Abstract
We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities, mean and variance of the system size at time t by employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.
Supported by the National Natural Science Foundation of China (11671204)
References
[1] Whitt W. Untold horrors of the waiting room: What the equilibrium distribution will never tell about the queue-length process. Management Science, 1983, 29(4): 395–408.10.1287/mnsc.29.4.395Search in Google Scholar
[2] Baccelli F, Boyer P, Hebuterne G. Single-server queues with impatient customers. Advances in Applied Probability, 1984, 16(4): 887–905.10.2307/1427345Search in Google Scholar
[3] Daley D J. General customer impatience in the queue GI/G/1. Journal of Applied Probability, 1965, 2(1): 186–205.10.2307/3211884Search in Google Scholar
[4] Gans N, Koole G, Mandelbaum A. Telephone call centers: Tutorial, review, and research prospects. Manufacturing Service Operations Management, 2003, 5(2): 79–141.10.1287/msom.5.2.79.16071Search in Google Scholar
[5] Takács L. A single-server queue with limited virtual waiting time. Journal of Applied Probability, 1974, 11(3): 612–617.10.2307/3212710Search in Google Scholar
[6] Palm C. Methods of judging the annoyance caused by congestion. Tele, 1953(2): 1–20.Search in Google Scholar
[7] Palm R C A. Research on telephone traffic carried by full availability groups. Tele, 1957.Search in Google Scholar
[8] Altman E, Yechiali U. Analysis of customers’ impatience in queues with server vacations. Queueing Systems, 2006, 52(4): 261–279.10.1007/s11134-006-6134-xSearch in Google Scholar
[9] Altman E, Yechiali U. Infinite–server queues with system’s additional tasks and impatient customers. Probability in the Engineering and Informational Sciences, 2008, 22(4): 477–493.10.1017/S0269964808000296Search in Google Scholar
[10] Ammar S I. Transient analysis of an M/M/1 queue with impatient behaviour and multiple vacations. Applied Mathematics and Computation, 2015, 260: 97–105.10.1016/j.amc.2015.03.066Search in Google Scholar
[11] Levy Y, Yechiali U. Utilization of idle time in an M/G/1 queueing system. Management Science, 1975, 22(2): 202–211.10.1287/mnsc.22.2.202Search in Google Scholar
[12] Doshi B T. Queueing systems with vacations — A survey. Queueing Systems, 1986, 1(1): 29–66.10.1007/BF01149327Search in Google Scholar
[13] Doshi B T. Single server queues with vacations. Stochastic Analysis of Computer and Communication Systems, 1990: 217–265.Search in Google Scholar
[14] Tian N S, Zhang G. Vacation queueing models: Theory and Applications. Springer Science & Business Media, New York, 2006.10.1007/978-0-387-33723-4Search in Google Scholar
[15] Ibe O C, Isijola O A. M/M/1 multiple vacation queueing systems with differentiated vacations. Modelling and Simulation in Engineering, 2014. http:dx.doi.org/10.1155/2014/158247.10.1155/2014/158247Search in Google Scholar
[16] Lorentzen L, Waadeland H. Continued fractions with applications. North-Holland, Amsterdam, 1992.Search in Google Scholar
[17] Gradshteyn I S, Ryzhik I M. Table of integrals, series, and products. Academic Press, Amsterdam, 2007.Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- A High-Moment Trapezoidal Fuzzy Random Portfolio Model with Background Risk
- Sequential First-Price Auction with Randomly Arriving Buyers
- Worst-Case Investment Strategy with Delay
- Research on Advertising and Pricing in E-Supply Chain Under Different Dominant Modes
- Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations
- Optimal Insurance-Package and Investment Problem for an Insurer
Articles in the same Issue
- A High-Moment Trapezoidal Fuzzy Random Portfolio Model with Background Risk
- Sequential First-Price Auction with Randomly Arriving Buyers
- Worst-Case Investment Strategy with Delay
- Research on Advertising and Pricing in E-Supply Chain Under Different Dominant Modes
- Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations
- Optimal Insurance-Package and Investment Problem for an Insurer