×

An application of Riordan arrays to the transient analysis of \(M/M/1\) queues. (English) Zbl 1334.90032

Summary: Riordan arrays are used extensively in many contexts as a combinatorial tool for solving enumeration problems such as lattice path counting problems. In this paper, we first show how Riordan array method can be applied to the transient analysis of \(M/M/1\) queue with zero customer at the initial point. Second, we extend this method to the generalized Riordan array with multiple support functions in order to deal with the transient analysis of \(M/M/1\) queue with non-zero customers at the initial point. Numerical examples are also given to show how easy and quick the transient probability obtained from the Riordan method can be computed.

MSC:

90B22 Queues and service in operations research
60K25 Queueing theory (aspects of probability theory)
PDFBibTeX XMLCite
Full Text: DOI

References:

[2] Böhm, W.; Krinik, a.; Mohanty, S. G., The combinatorics of birth-death processes and applications to queues, Queueing Syst., 26, 255-267 (1997) · Zbl 0892.90071
[3] Cheon, G.-S.; Jin, S.-T.; Kim, H.; Shapiro, L. W., Riordan group involutions and the \(\Delta \)-sequence, Discrete Appl. Math., 157, 1696-1701 (2009) · Zbl 1193.05013
[4] Cheon, G.-S.; Kim, H.; Shapiro, L. W., Combinatorics of Riordan arrays with identical A and Z sequences, Discrete Math., 312, 2040-2049 (2012) · Zbl 1243.05007
[5] Draief, M.; Mairesse, J., Services within a busy period of an \(M / M / 1\) queue and Dyck paths, Queueing Syst., 49, 73-84 (2005) · Zbl 1058.60076
[6] Jain, J. L.; Mohanty, S. G.; Bohm, W., A Course on Queueing Models (2007), Chapman and Hall: Chapman and Hall Boca Raton, London, New York
[7] Luzón, A.; Merlini, D.; Morón, M. A.; Sprugnoli, R., Identities induced by Riordan arrays, Linear Algebra Appl., 436, 631-647 (2012) · Zbl 1232.05011
[8] Merlini, D.; Rogers, D. G.; Sprugnoli, R.; Verri, M. C., On some alternative characterizations of Riordan arrays, Can. J. Math., 49, 2, 301-320 (1997) · Zbl 0886.05013
[9] Merlini, D.; Sprugnoli, R.; Verri, M. C., Combinatorial inversions and implicit Riordan arrays, Electron. Notes Discrete Math., 26, 103-110 (2006) · Zbl 1225.05024
[10] Merlini, D.; Sprugnoli, R.; Verri, M. C., Waiting patterns for a printer, Discrete Appl. Math., 144, 3, 359-373 (2004) · Zbl 1076.68019
[11] Merlini, D.; Sprugnoli, R., The relevant prefixes of coloured Motzkin walks: an average case analysis, Theor. Comput. Sci., 411, 1, 148-163 (2010) · Zbl 1189.68181
[13] Muto, K.; Miyazaki, H.; Seki, Y.; Kimura, Y.; Shibata, Y., Lattice path counting and \(M / M / c\) queueing systems, Queueing Syst., 19, 193-214 (1995) · Zbl 0836.60099
[14] Shapiro, L. W.; Davenport, D. E.; Woodson, L., The double Riordan group, Electron. J. Comb., 18, 2, #P33 (2012) · Zbl 1243.05009
[15] Shapiro, L. W.; Getu, S.; Woan, W.-J.; Woodson, L., The Riordan group, Discrete Appl. Math., 34, 229-239 (1991) · Zbl 0754.05010
[16] Sharma, O. P.; Tarabia, A. M.K., A simple transient analysis of an m/m/1/n queue, Sankhyā: Indian J. Stat., 62, 273-281 (2000), Series A Pt. 2 · Zbl 0982.60095
[17] Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete Math., 132, 267-290 (1994) · Zbl 0814.05003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.