Analysis of Stationary Fluid Queue Driven by State-Dependent Birth-Death Process Subject to Catastrophes

Document Type : Article


1 Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Egypt

2 Department of Mathematics, National Institute of Technology Raipur Raipur - 492010, Chhattisgarh, India

3 College of Economics and Management, Shandong University of Science and Technology, Qingdao - 266590, Shandong, China


This paper investigates an infinite buffer capacity fluid queueing model driven by a state-dependent birth-death process prone to catastrophes. We employ the Laplace-Stieltjes transform and the continued fraction techniques to determine explicit expressions for the joint probability of the number of customers in the background queueing model and the content of the buffer in the steady state. The importance of the proposed system is that, in numerous practical situation, the service facility possesses defence mechanisms versus long waiting lines. The servers may increase their rate of service under the pressure of a large backlog of work. Therefore, it is of interest to discuss queueing systems taking into regard the state dependent nature of the systems. For example, the congestion control mechanisms deny the formation of long queue in computer and communication systems by controlling the transmission rates of packets based on the queue length (of packets) at source or destination. Numerical illustrations are added to uphold the theoretical results.


[1]  Anick, D., Mitra, D., and Sondhi, M.M. “Stochastic theory of a data-handling system with multiple sources”, Bell Syst. Tech. J., 61, pp. 1871-1894 (1982).
[2] Elwalid, A.I., and Mitra, D. "Analysis and design of rate-based congestion control of high speed networks, I: stochastic fluid models, access regulation”, Queueing Syst., 9, pp. 29-63 (1991).
[3] Kapoor, S., and Dharmaraja, S. “Applications of fluid queues in rechargeable batteries”. In: Applied Probability and Stochastic Processes, Joshua, V., Varadhan, S., Vishnevsky, V. (Eds), Infosys Science Foundation Series, pp. 91-101, Springer, Singapore (2020).
[4] Mitra, D. “Stochastic theory of a fluid model of producers and consumers coupled by a buffer”, Adv. Appl. Probab., 20(3), pp. 646-676 (1988).
[5] Latouche, G., and Taylor, P.G. “A stochastic fluid model for an ad hoc mobile network”, Queueing Syst., 63, pp. 109-129 (2009).
[6] Bekker, R., and Mandjes, M. “A fluid model for a relay node in an ad hoc network: the case of heavy-tailed input”, Math. Methods Oper. Res., 70, pp. 357-384 (2009).
[7] Stern, T.E., and Elwalid, A.I. “Analysis of separable Markov-modulated rate models for information-handling systems”, Adv. Appl. Probab., 23(1), pp. 105-139 (1991).
[8] Knessl, C., and Morrison, J.A. “Heavy-traffic analysis of a data-handling system with many sources”, SIAM J. Appl. Math., 51(1), pp. 187-213 (1991).
[9] El-Baz, A.H., Tarabia, A.M.K., and Darwiesh, A.M. “Cloud storage facility as a fluid queue controlled by Markovian queue”, Probab. Eng. Inf. Sci., 36(2), pp. 1-17 (2020).
[10] Virtamo, J., and Norros, I. “Fluid queue driven by an  queue”, Queueing Syst., 16, pp. 373-386 (1994).
[11] Adan, I., and Resing, J. “Simple analysis of a fluid queue driven by an  queue”, Queueing Syst., 22, pp. 171-174 (1996).
[12] Parthasarathy, P.R., Vijayashree, K.V., and Lenin, R.B. “An  driven fluid queue - continued fraction approach”, Queueing Syst., 42, pp. 189-199 (2002).
[13] Barbot, N., and Sericola, B. “Stationary solution to the fluid queue fed by an  queue”, J. Appl. Probab., 39, pp. 359-369 (2002).
[14] Konovalov, V. “Fluid queue driven by a  queue, stable problems for stochastic models”, J. Math. Sci., 91, pp. 2971-2930 (1998).
[15] Kapoor, S., and Dharmaraja, S. “On the exact transient solution of fluid queue driven by a birth death process with specific rational rates and absorption”, Opsearch, 52(4), pp. 746-755 (2015).
