We consider M/P h/n + M queueing systems in steady state. We prove that the Wasserstein distance between the stationary distribution of the normalized system size process and that of a piecewise Ornstein-Uhlenbeck (OU) process is bounded by C/ √ λ, where the constant C is independent of the arrival rate λ and the number of servers n as long as they are in the Halfin-Whitt parameter regime. For each integer m > 0, we also establish a similar bound for the difference of the mth steady-state moments. For the proofs, we develop a modular framework that is based on Stein's method. The framework has three components: Poisson equation, generator coupling, and state space collapse. The framework, with further refinement, is likely applicable to steady-state diffusion approximations for other stochastic systems. *

This paper provides an introduction to the Stein method framework in the context of steady-state diffusion approximations. The framework consists of three components: the Poisson equation and gradient bounds, generator coupling, and moment bounds. Working in the setting of the Erlang-A and Erlang-C models, we prove that both Wasserstein and Kolmogorov distances between the stationary distribution of a normalized customer count process, and that of an appropriately defined diffusion process decrease at a rate of 1/ √ R, where R is the offered load. Futhermore, these error bounds are universal, valid in any load condition from lightly loaded to heavily loaded.

This paper considers a closed queueing network model of ridesharing systems such as Didi Chuxing, Lyft, and Uber. We focus on empty-car routing, a mechanism by which we control car flow in the network to optimize system-wide utility functions, e.g. the availability of empty cars when a passenger arrives. We establish both process-level and steady-state convergence of the queueing network to a fluid limit in a large market regime where demand for rides and supply of cars tend to infinity, and use this limit to study a fluid-based optimization problem. We prove that the optimal network utility obtained from the fluid-based optimization is an upper bound on the utility in the finite car system for any routing policy, both static and dynamic, under which the closed queueing network has a stationary distribution. This upper bound is achieved asymptotically under the fluid-based optimal routing policy. Simulation results with real-world data released by Didi Chuxing demonstrate the benefit of using the fluid-based optimal routing policy compared to various other policies.

This paper studies the steady-state properties of the Join the Shortest Queue model in the Halfin-Whitt regime. We focus on the process tracking the number of idle servers, and the number of servers with non-empty buffers. Recently, [10] proved that a scaled version of this process converges, over finite time intervals, to a two-dimensional diffusion limit as the number of servers goes to infinity. In this paper we prove that the diffusion limit is exponentially ergodic, and that the diffusion scaled sequence of the steady-state number of idle servers and non-empty buffers is tight. Combined with the process-level convergence proved in [10], our results imply convergence of steady-state distributions. The methodology used is the generator expansion framework based on Stein's method, also referred to as the drift-based fluid limit Lyapunov function approach in [36]. One technical contribution to the framework is to show how it can be used as a general tool to establish exponential ergodicity. arXiv:1801.05121v2 [math.PR]

In the seminal paper of Gamarnik and Zeevi [17], the authors justify the steady-state diffusion approximation of a generalized Jackson network (GJN) in heavy traffic. Their approach involves the so-called limit interchange argument, which has since become a popular tool employed by many others who study diffusion approximations. In this paper we illustrate a novel approach by using it to justify the steady-state approximation of a GJN in heavy traffic. Our approach involves working directly with the basic adjoint relationship (BAR), an integral equation that characterizes the stationary distribution of a Markov process. As we will show, the BAR approach is a more natural choice than the limit interchange approach for justifying steadystate approximations, and can potentially be applied to the study of other stochastic processing networks such as multiclass queueing networks. Introduction.This paper considers open single-class queueing networks that have d service stations. Each station has a single server operating under the first-in-first-out (FIFO) service discipline. Upon completing service at a particular station, customers are either routed to another station, or exit the network. There is a single class of customers at each station, meaning that all customers are homogenous in terms of service times and routing. A customer entering the network will exit in finite time with probability one, hence the term open network. For each station, the external interarrival times (possibly null), service times, and routing decisions are assumed to follow three separate i.i.d. sequences of random variables; the three sequences are assumed to be independent. Furthermore, the interarrival times, service times and routing decisions are assumed to be independent between different stations. Such a network is hereafter referred to as a generalized Jackson network (GJN).