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On the number of failures until a first success in $$n$$ Bernoulli trials containing $$m=1,\dots,n$$ successes. (English) Zbl 1038.90025
Summary: We describe a random variable $$D_{n,m}$$, $$n\geq m\geq 1$$, as the number of failures until the first success in a sequence of $$n$$ Bernoulli trials containing exactly $$m$$ successes, for which all possible sequences containing $$m$$ successes and $$n-m$$ failures are equally likely. We give the probability density function, the expectation, and the variance of $$D_{n,m}$$. We define a random variable $$D_n$$, $$n\geq 1$$, to be the mean of $$D_{n,1}, \dots, D_{n,n}$$. We show that $$E[D_n]$$ is a monotonically increasing function of $$n$$ and is bounded by $$\ln n$$. We apply these results to a practical application involving a video-on-demand system with interleaved movie files and a delayed start protocol for keeping a balanced workload.
##### MSC:
 90B25 Reliability, availability, maintenance, inspection in operations research