Performance of an Unreliable Retrial Queue With Two Types of Customer Arrivals and Service Orbit
Abstract
This research provides a comprehensive analysis of a complex retrial queue, specifically a M1,M2/G1,G2/1 model. The unique characteristic of this model is its consideration of customer impatience, which can manifest as either persistent or impatient behavior. The study explores the intricate dynamics of the system, including the interplay between customer impatience and the retrial, service, repair, and reserved processes. To enhance realistic modeling, the study introduces a service orbit and repair services that are activated when the server breaks down. The Chapman-Kolmogorov equations are established, and the supplementary variables method is used to present the steady-state solutions. We provide the necessary and sufficient condition for the system to be stable, along with several specific cases. Explicit closed-form expressions for various performance measures are provided, which are then used to construct an expected total cost function. Numerical results are also presented to demonstrate how system parameters affect performance measures and the total cost function.
References
G. I. Falin, “A survey of retrial queues,” Queueing Systems: Theory and Applications, vol. 7, pp. 127–168, 1990.
G. L. Falin and J. G. C. Templeton, Retrial Queues. Chapman & Hall, London, 1997.
J. G. C. Templeton, “Retrial queues,” TOP, vol. 7, pp. 351–353, 1999.
J. R. Artalejo and A. Gomez-Corral, ´ Retrial queueing system: A computational approach. Springer Edition, Berlin, 2008.
T. Yang and J. G. C. Templeton, “A survey on retrial queues,” Queueing Systems, vol. 2, pp. 201–233, 1987.
G. Ayyappan and J. Udayageetha, “Analysis of mixed priority retrial queueing system with two way communication, collisions, working breakdown, Bernoulli vacation, negative arrival, repair, immediate feedback and reneging,” Stochastic Modeling and Applications, vol. 21, pp. 67–83, 2017.
G. Ayyappan and J. Udayageetha, “Transient solution of M[X1] ,M[X2]/G1,G2/1 with priority services, modified Bernoulli vacation, bernoulli feedback, breakdown, delaying repair and reneging,” Applications and Applied Mathematics: An International Journal, vol. 12, pp. 633–657, 2017.
G. Ayyappan and P. Thamizhselvi, “Transient analysis of M[X1] ,M[X2]/G1,G2/1 retrial queueing system with priority services, working vacations and vacation interruption, emergency vacation, negative arrival and delayed repair,” International Journal of Applied and Computational Mathematics, vol. 4, pp. 2199–5796, 2018.
G. Ayyappan and J. Udayageetha, “Transient analysis of M[X1] ,M[X2]/G1,G2/1 retrial queueing system with priority services, working breakdown, start up/close down time, Bernoulli vacation, reneging and balking,” Pakistan Journal of Statistics and Operation Research, vol. 16, pp. 203–216, 2020.
M. Boualem, “Insensitive bounds for the stationary distribution of a single server retrial queue with server subject to active breakdowns,” Advances in Operations Research, vol. 2014, Article ID 985453, 2014.
S. Gao, J. Zhang, and X. Wang, “Analysis of a retrial queue with two-type breakdowns and delayed repairs,” IEEE Access, vol. 8, pp. 172 428–172 442, 2020.
W. Jinting and Z. Peng, “A discrete-time retrial queue with negative customers and unreliable server,” Computers & Industrial Engineering, vol. 56, pp. 1216–1222, 2009.
R. Tian and Y. Zhang, “Analysis of M/M/1 queueing systems with negative customers and unreliable repairers,” Communications in Statistics–Theory and Methods, pp. 1–14, 2023.
A. Divya, A. Radhika, and U. Shweta, “Detection of optimal working vacation service rate for retrial priority G-queue with immediate Bernoulli feedbac,” Results in Control and Optimization, vol. 14, Article ID 100397, 2024.
K. Kim, “Delay cycle analysis of finite-buffer M/G/1 queues and its application to the analysis of M/G/1 priority queues with finite and infinite buffers,” Performance Evaluation, vol. 143, Article ID 102133, 2020.
A. Aissani, F. Lounis, D. Hamadouche, and S. Taleb, “Analysis of customers impatience in a repairable retrial queue under postponed preventive actions,” American Journal of Mathematical and Management Sciences, vol. 38, pp. 125–150, 2019.
H. Hablal, N. Touche, L. M. Alem, A. A. Bouchentouf, and M. Boualem, “Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system,” Hacettepe Journal of Mathematics and Statistics, vol. 52, pp. 1438–1460, 2023.
X. Wu, P. Brill, M. Hlynka, and J. Wang, “An M/G/1 retrial queue with balking and retrials during service,” International Journal of Operational Research, vol. 1, pp. 30–51, 2005.
