# American Institute of Mathematical Sciences

October  2012, 8(4): 861-875. doi: 10.3934/jimo.2012.8.861

## On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations

 1 Department of Mathematics, Beijing Jiaotong University, 100044 Beijing 2 Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506, United States

Received  September 2011 Revised  July 2012 Published  September 2012

A single-server retrial queue with two types of customers in which the server is subject to vacations along with breakdowns and repairs is studied. Two types of customers arrive to the system in accordance with two different independent Poisson flows. The service times of the two types of customers have two different independent general distributions. We assume that when a service is completed, the server will take vacations after an exponentially distributed reserved time. It is assumed that the server has an exponentially distributed lifetime, a generally distributed vacation time and a generally distributed repair time. There is no waiting space in front of the server, therefore, if the server is found busy, or on vacation, or down, the blocked two types of customers form two sources of repeated customers. Explicit expressions are derived for the expected number of retrial customers of each type. Additionally, by assuming both types of customers face linear costs for waiting and retrial, we discuss and compare the optimal and equilibrium retrial rates regarding the situations in which the customers are cooperative or noncooperative, respectively.
Citation: Feng Zhang, Jinting Wang, Bin Liu. On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 861-875. doi: 10.3934/jimo.2012.8.861
##### References:

show all references

##### References:
 [1] Jinting Wang, Linfei Zhao, Feng Zhang. Analysis of the finite source retrial queues with server breakdowns and repairs. Journal of Industrial & Management Optimization, 2011, 7 (3) : 655-676. doi: 10.3934/jimo.2011.7.655 [2] Arnaud Devos, Joris Walraevens, Tuan Phung-Duc, Herwig Bruneel. Analysis of the queue lengths in a priority retrial queue with constant retrial policy. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-30. doi: 10.3934/jimo.2019082 [3] Tuan Phung-Duc, Ken'ichi Kawanishi. Multiserver retrial queue with setup time and its application to data centers. Journal of Industrial & Management Optimization, 2019, 15 (1) : 15-35. doi: 10.3934/jimo.2018030 [4] Dhanya Shajin, A. N. Dudin, Olga Dudina, A. Krishnamoorthy. A two-priority single server retrial queue with additional items. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2019085 [5] Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102 [6] Gang Chen, Zaiming Liu, Jinbiao Wu. Optimal threshold control of a retrial queueing system with finite buffer. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1537-1552. doi: 10.3934/jimo.2017006 [7] Tuan Phung-Duc, Ken’ichi Kawanishi. Multiserver retrial queues with after-call work. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 639-656. doi: 10.3934/naco.2011.1.639 [8] Tuan Phung-Duc. Single server retrial queues with setup time. Journal of Industrial & Management Optimization, 2017, (3) : 1329-1345. doi: 10.3934/jimo.2016075 [9] Jesus R. Artalejo, Tuan Phung-Duc. Markovian retrial queues with two way communication. Journal of Industrial & Management Optimization, 2012, 8 (4) : 781-806. doi: 10.3934/jimo.2012.8.781 [10] Tuan Phung-Duc, Wouter Rogiest, Sabine Wittevrongel. Single server retrial queues with speed scaling: Analysis and performance evaluation. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1927-1943. doi: 10.3934/jimo.2017025 [11] Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. M/M/3/3 and M/M/4/4 retrial queues. Journal of Industrial & Management Optimization, 2009, 5 (3) : 431-451. doi: 10.3934/jimo.2009.5.431 [12] Bara Kim. Stability of a retrial queueing network with different classes of customers and restricted resource pooling. Journal of Industrial & Management Optimization, 2011, 7 (3) : 753-765. doi: 10.3934/jimo.2011.7.753 [13] Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. State-dependent M/M/c/c + r retrial queues with Bernoulli abandonment. Journal of Industrial & Management Optimization, 2010, 6 (3) : 517-540. doi: 10.3934/jimo.2010.6.517 [14] Madhu Jain, Sudeep Singh Sanga. Admission control for finite capacity queueing model with general retrial times and state-dependent rates. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-25. doi: 10.3934/jimo.2019073 [15] Gopinath Panda, Veena Goswami, Abhijit Datta Banik, Dibyajyoti Guha. Equilibrium balking strategies in renewal input queue with Bernoulli-schedule controlled vacation and vacation interruption. Journal of Industrial & Management Optimization, 2016, 12 (3) : 851-878. doi: 10.3934/jimo.2016.12.851 [16] Sung-Seok Ko. A nonhomogeneous quasi-birth-death process approach for an $(s, S)$ policy for a perishable inventory system with retrial demands. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019009 [17] Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721 [18] Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076 [19] Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331 [20] Linlin Tian, Xiaoyi Zhang, Yizhou Bai. Optimal dividend of compound poisson process under a stochastic interest rate. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019047

2018 Impact Factor: 1.025