189 research outputs found

    A tight bound on the throughput of queueing networks with blocking

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    In this paper, we present a bounding methodology that allows to compute a tight lower bound on the cycle time of fork--join queueing networks with blocking and with general service time distributions. The methodology relies on two ideas. First, probability masses fitting (PMF) discretizes the service time distributions so that the evolution of the modified network can be modelled by a Markov chain. The PMF discretization is simple: the probability masses on regular intervals are computed and aggregated on a single value in the orresponding interval. Second, we take advantage of the concept of critical path, i.e. the sequence of jobs that covers a sample run. We show that the critical path can be computed with the discretized distributions and that the same sequence of jobs offers a lower bound on the original cycle time. The tightness of the bound is shown on computational experiments. Finally, we discuss the extension to split--and--merge networks and approximate estimations of the cycle time.queueing networks, blocking, throughput, bound, probability masses fitting, critical path.

    Stationary distributions of continuous-time Markov chains: a review of theory and truncation-based approximations

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    Computing the stationary distributions of a continuous-time Markov chain involves solving a set of linear equations. In most cases of interest, the number of equations is infinite or too large, and cannot be solved analytically or numerically. Several approximation schemes overcome this issue by truncating the state space to a manageable size. In this review, we first give a comprehensive theoretical account of the stationary distributions and their relation to the long-term behaviour of the Markov chain, which is readily accessible to non-experts and free of irreducibility assumptions made in standard texts. We then review truncation-based approximation schemes paying particular attention to their convergence and to the errors they introduce, and we illustrate their performance with an example of a stochastic reaction network of relevance in biology and chemistry. We conclude by elaborating on computational trade-offs associated with error control and some open questions

    Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

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    New algorithms for computing of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.Comment: 83 page

    Efficient rare-event simulation for the maximum of heavy-tailed random walks

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    Let (Xn:n0)(X_n:n\geq 0) be a sequence of i.i.d. r.v.'s with negative mean. Set S0=0S_0=0 and define Sn=X1+...+XnS_n=X_1+... +X_n. We propose an importance sampling algorithm to estimate the tail of M=max{Sn:n0}M=\max \{S_n:n\geq 0\} that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmund's (1979) algorithm. The rigorous analysis of efficiency requires new Lyapunov-type inequalities that can be useful in the study of more general importance sampling algorithms.Comment: Published in at http://dx.doi.org/10.1214/07-AAP485 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Nested Fork-Join Queuing Networks and Their Application to Mobility Airfield Operations Analysis

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    A single-chain nested fork-join queuing network (FJQN) model of mobility airfield ground processing is proposed. In order to analyze the queuing network model, advances on two fronts are made. First, a general technique for decomposing nested FJQNs with probabilistic forks is proposed, which consists of incorporating feedback loops into the embedded Markov chain of the synchronization station, then using Marie\u27s Method to decompose the network. Numerical studies show this strategy to be effective, with less than two percent relative error in the approximate performance measures in most realistic cases. The second contribution is the identification of a quick, efficient method for solving for the stationary probabilities of the λn/Ck/r/N queue. Unpreconditioned Conjugate Gradient Squared is shown to be the method of choice in the context of decomposition using Marie\u27s Method, thus broadening the class of networks where the method is of practical use. The mobility airfield model is analyzed using the strategies described above, and accurate approximations of airfield performance measures are obtained in a fraction of the time needed for a simulation study. The proposed airfield modeling approach is especially effective for quick-look studies and sensitivity analysis

    Collaborative Uploading in Heterogeneous Networks: Optimal and Adaptive Strategies

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    Collaborative uploading describes a type of crowdsourcing scenario in networked environments where a device utilizes multiple paths over neighboring devices to upload content to a centralized processing entity such as a cloud service. Intermediate devices may aggregate and preprocess this data stream. Such scenarios arise in the composition and aggregation of information, e.g., from smartphones or sensors. We use a queuing theoretic description of the collaborative uploading scenario, capturing the ability to split data into chunks that are then transmitted over multiple paths, and finally merged at the destination. We analyze replication and allocation strategies that control the mapping of data to paths and provide closed-form expressions that pinpoint the optimal strategy given a description of the paths' service distributions. Finally, we provide an online path-aware adaptation of the allocation strategy that uses statistical inference to sequentially minimize the expected waiting time for the uploaded data. Numerical results show the effectiveness of the adaptive approach compared to the proportional allocation and a variant of the join-the-shortest-queue allocation, especially for bursty path conditions.Comment: 15 pages, 11 figures, extended version of a conference paper accepted for publication in the Proceedings of the IEEE International Conference on Computer Communications (INFOCOM), 201

    Queues in a random environment

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    Exponential single server queues with state dependent arrival and service rates are considered which evolve under influences of external environments. The transitions of the queues are influenced by the environment's state and the movements of the environment depend on the status of the queues (bi-directional interaction). The structure of the environment is constructed in a way to encompass various models from the recent Operation Research literature, where a queue is coupled e.g. with an inventory or with reliability issues. With a Markovian joint queueing-environment process we prove separability for a large class of such interactive systems, i.e. the steady state distribution is of product form and explicitly given: The queue and the environment processes decouple asymptotically and in steady state. For non-separable systems we develop ergodicity criteria via Lyapunov functions. By examples we show principles for bounding throughputs of non-separable systems by throughputs of two separable systems as upper and lower bound
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