1,285 research outputs found

    Batch queues, reversibility and first-passage percolation

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    We consider a model of queues in discrete time, with batch services and arrivals. The case where arrival and service batches both have Bernoulli distributions corresponds to a discrete-time M/M/1 queue, and the case where both have geometric distributions has also been previously studied. We describe a common extension to a more general class where the batches are the product of a Bernoulli and a geometric, and use reversibility arguments to prove versions of Burke's theorem for these models. Extensions to models with continuous time or continuous workload are also described. As an application, we show how these results can be combined with methods of Seppalainen and O'Connell to provide exact solutions for a new class of first-passage percolation problems.Comment: 16 pages. Mostly minor revisions; some new explanatory text added in various places. Thanks to a referee for helpful comments and suggestion

    Large deviations of a modified Jackson network: stability and rough asymptotics

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    Consider a modified, stable, two node Jackson network where server 2 helps server 1 when server 2 is idle. The probability of a large deviation of the number of customers at node one can be calculated using the flat boundary theory of Schwartz and Weiss [Large Deviations Performance Analysis (1994), Chapman and Hall, New York]. Surprisingly, however, these calculations show that the proportion of time spent on the boundary, where server 2 is idle, may be zero. This is in sharp contrast to the unmodified Jackson network which spends a nonzero proportion of time on this boundary.Comment: Published at http://dx.doi.org/10.1214/105051604000000666 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Unreliable Retrial Queues in a Random Environment

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    This dissertation investigates stability conditions and approximate steady-state performance measures for unreliable, single-server retrial queues operating in a randomly evolving environment. In such systems, arriving customers that find the server busy or failed join a retrial queue from which they attempt to regain access to the server at random intervals. Such models are useful for the performance evaluation of communications and computer networks which are characterized by time-varying arrival, service and failure rates. To model this time-varying behavior, we study systems whose parameters are modulated by a finite Markov process. Two distinct cases are analyzed. The first considers systems with Markov-modulated arrival, service, retrial, failure and repair rates assuming all interevent and service times are exponentially distributed. The joint process of the orbit size, environment state, and server status is shown to be a tri-layered, level-dependent quasi-birth-and-death (LDQBD) process, and we provide a necessary and sufficient condition for the positive recurrence of LDQBDs using classical techniques. Moreover, we apply efficient numerical algorithms, designed to exploit the matrix-geometric structure of the model, to compute the approximate steady-state orbit size distribution and mean congestion and delay measures. The second case assumes that customers bring generally distributed service requirements while all other processes are identical to the first case. We show that the joint process of orbit size, environment state and server status is a level-dependent, M/G/1-type stochastic process. By employing regenerative theory, and exploiting the M/G/1-type structure, we derive a necessary and sufficient condition for stability of the system. Finally, for the exponential model, we illustrate how the main results may be used to simultaneously select mean time customers spend in orbit, subject to bound and stability constraints

    Learning Algorithms for Minimizing Queue Length Regret

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    We consider a system consisting of a single transmitter/receiver pair and NN channels over which they may communicate. Packets randomly arrive to the transmitter's queue and wait to be successfully sent to the receiver. The transmitter may attempt a frame transmission on one channel at a time, where each frame includes a packet if one is in the queue. For each channel, an attempted transmission is successful with an unknown probability. The transmitter's objective is to quickly identify the best channel to minimize the number of packets in the queue over TT time slots. To analyze system performance, we introduce queue length regret, which is the expected difference between the total queue length of a learning policy and a controller that knows the rates, a priori. One approach to designing a transmission policy would be to apply algorithms from the literature that solve the closely-related stochastic multi-armed bandit problem. These policies would focus on maximizing the number of successful frame transmissions over time. However, we show that these methods have Ω(logT)\Omega(\log{T}) queue length regret. On the other hand, we show that there exists a set of queue-length based policies that can obtain order optimal O(1)O(1) queue length regret. We use our theoretical analysis to devise heuristic methods that are shown to perform well in simulation.Comment: 28 Pages, 11 figure

    Content Based Status Updates

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    Consider a stream of status updates generated by a source, where each update is of one of two types: high priority or ordinary (low priority). These updates are to be transmitted through a network to a monitor. However, the transmission policy of each packet depends on the type of stream it belongs to. For the low priority stream, we analyze and compare the performances of two transmission schemes: (i) Ordinary updates are served in a First-Come-First-Served (FCFS) fashion, whereas, in (ii), the ordinary updates are transmitted according to an M/G/1/1 with preemption policy. In both schemes, high priority updates are transmitted according to an M/G/1/1 with preemption policy and receive preferential treatment. An arriving priority update discards and replaces any currently-in-service high priority update, and preempts (with eventual resume for scheme (i)) any ordinary update. We model the arrival processes of the two kinds of updates, in both schemes, as independent Poisson processes. For scheme (i), we find the arrival and service rates under which the system is stable and give closed-form expressions for average peak age and a lower bound on the average age of the ordinary stream. For scheme (ii), we derive closed-form expressions for the average age and average peak age of the high priority and low priority streams. We finally show that, if the service time is exponentially distributed, the M/M/1/1 with preemption policy leads to an average age of the low priority stream higher than the one achieved using the FCFS scheme. Therefore, the M/M//1/1 with preemption policy, when applied on the low priority stream of updates and in the presence of a higher priority scheme, is not anymore the optimal transmission policy from an age point of view
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