and $d$ values satisfy . The overall file size is denoted by $\mathcal{M}$. The storage size for each node is $\alpha$, and during the repair process, the newcomer requests $\beta$ amount of traffic from each of the helpers. The total repair-bandwidth is thus $\gamma\stackrel{\Delta}{=}d\beta$. We use the notation $(\cdot)^+$ to mean $(x)^+=\max(x,0)$. We also define the indicator function as follows $$\begin{aligned}
1_{\{B\}}=
\begin{cases}1, \mbox{ if condition $B$ is true} \\
0, \mbox{ otherwise}.
\end{cases}\end{aligned}$$

In this work, we consider exclusively single failure at any given time. The setting of multiple simultaneous failed nodes [@el2010fractional; @shum2013cooperative; @kamath2014codes] is beyond the scope of this work. We consider the multiple failures scenario in a separate work, see [@arxiv_multiple].

Dynamic Versus Stationary Helper Selection Schemes {#sec:hs}
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In general, the helper selection at current time $\tau$ can depend on the history of the failure patterns and the helper choices for all the previous time slots 1 to $(\tau-1)$. We call such a general helper selection scheme [*the dynamic helper selection (DHS)*]{}. Mathematically, the helper set decision at time $\tau$ can be written in function form as $D_\tau(\{F_l\}_{l=1}^{\tau})$ that takes $F_l$, the failed node at time $l$, for all $l=1$ to $\tau$ and returns the 