The impact of the $NT$-policy on the behaviour of a discrete-time queue with general service times

Bart Feyaerts; Stijn De Vuyst; Herwig Bruneel; Sabine Wittevrongel

doi:10.3934/jimo.2014.10.131

Article Contents

2014, Volume 10, Issue 1: 131-149. Doi: 10.3934/jimo.2014.10.131

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The impact of the $NT$-policy on the behaviour of a discrete-time queue with general service times

1.
SMACS Research Group, TELIN Department, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent
2.
Supply Networks and Logistics Research Center, Department of Industrial Management, Ghent University, Technologiepark 903, B-9052 Zwijnaarde

Received: September 2012

Revised: June 2013

Published: January 2014

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Abstract

In this paper, we analyse the behaviour of a discrete-time single-server queueing system with general service times, equipped with the $NT$-policy. This is a threshold policy designed to reduce the number of service unit activation/deactivation cycles, whilst ensuring an acceptable delay trade-off. Once the server is deactivated, reactivation will be postponed until either $N$ customers have accumulated in the queue or the first customer has been in the queue for $T$ slots, whichever happens first. Due to this modus operandi, the system circulates between three phases: empty, accumulating and serving.
We assume a Bernoulli arrival process of customers and independent and identically distributed service times. Using a probability generating functions approach, we obtain expressions for the steady-state distributions of the phase sojourn times, the cycle length, the system content and the customer delay. The influence of the threshold parameters $N$ and $T$ on the mean sojourn times and the expected delay is discussed by means of numerical examples.

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Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 90B22.

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