dependent QBD process can be studied using results that are highly analogous to the more well-established theory of level-dependent QBD processes that focuses primarily on the joint distribution of the level and phase. We also explain how our methods naturally extend to the study of level-dependent Markov processes of M/G/1 type, if we instead keep track of the running minimum level instead of the running maximum level.

Keywords

Markov process of M/G/1 typequasi-birth–death processtime-dependent behavior

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K25: Queueing theory
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 1 , March 2023 , pp. 14 - 29
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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