Variant impatient behavior of a Markovian queue with balking reserved idle time and working vacation

Azhagappan, Arumugam; Deepa, Thirunavukkarasu

RAIRO - Operations Research - Recherche Opérationnelle, Tome 54 (2020) no. 3, pp. 783-793.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

The customers’ impatience and its effect plays a major role in the economy of a country. It directly affects the sales of products and profit of a trading company. So, it is very important to study various impatient behaviors of customers and to analyze different strategies to hold such impatient customers. This situation is modeled mathematically in this research work along with working vacation and reserved idle time of server, balking and re-service of customers. This paper studies the transient analysis of an M/M/1 queueing model with variant impatient behavior, balking, re-service, reserved idle time and working vacation. Whenever the system becomes empty, the server resumes working vacation. When he is coming back from the working vacation and finding the empty system, he stays idle for a fixed time period known as reserved idle time and waits for an arrival. If an arrival occurs before the completion of reserved idle time, the server starts a busy period. Otherwise, he resumes another working vacation after the completion of reserved idle time. During working vacation, the arriving customers may either join or balk the queue. The customers waiting in the queue for service, during working vacation period, become impatient. But, the customer who is receiving the service in the slow service rate, does not become impatient. After each service, the customer may demand for immediate re-service. The transient system size probabilities for the proposed model are derived using generating function and continued fraction. The time-dependent mean and variance of system size are also obtained. Finally, numerical illustrations are provided to visualize the impact of various system parameters.

E. Altman and U. Yechiali, Analysis of customers’ impatience in queues with server vacation. Queueing Syst. 52 (2006) 261–279. | DOI | MR | Zbl

S.I. Ammar, Transient analysis of an $M/M/1$ queue with impatient behavior and multiple vacations. Appl. Math. Comput. 260 (2015) 97–105. | MR | Zbl

A. Azhagappan, Transient behavior of a Markovian queue with working vacation variant reneging and a waiting server. TOP 27 (2019) 351. | DOI

J.H. Dshalalow and A. Merie, Fluctuation analysis in queues with several operational modes and priority customers. TOP 26 (2018) 309–333. | DOI | MR

B. Ganguly, S. Pareek, B. Sarkar, M. Sarkar and M. Omair, Influence of controllable lead time, premium price, and unequal shipments under environmental effects in a supply chain management. RAIRO: OR 53 (2019) 1427–1451. | DOI | Numdam | MR

F.A. Haight, Queueing with balking. Biometrika 44 (1957) 360–369. | DOI | MR | Zbl

M. Jain, Priority queue with batch arrival, balking, threshold recovery, unreliable server and optimal service. RAIRO: OR 51 (2017) 417–432. | DOI | Numdam | MR

M. Kim and B. Sarkar, Multi-stage cleaner production process with quality improvement and lead time dependent ordering cost. J. Cleaner Prod. 144 (2017) 572–590. | DOI

S.J. Kim and B. Sarkar, Supply chain model with stochastic lead time, trade-credit financing, and transportation discounts. Math. Prob. Eng. 2017 (2017) 6465912. | MR | Zbl

B.K. Kumar, P.R. Parthasarthy and M. Sharafali, Transient solution of an $M/M/1$ queue with balking. Queueing Syst. 13 (1993) 441–447. | DOI | MR | Zbl

P.V. Laxmi and K. Jyothsna, Balking and reneging multiple working vacations queue with heterogeneous servers. J. Math. Model. Algorithms 14 (2015) 267–285. | DOI | MR | Zbl

A.I. Malik and B. Sarkar, Optimizing a multi-product continuous-review inventory model with uncertain demand, quality improvement, setup cost reduction, and variation control in lead time. IEEE Access 6 (2018) 36176–36187. | DOI

N. Perel and U. Yechiali, Queues with slow servers and impatient customers. Eur. J. Oper. Res. 201 (2010) 247–258. | DOI | MR | Zbl

B. Sarkar and A. Majumder, Integrated vendor-buyer supply chain model with vendor’s setup cost reduction. Appl. Math. Comput. 224 (2013) 362–371. | MR

B. Sarkar, H. Gupta, K. Chaudhuri and S.K. Goyal, An integrated inventory model with variable lead time, defective units and delay in payments. Appl. Math. Comput. 237 (2014) 650–658. | MR

B. Sarkar, B. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model. J. Manuf. Syst. 35 (2015) 26–36. | DOI

B. Sarkar, A. Majumder, M. Sarkar, B.K. Dey and G. Roy, Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction. J. Ind. Manage. Optim. 13 (2017) 1085–1104. | DOI | MR | Zbl

N. Selvaraju and C. Goswami, Impatient customers in an $M/M/1$ queue with single and multiple working vacations. Comput. Ind. Eng. 65 (2013) 207–215. | DOI

L.D. Servi and S.G. Finn, $M/M/1$ queue with working vacations $(M/M/1/WV)$. Perform. Eval. 50 (2002) 41–52. | DOI

R. Sudhesh and L.F. Raj, Computational analysis of stationary and transient distribution of single server queue with working vacation. Global Trends Comput. Commun. Syst. Commun. Comput. Inf. Sci. 269 (2012) 480–489.

R. Sudhesh and A. Azhagappan, Transient analysis of $M/M/1$ queue with server vacation customers impatient and a waiting server timer. Asian J. Res. Soc. Sci. Humanities 6 (2016) 1096–1104.

R. Sudhesh and A. Azhagappan, Transient analysis of an $M/M/1$ queue with variant impatient behavior and working vacations. Opsearch 55 (2018) 787–806. | DOI | MR

R. Sudhesh and A. Azhagappan, Transient solution of an $M/M{/}^{\infty }$ queue with system’s additional tasks and impatient customers. Int. J. Math. Oper. Res. 16 (2020) 82–97. | DOI | MR

R. Sudhesh, A. Azhagappan and S. Dharmaraja, Transient analysis of $M/M/1$ queue with working vacation heterogeneous service and customers’ impatience. RAIRO: OR 51 (2017) 591–606. | DOI | Numdam | MR | Zbl

S.L. Yu and Z.M. Liu, Analysis of queues in a random environment with impatient customers. Acta Math. Appl. Sinica, English Ser. 33 (2017) 837–850. | DOI | MR | Zbl

Y. Zhang and J. Wang, Equilibrium pricing in an $M/G/1$ retrial queue with reserved idle time and setup time. Appl. Math. Model. 49 (2017) 514–530. | DOI | MR