Batch arrival single-server queue with variable service speed and setup time

In this paper, we consider an M^x/M/1/SET-VARI queue which has batch arrivals, variable service speed and setup time. Our model is motivated by power‐aware servers in data centers where dynamic scaling techniques are used. The service speed of the server is proportional to the number of jobs in the system. The contribution of our paper is threefold. First, we obtain the necessary and sufficient condition for the stability of the system. Second, we derive an expression for the probability generating function of the number of jobs in the system. Third, our main contribution is the derivation of the Laplace–Stieltjes transform (LST) of the sojourn time distribution, which is obtained in series form involving infinite‐dimensional matrices. In this model, since the service speed varies upon arrivals and departures of jobs, the sojourn time of a tagged job is affected by the batches that arrive after it. This makes the derivation of the LST of the sojourn time complex and challenging. In addition, we present some numerical examples to show the trade‐off between the mean sojourn time (response time) and the energy consumption. Using the numerical inverse Laplace–Stieltjes transform, we also obtain the sojourn time distribution, which can be used for setting the service‐level agreement in data centers.