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

A strong form of the law of the iterated logarithm (LIL) is established for a two-stage tandem queue. The concerned LIL, with a stronger mathematical form, is a later generalization of Lévy's LIL, and it quantifies the magnitude of asymptotic stochastic fluctuations of stochastic processes compensated by their deterministic fluid limits. The LILs are established in twelve cases covering three regimes: the underloaded, critically loaded and overloaded divided by traffic intensity of the two stages, for five processes: the queue length, workload, busy, idle and departure processes. All the LILs are expressed into some simple analytic functions of the primitive data: the first and second moments of the interarrival and service times. The proofs are based on a fluid approximation and a strong approximation, which approximate discrete performance processes with their expected mean values and reflected Brownian motions, respectively.