Transform-free analysis of the GI/G/1/K queue through the decomposed Littleʹs formula

In this paper, we consider the steady-state queue length distribution of the GI/G/1/K queue. As a result, we obtain transform-free expressions for the steady-state queue length distributions at an arrival, at a departure and at an arbitrary time, all in product forms. The results are obtained by what we call the decomposed Littleʹs formula, which applies the Littleʹs formula L=λW to the nth waiting position in the queue. Utilizing the results, we improve and generalize existing bounds on the difference between the time average and arrival (departure) average mean queue lengths, and propose a two-moment approximation for the queue length. To evaluate the approximation, we focus on the probability of customer loss and the mean queue length, which are of great practical importance. Our approximations turn out to be remarkably simple yet fairly good especially in heavy traffic.