Transient analysis of a single server discrete-time queue with system disaster

Sudhesh, R.; Sebasthi Priya, R.; Lenin, R. B.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 1, pp. 123-134.

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.

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

W.M. Bohm, Markovian Queueing Systems in Discrete time, vol. 137. Hain, Frankfurt am Main (1993).

P.J. Brockwell, J. Gani and S.I. Resnick, 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).

X. Chao and Y. Zheng, Transient analysis of immigration-birth-death process with total catastrophes. Probab. Eng. Inf. Sci. 17 (2003) 83–106. | 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

F.W. Crawford and M.A. Suchard, Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution. J. Math. Biol. 65 (2012) 553–580. | DOI | MR | Zbl

A. Economou and D. Fakinos, 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

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

A. Economou and A. Gmez-Corral, The Batch Markovian arrival process subject to renewal generated Geometric Catastrophes. Stoch. Models 23 (2007) 211–233. | DOI | MR | Zbl

J. Gani and R.J. Swift, 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

G. Jain and K. Sigman, A Pollaczek-Khintchine formula for M/G/1 queues with disasters. J. Appl. Probab. 33 (1996) 1191–1200. | DOI | MR | Zbl

A. Krinik and C. Mortensen, Transient probability functions of finite birth-death processes with catastrophes. J. Stat. Plann. Inference 137 (2007) 1530–1543. | DOI | MR | Zbl

E.G. Kyriakidis, The transient probabilities of the simple immigration-catastrophe process. Math. Scientist 26 (2001) 56–58. | MR | Zbl

D.H. Lee and W.S. 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

D.H. Lee, W.S. Yang and H.M. Park, 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).

H.M. Park, W.S. Yang and K.C. Chae, Analysis of the GI/Geo/1 queue with disasters. Stochastic Anal. Appl. 28 (2009) 44–53. | DOI | MR | Zbl

H.M. Park, W.S. Yang and K.C. Chae, The GEO/G/1 queue with negative customers and disasters. Stochastic Model. 25 (2009) 673–688. | DOI | MR | Zbl

P.R. Parthasarathy and R.B. Lenin, Exact busy period distribution of a discrete queue with quadratic rates. Int. J. Comput. Math. 71 (1999) 427–436. | DOI | MR | Zbl

Y.W. Shin, 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

D. Stirzaker, Disasters. Mathematical Scientist 26 (2001) 59–62. | 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.J. Swift, 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

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

J. Wang and P. Zhang, A discrete-time retrial queue with negative customers and unreliable server. Comput. Ind. Eng. 56 (2009) 1216–1222. | DOI

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