Analysis of state-dependent discrete-time queue with system disaster
RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 5, pp. 1915-1927.

An explicit expression for time-dependent system size probabilities is obtained for the general state-dependent discrete-time queue with system disaster. Using generating function for the nth state transient probabilities, the underlying difference equation of system size probabilities are transformed into three-term recurrence relation which is then expressed as a continued fraction. The continued fractions are converted into formal power series which yield the time-dependent system size probabilities in closed form. Further, the busy period distribution is obtained for the considered model. As a special case, the system size probabilities and busy period distribution of Geo/Geo/1 queue are deduced. Finally, numerical illustrations are presented to visualize the system effect for various values of the parameters.

Reçu le :
Accepté le :
DOI : 10.1051/ro/2018078
Classification : 60k25
Mots-clés : Generating function, continued fraction, power series, Geo/Geo/1 queue, busy period distribution, ballot numbers
Sudhesh, Ramupillai 1 ; Vaithiyanathan, Arumugam 1

1 
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     title = {Analysis of state-dependent discrete-time queue with system disaster},
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Sudhesh, Ramupillai; Vaithiyanathan, Arumugam. Analysis of state-dependent discrete-time queue with system disaster. RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 5, pp. 1915-1927. doi : 10.1051/ro/2018078. http://archive.numdam.org/articles/10.1051/ro/2018078/

V. Anisimov and R. Artalejo, Analysis of Markov multi-server retrial queues with negative arrivals. Queueing Syst. 39 (2001) 157–182. | DOI | MR | Zbl

I. Atencia and P. Moreno, The discrete time Geo/Geo/1 queue with negative customers and disasters. Comput. Oper. Res. 31 (2004) 1537–1548. | DOI | Zbl

I. Atencia and P. Moreno, A single-server G-queue in discrete-time with geometrical arrival and service process. Perform. Eval. 59 (2005) 85–97. | DOI

H. Baumann and W. Sandmann, Steady state analysis of level dependent quasi-birth-and-death processes with catastrophes. Comput. Oper. Res. 39 (2012) 413–423. | DOI | MR | Zbl

P.P. Bocharov, C. D’Apice, R. Manzo and A.V. Pechinkin, Analysis of the multi-server Markov queuing system with unlimited buffer and negative customers. Queueing Syst. 68 (2007) 85–94. | MR | Zbl

O. Boudali and A.G. Economou, Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes. Eur. J. Oper. Res. 218 (2012) 708–715. | DOI | MR | Zbl

O. Boudali and A. Economou, The effect of catastrophes on the strategic customer behavior in queueing systems. Nav. Res. Log. 60 (2013) 571–587. | DOI | MR | Zbl

P.J. Brockwell, The extinction time of a general birth and death process with catastrophes. J. Appl. Probab. 23 (1986) 851–858. | DOI | MR | Zbl

A. Chen and E. Renshaw, The M/M/1 queue with mass exodus and mass arrivals when empty. J. Appl. Probab. 34 (1997) 192–207. | DOI | MR | Zbl

A. Di Crescenzo, V. Giorno, A.G. Nobile and L.M. Ricciardi, A note on birth death processes with catastrophes. Stat. Probab. Lett. 78 (2008) 2248–2257. | DOI | MR | Zbl

A. Di Crescenzo, A.V. Giorno, B.K. Kumar and A.G. Nobile, A double-ended queue with catastrophes and repairs and a jump-diffusion approximation. Method. Comput. Appl. Probab. 14 (2012) 937–954. | DOI | MR | Zbl

A. Di Crescenzo, V. Giorno and A.G. Nobile, Constructing transient birth-death processes by means of suitable transformations. Appl. Math. Comput. 281 (2016) 152–171. | DOI | MR | Zbl

A. Economou and D. Fakinos, Alternative approaches for the transient analysis of Markov chains with catastrophes. J. Stat. Theory Pract. 2 (2008) 183–197. | DOI | MR | Zbl

