Strong approximations for Markovian service networks

Abstract

Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system. Included in our modeling framework are networks of M_t/M_t/n_t queues with abandonment and retrials. The asymptotic limiting regime that we explore for these networks has a natural interpretation of scaling up the number of servers in response to a similar scaling up of the arrival rate for the customers. The individual service rates, however, are not scaled. We employ the theory of strong approximations to obtain functional strong laws of large numbers and functional central limit theorems for these networks. This gives us a tractable set of network fluid and diffusion approximations. A common theme for service network models with features like many servers, priorities, or abandonment is “non-smooth” state dependence that has not been covered systematically by previous work. We prove our central limit theorems in the presence of this non-smoothness by using a new notion of derivative.