Model

Retrying tasks, not jobs

Job-level success is not something you can buy from reliable parts; it comes from retrying at the task level. This works only if task failures are independent.

P (job succeeds) = (1 - p)^{n} \approx e^{- n p} \to (1 - p^{k})^{n} with k attempts per task

$n$: Tasks; tasks in the job
$p$: Task failure; failure probability of one task attempt
$k$: Attempts; attempts allowed per task

Recurring principle: Independence is assumed

How it behaves

One input on the horizontal axis, the others held fixed.

no retry (k = 1)k = 2k = 3

Data table

Chance the whole job succeeds, by tasks (p = 10⁻³ per attempt)
Tasks n	series	P(job succeeds)
10	no retry (k = 1)	0.99
11.9	no retry (k = 1)	0.988
14.1	no retry (k = 1)	0.986
16.8	no retry (k = 1)	0.983
20	no retry (k = 1)	0.98
23.7	no retry (k = 1)	0.977
28.2	no retry (k = 1)	0.972
33.5	no retry (k = 1)	0.967
39.8	no retry (k = 1)	0.961
47.3	no retry (k = 1)	0.954
56.2	no retry (k = 1)	0.945
66.8	no retry (k = 1)	0.935
79.4	no retry (k = 1)	0.924
94.4	no retry (k = 1)	0.91
112	no retry (k = 1)	0.894
133	no retry (k = 1)	0.875
158	no retry (k = 1)	0.853
188	no retry (k = 1)	0.828
224	no retry (k = 1)	0.799
266	no retry (k = 1)	0.766
316	no retry (k = 1)	0.729
376	no retry (k = 1)	0.687
447	no retry (k = 1)	0.64
531	no retry (k = 1)	0.588
631	no retry (k = 1)	0.532
750	no retry (k = 1)	0.472
891	no retry (k = 1)	0.41
1.06K	no retry (k = 1)	0.347
1.26K	no retry (k = 1)	0.284
1.5K	no retry (k = 1)	0.224
1.78K	no retry (k = 1)	0.169
2.11K	no retry (k = 1)	0.121
2.51K	no retry (k = 1)	0.081
2.99K	no retry (k = 1)	0.05
3.55K	no retry (k = 1)	0.029
4.22K	no retry (k = 1)	0.015
5.01K	no retry (k = 1)	0.007
5.96K	no retry (k = 1)	0.003
7.08K	no retry (k = 1)	0.001
8.41K	no retry (k = 1)	0
10K	no retry (k = 1)	0
11.9K	no retry (k = 1)	0
14.1K	no retry (k = 1)	0
16.8K	no retry (k = 1)	0
20K	no retry (k = 1)	0
23.7K	no retry (k = 1)	0
28.2K	no retry (k = 1)	0
33.5K	no retry (k = 1)	0
39.8K	no retry (k = 1)	0
47.3K	no retry (k = 1)	0
56.2K	no retry (k = 1)	0
66.8K	no retry (k = 1)	0
79.4K	no retry (k = 1)	0
94.4K	no retry (k = 1)	0
112K	no retry (k = 1)	0
133K	no retry (k = 1)	0
158K	no retry (k = 1)	0
188K	no retry (k = 1)	0
224K	no retry (k = 1)	0
266K	no retry (k = 1)	0
316K	no retry (k = 1)	0
376K	no retry (k = 1)	0
447K	no retry (k = 1)	0
531K	no retry (k = 1)	0
631K	no retry (k = 1)	0
750K	no retry (k = 1)	0
891K	no retry (k = 1)	0
1.06M	no retry (k = 1)	0
1.26M	no retry (k = 1)	0
1.5M	no retry (k = 1)	0
1.78M	no retry (k = 1)	0
2.11M	no retry (k = 1)	0
2.51M	no retry (k = 1)	0
2.99M	no retry (k = 1)	0
3.55M	no retry (k = 1)	0
4.22M	no retry (k = 1)	0
5.01M	no retry (k = 1)	0
5.96M	no retry (k = 1)	0
7.08M	no retry (k = 1)	0
8.41M	no retry (k = 1)	0
10M	no retry (k = 1)	0
10	k = 2	1
11.9	k = 2	1
14.1	k = 2	1
16.8	k = 2	1
