For the Weibull distribution, comparisons of the system total overheads between two-level incremental checkpoint recovery scheme and two-level checkpoint recovery scheme.

<p>(a) <i>u = O_i/O_m</i> = 10%; (b) <i>u = O_i/O_m</i> = 12.5%; (c) <i>u = O_i/O_m</i>  = 15%; (d) <i>u = O_i/O_m</i> = 17.5%.</p

The relationship between optimal number of <i>i</i>-checkpoints and <i>p</i>_n under different <i>u</i> = <i>O_i/O_m</i>.

<p>The relationship between optimal number of <i>i</i>-checkpoints and <i>p</i>_n under different <i>u</i> = <i>O_i/O_m</i>.</p

For exponential distribution, comparisons of the system total overheads between two-level incremental checkpoint recovery scheme and two-level checkpoint recovery scheme.

<p>(a) <i>u = O_i/O_m</i> = 10%; (b) <i>u = O_i/O_m</i> = 30%; (c) <i>u = O_i/O_m</i>  = 33%; (d) <i>u = O_i/O_m</i> = 40%.</p

Notation.

<p>Notation.</p

The comparison results between two-level incremental checkpoint recovery scheme and two-level checkpoint recovery scheme for the Weibull distribution.

<p>(a) The relationship between the total overheads of setting checkpoints and the number of failures; (b) The relationship between the total re-computing time and the number of failures; (c) The relationship between the total overheads of recovering from the failures and the number of failures; (d) The relationship between the system total overheads and the number of failures; Note: Time Unit depends on parameters in practical implementation, such as the practical value of <i>O_m</i>, so it is not given here, which is similar to the case in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0104591#pone-0104591-g006" target="_blank">Figs. 6</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0104591#pone-0104591-g010" target="_blank">10</a>.</p

The comparison results between two-level incremental checkpoint recovery scheme and two-level checkpoint recovery scheme for exponential distribution.

<p>(a) The relationship between the total overheads of setting checkpoints and the completion time; (b) The relationship between the total re-computing time and the completion time; (c) The relationship between the total overheads of recovering from the failures and the completion time; (d) The relationship between the system total overheads and the completion time.</p

The two-level incremental checkpoint model.

<p>The two-level incremental checkpoint model.</p

The relationship between optimal number of <i>i</i>-checkpoints and <i>u</i> under different <i>p_n</i>.

<p>The relationship between optimal number of <i>i</i>-checkpoints and <i>u</i> under different <i>p_n</i>.</p

The comparison results between two-level incremental checkpoint recovery scheme and two-level checkpoint recovery scheme for exponential distribution.

<p>(a) The relationship between the total overheads of setting checkpoints and the number of failures; (b) The relationship between the total re-computing time and the number of failures; (c) The relationship between the total overheads of recovering from the failures and the number of failures; (d) The relationship between the system total overheads and the number of failures.</p

The relationship between <i>T</i>_re-compute2 and the checkpoint interval.

<p>The relationship between <i>T</i>_re-compute2 and the checkpoint interval.</p