A common pitfall when hosting applications in today's cloud environments is that virtual servers often experience varying execution speeds due to the interference from co-located virtual servers degrading the tail sojourn times specified in service level agreements. Motivated by the significance of tail sojourn times for cloud clusters, we develop a model of N parallel virtual server queues, each of which processes jobs in a processor sharing fashion under varying execution speeds governed by Markov-modulated processes. We derive the tail distribution of the workloads for each server and the approximation for the tail sojourn times based on large deviation analysis. Furthermore, we optimize the cluster sizes that fulfill the requirements of target tail sojourn times. Extensive simulation experiments show very good matches to the derived analysis in a variety of scenarios, i.e., large numbers of servers experiencing a high number of different execution speeds, under various traffic intensities, workload variations and cluster sizes. Finally, we apply our proposed analysis to estimating the tail sojourn times of a Wikipedia system hosted in a private cloud, and the testbed results strongly confirm the applicability and accuracy of our analysis.