We model a selection process arising in certain storage problems. A sequence $(X_1,\ldots,X_n)$ of non-negative, independent and identically distributed random variables is given. $F(x)$ denotes the common distribution of the $X_i$'s. With $F(x)$ given we seek a decision rule for selecting a maximum number of $X_i$'s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constraint $c>0$, and (2) the $X_i$'s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected. we prove first that there exists such a rule of threshold type, i.e., the $i$th element inspected is accepted if and only if it is no larger than a threshold which depends only on $i$ and the sum of the elements already accepted. Next, we prove that if $F(x)\simAx^\alpha$ as $x\rightarrow 0$ for some $A$, $\alpha>0$, hen for fixed $c$ the expected number, $E_n(c)$, selected by an optimal threshold is characterized by \[ E_n(c) \sim \left[ A\left(\frac{\alpha+1} {\alpha} c\right)^\alpha n \right]^{1/(1+\alpha)} \quad\mathrm{as }n\rightarrow\infty. \] Asymptotics as $c\rightarrow\infty$ and $n\rightarrow\infty$ with $c/n$ help fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.