Today's data centers are designed to scale up to respond to the offered workload in a rapid, efficient, and effective manner, and at the same time, they must satisfy the Service Level Agreement (SLA) requirements. This opens up many interesting and challenging research issues and opportunities. The Software-as-a-Service (SaaS) is the most popular cloud service model being used these days, in which multi-core VMs are allocated efficiently to meet the offered workload, and in a way to avoid any violations to the agreed SLA. This entails the need to model SaaS services to predict the performance and overall system cost, and to estimate the required number of VM resources and their respective multi-core capacity prior to the actual deployment. To this end, we present in this paper a queuing mathematical model to study and analyze the performance of multi-core VMs hosting cloud SaaS applications. Our analytical model estimates under any offered workload the number of required multi-core VM instances needed to satisfy the Quality of Service (QoS) parameters. Our mathematical model is validated using DES (Discrete Event Simulator) simulations. Results obtained from our analysis as well as simulation models show that the proposed model is powerful and able to correctly and effectively predict the system performance and cost, and also to determine the number of VMs cores needed for SaaS services in order to achieve QoS targets under different workload conditions.