A multi-server synchronous vacation model with thresholds and a probabilistic decision rule

doi:10.1016/j.ejor.2006.07.037

European Journal of Operational Research

Volume 182, Issue 1, 1 October 2007, Pages 305-320

https://doi.org/10.1016/j.ejor.2006.07.037 Get rights and content

Abstract

In this paper a multi-server queueing model with Markovian arrivals and synchronous phase type vacations is studied using a probabilistic rule and controlled thresholds. The steady-state analysis of the model is presented. An optimization problem and some interesting numerical results are discussed.

Section snippets

Introduction and model description

Quality and efficiency in any service system dictate the resources are properly managed and utilized. In the current of era of stiff competition most servers are trained to handle multiple tasks (not necessarily all at the same time). This special training enables the management to optimize the servers’ utilization for betterment of the company in the form of increased productivity and market share, efficiency of the system, and above all exposing the employees to more career opportunities both

The steady state analysis

In this section we will analyze the model in steady state. Let N₁(t), N₂(t), and N₃(t) denote, respectively, the number of customers in the system, the state of the vacation process, and the phase of the arrival process at time t. The state N₂(t) = 0 will indicate that there is no vacation going and N₂(t) = j indicates that the vacation process is in phase j, 1 ⩽ j ⩽ n. The process {(N₁(t), N₂(t), N₃(t)) : t ⩾ 0} is a continuous-time Markov chain with state space given by $Ω = ⋃ {(i, j, k) : i ⩾ 0, 0 ⩽ j ⩽ n, 1 ⩽ k ⩽ m} .$ By level

Illustrative numerical examples

In this section we discuss some interesting numerical examples that qualitatively describe the model under study. The correctness and the accuracy of the code are verified by a number of accuracy checks. For example, we obtained the numerical solution for the Poisson arrivals in its simple form. Next, we implemented the general algorithm, but using the following MAP representation: Let D₀ be an irreducible, stable matrix with eigenvalue of maximum real part −κ < 0. Let ς denote the corresponding

Acknowledgements

Thanks are due to two anonymous referees for their careful reading and suggestions that improved the presentation of the paper.

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We analyze different scenarios by varying the parameters of the model including the arrival processes and service time distributions. For the arrival process, we consider the following five sets of values for D0 and D1, which are considered as input data in many published works in the literature (see, e.g., [3,5–7]). For sake of completeness, we display them here.
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Stochastics and StatisticsA multi-server synchronous vacation model with thresholds and a probabilistic decision rule

Abstract

Section snippets

Introduction and model description

The steady state analysis

Illustrative numerical examples

Acknowledgements

European Journal of Operational Research

Performance Evaluation

The batch Markovian arrival process: A review and future work

Queueing systems with vacations: A survey

Queueing System

Single server queues with vacations

Finite birth-and-death models in randomly changing environments

Advances in Applied Probability

Matrix Computations

A logarithmic reduction algorithm for quasi-birth-and-death processes

Journal of Applied Probability

New results on the single server queue with a batch Markovian arrival process

Stochastic Models

Stochastics and Statistics
A multi-server synchronous vacation model with thresholds and a probabilistic decision rule