S. Taleb, H. Saggou, and A. Aissani, “Unreliable M/G/1 retrial queue with geometric loss and random reserved time,” International Journal of Operational Research, vol. 7, pp. 171–191, 2010.
D. Zirem, M. Boualem, K. Adel-Aissanou, and D. Aissani, “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.
S. Karpagam, “Analysis of bulk service queuing system with rework, unreliable server, resuming service and two kinds of multiple vacation,” Yugoslav Journal of Operations Research, vol. 33, pp. 17–39, 2022.
G. Ayyappan and S. Nithya, “Analysis of M[X1] ,M[X2]/G1,G2/1 retrial queue with priority services, differentiate breakdown, delayed repair, Bernoulli feedback, balking and working vacation,” International Journal of Mathematics in Operational Research, vol. 28, pp. 491– 513, 2024.
P. Indumathi and K. Karthikeyan, “ANFIS-enhanced M/M/2 queueing model investigation in heterogeneous server systems with catastrophe and restoration,” Contemporary Mathematics, vol. 5, pp. 1323–1343, 2024.
L. M. Alem, M. Boualem, and D. A¨ıssani, “Bounds of the stationary distribution in M/G/1 retrial queue with two-way communication and n types of outgoing calls,” Yugoslav Journal of Operations Research, vol. 29, pp. 375–39, 2019.
L. M. Alem, M. Boualem, and D. A¨ıssani, “Stochastic comparison bounds for an M1,M2/G1,G2/1 retrial queue with two-way communication,” Hacettepe Journal of Mathematics and Statistics, vol. 48, pp. 1185–1200, 2019.
J. R. Artalejo and T. Phung-Duc, “Single server retrial queues with two-way communication,” Applied Mathematical Modelling, vol. 37, pp. 1811–1822, 2013.
A. Nazarov, J. Sztrik, and A. Kvach, “Asymptotic sojourn time analysis of finite-source M/M/1 retrial queueing system with two-way communication,” in Information Technologies and Mathematical Modelling. Queueing Theory and Applications, pp. 172–183, 2018.
S. Chakravarthy and A. Dudin, “Analysis of a retrial queueing model with map arrivals and two types of customers,” Mathematical and Computer Modelling, vol. 37, pp. 343–363, 2003.
S. Gao, “A preemptive priority retrial queue with two classes of customers and general retrial times,” Operational Research, vol. 15, pp. 233–251, 2015.
B. Kim and J. Kim, “Waiting time distributions in an M/G/1 retrial queue with two classes of customers,” Annals of Operations Research, vol. 252, pp. 121–134, 2017.
S. Abdollahi and M. R. Salehi Rad, “Analysis of a batch arrival retrial queue with two-phase services,” Bulletin of the Iranian Mathematical Society, vol. 48, pp. 791–804, 2022.
B. Liu, J. Min, and Y. Q. Zhao, “Refined tail asymptotic properties for the MX /G/1 retrial queue,” Queueing Systems: Theory and Applications, vol. 104, pp. 65–105, 2023.
V. Klimenok, A. Dudin, and V. Boksha, “Queueing system with two types of customers and limited processor sharing,” in Information Technologies and Mathematical Modelling. Queueing Theory and Applications, pp. 157–171, 2022.
R. Raina and J. Vidyottama, “Optimization of traffic control in MMAP
/PH
/S priority queueing model with PH retrial times and the preemptive repeat policy,” Journal of Industrial and Management Optimization 19, vol. 4, pp. 2333–2353, 2021.
C. D’Apice, A. Dudin, S. Dudin, and R. Manzo, “Priority queueing system with many types of requests and restricted processor sharing,” Journal of Ambient Intelligence and Humanized Computing, vol. 14, pp. 12 651–12 662, 2023.
V. Klimenok, A. Dudin, O. Dudina, and I. Kochetkova, “Queueing system with two types of customers and dynamic change of a priority,” Mathematics, vol. 8, Article ID 824, 2020.
L. Boutarfa and N. Djellab, “On the performance of the M1,M2/G1,G2/1 retrial queue with pre-emptive resume policy,” Yugoslav Journal of Operations Research, vol. 25, pp. 153–164, 2015.
T. Li and L. Zhang, “An M/G/1 retrial G-queue with general retrial times and working breakdowns,” Mathematical and Computational Applications, vol. 22, 2017.
P. Rajadurai, “A study on M/G/1 retrial queueing system with three different types of customers under working vacation policy,” International Journal of Mathematical Modelling and Numerical Optimisation, vol. 8, pp. 393–417, 2018.
S. Taleb and A. Aissani, “Preventive maintenance in an unreliable M/G/1 retrial queue with persistent and impatient customers,” Annals of Operations Research, vol. 247, pp. 291–317, 2016.
