We investigate the stability condition of redundancy- @math multi-server systems. Each server has its own queue and implements popular scheduling disciplines such as First-Come-First-Serve (FCFS), Processor Sharing (PS), and Random Order of Service (ROS). New jobs arrive according to a Poisson process and copies of each job are sent to @math servers chosen uniformly at random. The service times of jobs are assumed to be exponentially distributed. A job departs as soon as one of its copies finishes service. Under the assumption that all @math copies are i.i.d., we show that for PS and ROS (for FCFS it is already known) sending redundant copies does not reduce the stability region. Under the assumption that the @math copies are identical, we show that (i) ROS does not reduce the stability region, (ii) FCFS reduces the stability region, which can be characterized through an associated saturated system, and (iii) PS severely reduces the stability region, which coincides with the system where all copies have to be served. The proofs are based on careful characterizations of scaling limits of the underlying stochastic process. Through simulations we obtain interesting insights on the system's performance for non-exponential service time distributions and heterogeneous server speeds.
In a recent study, @cite_18 , the impact of the scheduling policy employed in the server is investigated for i.i.d. copies and exponential service. The authors show that for FCFS the performance might not improve as the number of redundant copies increases, while for other policies as proposed in that paper, redundancy does improve the performance.
@cite_18
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Abstract Redundancy is an increasingly popular technique for reducing response times in computer systems, and there is a growing body of theoretical work seeking to analyze performance in systems with redundancy. The idea is to dispatch a job to multiple servers at the same time and wait for the first copy to complete service. Redundancy can help reduce response time because redundant jobs get to experience the shortest of multiple queueing times and potentially of multiple service times—but it can hurt jobs that are not redundant and must wait behind the redundant jobs’ extra copies. Thus in designing redundancy systems it is critical to find ways to leverage the potential benefits without incurring the potential costs. Scheduling represents one tool for maximizing the benefits of redundancy. In this paper we study three scheduling policies: First-Come First-Served (FCFS), Least Redundant First (LRF, under which less-redundant jobs have priority over more-redundant jobs), and Primaries First (PF, under which each job designates a “primary” copy, and all other copies have lowest priority). Our goal for each of these policies is to understand the marginal impact of redundancy: how much redundancy is needed to get the biggest benefit? We study this question analytically for LRF and FCFS, and via simulation for all three policies. One of our primary contributions is a surprisingly intricate proof that mean response time is convex as well as decreasing as the proportion of jobs that are redundant increases under LRF for exponential services. While response time under PF is also decreasing and appears to be convex as well, we find that, surprisingly, FCFS may be neither decreasing nor convex, depending on the parameter values. Thus, the scheduling policy is key in determining both whether redundancy helps and the marginal effects of adding more redundancy to the system.