Time-dependent properties of symmetric queues

Fralix, Brian; Zwart, Bert

doi:10.1007/s11134-010-9202-1

Cited by 5 publications

(3 citation statements)

References 16 publications

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“…The drift and the variance of the RBM for PS and LCFS are identical, which is no coincidence, because it is shown in [2] that the distribution of…”

Section: Discussionmentioning

confidence: 95%

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Conjectures on symmetric queues in heavy traffic

Zwart

2022

Queueing Syst

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We consider the queue-length process in the M/G/1 queue with symmetric service discipline, which is defined as follows: With n customers in the system, the server works on the customer in position i at rate γ (n, i) ≥ 0. We assume i γ (n, i) = 1 for all n ≥ 1, to make the service discipline work conserving. The customer arrival process is Poisson with rate λ. A customer arriving to a queue of size n chooses position i with probability γ (n + 1, i), moving each customer in position k ≥ i to position k + 1. Conversely, when a customer at position i departs, each customer in position k > i moves to position k − 1. We refer to this system as a symmetric queue governed by γ . Important special cases of symmetric service disciplines are last-come-firstserved (LCFS), where γ (n, 1) = 1 for all n ≥ 1, and processor sharing (PS), which is modeled by taking γ (n, i) = 1/n for all 1 ≤ i ≤ n and all n ≥ 1. We refer to [4] for more background.Symmetric service disciplines have the appealing property that the stationary distribution of the queue-length process (Q γ (t), t ≥ 0), if it exists, is insensitive to the service-time distribution, apart from its mean m. In particular, if the load ρ = λm < 1, then, as t → ∞, Q γ (t) converges weakly to a random variable Q γ (∞) which satisfies(1)This note states two open problems related to the queue-length process in heavy traffic. Problem statementAssume first that the second moment of the service-time distribution is finite. Let r be a scaling parameter, and let λ r be a sequence of arrival rates such that λ r → 1/m and r (1 − ρ r ) = r (1 − λ r m) converges to a real-valued constant. Let Q γ r (t), t ≥ 0 be the corresponding queue-length process in the symmetric queue governed by γ B Bert Zwart

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“…The drift and the variance of the RBM for PS and LCFS are identical, which is no coincidence, because it is shown in [2] that the distribution of…”

Section: Discussionmentioning

confidence: 95%

“…It is possible to determine weak convergence of Qγ r (t) for fixed t as r → ∞, assuming Q γ r (0) = 0, using the result in [2]. This suggests that marginal distributions of a candidate limit process Q γ * (•) should be independent of γ .…”

Section: Discussionmentioning

confidence: 99%

Conjectures on symmetric queues in heavy traffic

Zwart

2022

Queueing Syst

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“…A. An allocation mechanism with the product-form property Despite the complexity of our stochastic model, we are able to identify a special case for which the entire network behaves like a multiclass processor-sharing queue, of which the invariant distribution is explicit, and for which even timedependent properties are known [6,48].…”

Section: Markovian Modelmentioning

confidence: 99%

A Stochastic Resource-Sharing Network for Electric Vehicle Charging

Aveklouris

Vlasiou

Zwart

2019

IEEE Trans. Control Netw. Syst.

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We consider a distribution grid used to charge electric vehicles such that voltage drops stay bounded. We model this as a class of resource-sharing networks, known as bandwidth-sharing networks in the communication network literature. We focus on resource-sharing networks that are driven by a class of greedy control rules that can be implemented in a decentralized fashion. For a large number of such control rules, we can characterize the performance of the system by a fluid approximation. This leads to a set of dynamic equations that take into account the stochastic behavior of EVs. We show that the invariant point of these equations is unique and can be computed by solving a specific ACOPF problem, which admits an exact convex relaxation. We illustrate our findings with a case study using the SCE 47-bus network and several special cases that allow for explicit computations.

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On Lattice Path Counting and the Random Product Representation, with Applications to the Er/M/1 Queue and the M/Er/1 Queue

Liu

Fralix

2018

Methodol Comput Appl Probab

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Time-dependent properties of symmetric queues

Cited by 5 publications

References 16 publications

Conjectures on symmetric queues in heavy traffic

Conjectures on symmetric queues in heavy traffic

A Stochastic Resource-Sharing Network for Electric Vehicle Charging

On Lattice Path Counting and the Random Product Representation, with Applications to the Er/M/1 Queue and the M/Er/1 Queue

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