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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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