Fork-join and redundancy systems with heavy-tailed job sizes
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We investigate the tail asymptotics of the response time distribution for the cancel-on-start (c.o.s.) and cancel-on-completion (c.o.c.) variants of redundancy-d scheduling and the fork-join model with heavy-tailed job sizes. We present bounds, which only differ in the pre-factor, for the tail probability of the response time in the case of the first-come first-served (FCFS) discipline. For the c.o.s. variant we restrict ourselves to redundancy-d scheduling, which is a special case of the fork-join model. In particular, for regularly varying job sizes with tail index -ν the tail index of the response time for the c.o.s. variant of redundancy-d equals -min{d_cap(ν-1),ν}, where d_cap = min{d,N-k}, N is the number of servers and k is the integer part of the load. This result indicates that for d_cap < ν/ν-1 the waiting time component is dominant, whereas for d_cap > ν/ν-1 the job size component is dominant. Thus, having d = ⌈min{ν/ν-1,N-k}⌉ replicas is sufficient to achieve the optimal asymptotic tail behavior of the response time. For the c.o.c. variant of the fork-join(n_F,n_J) model the tail index of the response time, under some assumptions on the load, equals 1-ν and 1-(n_F+1-n_J)ν, for identical and i.i.d. replicas, respectively; here the waiting time component is always dominant.
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