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The ratio of the extreme to the sum in a random sequence. (English) Zbl 1164.60021
The ratio $$R_n=X_{(n)}/S_n$$ is considered, where $$S_n=\sum_{i=1}^n X_i$$, $$X_{(n)}=\max_{1\leq i\leq n} X_i$$, $$X_i$$ are i.i.d. random variables. It is shown that $${\mathbf E}R_n={{\mathbf E} X{(n)}\over {\mathbf E}S_n}(1+o(1))$$ as $$n\to\infty$$ if $${\mathbf E}X_i^2<\infty$$ or if the survival function of $$X_i$$ is regularly varying with the index of variation less then -1. The proof is based on an integral representation for $${\mathbf E}R_n$$. The results are applied to a multiprocessor scheduling asymptotical analysis.

##### MSC:
 60F99 Limit theorems in probability theory 68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
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##### References:
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