The threshold based queueing system with hysteresis for performance analysis of clouds

In this paper, we propose to model a node cloud by a threshold based queueing system with hysteresis. The client requests (or jobs) arrive into the buffer, and are executed by service centers or Virtual Machines (VMs). We suppose that virtual machines are activated and deactivated according to the occupation of the queue, in order to model the dynamic behavior of the system. The queueing model is represented by a forward threshold sequence which has different values than the reverse threshold sequence. The forward thresholds represent the numbers of the customers in the queue from which we increase by one the number of VMs. Similarly, the backward thresholds are the values from which we decrease by one the number of VMs. When the forward thresholds are different then the backward thresholds then hysteresis is present. The relevance of hysteresis for the cloud model is to reduce the frequent transitions between activation and deactivation states of the VMs. We will use stochastic comparison method in order to prove and guarantee that the hysteresis model can be bounded by models with the same sequences of forward and backward threshold. The advantage of the bounding models is that the stationary probability distribution can be computed exactly and easily from a mathematical formula. We present some numerical results for the performance measures in order to show that the bounding values provide an accurate coverage for the exact values.