We evaluate the performance of a cloud system using 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 according to the threshold values of the queue length. We suppose by batches, and follow a phase-type process. This system is analyze, especially when the size of the state space increases and the length of batch arrival distribution is large. So, to solve this problem, we propose to use stochastic bounds and the performance measures and compare the proposed bounding models with the exact one. The relevance of our methodology is to offer a trade-off between computational complexity and accuracy of the results and provide very interesting solutions for network dimensioning