A unifying framework for redundancy models: product form and impact of independence assumption

Urtzi Ayesta (1, 2) , Tejas Bodas (3) , Ina Maria Maaike Verloop (2)

1 Ikerbasque - Basque Foundation for Science
2 IRIT - Institut de recherche en informatique de Toulouse
3 LAAS-SARA - Équipe Services et Architectures pour Réseaux Avancés

In this paper, we present a unifying analysis for redundancy systems with cancel-on-start (c.o.s.) and cancel-on-complete (c.o.c.) with exponentially distributed service requirements. With c.o.s. (c.o.c.) all redundant copies are removed as soon as one of the copies starts (completes) service. As a consequence, c.o.s. does not waste any computing resources, as opposed to c.o.c.. We show that the c.o.s. model is equivalent to a queue-ing system with multi-type jobs and servers, which was analyzed in [1], and show that c.o.c. (under the assumption of i.i.d. copies) can be analyzed by a generalization of [1] where state-dependent departure rates are permitted. This allows us to show that the stationary distribution for both the c.o.c. and c.o.s. models have a product form. We give a detailed first-time analysis for c.o.s and derive a closed form expression for important metrics like mean number of jobs in the system, and probability of waiting. Comparing the performance of c.o.s. with that of c.o.c. gives the unexpected result (since c.o.s. does not waste any resources), that c.o.s. is worse in terms of mean number of jobs. The latter illustrates that the i.i.d. assumption (together with exponentially distributed requirements) might lead to conclusions that are qualitatively different from that observed in practice. We also show that the c.o.s. model is equivalent to Join-Shortest-Work queue with redundancy (JSW(d)). In the latter, an incoming job is dispatched to the server with smallest workload among d randomly chosen ones. Thus, all our results apply mutatis-mutandis to JSW(d).

Domaines: Performance et fiabilité [cs.PF]; Recherche opérationnelle [math.OC]; Probabilités [math.PR]

Urtzi Ayesta, Tejas Bodas, Ina Maria Maaike Verloop. A unifying framework for redundancy models: product form and impact of independence assumption. 2018. ⟨hal-01713937v1⟩