Analysis of a bulk arrival $N$-policy queue with two-service genre, breakdown, delayed repair under Bernoulli vacation and repeated service policy

Begum, Anjana; Choudhury, Gautam

doi:10.1051/ro/2021167

Analysis of a bulk arrival

N

-policy queue with two-service genre, breakdown, delayed repair under Bernoulli vacation and repeated service policy

Begum, Anjana ; Choudhury, Gautam

RAIRO. Operations Research, Tome 56 (2022) no. 2, pp. 979-1012

Résumé

This article deals with an unreliable bulk arrival single server queue rendering two-heterogeneous optional repeated service (THORS) with delayed repair, under Bernoulli Vacation Schedule (BVS) and N-policy. For this model, the joint distribution of the server’s state and queue length are derived under both elapsed and remaining times. Further, probability generating function (PGF) of the queue size distribution along with the mean system size of the model are determined for any arbitrary time point and service completion epoch, besides various pivotal system characteristics. A suitable linear cost structure of the underlying model is developed, and with the help of a difference operator, a locally optimal N-policy at a lower cost is obtained. Finally, numerical experiments have been carried out in support of the theory.

MR Zbl

DOI : 10.1051/ro/2021167

Classification : 60K25, 90B22
Keywords: Two genres of service, Re-service, elapsed time, remaining time, double transform, BVS, $$-policy

@article{RO_2022__56_2_979_0,
     author = {Begum, Anjana and Choudhury, Gautam},
     title = {Analysis of a bulk arrival $N$-policy queue with two-service genre, breakdown, delayed repair under {Bernoulli} vacation and repeated service policy},
     journal = {RAIRO. Operations Research},
     pages = {979--1012},
     year = {2022},
     publisher = {EDP-Sciences},
     volume = {56},
     number = {2},
     doi = {10.1051/ro/2021167},
     mrnumber = {4407577},
     zbl = {1491.60154},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ro/2021167/}
}

TY  - JOUR
AU  - Begum, Anjana
AU  - Choudhury, Gautam
TI  - Analysis of a bulk arrival $N$-policy queue with two-service genre, breakdown, delayed repair under Bernoulli vacation and repeated service policy
JO  - RAIRO. Operations Research
PY  - 2022
SP  - 979
EP  - 1012
VL  - 56
IS  - 2
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/ro/2021167/
DO  - 10.1051/ro/2021167
LA  - en
ID  - RO_2022__56_2_979_0
ER  -

%0 Journal Article
%A Begum, Anjana
%A Choudhury, Gautam
%T Analysis of a bulk arrival $N$-policy queue with two-service genre, breakdown, delayed repair under Bernoulli vacation and repeated service policy
%J RAIRO. Operations Research
%D 2022
%P 979-1012
%V 56
%N 2
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/ro/2021167/
%R 10.1051/ro/2021167
%G en
%F RO_2022__56_2_979_0

Begum, Anjana; Choudhury, Gautam. Analysis of a bulk arrival $N$-policy queue with two-service genre, breakdown, delayed repair under Bernoulli vacation and repeated service policy. RAIRO. Operations Research, Tome 56 (2022) no. 2, pp. 979-1012. doi: 10.1051/ro/2021167

Bibliographie
Cité par

[1] K. Abbas and D. Aissani, Approximation of performance measures in an $M / G / 1$ queue with breakdowns. Qual. Technol. Quant. Manage. 7 (2010) 353–363. | DOI

[2] M. Agiwal, A. Roy and N. Saxena, Next generation 5G wireless networks: a comprehensive survey. IEEE Commun. Surv. Tutorials 18 (2016) 1617–1655. | DOI

[3] M. Baruah, K. C. Madan and T. Eldabi, Balking and re-service in a vacation queue with arrival and two types of heterogeneous service. J. Math. Res. 4 (2012) 1114–1124. | DOI

[4] A. Begum and G. Choudhury, Analysis of an unreliable single server batch arrival queue with two types of services under Bernoulli vacation policy. Commun. Stat. Theory Methods 50 (2021) 2136–2160. | MR | Zbl | DOI

[5] G. Choudhury and M. Deka, A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation. Appl. Math. Modell. 36 (2012) 6050–6060. | MR | Zbl | DOI

[6] G. Choudhury and M. Deka, A batch arrival unreliable server delaying repair queue with two phases of service and Bernoulli vacation under multiple vacation policy. Qual. Technol. Quant. Manage. 15 (2018) 157–186. | DOI

