Analysis of a geometric catastrophe model with discrete-time batch renewal arrival process

Kumar, Nitin; Barbhuiya, Farida P.; Gupta, Umesh C.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 54 (2020) no. 5, pp. 1249-1268.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

Discrete-time stochastic models have been extensively studied since the past few decades due to its huge application in areas of computer-communication networks and telecommunication systems. However, the growing use of the internet often makes these systems vulnerable to catastrophe/ virus attack leading to the removal of some or all the elements from the system. Taking note of this, we consider a discrete-time model where the population (in the form of packets, data, etc.) is assumed to grow in batches according to renewal process and is likely to be affected by catastrophes which occur according to Bernoulli process. The catastrophes have a sequential impact on the population and it destroys each individual at a time with probability p. This destruction process stops as soon as an individual survives or when the entire population becomes extinct. We analyze both late and early arrival systems independently and using supplementary variable and shift operator methods obtain explicit expressions of steady-state population size distribution at pre-arrival and arbitrary epochs. We deduce some important performance measures and further show that for both the systems the tail probabilities at pre-arrival epoch can be well approximated using a single root of the characteristic equation. In order to illustrate the computational procedure, we present some numerical results and also investigate the change in the behavior of the model with the change in parameter values.

L. Abolnikov and A. Dukhovny, Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications. Int. J. Stochastic Anal. 4 (1900) 333–355. | DOI | MR | Zbl

A.S. Alfa, Queueing Theory for Telecommunications: Discrete Time Modelling of a Single Node System. Springer Science & Business Media (2010). | 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

F.P. Barbhuiya, N. Kumar and U.C. Gupta, Batch renewal arrival process subject to geometric catastrophes. Method. Comput. Appl. Probab. 21 (2019) 69–83. | DOI | MR | Zbl

R. Bartoszynski, W.J. Buehler, W. Chan and D.K. Pearl, Population processes under the influence of disasters occurring independently of population size. J. Math. Biol. 27 (1989) 167–178. | DOI | MR | Zbl

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

O. Boudali and A. 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

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 Acadmic, Boston (1993). | DOI

H. Bruneel and T. Maertens, A discrete-time queue with customers with geometric deadlines. Perform. Eval. 85 (2015) 52–70. | DOI

H. Bruneel, T. Maertens, B. Steyaert, D. Claeys, D. Fiems and J. Walraevens, Analysis of a two-class single-server discrete-time fcfs queue: the effect of interclass correlation. TOP 26 (2018) 403–436. | DOI | MR

B. Cairns and P.K. Pollett, Extinction times for a general birth, death and catastrophe process. J. Appl. Probab. 41 (2004) 1211–1218. | DOI | MR | Zbl

M.L. Chaudhry, On numerical computations of some discrete-time queues. In: Computational Probability, edited by W.K. Grassmann, Springer (2000) 365–408. | DOI | Zbl

M.L. Chaudhry, U.C. Gupta and J.G.C. Templeton, On the relations among the distributions at different epochs for discrete-time $GI/Geom/1$ queues. Oper. Res. Lett. 18 (1996) 247–255. | DOI | MR | Zbl

D. Claeys, B. Steyaert, J. Walraevens, K. Laevens and H. Bruneel, Tail probabilities of the delay in a batch-service queueing model with batch-size dependent service times and a timer mechanism. Comput. Oper. Res. 40 (2013) 1497–1505. | DOI | MR

C. Dabrowski, Catastrophic event phenomena in communication networks: a survey. Comput. Sci. Rev. 18 (2015) 10–45. | DOI | MR | Zbl

A. Economou, On the control of a compound immigration process through total catastrophes. Eur. J. Oper. Res. 147 (2003) 522–529. | DOI | MR | Zbl

A. Economou, The compound poisson immigration process subject to binomial catastrophes. J. Appl. Probab. 41 (2004) 508–523. | DOI | MR | Zbl

A. Economou and D. Fakinos, A continuous-time markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes. Eur. J. Oper. Res. 149 (2003) 625–640. | DOI | MR | Zbl

A. Economou and A. Gómez-Corral, The batch markovian arrival process subject to renewal generated geometric catastrophes. Stochastic Models 23 (2007) 211–233. | DOI | MR | Zbl

S.N. Elaydi, An Introduction to Difference Equations. Springer, New York (2005). | MR | Zbl

R.G. Gallager, Stochastic Processes: Theory for Applications. Cambridge University Press, Cambridge (2013). | DOI | MR | Zbl

J.J. Hunter, In: Vol. 1 of Mathematical Techniques of Applied Probability: Discrete Time Models: Basic Theory. Academic Press, London (2014). | MR

S. Jeyakumar and P. Gunasekaran, An analysis of discrete queue with disaster and single vacation. Int. J. Pure Appl. Math. 113 (2017) 82–90.

C. Lee, The density of the extinction probability of a time homogeneous linear birth and death process under the influence of randomly occurring disasters. Math. Biosci. 164 (2000) 93–102. | DOI | MR | Zbl

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

C.L. Shafer, Inter-reserve distance. Biol. Conserv. 100 (2001) 215–227. | DOI

H. Takagi, In: Vol. 3 of Queuing Analysis: A Foundation of Performance Evaluation. Discrete Time Systems. North-Holland, Amsterdam (1993). | MR

M.E. Woodward, Communication and Computer Networks: Modelling with Discrete-time Queues. Wiley-IEEE Computer Society Pr (1994). | Zbl

X.W. Yi, J.D. Kim, D.W. Choi and K.C. Chae, The $Geo/G/1$ queue with disasters and multiple working vacations. Stochastic Models 23 (2007) 537–549. | DOI | MR | Zbl