Study of a two types of general heterogeneous service queueing system in a single server with optional repeated service and feedback queue

Abstract

This paper addresses a model on a single server queue and two service representatives. After a customer is served, he/she has the three options: opting for receive the same service again (re-service), joining as a new customer for another regular service (feedback), or leaving the service system altogether. To ensure the queueing system is Markovian, we introduce an additional variable (supplementary variable) and using this approach, we derive the explicit distribution of queue size at random and departure epochs. Additionally, we determine the distribution of response time, inter-departure time, and busy period. By using the embedded Markov chain technique we have also derived the queue size distribution at departure epoch. We have also presented the cost analysis of the model with some numerical examples. The numerical illustration validates our findings and provides valuable insights into the queuing system.

Keywords

• Bernoulli feedback
• busy period
• queueing system
• re-service

References

1. [1] L. Abolnikov and A. Dukhovny, Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications, Int. J. Stoch. Anal. 4, 333–355, 1991.
2. [2] R.F. Anabosi and K.C. Madan, A single server queue with two types of service, Bernoulli schedule server vacations and a single vacation policy, Pakistan J. Statist. 19 (3), 331–342, 2003.
3. [3] A. Begum and G. Choudhury, Analysis of an $M/\binom{G_1}{G_2}$/1 queue with Bernoulli vacation and server breakdown, Int. J. Appl. Comput. Math. 9 (9), 1–32, 2023.
4. [4] P.J. Burke, Delays in single-server queues with batch input, Oper. Res. 23(4), 830– 833, 1975.
5. [5] B.D. Choi, B. Kim and S.H. Choi, On the M/G/1 Bernoulli feedback queue with multi-class customers, Comput. Oper. Res. 27 (3), 269–286, 2000.
6. [6] G. Choudhury and C.R. Kalita, An M/G/1 queue with two types of general heterogeneous service and optional repeated service subject to servers breakdown and delayed repair, Qual. Technol. Quant. Manag. 15 (5), 622–654, 2018.
7. [7] G. Choudhury and M. Paul, A two phase queueing system with Bernoulli feedback, Int. J. Inf. Manag. Sci. 16 (1), 35–52, 2005.
8. [8] D.R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, in: Math. Proc. Cambridge Philos. Soc., Cambridge University Press, 51, 433–441, 1955.

9. [9] M.M.N. GnanaSekar and I. Kandaiyan, Nonlinear metaheuristic cost optimization and ANFIS computing of feedback retrial queue with two dependent phases of service under Bernoulli working vacation, Int. J. Modern Phys. B, Doi: 10.1142/S0217979224400046, 2023.
10. [10] M.M.N. GnanaSekar and I. Kandaiyan, Analysis of an $M/G/1$retrial queue with delayed repair and feedback under working vacation policy with impatient customers, Symmetry 14 (10), 1–18, 2024.
11. [11] M. Jain and S. Kaur, (p, N)-Policy for unreliable server bulkqueue with Bernoulli feedback, Int. J. Appl. Comput. Math. 6, 1–28, 2020.
12. [12] C.R. Kalita and G. Choudhury, Analysis of an unreliable $M/\binom{G_1}{G_2}$/1 repeated service queue with delayed repair under randomized vacation policy, Comm. Statist. Theory Methods 48 (21), 5336–5369, 2019.
13. [13] I.E. Khan and R. Paramasivam, Reduction in waiting time in an $M/M/1/N$ encouraged arrival queue with feedback, balking and maintaining of reneged customers, Symmetry 14 (8), 1–18, 2022.
14. [14] B. Krishna Kumar, R. Rukmani, V. Thangaraj and U.R. Krieger, A single server retrial queue with Bernoulli feedback and collisions, J. Stat. Theory Pract. 4 (2), 243–260, 2010.
15. [15] S. Lan and Y. Tang, Performance and reliability analysis of a repairable discrete-time Geo/G/1 queue with Bernoulli feedback and randomized policy, Appl. Stoch. Models Bus. Ind. 33 (5), 522–543, 2017.
16. [16] K.C. Madan, A.D. Al-Nasser and A.Q. Al-Masri, On ${M^{[x]}/G_1G_2/1}$ queue with optional re-service, Appl. Math. Comput. 152 (1), 71–88, 2004.
17. [17] K.C. Madan, Z.R. Al-Rawi and A.D. Al-Nasser, On $M^x/\binom{G_1}{G_2}/1/G(BS)/Vs$ vacation queue with two types of general heterogeneous service, J. Appl. Math. Decis. Sci. 3, 123135, 2005.
18. [18] S. Mahanta and G. Choudhury, On ${M/\binom{G_1}{G_2}/1/V(MV)}$ queue with two types of general heterogeneous service with Bernoulli feedback, Cogent Math. Stat. 5 (1), 1–9, 2018. [19] S. Mahanta, N. Kumar and G. Choudhury, An analytical approach of Markov modulated Poisson input with feedback queue and repeated service under N-policy with setup time, Qual. Technol. Quant. Manag. 21 (2), 257–285, 2024.
19. [20] J. Medhi and J.G.C. Templeton, A Poisson input queue under N-policy and with a general start up time, Comput. Oper. Res. 19 (1), 35–41, 1992.
20. [21] A. Melikov, S. Aliyeva, S.S. Nair and B. Krishna Kumar, Retrial queuing-inventory systems with delayed feedback and instantaneous damaging of items, Axioms 11 (5), 1–17, 2022.
21. [22] S.P. Niranjan, S. Devi Latha, M. Mahdal and K. Karthik, Multiple control policy in unreliable two-phase bulk queueing system with active Bernoulli feedback and vacation, Mathematics 12 (1), 1–20, 2023.
22. [23] K. Rege, On the M/G/1 queue with Bernoulli feedback, Oper. Res. Lett. 14 (3), 163–170, 1993.
23. [24] L. Takács, Introduction to the Theory of Queues, Oxford University Press, New York, 1962.
24. [25] L. Takacs, A single-server queue with feedback, Bell Syst. Tech. J. 42 (2), 505–519, 1963.
25. [26] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Volume I: Vacation and Priority Systems, Elsevier Science Pub. Co. 1991.
26. [27] H. Takagi, A note on the response time in M/G/1 queues wuth service in random order and Bernoulli feedback, J. Oper. Res. Soc. Japan 39 (4), 486–500, 1996.
27. [28] S. Upadhyaya, Performance prediction of a discrete-time batch arrival retrial queue with Bernoulli feedback, Appl. Math. Comput. 283, 108–119, 2016.
28. [29] R.W. Wolff, Poisson arrivals see time averages, Oper. Res. 30 (2), 223–231, 1982.