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Mathematics
Volume 14
Issue 12
10.3390/math14122201
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Performance Evaluation of a Single-Server Queueing System with Correlated Arrivals, Two-Tier Service Structure, Random Breakdowns and Phase-Type Repairs

by
G. Archana Alias Gurulakshmi
1,*,
Aliakbar Montazer Haghighi
2,
G. Ayyappan
3,
N. Arulmozhi
4 and
Natarajan Aishwarya
3
1
Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi 600062, India
2
Department of Mathematics, Prairie View A&M University, Prairie View, TX 77446, USA
3
Department of Mathematics, Puducherry Technological University, Puducherry 605014, India
4
Department of Mathematics, Sri Venkateshwaraa College of Engineering and Technology, Puducherry 605102, India
*
Author to whom correspondence should be addressed.
Mathematics 2026, 14(12), 2201; https://doi.org/10.3390/math14122201
Submission received: 13 May 2026 / Revised: 10 June 2026 / Accepted: 11 June 2026 / Published: 18 June 2026
Versions Notes

Abstract

This paper analyzes a single-server queueing system with infinite capacity, where arrivals follow a Markovian arrival process and service and repair times are modeled by phase-type distributions. The service mechanism is two-tier: every customer undergoes a mandatory primary service, after which an optional secondary service is available upon request. When the system is empty, the server initiates a closedown process before taking successive multiple vacations; upon return, the server goes through a setup process before beginning service again. Service can be interrupted by random breakdowns in either mode, triggering a phase-type repair. Matrix-analytic methods are used for the steady-state analysis, yielding the stability condition, stationary probability vectors, busy period analysis and key performance measures. A cost analysis framework is also developed. Numerical experiments validate the analytical results and illustrate the practical applicability of the model.
Keywords: Markovian arrival process; phase-type distribution; optional service; multiple vacations; closedown; setup; server breakdown; matrix-analytic method Markovian arrival process; phase-type distribution; optional service; multiple vacations; closedown; setup; server breakdown; matrix-analytic method

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MDPI and ACS Style

Gurulakshmi, G.A.A.; Haghighi, A.M.; Ayyappan, G.; Arulmozhi, N.; Aishwarya, N. Performance Evaluation of a Single-Server Queueing System with Correlated Arrivals, Two-Tier Service Structure, Random Breakdowns and Phase-Type Repairs. Mathematics 2026, 14, 2201. https://doi.org/10.3390/math14122201

AMA Style

Gurulakshmi GAA, Haghighi AM, Ayyappan G, Arulmozhi N, Aishwarya N. Performance Evaluation of a Single-Server Queueing System with Correlated Arrivals, Two-Tier Service Structure, Random Breakdowns and Phase-Type Repairs. Mathematics. 2026; 14(12):2201. https://doi.org/10.3390/math14122201

Chicago/Turabian Style

Gurulakshmi, G. Archana Alias, Aliakbar Montazer Haghighi, G. Ayyappan, N. Arulmozhi, and Natarajan Aishwarya. 2026. "Performance Evaluation of a Single-Server Queueing System with Correlated Arrivals, Two-Tier Service Structure, Random Breakdowns and Phase-Type Repairs" Mathematics 14, no. 12: 2201. https://doi.org/10.3390/math14122201

APA Style

Gurulakshmi, G. A. A., Haghighi, A. M., Ayyappan, G., Arulmozhi, N., & Aishwarya, N. (2026). Performance Evaluation of a Single-Server Queueing System with Correlated Arrivals, Two-Tier Service Structure, Random Breakdowns and Phase-Type Repairs. Mathematics, 14(12), 2201. https://doi.org/10.3390/math14122201

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