20,075 research outputs found

    Loss systems in a random environment

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    We consider a single server system with infinite waiting room in a random environment. The service system and the environment interact in both directions. Whenever the environment enters a prespecified subset of its state space the service process is completely blocked: Service is interrupted and newly arriving customers are lost. We prove an if-and-only-if-condition for a product form steady state distribution of the joint queueing-environment process. A consequence is a strong insensitivity property for such systems. We discuss several applications, e.g. from inventory theory and reliability theory, and show that our result extends and generalizes several theorems found in the literature, e.g. of queueing-inventory processes. We investigate further classical loss systems, where due to finite waiting room loss of customers occurs. In connection with loss of customers due to blocking by the environment and service interruptions new phenomena arise. We further investigate the embedded Markov chains at departure epochs and show that the behaviour of the embedded Markov chain is often considerably different from that of the continuous time Markov process. This is different from the behaviour of the standard M/G/1, where the steady state of the embedded Markov chain and the continuous time process coincide. For exponential queueing systems we show that there is a product form equilibrium of the embedded Markov chain under rather general conditions. For systems with non-exponential service times more restrictive constraints are needed, which we prove by a counter example where the environment represents an inventory attached to an M/D/1 queue. Such integrated queueing-inventory systems are dealt with in the literature previously, and are revisited here in detail

    Bits Through Bufferless Queues

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    This paper investigates the capacity of a channel in which information is conveyed by the timing of consecutive packets passing through a queue with independent and identically distributed service times. Such timing channels are commonly studied under the assumption of a work-conserving queue. In contrast, this paper studies the case of a bufferless queue that drops arriving packets while a packet is in service. Under this bufferless model, the paper provides upper bounds on the capacity of timing channels and establishes achievable rates for the case of bufferless M/M/1 and M/G/1 queues. In particular, it is shown that a bufferless M/M/1 queue at worst suffers less than 10% reduction in capacity when compared to an M/M/1 work-conserving queue.Comment: 8 pages, 3 figures, accepted in 51st Annual Allerton Conference on Communication, Control, and Computing, University of Illinois, Monticello, Illinois, Oct 2-4, 201

    Defective and Clustered Choosability of Sparse Graphs

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    An (improper) graph colouring has "defect" dd if each monochromatic subgraph has maximum degree at most dd, and has "clustering" cc if each monochromatic component has at most cc vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than 2d+2d+2k\frac{2d+2}{d+2} k is kk-choosable with defect dd. This improves upon a similar result by Havet and Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with maximum average degree mm, no (1−ϵ)m(1-\epsilon)m bound on the number of colours was previously known. The above result with d=1d=1 solves this problem. It implies that every graph with maximum average degree mm is ⌊34m+1⌋\lfloor{\frac{3}{4}m+1}\rfloor-choosable with clustering 2. This extends a result of Kopreski and Yu [Discrete Math., 2017] to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degree mm is ⌊710m+1⌋\lfloor{\frac{7}{10}m+1}\rfloor-choosable with clustering 99, and is ⌊23m+1⌋\lfloor{\frac{2}{3}m+1}\rfloor-choosable with clustering O(m)O(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth-moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented

    Discrete-time queues with zero-regenerative arrivals: moments and examples

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    In this paper we investigate a single-server discrete-time queueing system with single-slot service times. The stationary ergodic arrival process this queueing system is subject to, satisfies a regeneration property when there are no arrivals during a slot. Expressions for the mean and the variance of the queue content in steady state are obtained for this broad class which includes among others autoregressive arrival processes and M/G/infinity-input or train arrival processes. To illustrate our results, we then consider a number of numerical examples

    Greedy Algorithms for Optimal Distribution Approximation

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    The approximation of a discrete probability distribution t\mathbf{t} by an MM-type distribution p\mathbf{p} is considered. The approximation error is measured by the informational divergence D(t∥p)\mathbb{D}(\mathbf{t}\Vert\mathbf{p}), which is an appropriate measure, e.g., in the context of data compression. Properties of the optimal approximation are derived and bounds on the approximation error are presented, which are asymptotically tight. It is shown that MM-type approximations that minimize either D(t∥p)\mathbb{D}(\mathbf{t}\Vert\mathbf{p}), or D(p∥t)\mathbb{D}(\mathbf{p}\Vert\mathbf{t}), or the variational distance ∥p−t∥1\Vert\mathbf{p}-\mathbf{t}\Vert_1 can all be found by using specific instances of the same general greedy algorithm.Comment: 5 page
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