Asymptotic independence of servers' activity in queueing systems with limited resource pooling

We consider multi-class multi-server queuing systems where a subset of servers, called a server pool, may collaborate in serving jobs of a given class. The pools of servers associated with different classes may overlap, so the sharing of server resources across classes is done via a dynamic allocation policy based on a fairness criterion. We consider an asymptotic regime where the total load increases proportionally with the system size. We show that under limited scaling in size of server pools the stationary distribution for activity of a fixed finite subset of servers has asymptotically a product form, which in turn implies a concentration result for server activity. In particular, we establish a clear connection between the scaling of server pools' size and asymptotic independence. Further, these results are robust to the service requirement distribution of jobs. For large-scale cloud systems where heterogeneous pools of servers collaborate in serving jobs of diverse classes, a concentration in server activity indicates that the overall power and network capacity that need to be provisioned may be substantially lower than the worst case, thus reducing costs.