Fluid approximation and its convergence Rate for GI/G/1 queue with vacations

A GI/G/1 queue with vacations is considered in this paper. We develop an approximating technique on max function of independent and identically distributed (i.i.d.) random variables, that is max{eta (i) , 1 a parts per thousand currency sign i a parts per thousand currency sign n}. The approximating technique is used to obtain the fluid approximation for the queue length, workload and busy time processes. Furthermore, under uniform topology, if the scaled arrival process and the scaled service process converge to the corresponding fluid processes with an exponential rate, we prove by the approximating technique that the scaled processes characterizing the queue converge to the corresponding fluid limits with the exponential rate only for large N. Here the scaled processes include the queue length process, workload process and busy time process.