A Markovian Batch Arrival Queueing System With Disasters, Working Breakdowns, and Impatience: Mathematical Modeling and Economic Analysis

Hayat Ramdani; Amina Angelika Bouchentouf; Lahcene Yahiaoui

doi:10.2298/YJOR230515037R

Authors

Hayat Ramdani Laboratory of Mathematics of Sidi Bel Abbes, Kasdi Merbah University of Ouargla, Algeria https://orcid.org/0009-0009-2970-9105
Amina Angelika Bouchentouf Laboratory of Mathematics, Djillali Liabes University of Sidi Bel Abbes, Sidi Bel Abbes, Algeria https://orcid.org/0000-0001-8972-4221
Lahcene Yahiaoui Laboratory of Stochastic Models, Statistic and Applications, University of Saida-Dr. Moulay Tahar, Saida, Algeria https://orcid.org/0000-0001-6573-672X

DOI:

https://doi.org/10.2298/YJOR230515037R

Keywords:

Markovian queueing models, disasters, impatience

Abstract

Machining systems are essential for many industrial applications, such as manufacturing, processing, and assembly. However, these systems are often exposed to various sources of uncertainty and disruption, such as disasters and customer impatience. These factors can adversely affect the performance, reliability, and profitability of the machining systems. Consequently, modeling and analyzing machining systems under these conditions becomes crucial. In this paper, we deal with a Markovian multi-server queueing system with batch arrival, Bernoulli feedback, and customers’ impatience (balking and reneging). The system undergoes disastrous interruptions that force all customers– whether waiting or currently in service–to exit, leading to server failures. Moreover, the system dynamically alternates between main servers and substitute servers based on the occurrence of disasters. These substitute servers operate at reduced rates compared to the main servers. Our contributions include deriving the stability condition for the system and obtaining the probability generating function of steady-state probabilities, enabling us to derive essential performance measures. Additionally, we develop a cost model and conduct an economic analysis for the system.

References

B. Kumar and D. Arivudainambi, “Transient solution of an M/M/1 queue with catastrophes,” Computers & Mathematics with Applications, vol. 40, pp. 1233-1240, 2000.

Y. Shin, “Multi-server retrial queue with negative customers and disasters,” Queueing Systems, vol. 55, pp. 223-237, 2007.

F. Jolai, S. Asadzadeh, and M. Taghizadeh, “Performance estimation of an email contact center by a finite source discrete time Geo/Geo/1 queue with disasters,” Computers & Industrial Engineering, vol. 55, pp. 543-556, 2008.

H. Park, W. Yang, and K. Chae, “Analysis of the GI/Geo/1 queue with disasters,” Stochastic Analysis & Applications, vol. 28, pp. 44-53, 2009.

B. Kim and D. Lee, “The M/G/1 queue with disasters and working breakdowns,” Applied Mathematical Modelling, vol. 38, pp. 1788-1798, 2014.

T. Jiang, L. Liu, and J. Li, “Analysis of the M/G/1 queue in multi-phase random environment with disasters,” Journal of Mathematical Analysis & Applications, vol. 430, pp. 857-873, 2015.

T. Jiang and L. Liu, “The GI/M/1 queue in a multi-phase service environment with disasters and working breakdowns,” International Journal of Computer Mathematics, vol. 94, pp. 707- 726, 2017.

R. Sudesh and R. Priya, “An analysis of discrete-time Geo/Geo/1 queue with feedback, repair and disaster,” International Journal of Operational Research, vol. 35, pp. 37-53, 2019.

U. Yechiali, “Queues with system disasters and impatient customers when system is down,” Queueing Syst, vol. 56, pp. 195-202, 2007.

O. Boudali and A. Economou, “Optimal and equilibrium balking strategies in the single server markovian queue with catastrophes,” European Journal of Operational Research, vol. 218, pp. 708-715, 2012.

--, “The effect of catastrophes on the strategic customer behavior in queueing systems,” Naval Research Logistics, vol. 60, pp. 571-587, 2013.

S. I. Ammar, T. Jiang, and Q. Ye, “Transient analysis of impatient customers in an M/M/1 disasters queue in random environment,” Engineering Computations, vol. 37, pp. 1945-1965, 2020.

R. Sudesh, “Transient analysis of a queue with system disasters and customer impatience,” Queueing Syst, vol. 66, pp. 95-105, 2010.

S. Dimou and A. Economou, “The single server queue with catastrophes and geometric reneging,” Methodology & Computing in Applied Probability, vol. 15, pp. 595-621, 2013.

D. Zirem, M. Boualem, K. Adel-Aissanou, and D. A¨ıssani, “Analysis of a single server batch arrival unreliable queue with balking and general retrial time,” Quality Technology & Quantitative Management, vol. 16, pp. 672-695, 2019.

M. Zhang and S. Gao, “Queue with working breakdowns and impatient customers,” RAIROOper. Res, vol. 54, pp. 815-825, 2020.

M. Boualem, “Stochastic analysis of a single server unreliable queue with balking and general retrial time,” Discrete and continuous models and applied computational science, vol. 4, pp. 319-326, 2020.

