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
2912554127 2149912409 2971432649 2757172027
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. Load sharing is a technique to improve the performance of distributed systems by distributing the system workload from heavily loaded nodes, where service is poor, to lightly loaded nodes in the system. Previous studies have considered two adaptive load sharing policies: sender-initiated and receiver-initiated. In the sender-initiated policy, a heavily loaded node attempts to transfer work to a lightly loaded node and in the receiver-initiated policy a lightly loaded node attempts to get work from a heavily loaded node. Almost all the previous studies assumed the first-come first-served node scheduling policy; furthermore, analysis and simulations in these studies have been done under the assumption that the job service times are exponentially distributed and the job arrivals form a Poisson process (i.e., job inter-arrival times are exponentially distributed). The goal of this paper is to fill the void in the existing literature. We study the impact of these assumptions on the performance of the sender-initiated and receiver initiated policies. We consider three node scheduling policies-first-come first-served (FCFS), shortest job first (SJF), and round robin (RR) policies. Furthermore, we also look at the impact of variance in the inter-arrival times and in the job service times. Our results show that: (i) When non-preemptive node scheduling policies (FCFS and SJF) are used, the receiver-initiated policy is (substantially) more sensitive to variance in inter-arrival times than the sender-initiated policies and the sender-initiated policies are relatively more sensitive to the variance in job service times; (ii) When the preemptive node scheduling policy (RR) is used, the sender-initiated policy provides a better performance than the receiver-initiated policy. Abstract 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 queueing system with multi-type jobs and servers, which was analyzed in , (2012), and show that c . o . c . (under the assumption of i.i.d. copies) can be analyzed by a generalization of , (2012) 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 has 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. We also note that the c . o . s . model is equivalent to Join-Shortest-Work queue with power of d (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 ). Comparing the performance of c . o . s . with that of c . o . c . with i.i.d. copies gives the unexpected conclusion (since c . o . s . does not waste any resources) that c . o . s . is worse in terms of mean number of jobs. As part of ancillary results, we illustrate that this is primarily due to the assumption of i.i.d. copies in case of c . o . c . (together with exponentially distributed requirements) and that such assumptions might lead to conclusions that are qualitatively different from that observed in practice. Recent computer systems research has proposed using redundant requests to reduce latency. The idea is to replicate a request so that it joins the queue at multiple servers. The request is considered complete as soon as any one of its copies completes. Redundancy allows us to overcome server-side variability–the fact that a server might be temporarily slow due to factors such as background load, network interrupts, and garbage collection to reduce response time. In the past few years, queueing theorists have begun to study redundancy, first via approximations, and, more recently, via exact analysis. Unfortunately, for analytical tractability, most existing theoretical analysis has assumed an Independent Runtimes (IR) model, wherein the replicas of a job each experience independent runtimes (service times) at different servers. The IR model is unrealistic and has led to theoretical results that can be at odds with computer systems implementation results. This paper introduces a much more realistic model of redundancy. Our model decouples the inherent job size ( @math ) from the server-side slowdown ( @math ), where we track both @math and @math for each job. Analysis within the @math model is, of course, much more difficult. Nevertheless, we design a dispatching policy, Redundant-to-Idle-Queue, which is both analytically tractable within the @math model and has provably excellent performance.