Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: $\mathrm{D-MAP} / G_n^{(a , b)} / 1$

Gupta, Umesh Chandra; Kumar, Nitin; Pradhan, Sourav; Barbhuiya, Farida Parvez; Chaudhry, Mohan L.

doi:10.1051/ro/2021054

Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process:

D - MAP / G_{n}^{(a, b)} / 1

Gupta, Umesh Chandra ; Kumar, Nitin ; Pradhan, Sourav ; Barbhuiya, Farida Parvez ; Chaudhry, Mohan L.

RAIRO. Operations Research, Tome 55 (2021) no. 3, pp. 1231-1256

Résumé

Discrete-time queueing models find a large number of applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems and computer networks. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain the vector generating function at the departure epoch. The generating function is in turn used to extract the joint distribution of queue and server content in terms of the roots of the characteristic equation. Further, we develop the relationship between the distribution at the departure epoch and the distribution at arbitrary, pre-arrival and outside observer’s epochs, where the first is used to obtain the latter ones. We evaluate some essential performance measures of the system and also discuss the computing process extensively which is demonstrated by some numerical examples.

MR Zbl

DOI : 10.1051/ro/2021054

Classification : 60K25, 68M20
Keywords: Batch-size dependent, discrete-Markovian arrival process, discrete-time, general bulk service, phase-type distribution

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     author = {Gupta, Umesh Chandra and Kumar, Nitin and Pradhan, Sourav and Barbhuiya, Farida Parvez and Chaudhry, Mohan L.},
     title = {Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: $\mathrm{D-MAP} / G_n^{(a , b)} / 1$},
     journal = {RAIRO. Operations Research},
     pages = {1231--1256},
     year = {2021},
     publisher = {EDP-Sciences},
     volume = {55},
     number = {3},
     doi = {10.1051/ro/2021054},
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     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ro/2021054/}
}

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Gupta, Umesh Chandra; Kumar, Nitin; Pradhan, Sourav; Barbhuiya, Farida Parvez; Chaudhry, Mohan L. Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: $\mathrm{D-MAP} / G_n^{(a , b)} / 1$. RAIRO. Operations Research, Tome 55 (2021) no. 3, pp. 1231-1256. doi: 10.1051/ro/2021054

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