Asymptotic variability analysis for a two-stage tandem queue, part I: The functional law of the iterated logarithm

Abstract

A functional law of the iterated logarithm (LIL) and its corresponding LIL are established for a two-stage tandem queue. The functional LIL and the LIL limits quantify the magnitude of asymptotic stochastic fluctuations of stochastic processes compensated by their deterministic fluid limits in two forms: the functional and numerical, respectively. The earliest functional LIL and LIL concerned are both developed for Brownian motion by Volker Strassen and Lévy respectively. We establish the functional LILs and their corresponding LILs in twelve cases covering three regimes divided by the traffic intensity: the underloaded, critically loaded and overloaded, for five processes: the queue length, workload, busy, idle and departure processes. By the primitive data of the first and second moments of the interarrival and service times, all the functional LILs are expressed into some compact sets of continuous functions and all the corresponding LILs are some analytic functions. The proofs are based on the fluid approximation and the strong approximation of the queueing system, with the fluid approximation characterizing the expected values of the performance functions and the strong approximation approximating discrete performance processes with reflected Brownian motions. A companion paper (“Asymptotic variability analysis for a two-stage tandem queue, part II: The law of the iterated logarithm”) develops another different version of LIL, which is a later generalization of Lévy's LIL, and also quantifies the magnitude of the asymptotic stochastic fluctuations of the performance functions.