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Abstract. In this paper, a single-server queue with negative customers is considered. The arrival of a negative customer will remove one positive customer that is being served, if any is present. An alternative approach will be introduced to derive a set of equations which will be solved to obtain the stationary queue length distribution. We assume that the service time distribution tends to a constant asymptotic rate when time t goes to infinity. This assumption will allow for finding the stationary queue length of queueing systems with non-exponential service time distributions. Numerical examples for gamma distributed service time with fractional value of shape parameter will be presented in which the steady-state distribution of queue length with such service time distributions may not be easily computed by most of the existing analytical methods.

The vector t 1 m  of N age-specific mortality rates in year t+1 is modeled to be dependent on the vector t m in the present year t and l-1 other vectors before year t via a conditional distribution which is derived from an N(l+1)-dimensional power-normal distribution. The marginal distribution of the mortality rate at age x is computed from the conditional distribution. The prediction interval with end points given by the 100(α/2) and 100(1-α/2) percentage points of the marginal distribution is used to predict the mortality rate at age x in year t+1. The similar idea is used to find a prediction interval for the mortality rate at age x in year t+d where d >1. The United States mortality data from 1933 to 2000 are used to estimate the N(l+1)-dimensional power-normal distribution. The prediction intervals based on the distributions when N=19, l=1 and 2 are found to have good ability of covering the observed future mortality rates in the years from 2001 to 2010. The length of the prediction interval may be shortened by choosing small values of N and using more recent historical data. Mathematics Subject Classification: 62M10, 62P05

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