The stationary distribution of the redundancy-d model with random order of service

Redundancy has gained considerable attention as a dispatching paradigm that promises the potential for significant response time improvements, see [4, 6] and the references therein. The premise of redundancy is that upon a job's arrival, multiple copies of the job are dispatched to different servers. A job's class is defined by the set of servers to which its copies are dispatched, and there is a bipartite graph specifying the relationships between job classes and servers. A job is considered complete, and departs from the system, as soon as any one of its copies has completed service. The additional copies of a job are removed either (i) when the first copy enters service, known as the cancel-onstart (c.o.s.) model, or (ii) when the first copy completes service, known as the cancel-on-complete (c.o.c.) model. In both models, redundancy has the potential to significantly reduce response time by exploiting the variability of queue lengths and server capacities.