Two-Dimensional Nearest-Neighbour Queueing Models, a Review and an Example

Cohen, J. W.

doi:10.1007/978-3-642-79917-4_9

Cited by 5 publications

(14 citation statements)

References 10 publications

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“…Weak convergence of the finite capacity stationary distribution, as K goes to infinity, is established under ergodicity of the infinite capacity process. The alternative proof to the result of [8] is then provided. Finally, Section 4 states similar characterizations of the invariant distribution for the asymmetric, finite or infinite capacity models.…”

Section: Introductionmentioning

confidence: 95%

“…Stationary blocking probability. The classical approach to the invariant distribution for infinite K (see [8,16,25]) is through the bivariate generating function of the stationary queue-length vector. The same method will here be used for K < ∞, leading to the determination of the stationary blocking probability π K (K, K).…”

Section: The Finite Capacity Modelmentioning

confidence: 99%

“…The most explicit formulation of A(y) is given in [8], through an infinite product. It is recalled below as Theorem 6.…”

Section: The Infinite Capacity Modelmentioning

confidence: 99%

“…The simplest model with two queues, known as two queues in parallel or join-the-shortest-queue model, has itself given rise to an abundant literature. Most of it, from the first paper by Haight [20] to the complete solution by Flatto and MacKean [16] and by Cohen [8], is devoted to the infinite capacity model. The latter indeed raises interesting issues of complex analysis.…”

Section: Introductionmentioning

confidence: 99%

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Stationary analysis of the shortest queue problem

2017

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Abstract. A simple analytical solution is proposed for the stationary loss system of two parallel queues with finite capacity K, in which new customers join the shortest queue, or one of the two with equal probability if their lengths are equal. The arrival process is Poisson, service times at each queue have exponential distributions with the same parameter, and both queues have equal capacity. Using standard generating function arguments, a simple expression for the blocking probability is derived, which as far as we know is original. Using coupling arguments and explicit formulas, comparisons with related loss systems are then provided. Bounds are similarly obtained for the average total number of customers, with the stationary distribution explicitly determined on {K, . . . , 2K}, and elsewhere upper bounded. Furthermore, from the balance equations, all stationary probabilities are obtained as explicit combinations of their values at states (0, k) for 0 ≤ k ≤ K. These expressions extend to the infinite capacity and asymmetric cases, i.e., when the queues have different service rates. For the initial symmetric finite capacity model, the stationary probabilities of states (0, k) can be obtained recursively from the blocking probability. In the other cases, they are implicitly determined through a functional equation that characterizes their generating function. The whole approach shows that the stationary distribution of the infinite capacity symmetric process is the limit of the corresponding finite capacity distributions. For the infinite capacity symmetric model, we provide an elementary proof of a result by Cohen which gives the solution of the functional equation in terms of an infinite product with explicit zeroes and poles.

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Section: Introductionmentioning

confidence: 95%

Section: The Finite Capacity Modelmentioning

confidence: 99%

“…The most explicit formulation of A(y) is given in [8], through an infinite product. It is recalled below as Theorem 6.…”

Section: The Infinite Capacity Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stationary analysis of the shortest queue problem

2017

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“…We refer to [17] for a more detailed review. The issues so far addressed go from stability criteria, as for example in [9,16,25,34], and steady-state analysis [1,2,[13][14][15]20,23,31], including bounds and comparisons with other policies [5,29,[48][49][50], to asymptotic approaches, such as steady-state tail asymptotics [35,36], mean-field and other scaling limits [10,18,32,37,45,47] or large deviations and rare events [24,39,41,46]. A significant part of the literature is also dedicated to numerical evaluation, through different algorithmic approaches, such as the matrix-analytic method [26,40,44] or the power series algorithm [8].…”

Section: Introductionmentioning

confidence: 99%

Martingales and buffer overflow for the symmetric shortest queue model

Tibi

2019

Queueing Syst

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A variant of the standard symmetric system of two parallel queues under the jointhe-shortest-queue policy is introduced. Here, the shortest queue has service rate μ 1 , while the longest queue has rate μ 2 , where μ 1 + μ 2 = 1. In the case of equality, both queues are served at rate 1/2. Each queue has capacity K , which may be finite or infinite, and the global Poisson arrival rate is ρ. Couplings show that, as μ 2 varies, the model is totally ordered, in terms of both total number N (t) of customers in the system and longest queue length. The two extreme cases μ 2 = 0 and μ 2 = 1 then provide simple stochastic bounds for N (t) for arbitrary μ 2 . The ordering partially extends to the enlarged model in which, whenever the shortest queue is empty, the idle server at that queue helps-or on the contrary disturbs-the other server. The previous bounds remain valid. In the extended setup, different martingales are next built from the infinite capacity process. Those lead to simple explicit formulations of the means and Laplace transforms of the hitting time of saturation, for the process with finite K started from any initial state. Asymptotics are then derived, as K gets large. Using one particular mean time, the stationary blocking probability is explicitly obtained, extending the result by Dester et al. regarding the standard symmetric model. Finally, the joint distribution of the time and state of the system at the first queue-length equality is expressed as a function of the inverse of a simple explicit matrix.

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A finite compensation procedure for a class of two-dimensional random walks

Adan

Dimitriou

2023

Indagationes Mathematicae

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Two-Dimensional Nearest-Neighbour Queueing Models, a Review and an Example

Cited by 5 publications

References 10 publications

Stationary analysis of the shortest queue problem

Stationary analysis of the shortest queue problem

Martingales and buffer overflow for the symmetric shortest queue model

A finite compensation procedure for a class of two-dimensional random walks

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