Modelling large timescale and small timescale service variability
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Modelling large timescale and small timescale service variability Marco Gribaudo1 · Illés Horváth2
· Daniele Manini3 · Miklós Telek4
© The Author(s) 2019
Abstract The performance of service units may depend on various randomly changing environmental effects. It is quite often the case that these effects vary on different timescales. In this paper, we consider small and large scale (short and long term) service variability, where the short term variability affects the instantaneous service speed of the service unit and a modulating background Markov chain characterizes the long term effect. The main modelling challenge in this work is that the considered small and long term variation results in randomness along different axes: short term variability along the time axis and long term variability along the work axis. We present a simulation approach and an explicit analytic formula for the service time distribution in the double transform domain that allows for the efficient computation of service time moments. Finally, we compare the simulation results with analytic ones. Keywords Short and long term service variability · Brownian motion · Markov modulation · Performance analysis
This work is partially supported by the OTKA K-123914 and the TUDFO/51757/2019-ITM grants.
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Illés Horváth [email protected] Marco Gribaudo [email protected] Daniele Manini [email protected] Miklós Telek [email protected]
1
Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milan, Italy
2
MTA-BME Information Systems Research Group, Budapest, Hungary
3
Dipartimento di Informatica, Università di Torino, Turin, Italy
4
Department of Networked Systems and Services, Budapest University of Technology and Economics, Budapest, Hungary
123
Annals of Operations Research
1 Introduction Service speed variability is a problem that has been observed in many practical application scenarios. For example, in Kimber and Daly (1986), it has been observed for vehicular traffic. More recently this problem has been recognized in data centers (Guo et al. 2014). The effect of variability was also studied in Anjum and Perros (2015) with application to video-streaming. Most of the previous literature, however, focused only on large-timescale variability, where Markov-modulating models represent the random fluctuations of the environment. These set of models are commonly referred to as reward models and have been studied for a long time (Howard 1971). The variation in the service speed can be modelled by dividing the jobs into “infinitesimal quantities of work to be done” and considering the “speed at which this infinitesimal work is performed”, i.e., the random amount of time needed to execute the infinitesimal amount of work. Then, once a model defines how speed changes over time, the complete system can be modelled in a straight-forward way where the amount of work increases gradually along the analysis and the time required to execute the given amount of work is a random process. If the service pr
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