We consider a tandem queue with coupled processors, which is subject to global breakdowns. When the network is in the operating mode and both queues are non empty, the total service capacity is shared among the stations according to fixed proportions. When one of the stations becomes empty, the total service capacity is given to the non-empty station. Moreover, arrival rates depend on the state of the network. The system is described by a Markov modulated random walk in the quarter plane representing the number of jobs in the two stations and the state of the network. By applying the generating function approach, we first apply the power series approximation method to obtain power series expansions of the generating function of the stationary queue lengths for both network states. Then, we also provide a way to derive the generating function of the stationary queue lengths for both network states in terms of the solution of a Riemann-Hilbert boundary value problem. Numerical results are obtained to show insights in the system performance.
Most of the existing studies involving breakdowns have concentrated on models with a single job queue served by one or more processors, e.g. @cite_23 @cite_26 . Other related results where jobs from a single source are directed to one of several parallel queues, and where breakdowns result in the loss of jobs, or the direct transfer to other queues are given in @cite_17 @cite_12 @cite_32 . A network of two nodes subject to breakdowns and repairs was also analysed in @cite_10 . Approximate solutions to obtain performance measures, based on replacing an interruptable server with an uninterruptable but slower one, choosing the new service rate without affecting the overall service capacity was given in @cite_2 @cite_34 .
@cite_26 @cite_32 @cite_23 @cite_2 @cite_34 @cite_10 @cite_12 @cite_17
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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. In a Fork-Join (FJ) queueing system an upstream fork station splits incoming jobs into N tasks to be further processed by N parallel servers, each with its own queue; the response time of one job is determined, at a downstream join station, by the maximum of the corresponding tasks' response times. This queueing system is useful to the modelling of multi-service systems subject to synchronization constraints, such as MapReduce clusters or multipath routing. Despite their apparent simplicity, FJ systems are hard to analyze. This paper provides the first computable stochastic bounds on the waiting and response time distributions in FJ systems. We consider four practical scenarios by combining 1a) renewal and 1b) non-renewal arrivals, and 2a) non-blocking and 2b) blocking servers. In the case of non blocking servers we prove that delays scale as O(logN), a law which is known for first moments under renewal input only. In the case of blocking servers, we prove that the same factor of log N dictates the stability region of the system. Simulation results indicate that our bounds are tight, especially at high utilizations, in all four scenarios. A remarkable insight gained from our results is that, at moderate to high utilizations, multipath routing 'makes sense' from a queueing perspective for two paths only, i.e., response times drop the most when N = 2; the technical explanation is that the resequencing (delay) price starts to quickly dominate the tempting gain due to multipath transmissions. We study a processing system comprised of parallel queues, whose individual service rates are specified by a global service mode (configuration). The issue is how to switch the system between various possible service modes, so as to maximize its throughput and maintain stability under the most workload-intensive input traffic traces (arrival processes). Stability preserves the job inflow–outflow balance at each queue on the traffic traces. Two key families of service policies are shown to maximize throughput, under the mild condition that traffic traces have long-term average workload rates. In the first family of cone policies, the service mode is chosen based on the system backlog state belonging to a corresponding cone. Two distinct policy classes of that nature are investigated, MaxProduct and FastEmpty. In the second family of batch policies (BatchAdapt), jobs are collectively scheduled over adaptively chosen horizons, according to an asymptotically optimal, robust schedule. The issues of nonpreemptive job processing and non-negligible switching times between service modes are addressed. The analysis is extended to cover feed-forward networks of such processing systems nodes. The approach taken unifies and generalizes prior studies, by developing a general trace-based modeling framework (sample-path approach) for addressing the queueing stability problem. It treats the queueing structure as a deterministic dynamical system and analyzes directly its evolution trajectories. It does not require any probabilistic superstructure, which is typically used in previous approaches. Probability can be superposed later to address finer performance questions (e.g., delay). The throughput maximization problem is seen to be primarily of structural nature. The developed methodology appears to have broader applicability to other queueing systems. Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type-i job (i = 1,2) finds the server busy, it is blocked and routed to a separate type-i retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a Riemann- Hilbert boundary value problem. Various performance measures are calculated and numerical results are presented. In particular, numerical results demonstrate that the proposed multiple access system with two types of jobs and constant retrial rates provides incentives for the users to respect their contracts. Jobs generated by a single Poisson source can be routed through N alternative gateways, modelled as parallel M M 1 queues. The servers are subject to random breakdowns which leave their corresponding queues intact, but may affect the routing of jobs during the subsequent repair periods. The marginal equilibrium queue size distributions are determined by spectral expansion. This can be done, at least in principle, for any number of queues. Several routing strategies are evaluated and compared empirically. Numerical results, including optimal routine are presented and possible generalizations are considered. > 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. In large-scale data enters, software and hardware failures are frequent, resulting in failures of job executions that may cause significant resource waste and performance deterioration. To proactively minimize the resource inefficiency due to job failures, it is important to identify them in advance using key job attributes. However, so far, prevailing research on datacenter workload characterization has overlooked job failures, including their patterns, root causes, and impact. In this paper, we aim to develop prediction models and mitigation policies for unsuccessful jobs, so as to reduce the resource waste in big data enters. In particular, we base our analysis on Google cluster traces, consisting of a large number of big-data jobs with a high task fan-out. We first identify the time-varying patterns of failed jobs and the contributing system features. Based on our characterization study, we develop an on-line predictive model for job failures by applying various statistical learning techniques, namely Linear Discriminate Analysis (LDA), Quadratic Discriminate Analysis (QDA), and Logistic Regression (LR). Furthermore, we propose a delay-based mitigation policy which, after a certain grace period, proactively terminates the execution of jobs that are predicted to fail. The particular objective of postponing job terminations is to strike a good tradeoffs between resource waste and false prediction of successful jobs. Our evaluation results show that the proposed method is able to significantly reduce the resource waste by 41.9 on average, and keep false terminations of jobs low, i.e., only 1 . We consider the problem of load balancing in dynamic distributed systems in cases where new incoming tasks can make use of old information. For example, consider a multiprocessor system where incoming tasks with exponentially distributed service requirements arrive as a Poisson process, the tasks must choose a processor for service, and a task knows when making this choice the processor queue lengths from T seconds ago. What is a good strategy for choosing a processor in order for tasks to minimize their expected time in the system? Such models can also be used to describe settings where there is a transfer delay between the time a task enters a system and the time it reaches a processor for service. Our models are based on considering the behavior of limiting systems where the number of processors goes to infinity. The limiting systems can be shown to accurately describe the behavior of sufficiently large systems and simulations demonstrate that they are reasonably accurate even for systems with a small number of processors. Our studies of specific models demonstrate the importance of using randomness to break symmetry in these systems and yield important rules of thumb for system design. The most significant result is that only small amounts of queue length information can be extremely useful in these settings; for example, having incoming tasks choose the least loaded of two randomly chosen processors is extremely effective over a large range of possible system parameters. In contrast, using global information can actually degrade performance unless used carefully; for example, unlike most settings where the load information is current, having tasks go to the apparently least loaded server can significantly hurt performance.