A Function Law Of The Iterated Logarithm For A Batch-Arrival Single Server Queue With Bernoul Feedback

In this paper, we firstly research the strong approximation for a batch-arrival single-server queue (GIB/Gl/1) with Bernoulli feedback, and secondly we develop a functional law of the iterated logarithm (FLIL) and its corresponding law of iterated logarithm (LIL) for the queue model on the basis of the strong approximation approach results.Strong approximation is an important approximation methods in stochastic process, the idea is to approximate the stochastic process to the Brownian motion network. For strong approximation of the batch-arrival single-server queue with Bernoulli feedback, we do not restrict the condition of traffic intensity and obtain strong approximation results of queue length, workload, idle time, busy time and departure processes by taking advantage of stochastic process limit theory, and this provides necessary preparations for getting the functional law of the iterated logarithm.The FLIL and LIL are used to describe the asymptotic behavior of stochastic processes. In the functional set version and the numerical version respectively, the FLIL and LIL quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensat-ed by their deterministic fluid limits. For the FLIL of the batch-arrival single-server queue with Bernoulli feedback, we obtain the FLIL results in three cases, underload (ρ<l), critically loaded (ρ=l) and overloaded (ρ>l) for five performance measures:queue length, workload, idle time, busy time and departure processes. We first relate the FLIL of performance functions to the FLIL of their strong approximations, and second we obtain the desired FLIL limits by analyzing the Brownian motions given by the strong approximation and its FLIL. The LIL can be seen as the accurate results of its FLIL and obtained directly from their corresponding supremums and infimums of the FLIL sets which consist of continuous function. We give some intuitive analysis for the results,and also give examples of LIL and plot corresponding graphics.