[7] G. Choudhury and C. R. Kalita, An $M / G / 1$ queue with two types of general heterogeneous service and optional repeated service subject to server’s breakdown and delayed repair. Qual. Technol. Quant. Manage. 15 (2017) 622–654. | DOI

[8] G. Choudhury and M. Paul, A batch arrival queue with a second optional service channel under $N$ -policy. Stoch. Anal. App. 24 (2006) 1–21. | MR | Zbl | DOI

[9] G. Choudhury, J. C. Ke and L. Tadj, The $N$ -policy for an unreliable server with delaying repair and two phases of service. J. Comput. Appl. Math. 231 (2009) 349–364. | MR | Zbl | DOI

[10] D. R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Math. Proc. Cambridge Philos. Soc. 51 (1955) 433–441. | MR | Zbl | DOI

[11] I. Dimitriou, A modified vacation queueing model and its application on the Discontinuous Reception power saving mechanism in unreliable Long Term Evolution networks. Perform Eval. 77 (2014) 37–56. | DOI

[12] B. T. Doshi, Queueing systems with vacations a survey. Queueing Syst. 1 (1986) 29–66. | MR | Zbl | DOI

[13] S. A. Fowler, A. Mellouk and N. Yamada, LTE-Advanced DRX Mechanism for Power Saving. Wiley, New York (2013). | DOI

[14] A. Gautam, G. Choudhury and S. Dharmaraja, Performance analysis of DRX mechanism using batch arrival vacation queueing system with $N$ -policy in LTE-A networks. Ann. Telecommun. 75 (2020) 353–367. | DOI

[15] C. R. Kalita and G. Choudhury, A note on reliability analysis of an $N$ -policy unreliable $M^{X} / (\begin{matrix} G_{1} \\ G_{2} \end{matrix}) / 1$ queue with optional repeated service. RAIRO-Oper. Res. 52 (2018) 713–724. | MR | Zbl | Numdam | DOI

[16] C. R. Kalita and G. Choudhury, Analysis of an unreliable $M^{X} / (\begin{matrix} G_{1} \\ G_{2} \end{matrix}) / 1$ repeated service queue with delayed repair under randomized vacation policy. Commun. Stat. Theory Methods 48 (2019) 5336–5369. | MR | Zbl | DOI

[17] J. C. Ke, H. I. Huang and Y. K. Chu, Batch arrival queue with $N$ -policy and at most $J$ vacations. Appl. Math. Modell. 34 (2010) 451–466. | MR | Zbl | DOI

[18] J. Keilson and L. D. Servi, Oscillating random walk models for $G I / G / 1$ vacation systems with Bernoulli schedules. J. Appl. Probab. 23 (1986) 790–802. | MR | Zbl | DOI

[19] R. F. Khalaf, K. C. Madan and C. A. Lukas, An $M^{[X]} / G / 1$ queue with Bernoulli schedule, general vacation times, random breakdowns, general delay times and general repair times. Appl. Math. Sci. 5 (2011) 35–51. | MR | Zbl

[20] A. Krishnamoorthy, P. K. Pramod and T. G. Deepak, On a queue with interruptions and repeat or resumption of service. Nonlinear Anal. 71 (2009) 1673–1683. | MR | Zbl | DOI

[21] M. S. Kumar and R. Arumuganathan, An $M^{X} / G / 1$ unreliable retrial queue with two phase service and persistence behaviour of customers in service. RAIRO-Oper. Res. 47 (2013) 9–32. | MR | Zbl | Numdam | DOI

[22] S. Lan and Y. Tang, An $N$ -Policy discrete-time $G e o / G / 1$ queue with modified multiple server vacations and Bernoulli feedback. RAIRO-Oper. Res. 53 (2019) 367–387. | MR | Zbl | Numdam | DOI

[23] H. S. Lee and M. M. Srinivasan, Control policies for the $M^{[X]} / G / 1$ queueing system. Manage. Sci. 35 (1989) 708–721. | MR | Zbl | DOI

[24] H. W. Lee, S. S. Lee and K. C. Chae, Operating characteristic of $M^{X} / G / 1$ queue with $N$ -policy. Queueing Syst. 15 (1994) 387–399. | MR | Zbl | DOI

[25] Q. Li, D. Xu and J. Cao, Reliability approximation of a Markov queueing system with server breakdown and repair. Microelectron. Reliab. 37 (1997) 1203–1212. | DOI

[26] W. Li, D. Shi and X. Chao, Reliability analysis of $M / G / 1$ queueing systems with server breakdowns and vacations. J. Appl. Probab. 34 (1997) 546–555. | MR | Zbl | DOI

[27] T. Li, L. Zhang and S. Gao, An $M / G / 1$ retrial queue with single working vacation under Bernoulli. RAIRO-Oper. Res. 54 (2020) 471–488. | MR | Zbl | Numdam | DOI

[28] K. C. Madan, A queueing system with random failures and delayed repairs. J. Indian Stat. Assoc. 32 (1994) 39–48.

