Volume 55, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Page(s)||S2259 - S2290|
|Published online||02 March 2021|
Alternative approach based on roots for computing the stationary queue-length distributions in GIX/M(1,b)/1 single working vacation queue
School of Mathematical Science, Sichuan Normal University, 610066 Chengdu, Sichuan, P.R. China
2 School of Mathematics and Statistics, Sichuan University of Science and Engineering, 643000 Zigong, Sichuan, P.R. China
* Corresponding author: firstname.lastname@example.org
Accepted: 22 July 2020
The purpose of this paper is to present an alternative algorithm for computing the stationary queue-length and system-length distributions of a single working vacation queue with renewal input batch arrival and exponential holding times. Here we assume that a group of customers arrives into the system, and they are served in batches not exceeding a specific number b. Because of batch arrival, the transition probability matrix of the corresponding embedded Markov chain for the working vacation queue has no skip-free-to-the-right property. Without considering whether the transition probability matrix has a special block structure, through the calculation of roots of the associated characteristic equation of the generating function of queue-length distribution immediately before batch arrival, we suggest a procedure to obtain the steady-state distributions of the number of customers in the queue at different epochs. Furthermore, we present the analytic results for the sojourn time of an arbitrary customer in a batch by utilizing the queue-length distribution at the pre-arrival epoch. Finally, various examples are provided to show the applicability of the numerical algorithm.
Mathematics Subject Classification: 60K25 / 90B22
Key words: Working vacation queue / renewal input / batch arrival bulk service / roots / Rouché’s theorem
© EDP Sciences, ROADEF, SMAI 2021
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