A new formula for the transient solution of the Erlang queueing model

Leonenko, Ganna

doi:10.1016/j.spl.2008.09.014

Cited by 11 publications

(7 citation statements)

References 18 publications

(19 reference statements)

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“…The kinetic description of the Markovian model is given by a set of differential equations for the time evolution of the state probabilities P (i, t) with i ∈ [0 · · · N ] giving the number of particles in the channel. Unlike the non-Markovian model, analytic solutions for the steady state properties can be obtained for arbitrary N (some generalizations of the Markovian models for which time-dependent solutions can be obtained and could be investigated in the future [31,32]).…”

Section: Markovian Versus Non-markovian Modelsmentioning

confidence: 99%

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Recurrence dynamics of particulate transport with reversible blockage: From a single channel to a bundle of coupled channels

et al. 2019

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We model a particulate flow of constant velocity through confined geometries, ranging from a single channel to a bundle of Nc identical coupled channels, under conditions of reversible blockage. Quantities of interest include the exiting particle flux (or throughput) and the probability that the bundle is open. For a constant entering flux, the bundle evolves through a transient regime to a steady state. We present analytic solutions for the stationary properties of a single channel with capacity N ≤ 3 and for a bundle of channels each of capacity N = 1. For larger values of N and Nc, the system's steady state behavior is explored by numerical simulation. Depending on the deblocking time, the exiting flux either increases monotonically with intensity or displays a maximum at a finite intensity. For large N we observe an abrupt change from a state with few blockages to one in which the bundle is permanently blocked and the exiting flux is due entirely to the release of blocked particles. We also compare the relative efficiency of coupled and uncoupled bundles. For N = 1 the coupled system is always more efficient, but for N > 1 the behavior is more complex.

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Section: Markovian Versus Non-markovian Modelsmentioning

confidence: 99%

“…The same procedure can be carried out for N = 3 using Eqs. (30) and (31), but the resulting expression for µ is considerably more complex. For general N we therefore propose the following ansatz, taking a similar form as the mapping for N = 2:…”

Section: Markovian Versus Non-markovian Modelsmentioning

confidence: 99%

Recurrence dynamics of particulate transport with reversible blockage: From a single channel to a bundle of coupled channels

et al. 2019

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“…Truslove [28] considers this queue with finite waiting room. Leonenko [15] studies the transient solution to the M/E k /1 queue following an approach due to Parthasarathy, [21]. A paper by Griffiths, Leonenko and Williams [10] also provides an exact solution to the transient distribution of the M/E k /1 queue.…”

Section: Introductionmentioning

confidence: 99%

The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates

Margolius¹

2021

Preprint

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We study a queueing system with Erlang arrivals with k phases and Erlang service with m phases. Transition rates among phases vary periodically with time. For these systems, we derive the asymptotic periodic distribution of the level and phase as a function of time within the period. The asymptotic periodic distribution is analogous to a steady-state distribution for a system with constant rates. If the time within the period is considered part of the state, then it is a steady-state distribution. We also obtain waiting time and busy period distributions. These solutions are expressed as infinite series. We provide bounds for the error of the estimate obtained by truncating the series. Examples are provided comparing the solution of the system of ordinary differential equation with a truncated state space to these asymptotic solutions involving remarkably few terms of the infinite series.

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“…For M/E k /1 queues, several results have been presented. Griffiths et al [1,9] and Leonenko [10] derived transient solutions for M/E k /1 queues starting with a positive number of initial customers, and Baek et al [11] extended their results to be applied to analysis in a single busy period. Furthermore, Kapodistria et al [12] presented the time-dependent solutions to a linear birth/immigration-death process with binomial catastrophes.…”

Section: Introductionmentioning

confidence: 99%

Exact Time-Dependent Queue-Length Solution to a Discrete-Time Geo/D/1 Queue

Baek

Bae

2019

Mathematics

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Time-dependent solutions to queuing models are beneficial for evaluating the performance of real-world systems such as communication, transportation, and service systems. However, restricted results have been reported due to mathematical complexity. In this study, we present a time-dependent queue-length formula for a discrete-time G e o / D / 1 queue starting with a positive number of initial customers. We derive the time-dependent formula in closed form.

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A new formula for the transient solution of the Erlang queueing model

Cited by 11 publications

References 18 publications

Recurrence dynamics of particulate transport with reversible blockage: From a single channel to a bundle of coupled channels

Recurrence dynamics of particulate transport with reversible blockage: From a single channel to a bundle of coupled channels

The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates

Exact Time-Dependent Queue-Length Solution to a Discrete-Time Geo/D/1 Queue

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