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Some aspects of stationary characteristics and optimal control of the BMAP ∕ G − G ∕ 1 ∕ N( ∞ ) oscillating queueing system

Authors

  • A.D. Banik

    Corresponding author
    1. School of Basic Sciences, Indian Institute of Technology, Samantapuri, Nandan Kanan Road, Bhubaneswar-751 013, India
    • Correspondence to: A. D. Banik, School of Basic Sciences, Indian Institute of Technology, Samantapuri, Nandan Kanan Road, Bhubaneswar-751 013, India.

      E-mail: banikad@gmail.com

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Abstract

We consider a finite-buffer queue where arrivals occur according to a batch Markovian arrival process (BMAP), and there are two servers present in the system. At the beginning of a busy period, the low performance server serves till queue length reaches a critical level inline image, and when queue length is greater than or equal to b, the high performance server starts working. High performance server serves till queue length drops down to a satisfactory level a ( < b) and then low performance server begins to serve again, and the process continues in this manner. The analysis has been carried out using a combination of embedded Markov chain and supplementary variable method. We obtain queue length distributions at pre-arrival-, arbitrary- and post-departure-epochs, and some important performance measures, such as probability of loss for the first-, an arbitrary- and the last-customer of a batch, mean queue length and mean waiting time. The total expected cost function per unit time is derived in order to determine locally optimal values for N, a and b at a minimum cost. Both partial- and total-batch rejection strategies have been analyzed. Also, we investigate the corresponding BMAP ∕ G − G ∕ 1 ∕ ∞ queue using matrix-analytic- and supplementary variable-method. We calculate previously described probabilities with performance measures for infinite-buffer model as well. In the end, some numerical results have been presented to show the effect of model parameters on the performance measures. Copyright © 2014 John Wiley & Sons, Ltd.

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