Abstract : We propose to evaluate the performance of a cloud system in terms of buffer occupancy, using mathematical analysis. The client requests (or jobs) arrive by batch, and follow a phase-type process in order to represent the variability of the traffic intensity. The PM (Physical Machine) is modeled by a hysteresis queueing system with phase-type and batch arrivals. To represent the dynamic allocation of the resources, the hysteresis queue activates and deactivates the Virtual Machines (VMs) according to the threshold values of the queue length. This system is represented by a complex Markov chain which is difficult to analyze especially when the size of the state space increases and the length of batch arrival distribution is large. We propose to use stochastic bounds in order to define bounding systems less complex. We derive performance measure bounds as mean buffer length, and blocking probabilities. The relevance of the results is to offer a trade-off between computational complexity and accuracy of the results, providing very interesting solutions in network dimensioning.