Stabilizing the virtual response time in single-server processor sharing queues with slowly time-varying arrival rates
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Stabilizing the virtual response time in single-server processor sharing queues with slowly time-varying arrival rates Yongkyu Cho1 · Young Myoung Ko2 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Motivated by the work of Whitt (Queueing Syst 81(4):341–378, 2015) and Ma and Whitt (in: Winter simulation conference, pp 2598–2609, 2015), that studied performance stabilization in a G It /G It /1 queue, this research study investigates the stabilization of the mean virtual response time in a single-server processor sharing (PS) queueing system with a time-varying arrival rate and a service rate control (a G It /G It /1/P S queue). We propose and compare a modified square-root (SR) control and a difference-matching (DM) control to stabilize the mean virtual response time of a G It /G It /1/P S queue. Extensive simulation studies with various settings of arrival processes and service times show that the DM control outperforms the SR control for heavy-traffic conditions, and that the SR control performs better for lighttraffic conditions. Keywords Stabilizing performance · Nonstationary queues · Processor sharing · Service rate control · Queueing simulation
1 Introduction Modern data centers consume tremendous amounts of energy to supply networking, computing, and storage services to global IT companies. Concerns about energy consumption have prompted researchers to explore operational methods that maximize energy efficiency and satisfy a certain level of quality of service (QoS) (Anselmi and Verloop 2011; Ko and Cho 2014; Liao et al. 2015). QoS can be achieved by adding constraints that impose upper bounds for response time-related metrics, e.g., the mean (virtual) response time and the tail probability of the response time. In general, these constraints are binding, because of the
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Young Myoung Ko [email protected]; [email protected] Yongkyu Cho [email protected]
1
Samsung Electronics, 129, Samsung-ro, Yeongtong-gu, Suwon, Gyeonggi 16677, Korea
2
Department of Industrial and Management Engineering, Pohang University of Science and Technology (POSTECH), 77, Cheongam-ro, Nam-gu, Pohang, Gyeongbuk 37673, Korea
123
Annals of Operations Research
conflict between the QoS-related metrics and energy consumption. Binding the QoS-related constraints implies that the metrics are maintained as a constant value, and suggests the need to investigate the stabilization of response times. Although some proposed methodologies (Anselmi and Verloop 2011; Ko and Cho 2014; Liao et al. 2015) assume the stationarity of data traffic arrival processes, nonstationary properties, such as time-varying arrival rates from real data (CAIDA 2016), make it difficult to analyze queueing system performance. Due to such difficulties, stabilizing performance has been suggested so that it makes the systems look stationary in terms of the performance measures. The approach may give us a chance to design, operate, and manage large-scale stochastic systems in which statistical performance guarantees
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