Paper Title
Mean Access Over A Threshold for Number of Customers in Discrete-Time Queueing System

Abstract
This paper considers a discrete-time queueing system where customers arrive at a facility according to a batch geometric process with customer service times assumed to be independent and identically distributed. The random variable N_k is defined as the number of customers in the queueing system at the k-th slot boundary. Let τ(T) be the slot boundary of first entry into the interval (T,∞), T>0, of the number of customers. The distribution of the excess L(T) at the first-passage time over the threshold T for the number of customers is defined as L(T)≡N_τ(T) -T. That is, P(L(T)=l)≡P(N_τ(T) -T=l). Ghost and Resnick [1] has investigated some theoretical and practical aspects of the use of the mean excess plot, a widely used tool in the study of risk, insurance and extreme values. Mijatovic and Pistorius [2] has established the existence of the weak limit of undershoots and overshoots of the reflected process of a Levy process as a threshold level tends to infinity and has provided explicit formulas for their joint cumulative distribution functions. They have applied their results to analyze the behavior of the classical continuous-time M/G/1 queueing system at buffer-overflow, both in a stable and unstable case. In M/G/1 queueing model with Weibull service time distribution, Bae and Park [3] has proposed the approximation of the distribution of the excess for the workload over a threshold at the moment where the workload process exceeds the threshold. In this paper we investigate the mean excessE(L(T)) over the thresholdT for the number of customers in adiscrete-time queueing system. This paper investigates the mean excess over a threshold for the number of customers in the queueing system. Keywords - Queueing System, Batch Geometric Process, Number of Customers, Overshoot, Mean Excess.