ScienceDirect - Performance Evaluation : A queueing system with discrete autoregressive arrivals

Performance Evaluation
Volume 64, Issue 2, February 2007, Pages 148-161

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doi:10.1016/j.peva.2006.05.008

A queueing system with discrete autoregressive arrivals

Bara Kim ^a^,^,, Yong Chang ^b^,, Yeong Cheol Kim ^c^, and Bong Dae Choi ^a^,

^aDepartment of Mathematics and Telecommunication Mathematics Research Center, Korea University, 1, Anam-dong, Sungbuk-ku, Seoul, 136-701, Republic of Korea ^bTelecommunication Systems Division, Information and Communication Business, Samsung Electronics Co., Ltd, Suwon-si, Gyeonggi-do, 442-742, Republic of Korea ^cDepartment of Mathematics, Mokpo National University, Muan-gun, Chonnam, 534-729, Republic of Korea

Received 29 April 2003;

revised 3 March 2006.

Available online 7 July 2006.

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Abstract

We consider a discrete time single server queue with discrete autoregressive process of order 1 (DAR(1)) input. By extracting a Markov process from the queue size process and applying the BASTA property, we derive simple recursive formulae for the stationary distributions of the queue size and the waiting time. These formulae are simple, numerically stable and transform-free. A stochastic decomposition property is given for the stationary waiting time, and relations between the distributions of the stationary queue size and the stationary waiting time are discussed. Numerical examples are given for stationary distributions of the queue size and the waiting time for various DAR(1) inputs.

Keywords: DAR(1); Discrete time queue; Queue size; Waiting time; Recursive formula; Stochastic decomposition