Fork–join and redundancy systems with heavy-tailed job sizes

Abstract

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 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-nu the tail index of the response time for the c.o.s. variant of redundancy-d equals -min {d_{mathrm {cap}}(nu -1),nu }, where d_{mathrm {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_{mathrm {cap}} < frac{nu }{nu -1} the waiting time component is dominant, whereas for d_{mathrm {cap}} > frac{nu }{nu -1} the job size component is dominant. Thus, having d = lceil min {frac{nu }{nu -1},N-k} rceil 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_{mathrm {F}},n_{mathrm {J}}) model, the tail index of the response time, under some assumptions on the load, equals 1-nu and 1-(n_{mathrm {F}}+1-n_{mathrm {J}})nu , for identical and i.i.d. replicas, respectively; here, the waiting time component is always dominant.