A. A. Bouchentouf, M. Cherfaoui, and M. Boualem, “Analysis and performance evaluation of markovian feedback multi-server queueing model with vacation and impatience,” American Journal of Mathematical and Management Sciences, vol. 40, pp. 261-282, 2021.

A. A. Bouchentouf, M. Boualem, L. Yahiaoui, and H. Ahmad, “A multi-station unreliable machine model with working vacation policy and customers’ impatience,” Quality Technology & Quantitative Management, vol. 19, p. 766-796, 2022.

M. Cherfaoui, A. A. Bouchentouf, and M. Boualem, “Modelling and simulation of bernoulli feedback queue with general customers’ impatience under variant vacation policy,” IJOR, vol. 46, pp. 451-480, 2023.

A. Chettouf, A. Bouchentouf, and M. Boualem, “A markovian queueing model for telecommunications support center with breakdowns and vacation periods,” SN Operations Research Forum, vol. 5, pp. 1-26, 2024.

G. Rupert and J. Miller, “A contribution to the theory of bulk queues,” Journal of the Royal Statistical Society: Series B (Methodological), vol. 21, pp. 320-337, 1959.

F. Foster, “Batched queuing processes,” Operations Research, vol. 12, pp. 441-449, 1964.

D. Shanbhag, “On infinite server queues with batch arrivals,” Journal of Applied Probability, vol. 3, pp. 274-279, 1966.

M. Brown and S. M. Ross, “Some results for infinite server poisson queues,” Journal of Applied Probability, vol. 6, pp. 604-611, 1969.

D. Lucantoni, “New results on the single server queue with a batch markovian arrival process,” Communications in Statistics. Stochastic Models, vol. 7, pp. 1-46, 1991.

L. Liu and J. Templeton, “Autocorrelations in infinite server batch arrival queues,” Queueing Syst, vol. 14, pp. 313-337, 1993.

H. Masuyama and T. Takine, “Analysis of an infinite-server queue with batch markovian arrival streams,” Queueing Syst., vol. 42, pp. 269-296, 2002.

H. Takagi and Y. Takahashi, “Priority queues with batch poisson arrivals,” Operations Research Letters, vol. 10, pp. 225-232, 1991.

S. Lee, H. Lee, S. Yoon, and K. Chae, “Batch arrival queue with N-policy and single vacation,” Computers & Operations Research, vol. 22, pp. 173-189, 1995.

S. Chiamsiri and M. S. Leonard, “A diffusion approximation for bulk queues,” Management Science, vol. 27, pp. 1188-1199, 1981.

G. Pang and W. Whitt, “Infinite-server queues with batch arrivals and dependent service times,” Probability in the Engineering & Informational Sciences, vol. 26, pp. 197-220, 2012.

“Poisson and gaussian approximations for multi-server queues with batch arrivals and batch abandonment,” Mathematics. Computer Science.

Y. Baba, “The MX /M/1 queue with multiple working vacation,” American Journal of Operations Research, vol. 2, pp. 217-224, 2012.

G. Mytalas and M. Zazanis, “An MX /G/1 queueing system with disasters and repairs under a multiple adapted vacation policy,” Naval Research Logistics, vol. 62, pp. 171-189, 2015.

S. Upadhyaya, “Performance prediction of a discrete-time batch arrival retrial queue with bernoulli feedback,” Applied Mathematics & Computations, vol. 283, pp. 108-119, 2016.

W. Graaf, W. Scheinhardt, and R. Boucherie, “Shot-noise fluid queues and infinite-server systems with batch arrivals,” Performance evaluation, vol. 116, pp. 143-155, 2017.

G. Ayyappan and S. Karpagam, “An M[X]/G(a,b)/1 queueing system with breakdown and repair, standby server, multiple vacation and control policy on request for re-service,” Mathematics, vol. 6, pp. 2227-7390, 2018.

G. Ayyappan and M. Nirmala, “An M[X]/G(a,b)/1 queue with breakdown and delay time to two phase repair under multiple vacation,” Applications & Applied Mathematics. An International Journal, vol. 13, pp. 639-663, 2018.

A. A. Bouchentouf and A. Guendouzi, “Cost optimization analysis for an MX /M/c vacation queueing system with servers and impatient customers,” SeMA Journal, vol. 76, pp. 309-341, 2019.

--, “The MX /M/c bernoulli feedback queue with variant multiple working vacations and impatient customers performance and economic analysis,” Arabian Journal Of Mathematics, vol. 9, pp. 309-327, 2020.

--, “Single server batch arrival bernoulli feedback queueing system with waiting server’, k-variant vacations and impatient customers,” Operations Research Forum, vol. 2, pp. 1-23, 2021.

S. R. Chakravarthy, Shruti, and A. Rumyantsev, “Analysis of a queueing model with batch markovian arrival process and general distribution for group clearance,” Methodol Comput Appl Probab., vol. 23, pp. 1551-1579, 2021.

M. F. Neuts, Matrix geometric solutions in stochastic models: An Algorithmic approach. Courier Corporation, 1981.

A Markovian Batch Arrival Queueing System With Disasters, Working Breakdowns, and Impatience: Mathematical Modeling and Economic Analysis

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