A novel probabilistic multi-tenant model is developed to characterize the service performance of cloud systems. The model considers essential cloud-system characteristics including virtualization, multi-tenancy and heterogeneity of the physical servers. Given the probabilistic multi-tenant model, three virtual machine mapping algorithms are proposed. Of particular interest is the max-load-first algorithm, which firstly maps the largest VM, in terms of the workload size of a user's request it serves, to the fastest physical server in the system. Monte-Carlo simulation results show that the max-load-first algorithm outperforms the other two algorithms based on the mean of stochastic completion time of a group of arbitrary users' requests. The simulation results also provide insight on how the initial loads of servers affect the performance of the cloud system.