A. Gomez-Corral, “Stochastic analysis of a single server retrial queue with general retrial ´ times,” Naval Research Logistics, vol. 46, pp. 561–581, 1999.
G. L. Falin and J. G. C. Templeton, Retrial Queues. Chapman & Hall, London, 1997.
J. G. C. Templeton, “Retrial queues,” TOP, vol. 7, pp. 351–353, 1999.
J. R. Artalejo and A. Gomez-Corral, ´ Retrial queueing system: A computational approach. Springer Edition, Berlin, 2008.
T. Yang and J. G. C. Templeton, “A survey on retrial queues,” Queueing Systems, vol. 2, pp. 201–233, 1987.
G. Ayyappan and J. Udayageetha, “Analysis of mixed priority retrial queueing system with two way communication, collisions, working breakdown, Bernoulli vacation, negative arrival, repair, immediate feedback and reneging,” Stochastic Modeling and Applications, vol. 21, pp. 67–83, 2017.
G. Ayyappan and J. Udayageetha, “Transient solution of M[X1] ,M[X2]/G1,G2/1 with priority services, modified Bernoulli vacation, bernoulli feedback, breakdown, delaying repair and reneging,” Applications and Applied Mathematics: An International Journal, vol. 12, pp. 633–657, 2017.
G. Ayyappan and P. Thamizhselvi, “Transient analysis of M[X1] ,M[X2]/G1,G2/1 retrial queueing system with priority services, working vacations and vacation interruption, emergency vacation, negative arrival and delayed repair,” International Journal of Applied and Computational Mathematics, vol. 4, pp. 2199–5796, 2018.
G. Ayyappan and J. Udayageetha, “Transient analysis of M[X1] ,M[X2]/G1,G2/1 retrial queueing system with priority services, working breakdown, start up/close down time, Bernoulli vacation, reneging and balking,” Pakistan Journal of Statistics and Operation Research, vol. 16, pp. 203–216, 2020.
M. Boualem, “Insensitive bounds for the stationary distribution of a single server retrial queue with server subject to active breakdowns,” Advances in Operations Research, vol. 2014, Article ID 985453, 2014.
S. Gao, J. Zhang, and X. Wang, “Analysis of a retrial queue with two-type breakdowns and delayed repairs,” IEEE Access, vol. 8, pp. 172 428–172 442, 2020.
W. Jinting and Z. Peng, “A discrete-time retrial queue with negative customers and unreliable server,” Computers & Industrial Engineering, vol. 56, pp. 1216–1222, 2009.
R. Tian and Y. Zhang, “Analysis of M/M/1 queueing systems with negative customers and unreliable repairers,” Communications in Statistics–Theory and Methods, pp. 1–14, 2023.
A. Divya, A. Radhika, and U. Shweta, “Detection of optimal working vacation service rate for retrial priority G-queue with immediate Bernoulli feedbac,” Results in Control and Optimization, vol. 14, Article ID 100397, 2024.
K. Kim, “Delay cycle analysis of finite-buffer M/G/1 queues and its application to the analysis of M/G/1 priority queues with finite and infinite buffers,” Performance Evaluation, vol. 143, Article ID 102133, 2020.
A. Aissani, F. Lounis, D. Hamadouche, and S. Taleb, “Analysis of customers impatience in a repairable retrial queue under postponed preventive actions,” American Journal of Mathematical and Management Sciences, vol. 38, pp. 125–150, 2019.
H. Hablal, N. Touche, L. M. Alem, A. A. Bouchentouf, and M. Boualem, “Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system,” Hacettepe Journal of Mathematics and Statistics, vol. 52, pp. 1438–1460, 2023.
X. Wu, P. Brill, M. Hlynka, and J. Wang, “An M/G/1 retrial queue with balking and retrials during service,” International Journal of Operational Research, vol. 1, pp. 30–51, 2005.
S. Taleb, H. Saggou, and A. Aissani, “Unreliable M/G/1 retrial queue with geometric loss and random reserved time,” International Journal of Operational Research, vol. 7, pp. 171–191, 2010.
D. Zirem, M. Boualem, K. Adel-Aissanou, and D. Aissani, “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.
S. Karpagam, “Analysis of bulk service queuing system with rework, unreliable server, resuming service and two kinds of multiple vacation,” Yugoslav Journal of Operations Research, vol. 33, pp. 17–39, 2022.
G. Ayyappan and S. Nithya, “Analysis of M[X1] ,M[X2]/G1,G2/1 retrial queue with priority services, differentiate breakdown, delayed repair, Bernoulli feedback, balking and working vacation,” International Journal of Mathematics in Operational Research, vol. 28, pp. 491– 513, 2024.
