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A necessary and sufficient condition for the tail of an infinitely divisible distribution on the real line to be estimated by the tail of its Lévy measure is found. The lower limit and the upper limit of the ratio of the right tail µ(r) of an infinitely divisible distribution µ to the right tail ν(r) of its Lévy measure ν as r → ∞ are estimated from above and below by reviving Teugels's classical method. The exponential class and the dominated varying class are studied in detail.

An M/G/1 retrial queue with batch arrivals is studied. The queue length K μ is decomposed into the sum of two independent random variables. One corresponds to the queue length K ∞ of a standard M/G/1 batch arrival queue, and another is compound-Poisson distributed. In the case of the distribution of the batch size being light-tailed, the tail asymptotics of K μ are investigated through the relation between K ∞ and its service times.

Let Rd be the d-dimensional Euclidean space where each point is expressed by a column vector. Let | x | and ‹x, y› denote the norm and the inner product in Rd. Let Q = (Qjk) be a real d × d-matrix of which all eigenvalues have positive real parts. Let X be a process of Ornstein-Uhlenbeck type (OU type process) on Rd associated with a Levy process {Z: t ≥ 0} and the matrix Q. Main purpose of this paper is to give a recurrence-transience criterion for the process X when Q is a Jordan cell matrix and to compare it with the case when Q is diagonalizable. Here by a Levy process we mean a stochastically continuous process with stationary independent increments, starting at 0.

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