E. Gelenbe, G-networks: a unifying model for neural and queueing networks. Ann. Oper. Res. 48 (1994) 433–461. | 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, Wiley, New York (1996) 1021–1069. | MR | Zbl

V. Giorno, A.G. Nobile and S. Spina, On some time non-homogeneous queueing systems with catastrophes. Appl. Math. Comput. 245 (2014) 220–234. | DOI | MR | Zbl

J.J. Hunter, Mathematical Techniques of Applied Probability – Discrete Time Models: Techniques and Applications 2, Academic Press, New York (1983). | MR | Zbl

T. Jiang, L. Liu and L. Jianjun, Analysis of the M/G/1 queue in multi-phase random environment with disasters. J. Math. Anal. App. 430 (2015) 857–873. | DOI | MR | Zbl

C.S. Kim, V.I. Klimenok and D.S. Orlovskii, Multi-server queueing system with a batch Markovian arrival process and negative customers. Queueing Syst. 12 (2006) 106–122.

B.K. Kumar and D. Arivudainambi, Transient solution of an M/M/1 queue with catastrophes. Comput. Math. Appl. 40 (2000) 1233–1240. | DOI | MR | Zbl

E.G. Kyriakidis, Stationary probabilities for a simple immigration – birth-death process under the influence of total catastrophes. Stat. Probab. Lett. 20 (1994) 239–240. | DOI | MR | Zbl

E.G. Kyriakidis, Transient solution for a simple immigration birth–death catastrophe process. Probab. Eng. Inf. Sci. 18 (2004) 233–236. | DOI | MR | Zbl

D. Lee and W. Yang, 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

P.R. Parthasarathy and R. Sudhesh, Exact transient solution of a discrete time queue with state dependent rates. Am. J. Math. Manag. Sci. 26 (2006) 253–276. | MR | Zbl

R. Sudhesh, Transient analysis of a queue with system disasters and customer impatience. Queueing Syst. 66 (2010) 95–105. | DOI | MR | Zbl

R. Sudhesh, R. Sebasthi Priya and R.B. Lenin, Transient analysis of a single server discrete – time queue with system disaster. RAIRO: OR 51 (2017) 123–134. | DOI | Numdam | MR | Zbl

H. Takagi, Queueing analysis: a foundation of performance evaluation. In: Discrete-time Systems 3. Amsterdam, North-Holland (1993). | MR

D. Towsley and S.K. Tripathi, A single server priority queue with server failures and queue flushing. Oper. Res. Lett. 10 (1991) 353–362. | DOI | MR | Zbl

J. Walraevens, D. Claeys and H. Bruneel, Analysis of steady-state and transient delay in discrete-time single-arrival and batch-arrival systems. Appl. Math. Comput. 240 (2014) 62–71. | DOI | MR | Zbl

J. Wang, Y. Huang and D. Zhangmin, A discrete-time on-off source queueing system with negative customers. Comput. Ind. Eng. 61 (2011) 1226–1232. | DOI

J. Wang, Y. Huang and T. Van Do, A single-server discrete-time queue with correlated positive and negative customer arrivals Appl. Math. Model. 37 (2013) 6212–6224. | DOI | MR | Zbl

W.S. Yang, J.D. Kim and K.C. Chae, Analysis of M/G/1 stochastic clearing systems. Stoch. Anal. App. 20 (2002) 1083–1100. | DOI | MR | Zbl

U. Yechiali, Queues with system disasters and impatient customers when system is down. Queueing Syst. 56 (2007) 195–202. | DOI | MR | Zbl

U. Yechiali and N. Paz, An M/M/1 Queue in random environment with disasters. Asia-Pac. J. Oper. Res. 31 (2014) 1450016. | DOI | MR | Zbl

Y.W. Shin, Multi-server retrial queue with negative customers and disasters. Queueing Syst. 55 (2007) 223–237. | DOI | MR | Zbl

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