Large-scale Service Systems with Packing Constraints and Heterogeneous Servers

Abstract

A service system with multiple types of arriving customers and multiple types of servers is considered. Several customers (possibly of different types) can be placed for concurrent service into same 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 largescale asymptotic regime is considered such that the customer arrival rates grow to infinity. We consider two variants of the model. For the infinite-server model, we prove asymptotic optimality of the Greedy Random (GRAND) algorithm in the sense of minimizing the weighted (by type) number of occupied servers in steadystate. (This version of GRAND generalizes that introduced in [1] for the homogeneous systems, with all servers of same type.) We then introduce a natural extension of GRAND algorithm for 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.