20	k = 2	1
23.7	k = 2	1
28.2	k = 2	1
33.5	k = 2	1
39.8	k = 2	1
47.3	k = 2	1
56.2	k = 2	1
66.8	k = 2	1
79.4	k = 2	1
94.4	k = 2	1
112	k = 2	1
133	k = 2	1
158	k = 2	1
188	k = 2	1
224	k = 2	1
266	k = 2	1
316	k = 2	1
376	k = 2	1
447	k = 2	1
531	k = 2	0.999
631	k = 2	0.999
750	k = 2	0.999
891	k = 2	0.999
1.06K	k = 2	0.999
1.26K	k = 2	0.999
1.5K	k = 2	0.999
1.78K	k = 2	0.998
2.11K	k = 2	0.998
2.51K	k = 2	0.997
2.99K	k = 2	0.997
3.55K	k = 2	0.996
4.22K	k = 2	0.996
5.01K	k = 2	0.995
5.96K	k = 2	0.994
7.08K	k = 2	0.993
8.41K	k = 2	0.992
10K	k = 2	0.99
11.9K	k = 2	0.988
14.1K	k = 2	0.986
16.8K	k = 2	0.983
20K	k = 2	0.98
23.7K	k = 2	0.977
28.2K	k = 2	0.972
33.5K	k = 2	0.967
39.8K	k = 2	0.961
47.3K	k = 2	0.954
56.2K	k = 2	0.945
66.8K	k = 2	0.935
79.4K	k = 2	0.924
94.4K	k = 2	0.91
112K	k = 2	0.894
133K	k = 2	0.875
158K	k = 2	0.853
188K	k = 2	0.828
224K	k = 2	0.799
266K	k = 2	0.766
316K	k = 2	0.729
376K	k = 2	0.687
447K	k = 2	0.64
531K	k = 2	0.588
631K	k = 2	0.532
750K	k = 2	0.472
891K	k = 2	0.41
1.06M	k = 2	0.347
1.26M	k = 2	0.284
1.5M	k = 2	0.224
1.78M	k = 2	0.169
2.11M	k = 2	0.121
2.51M	k = 2	0.081
2.99M	k = 2	0.051
3.55M	k = 2	0.029
4.22M	k = 2	0.015
5.01M	k = 2	0.007
5.96M	k = 2	0.003
7.08M	k = 2	0.001
8.41M	k = 2	0
10M	k = 2	0
10	k = 3	1
11.9	k = 3	1
14.1	k = 3	1
16.8	k = 3	1
20	k = 3	1
23.7	k = 3	1
28.2	k = 3	1
33.5	k = 3	1
39.8	k = 3	1
47.3	k = 3	1
56.2	k = 3	1
66.8	k = 3	1
79.4	k = 3	1
94.4	k = 3	1
112	k = 3	1
133	k = 3	1
158	k = 3	1
188	k = 3	1
224	k = 3	1
266	k = 3	1
316	k = 3	1
376	k = 3	1
447	k = 3	1
531	k = 3	1
631	k = 3	1
750	k = 3	1
891	k = 3	1
1.06K	k = 3	1
1.26K	k = 3	1
1.5K	k = 3	1
1.78K	k = 3	1
2.11K	k = 3	1
2.51K	k = 3	1
2.99K	k = 3	1
3.55K	k = 3	1
4.22K	k = 3	1
5.01K	k = 3	1
5.96K	k = 3	1
7.08K	k = 3	1
8.41K	k = 3	1
10K	k = 3	1
11.9K	k = 3	1
14.1K	k = 3	1
16.8K	k = 3	1
20K	k = 3	1
23.7K	k = 3	1
28.2K	k = 3	1
33.5K	k = 3	1
39.8K	k = 3	1
47.3K	k = 3	1
56.2K	k = 3	1
66.8K	k = 3	1
79.4K	k = 3	1
94.4K	k = 3	1
112K	k = 3	1
133K	k = 3	1
158K	k = 3	1
188K	k = 3	1
224K	k = 3	1
266K	k = 3	1
316K	k = 3	1
376K	k = 3	1
447K	k = 3	1
531K	k = 3	0.999
631K	k = 3	0.999
750K	k = 3	0.999
891K	k = 3	0.999
1.06M	k = 3	0.999
1.26M	k = 3	0.999
1.5M	k = 3	0.999
1.78M	k = 3	0.998
2.11M	k = 3	0.998
2.51M	k = 3	0.997
2.99M	k = 3	0.997
3.55M	k = 3	0.996
4.22M	k = 3	0.996
5.01M	k = 3	0.995
5.96M	k = 3	0.994
7.08M	k = 3	0.993
8.41M	k = 3	0.992
10M	k = 3	0.99

Worked examples, recomputed

Every number the mechanisms quote for this relationship is recomputed from it, so none of them disagrees with it.

Where it appears

Each mechanism states it in its own names. The general symbol each name stands for is shown with it.

Fault Tolerance & Delivery Semantics
What is the cheapest thing that can be safely done twice?
$(1 - p)^{n} \to (1 - p^{k})^{n}$
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