A generalized SBBP/G/1 queue and its applications

doi:10.1016/0166-5316(94)90033-7

Performance Evaluation

Volume 21, Issues 1–2, November 1994, Pages 163-181

https://doi.org/10.1016/0166-5316(94)90033-7 Get rights and content

Abstract

In this paper, we consider a discrete-time single-server queueing system with a generalized switched batch Bernoulli arrival and a general service time processes (a generalized SBBP/G/1 queue). Customers arrive to the system in batches and service times of customers are generally distributed. The batch size and the service time distributions are governed by a discrete-time alternating renewal process with states 1 and 2. The arrival and the service processes in this system are semi-Markovian in the sense that their distributions depend not only on the state in the current slot but also on the state in the next slot in the alternating renewal process. Sojourn times in each state have a general distribution whose generating function is represented as a rational function. The purpose of this paper is two-fold. The first is to provide the analytical results in a fairly general assumption on discrete-time queues with two-state Markov modulation. The second is to show a rich applicability of the mathematical model to important queueing systems such as a discrete-time GI^[X]/G/1 queue, a discrete-time queue with independent GI^[X]/G and BBP/G input streams and a discrete-time queue with service interruptions. Using the semi-Markovian nature of the arrival and the service mechanisms in our model, we show that our model is readily applied to analyze those systems.

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There are more references available in the full text version of this article.

Cited by (10)

Loss probability in a finite queue with service interruptions and queue length distribution in the corresponding infinite queue
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Citation Excerpt :
Also, the arrival process considered in this paper can have some correlations and it generalizes the one considered in [10]. Queueing systems with correlated arrivals and service interruptions have a wide range of applications to manufacturing, computer and telecommunication systems where the server is subject to breakdown or some scheduling mechanism such as round-robin (see, e.g. [9,2,18] and references therein). In packet/cell networks, the arrival process at statistical multiplexer is usually a superposition of sources which typically generate time-correlated and bursty traffic due to their origin (e.g., periodic sampling of voice traffic or MPEG encoded real-time video traffic) or traffic shaping.
In this paper, we consider a discrete-time finite-buffer queue with correlated arrivals and service interruptions and the corresponding infinite-buffer queue. Under some assumptions, we derive an exact relation holding between the loss probability in the finite-buffer queue and the queue length distribution in the corresponding infinite-buffer queue. The exact relation is considered as an integration/generalization of the exact relations which have been derived in previous papers. By applying the exact relation, we also develop formulas to estimate the loss probability.
A simple eigenvalue method for low-order D-BMAP/G/1 queues
2005, Applied Mathematical Modelling
This paper is aimed at those engineers and practitioners who need a simple and understandable non-matrix-analytic procedure to compute the performance measures of the discrete-time BMAP/G/1 queueing system when the order of parameter matrices is very low. We develop a set of system equations and derive the vector generating function of the queue length. Starting from the generating function, we propose a eigenvalue approach that can be implemented by those who have basic knowledge on M/G/1 queues and eigenvalue algebra.
Analysis of a discrete-time queue with gated priority
1995, Performance Evaluation
This paper considers a discrete-time queue with gated priority. Low priority customers arrive at the first queue at the gate in batches according to a batch Bernoulli process (BBP). When the gate opens, all low priority customers at the first queue move to the second queue at a single server. On the other hand, high priority customers directly join the second queue upon arrival. The arrival process of high priority customers is assumed to be a BBP. The server serves only customers in the second queue. For this queue, we derive the probability generating functions for the amount of work in the system, the waiting times and the queue lengths of high and low priority customers under the assumption of bounded gate opening intervals. We also provide some numerical examples.
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2016, International Journal of Mathematics in Operational Research
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^☆: This paper was presented in part at Modelling and Performance Evaluation of ATM Technology, La Martinique, French Carribean Island, January 1993.

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A generalized SBBP/G/1 queue and its applications☆

Abstract

Computing waiting time distribution for the G/G/1 queue by signal processing methods

IEEE Trans. Comm.

Analysis of discrete-time buffer with one single output channel, subject to a general interruption process

Delays in single-server queues with batch inputs

Oper. Res.

Switched batch Bernoulli process (SBBP) and the discrete-time SBBP/G/1 queue with application to statistical multiplexer performance

IEEE J. Select. Areas Comm.

A priority queue with low-priority arrivals general

Oper. Res.

Generalized SBBP/G/1 queue and its application to ATM networks

A discrete-time single server queue with a two-level modulated input and its applications

GLOBECOM '89