[16] Kapoor, S., Dharmaraja, S., and Arunachalam, V. “Transient solution of fluid queue modulated by two independent birth-death processes”, Int. J. Oper. Res., 36(1), pp. 1-11, (2019).
[17] Maki, D.P. “On birth-death processes with rational growth rates”, SIAM J. Math. Anal., 7, pp. 29-36 (1976).
[18] Lenin, R.B., and Parthsarathy, P.R. “A computational approach for fluid queues driven by truncated birth-death processes”, Methodol. Comput. Appl. Probab., 2, pp. 373-392 (2000).
[19] Kapoor, S., and Dharmaraja, S. “Steady state analysis of fluid queues driven by birth death processes with rational rates”, Int. J. Oper. Res., 37(4), pp. 562-578 (2020).
[20] Doorn, E.A. van, and Scheinhardt, W.R.W. “A fluid queue driven by an infinite-state birth-death process”, Teletraffic Science and Engineering, 2, Part A, pp. 465-475 (1997).
[21] Parthasarathy, P.R., and Vijayashree, K.V. “Fluid queues driven by birth and death processes with quadratic rates”, Int. J. Comput. Math., 80, pp. 1385-1395 (2003).
[22] Arunachalam, V., Gupta, V., and Dharmaraja, S. “A fluid queue modulated by two independent birth-death processes”, Comput. Math. with Appl., 60(8), pp. 2433-2444 (2010).
[23] Ammar, S.I. “Analysis of an  driven fluid queue with multiple exponential vacations”, Appl. Math. Comput., 227, pp. 227- 334 (2014).
[24] Zhang, H., and Shi, D. “The M/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption”, Int. J. Inf. Manage., 20, pp. 579-587 (2009).
[25] Xu, X., Geng, J., Liu, M., Guo, H. “Stationary analysis for fluid model driven by the M/M/C working vacation queue”, J. Math. Anal. Appl., 403, pp. 423-433 (2013).
[26] Mao, B.W., Wang, F.W., and Tian, N.S. “Fluid model driven by an M/M/1 queue with multiple vacations and N-policy”, J. Appl. Math. Comput., 38, pp. 119-131 (2012).
[27] Xu, X., Guo, H., Zhao, Y., Geng J. “The fluid model driven by the M/M/1 queue with working vacations and vacation interruption”, J. Comput. Inf. Syst., 8, pp. 7643-7651 (2012).
[28] Mao, B., Wang, F., and Tian, N. “Fluid model driven by an M/M/1/N queue with multiple exponential vacations”, J. Comput. Inf. Syst., 6, pp. 1809-1816 (2010).
[29] Vijayashree, K.V., and Anjuka, A. “Stationary analysis of a fluid queue driven by an M / M / 1 queue with working vacation”, Qual. Technol. Quant. Manag., 13, pp. 1-22 (2016).
[30] Vijayalakshmi, T., and Thangaraj, V. “Transient analysis of a fluid queue driven by a chain sequenced birth and death process with catastrophes”, Int. J. Math. Oper., 8(2), pp. 164-184 (2016).
[31] Vijayalakshmi, T., and Thangaraj, V. “A fluid model driven by an  queue with catastrophe and restoration time”, Int. J. Appl. Math., 26, pp. 123-135 (2013).
[32] Vijayashree, K.V., and Anjuka, A. “Stationary analysis of a fluid queue driven by an  queue with disaster and subsequent repair”, Int. J. Oper. Res., 31(4), pp. 461-477 (2018).
[33] Ammar, S.I. “Fluid queue driven by an  disasters queue”, Int. J. Comput. Math., 91, pp. 1497-1506 (2014).
[34] Ammar, S.I. “Fluid  catastrophic queue in a random environment”, RAIRO Oper. Res., 55, pp. 2677-2690 (2021).
[35] Kumar, B.K., Vijayakumar, A., and Sophia, S. “Transient analysis for state-dependent queues with catastrophes”, Stoch. Anal. Appl., 26, pp. 1201-1217 (2008).
[36] Guillemin, F., Sericola, B. “On the fluid queue driven by an ergodic birth and death process”, Telecommunications Networks Current Status and Future Trends, 379–404 (2012).
[37] Henrici, P. “Applied and Computational Complex Analysis”, Special Functions, Integral Transforms, Asymptotics, Continued Fractions, Volume 2: John Wiley & Sons, Inc. (1977).
[38] Flajolet, P., and Guillemin, F. “The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions”, Adv. Appl. Probab., 32(3), pp. 750-778 (2000).
[39] Guillemin, F., and Pinchon, D. “Excursions of birth and death processes, orthogonal polynomials, and continued fractions”, J. Appl. Probab., 36(3), 752-770 (1999).