P. Indumathi and K. Karthikeyan, “ANFIS-enhanced M/M/2 queueing model investigation in heterogeneous server systems with catastrophe and restoration,” Contemporary Mathematics, vol. 5, pp. 1323–1343, 2024.
L. M. Alem, M. Boualem, and D. A¨ıssani, “Bounds of the stationary distribution in M/G/1 retrial queue with two-way communication and n types of outgoing calls,” Yugoslav Journal of Operations Research, vol. 29, pp. 375–39, 2019.
L. M. Alem, M. Boualem, and D. A¨ıssani, “Stochastic comparison bounds for an M1,M2/G1,G2/1 retrial queue with two-way communication,” Hacettepe Journal of Mathematics and Statistics, vol. 48, pp. 1185–1200, 2019.
J. R. Artalejo and T. Phung-Duc, “Single server retrial queues with two-way communication,” Applied Mathematical Modelling, vol. 37, pp. 1811–1822, 2013.
A. Nazarov, J. Sztrik, and A. Kvach, “Asymptotic sojourn time analysis of finite-source M/M/1 retrial queueing system with two-way communication,” in Information Technologies and Mathematical Modelling. Queueing Theory and Applications, pp. 172–183, 2018.
S. Chakravarthy and A. Dudin, “Analysis of a retrial queueing model with map arrivals and two types of customers,” Mathematical and Computer Modelling, vol. 37, pp. 343–363, 2003.
S. Gao, “A preemptive priority retrial queue with two classes of customers and general retrial times,” Operational Research, vol. 15, pp. 233–251, 2015.
B. Kim and J. Kim, “Waiting time distributions in an M/G/1 retrial queue with two classes of customers,” Annals of Operations Research, vol. 252, pp. 121–134, 2017.
S. Abdollahi and M. R. Salehi Rad, “Analysis of a batch arrival retrial queue with two-phase services,” Bulletin of the Iranian Mathematical Society, vol. 48, pp. 791–804, 2022.
B. Liu, J. Min, and Y. Q. Zhao, “Refined tail asymptotic properties for the MX /G/1 retrial queue,” Queueing Systems: Theory and Applications, vol. 104, pp. 65–105, 2023.
V. Klimenok, A. Dudin, and V. Boksha, “Queueing system with two types of customers and limited processor sharing,” in Information Technologies and Mathematical Modelling. Queueing Theory and Applications, pp. 157–171, 2022.
R. Raina and J. Vidyottama, “Optimization of traffic control in MMAP
/PH
/S priority queueing model with PH retrial times and the preemptive repeat policy,” Journal of Industrial and Management Optimization 19, vol. 4, pp. 2333–2353, 2021.
C. D’Apice, A. Dudin, S. Dudin, and R. Manzo, “Priority queueing system with many types of requests and restricted processor sharing,” Journal of Ambient Intelligence and Humanized Computing, vol. 14, pp. 12 651–12 662, 2023.
V. Klimenok, A. Dudin, O. Dudina, and I. Kochetkova, “Queueing system with two types of customers and dynamic change of a priority,” Mathematics, vol. 8, Article ID 824, 2020.
L. Boutarfa and N. Djellab, “On the performance of the M1,M2/G1,G2/1 retrial queue with pre-emptive resume policy,” Yugoslav Journal of Operations Research, vol. 25, pp. 153–164, 2015.
T. Li and L. Zhang, “An M/G/1 retrial G-queue with general retrial times and working breakdowns,” Mathematical and Computational Applications, vol. 22, 2017.
P. Rajadurai, “A study on M/G/1 retrial queueing system with three different types of customers under working vacation policy,” International Journal of Mathematical Modelling and Numerical Optimisation, vol. 8, pp. 393–417, 2018.
S. Taleb and A. Aissani, “Preventive maintenance in an unreliable M/G/1 retrial queue with persistent and impatient customers,” Annals of Operations Research, vol. 247, pp. 291–317, 2016.
A. Gomez-Corral, “Stochastic analysis of a single server retrial queue with general retrial ´ times,” Naval Research Logistics, vol. 46, pp. 561–581, 1999.
Published
2025-05-12
How to Cite
DEHAMNIA, Nasreddine; BOUALEM, Mohamed; AISSANI, Djamil.
Performance of an Unreliable Retrial Queue With Two Types of Customer Arrivals and Service Orbit.
Yugoslav Journal of Operations Research, [S.l.], may 2025.
ISSN 2334-6043.
Available at: <https://yujor.fon.bg.ac.rs/index.php/yujor/article/view/1340>. Date accessed: 13 may 2025.
doi: https://doi.org/10.2298/YJOR240217016D.
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Section
Research Articles
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