[29] K. C. Madan, An $M / G / 1$ type queue with time-homogeneous breakdowns and deterministic repair times. Soochow J. Math. 29 (2003) 103–110. | MR | Zbl

[30] K. C. Madan, A. D. Al-Nasser and A. Q. Al-Masri, On $M^{X} / (\begin{matrix} G_{1} \\ G_{2} \end{matrix}) / 1$ queue with optional re-service. Appl. Math. Comput. 152 (2004) 71–88. | MR | Zbl

[31] K. C. Madan, Z. R. Al-Rawi and A. D. Al-Nasser, On $M^{X} / (\begin{matrix} G_{1} \\ G_{2} \end{matrix}) / 1 / G (B S) / V_{s}$ vacation queue with two types of general heterogeneous service. J. Appl. Math. Decis. Sci. 3 (2005) 123–135. | MR | Zbl | DOI

[32] M. K. Maheshwari, A. Roy and N. Saxena, Analytical modeling of DRX with flexible TTI for 5G communications. Trans. Emerging Telecommun. Technol. 29 (2018) e3275. | DOI

[33] F. A. Maraghi, K. C. Madan and K. Darby-Dowman, Batch arrival queueing system with random breakdowns and Bernoulli schedule server vacations having general vacation time distribution. Int. J. Inf. Manage. Sci. 20 (2009) 55–70. | MR | Zbl

[34] J. Medhi, Stochastic Models in Queueing Theory, 2nd edition. Academic Press, Burlington (2003). | MR | Zbl

[35] Y. Peng, Z. Liu and J. Wu, An $M / G / 1$ retrial $G$ -queue with preemptive resume priority and collisions subject to the server breakdowns and delayed repairs. J. Appl. Math. Comput. 44 (2014) 187–213. | MR | Zbl | DOI

[36] H. Saggou, T. Lachemot and M. Ourbih-Tari, Performance measures of $M / G / 1$ retrial queues with recurrent customers, breakdowns, and general delays. Commun. Stat. Theory Methods 46 (2017) 7998–8015. | MR | Zbl | DOI

[37] J. F. Shortle, J. M. Thompson, D. Gross and C. M. Harris, General arrival or service patterns. In: Fundamentals of Queueing Theory. Wiley (2018). | MR | Zbl | DOI

[38] C. J. Singh, M. Jain and S. Kaur, Performance analysis of bulk arrival queue with balking, optional service, delayed repair and multi-phase repair. Ain Shams Eng. J. 9 (2018) 2067–2077. | DOI

[39] L. Tadj and G. Choudhury, Optimal design and control of queues. TOP 13 (2005) 359–412. | MR | Zbl | DOI

[40] L. Tadj and J. C. Ke, A hysheretic bulk quoram queue with a choice of service and optional re service. Qual. Technol. Quant. Manage. 5 (2008) 161–178. | MR | DOI

[41] L. Tadj and K. P. Yoon, Binomial schedule for an $M / G / 1$ type queueing system with an Unreliable server under $N$ -policy, Adv. Decis. Sci. 2014 (2014) 819718. | MR | Zbl

[42] H. Takagi, Time-dependent analysis of $M / G / 1$ vacation models with exhaustive service. Queueing Syst. 6 (1990) 369–389. | MR | Zbl | DOI

[43] H. Takagi, Queueing Analysis – A Foundation of Performance Evaluation. Vol 1. Elsevier, Amsterdam (1991). | Zbl

[44] H. Takagi, Time-dependent process of $M / G / 1$ vacation models with exhaustive service. J. Appl. Probab. 29 (1992) 418–429. | MR | Zbl | DOI

[45] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications. Springer, New York (2006). | MR | Zbl | DOI

[46] J. Wang, An $M / G / 1$ queue with second optional service and server breakdowns. Comput. Math. App. 47 (2004) 1713–1723. | MR | Zbl

[47] H. White and L. S. Christie, Queueing with preemptive priorities or with breakdown. Oper. Res. 6 (1958) 79–96. | MR | Zbl | DOI

[48] R. W. Wolff, Poisson arrivals see time averages. Oper. Res. 30 (1982) 223–231. | MR | Zbl | DOI

Cité par Sources :