Many processes must complete in the presence of failures. Different systems respond to task failure in different ways. The system may resume a failed task from the failure point (or a saved checkpoint shortly before the failure point), it may give up on the task and select a replacement task from the ready queue, or it may restart the task. The behavior of systems under the first two scenarios is well documented, but the third (RESTART) has resisted detailed analysis. In this paper we derive tight asymptotic relations between the distribution of task times without failures to the total time when including failures, for any failure distribution. In particular, we show that if the task time distribution has an unbounded support then the total time distribution H is always heavy-tailed. Asymptotic expressions are given for the tail of H in various scenarios. The key ingredients of the analysis are the Cramér-Lundberg asymptotics for geometric sums and integral asymptotics, that in some cases are obtained via Tauberian theorems and in some cases by bare-hand calculations.

For many systems, failure is so common that the design choice of how to deal with it may have a significant impact on the performance of the system. There are many specific and distinct failure recovery schemes, but they can be grouped into three broad classes: RESUME , also referred to as preemptive resume (prs), or check-pointing; REPLACE , also referred to as preemptive repeat different (prd); and RESTART , also referred to as preemptive repeat identical (pri). The following describes the three recovery schemes: (1) RESUME: when a task is fails, it knows exactly where it stops, and can continue from that point when allowed to resume; (2) REPLACE: given a task fails, then when it begins processing again, it starts with a brand new task sampled from the same task time distribution; and, (3) RESTART: When a task fails, it loses all that it had acquired to up to that point and must start anew when upon continuing later. This is distinctly different from (2) since the task must run at least as long as it did before it failed, whereas a new sample, selected at random, might run for a shorter or longer time.

For some computing systems, failure is rare enough that it can be ignored. In other systems, failure is so common that how to handle it can have a significant impact on the performance of the system. There are many different recovery schemes for tasks, however, they can be classified into three broad categories: 1) Resume: when a task fails, it knows exactly where it stops and can continue at that point when allowed to resume (i.e., preemptive resume - prs); 2) Replace : when a task fails, then later when the processor continues, it begins with a brand new task (i.e., preemptive repeat different prd); and, 3) Restart: when a task fails it loses all work done to that point and must start anew upon continuing later (i.e., preemptive repeat identical - pri ).In this paper, assuming a computing system is unreliable, we discuss how heavy-tail (hereafter referred to as power-tail - PT) distributions can appear in a job's task stream given the Restart recovery procedure. This is an important consideration since it is known that power-tails can lead to unstable systems [4], We then demonstrate how to obtain performance and dependablity measures for a class of computing systems comprised of P unreliable processors and a finite number of tasks N given the above recovery procedures.

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