An unexpected limit of expected values

Let t⩾0t⩾0. Select numbers randomly from the interval [0,1][0,1] until the sum is greater than t  . Let α(t)α(t) be the expected number of selections. We prove that α(t)=etα(t)=et for 0⩽t⩽10⩽t⩽1. Moreover, limt→+∞α(t)-2t=23. This limit is a special case of our asymptotic results for solutions of the delay differential equation f′(t)=f(t)-f(t-1)f′(t)=f(t)-f(t-1) for t>1t>1. We also consider four other solutions of this equation that are related to the above selection process.