Large-scale heterogeneous service systems with general packing constraints

A service system with multiple types of customers, arriving according to Poisson processes, is considered. The system is heterogeneous in that the servers also can be of multiple types. Each customer has an independent exponentially distributed service time, with the mean determined by its type. Multiple customers (possibly of different types) can be placed for service into one server, subject to "packing" constraints, which depend on the server type. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered such that the customer arrival rates grow to infinity. We consider two variants of the model. For the {\em infinite-server} model, we prove asymptotic optimality of the {\em Greedy Random} (GRAND) algorithm in the sense of minimizing the weighted (by type) number of occupied servers in steady-state. (This version of GRAND generalizes that introduced in [15] for the homogeneous systems, with all servers of same type.) We then introduce a natural extension of GRAND algorithm for {\em finite-server} systems with blocking. Assuming subcritical system load, we prove existence, uniqueness, and local stability of the large-scale system equilibrium point such that no blocking occurs. This result strongly suggests a conjecture that the steady-state blocking probability under the algorithm vanishes in the large-scale limit.