A discrete-time Geo/Geo/1 queue with system disaster is considered in this paper. The time-dependent and steady state probabilities of number of customers present in the system are obtained in terms of ballot numbers by solving the underlying system of difference equations using the generating function and continued fractions. Further, the busy period distribution is derived in terms of Catalan numbers. For special cases, time-dependent system size probabilities and busy period distribution are verified with the existing results in the literature. Numerical illustrations are provided for different parameter values to see their effect on performance measures and to get more insight of the model behavior.
Accepté le :
DOI : 10.1051/ro/2016008
Mots clés : Catastrophes, busy period, Catalan numbers, ballot numbers, continued fractions
@article{RO_2017__51_1_123_0,
author = {Sudhesh, R. and Sebasthi Priya, R. and Lenin, R. B.},
title = {Transient analysis of a single server discrete-time queue with system disaster},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {123--134},
publisher = {EDP-Sciences},
volume = {51},
number = {1},
year = {2017},
doi = {10.1051/ro/2016008},
mrnumber = {3590465},
zbl = {1360.60170},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro/2016008/}
}
TY - JOUR AU - Sudhesh, R. AU - Sebasthi Priya, R. AU - Lenin, R. B. TI - Transient analysis of a single server discrete-time queue with system disaster JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2017 SP - 123 EP - 134 VL - 51 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro/2016008/ DO - 10.1051/ro/2016008 LA - en ID - RO_2017__51_1_123_0 ER -
%0 Journal Article %A Sudhesh, R. %A Sebasthi Priya, R. %A Lenin, R. B. %T Transient analysis of a single server discrete-time queue with system disaster %J RAIRO - Operations Research - Recherche Opérationnelle %D 2017 %P 123-134 %V 51 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro/2016008/ %R 10.1051/ro/2016008 %G en %F RO_2017__51_1_123_0
Sudhesh, R.; Sebasthi Priya, R.; Lenin, R. B. Transient analysis of a single server discrete-time queue with system disaster. RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 1, pp. 123-134. doi : 10.1051/ro/2016008. http://www.numdam.org/articles/10.1051/ro/2016008/
and , The discrete-time queue with negative customers and disasters. Comput. Oper. Res. 31 (2004) 1537–1548. | DOI | Zbl
W.M. Bohm, Markovian Queueing Systems in Discrete time, vol. 137. Hain, Frankfurt am Main (1993).
, and , Birth, immigration and catastrophe processes. Adv. Appl. Probab. 14 (1982) 709–731. | DOI | MR | Zbl
H. Bruneel and B.G. Kim, Discrete-Time Models for Communication systems Including ATM, Kluwer Academic Publishers (1993).
and , Transient analysis of immigration-birth-death process with total catastrophes. Probab. Eng. Inf. Sci. 17 (2003) 83–106. | DOI | MR | Zbl
and , The M/M/1 queue with mass exodus and mass arrivals when empty. J. Appl. Probab. 34 (1997) 192–207. | DOI | MR | Zbl
and , Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution. J. Math. Biol. 65 (2012) 553–580. | DOI | MR | Zbl
and , A continuous-time Markov chain under the influence of a regulating point process and applications. Eur. J. Oper. Res. 149 (2003) 625–640. | DOI | MR | Zbl
and , Alternative approaches for the transient analysis of Markov chains with catastrophes. J. Statist. Theory Practice 2 (2008) 183–197. | DOI | MR | Zbl
and , The Batch Markovian arrival process subject to renewal generated Geometric Catastrophes. Stoch. Models 23 (2007) 211–233. | DOI | MR | Zbl
and , Death and birth-death and immigration processes with catastrophes. J. Statist. Theory Practice 1 (2007) 39–48. | DOI | MR | Zbl
I.M. Gessel and R.P. Stanley, Algebraic Enumeration: Handbook of Combinatorics, edited by R.L. Graham, M. Grotschel and L. Lovasz. Elsevier, Amsterdam (1995). | MR | Zbl
I.P. Goulden and D.M. Jackson, Combinatorial enumeration. John Wiley and Sons, New York (1983). | MR | Zbl
J.J. Hunter, Mathematical Techniques of Applied Probability-Discrete time models: Techniques and Applications, 1st edition, Academic Press (1983). | MR | Zbl
and , A Pollaczek-Khintchine formula for M/G/1 queues with disasters. J. Appl. Probab. 33 (1996) 1191–1200. | DOI | MR | Zbl
and , Transient probability functions of finite birth-death processes with catastrophes. J. Stat. Plann. Inference 137 (2007) 1530–1543. | DOI | MR | Zbl
, The transient probabilities of the simple immigration-catastrophe process. Math. Scientist 26 (2001) 56–58. | MR | Zbl
and , The n-policy of a discrete time Geo/G/1 queue with disasters and its application to wireless sensor networks. Appl. Math. Model. 37 (2013) 9722–9731. | DOI | MR | Zbl
, and , Geo/G/1 queues with disasters and general repair times. Appl. Math. Model. 35 (2011) 1561–1570. | DOI | MR | Zbl
S. Mohanty and W. Panny, A discrete-time analogue of the M/M/1 queue and the transition solution. an analytic approach, in Vol. 77 of Limit Theorems in Probability and Statistics - Colloquium of Mathematical Society, edited by Janos Bolyai, P. Revesz. North Holland, Amsterdam (1989) 417–424. | MR | Zbl
S. Mohanty and P.R. Parthasarathy, On the transient solution of a discrete queue. Statistics Research Repository, vol. 15. McMaster University (1990).
, and , Analysis of the GI/Geo/1 queue with disasters. Stochastic Anal. Appl. 28 (2009) 44–53. | DOI | MR | Zbl
, and , The GEO/G/1 queue with negative customers and disasters. Stochastic Model. 25 (2009) 673–688. | DOI | MR | Zbl
and , Exact busy period distribution of a discrete queue with quadratic rates. Int. J. Comput. Math. 71 (1999) 427–436. | DOI | MR | Zbl
, Multi-server retrial queue with negative customers and disasters. Queueing Syst. 55 (2007) 223–237. | DOI | MR | Zbl
R.P. Stanley, Vol. 1 of Enumerative Combinatorics, 2nd edition. Cambridge University Press (2011). | MR | Zbl
, Disasters. Mathematical Scientist 26 (2001) 59–62. | MR | Zbl
, Transient analysis of a queue with system disasters and customer impatience. Queueing Syst. 66 (2010) 95–105. | DOI | MR | Zbl
, A simple immigration-catastrophe process. Math. Sci. 25 (2000) 32–36. | MR | Zbl
H. Takagi, Queueing Analysis – Foundation of Performance Evaluation, Discrete-Time Systems, vol. 3. North Holland (1993). | MR
, A single server priority queue with server failures and queue flushing. Oper. Res. Lett. 10 (1991) 353–362. | DOI | MR | Zbl
and , A discrete-time retrial queue with negative customers and unreliable server. Comput. Ind. Eng. 56 (2009) 1216–1222. | DOI
, and , A single-server discrete-time queue with correlated positive and negative customer arrivals. Appl. Math. Mod. 37 (2013) 6212–6224. | DOI | MR | Zbl
, Queues with system disasters and impatient customers when system is down. Queueing Syst. 56 (2007) 195–202. | DOI | MR | Zbl
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