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A non-exponential generalization of an inequality arising in queueing and insurance risk. (English) Zbl 0848.60082
Let \(\{X_i\}\) be a sequence of i.i.d. nonnegative random variables independent of \(\{Y_i\}\), which is also a sequence of i.i.d. nonnegative variables with \(E(Y) < E(X)\). If \(S_n = \sum^n_{i = 1} (Y_i - X_i)\), then \(\psi (x) : = P (\bigcup^\infty_{n = 1} \{S_n > x \}) \leq e^{- kx}, x \geq 0\), for some \(k > 0\), has an exponential bound. A generalization of this inequality is given.

MSC:
60K25 Queueing theory (aspects of probability theory)
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
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