2017/04/27 by Anton Braverman, Braverman, Anton
Business, Management and Accounting · Mathematics · #Advanced Queuing Theory Analysis #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #math.PR
paper · pdf · doi:10.48550/arxiv.1704.08398
PhD Thesis
arxiv created 2017/04/27 · openalex publication_date 2017/04/27 · arxiv updated 2017/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Diffusion approximations have been a popular tool for performance analysis in queueing theory, with the main reason being tractability and computational efficiency. This dissertation is concerned with establishing theoretical guarantees on the performance of steady-state diffusion approximations of queueing systems. We develop a modular framework based on Stein's method that allows us to establish error bounds, or convergence rates, for the approximations. We apply this framework three queueing systems: the Erlang-C, Erlang-A, and M/Ph/n+M systems. The former two systems are simpler and allow us to showcase the full potential of the framework. Namely, 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. For the Erlang-C model, we also show that a diffusion approximation with state-dependent diffusion coefficient can achieve a rate of convergence of 1/R, which is an order of magnitude faster when compared to approximations with constant diffusion coefficients.