Asymptotic Analysis of a Closed Queueing Network in a Markov Random Environment

Tatiana Rusilko () and Dmitry Salnikov ()
Additional contact information
Tatiana Rusilko: Yanka Kupala State University of Grodno
Dmitry Salnikov: Yanka Kupala State University of Grodno

SN Operations Research Forum, 2025, vol. 6, issue 4, 1-15

Abstract: Abstract The primary purpose of this paper is an asymptotic analysis of a closed homogeneous exponential queueing network in a Markov random environment. The zero node acts as a generator of finite arrivals to the network and is loaded with departures from the network nodes. The arrival rate depends on the state of the environment. The Markov random environment is defined by an infinitesimal generator matrix on a finite set of states. The asymptotic analysis is performed under the critical assumption that a large number of customers circulating in the network. The mathematical approach used is based on the approximation of a discrete random process describing the state of the environment and the network by a mixed discrete-continuous process. The theorem is formulated, and it is proved that the conditional probability density function of a mixed process satisfies the generalized Fokker–Planck–Kolmogorov equation; the precision of the asymptotic approximation is indicated. An analytical form of its drift and diffusion coefficients is found. The ordinary differential equations are given for the first- and second-order moments of the number of customers at network nodes. The presented asymptotic method makes it possible to calculate the main statistical characteristics of queue lengths in the network both in steady state and, importantly, in transient state. The study of a closed queueing network in a Markov random environment significantly expands the scope of practical application of network queueing models.

Keywords: Queueing network; Random environment; Markov-modulated process; Asymptotic analysis method; Diffusion approximation method (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s43069-025-00572-4 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:snopef:v:6:y:2025:i:4:d:10.1007_s43069-025-00572-4

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/43069

DOI: 10.1007/s43069-025-00572-4

Access Statistics for this article

SN Operations Research Forum is currently edited by Marco Lübbecke

More articles in SN Operations Research Forum from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().