Modelling large timescale and small timescale service variability

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.