Upper and Lower Bounds of Performance Metrics in Hybrid Systems with Setup Time

Ken’ichi Kawanishi () and Yuki Ino
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Ken’ichi Kawanishi: Faculty of Informatics, Gunma University, 4-2 Aramaki, Maebashi 371-8510, Gunma, Japan
Yuki Ino: Graduate School of Informatics, Gunma University, 4-2 Aramaki, Maebashi 371-8510, Gunma, Japan

Mathematics, 2025, vol. 13, issue 16, 1-40

Abstract: To address the increasing demand for computational and communication resources, modern networked systems often rely on heterogeneous servers, including those requiring setup times, such as virtual machines or servers, and others that are always active. In this paper, we model and analyze the performance of such hybrid systems using a level-dependent quasi-birth-and-death (LDQBD) process. Building upon an existing queueing model, we extend the analysis by considering scalable approximation methods. Since matrix analytic methods become computationally expensive in large-scale settings, we propose a stochastic bounding approach that derives upper and lower bounds for the stationary distribution, thereby significantly reducing computational cost. This approach further provides bounds on the performance metrics of the hybrid system.

Keywords: stochastic bounds; level-dependent QBD process; hybrid systems; setup times; performance evaluation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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