American Institute of Mathematical Sciences

Advanced
July  2009, 5(3): 431-451. doi: 10.3934/jimo.2009.5.431

M/M/3/3 and M/M/4/4 retrial queues

Tuan Phung-Duc 1, , Hiroyuki Masuyama 1, , Shoji Kasahara 1, and  Yutaka Takahashi 2,

1. 

Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan, Japan, Japan
2. 

Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto 606-8501

Received  September 2008 Revised  December 2008 Published  June 2009

This paper studies M/M/$c$/$c$ retrial queues, where $c$ servers are all identical. In the retrial queues, an arriving customer is served immediately if it finds an idle server upon arrival, otherwise the customer tries to enter the system after an exponentially distributed time independently of other customers. As is well known, it is a challenging problem to obtain an analytical solution for the stationary joint distribution of the numbers of retrial customers and busy servers in the M/M/$c$/$c$ retrial queue especially for $c \ge 3$. Under some technical assumptions, a few analytical solutions have been presented for $c \ge 3$. This paper derives analytical solutions for M/M/3/3 and M/M/4/4 retrial queues without such technical assumptions. Through many numerical examples, we show that the derived analytical solutions can be computed by a numerically stable algorithm.
Keywords: analytical solution, minimal solution, Multiserver retrial queue, continued fractions., three-term recurrence relations.
Mathematics Subject Classification: Primary: 68M20, 90B22; Secondary: 60K2.
Citation: 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
[1]

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
[2]

Laura Luzzi, Stefano Marmi. On the entropy of Japanese continued fractions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 673-711. doi: 10.3934/dcds.2008.20.673
[3]

Pierre Arnoux, Thomas A. Schmidt. Commensurable continued fractions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4389-4418. doi: 10.3934/dcds.2014.34.4389
[4]

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
[5]

Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313
[6]

Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1477-1498. doi: 10.3934/dcds.2013.33.1477
[7]

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
[8]

Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121
[9]

Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098
[10]

Marc Kessböhmer, Bernd O. Stratmann. On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2437-2451. doi: 10.3934/dcds.2012.32.2437
[11]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417
[12]

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
[13]

Alain Haraux. On the fast solution of evolution equations with a rapidly decaying source term. Mathematical Control & Related Fields, 2011, 1 (1) : 1-20. doi: 10.3934/mcrf.2011.1.1
[14]

Hua Zhong, Chunlai Mu, Ke Lin. Global weak solution and boundedness in a three-dimensional competing chemotaxis. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3875-3898. doi: 10.3934/dcds.2018168
[15]

Richard D. Neidinger. Efficient recurrence relations for univariate and multivariate Taylor series coefficients. Conference Publications, 2013, 2013 (special) : 587-596. doi: 10.3934/proc.2013.2013.587
[16]

Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060
[17]

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
[18]

Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571
[19]

Taebeom Kim, Sunčica Čanić, Giovanna Guidoboni. Existence and uniqueness of a solution to a three-dimensional axially symmetric Biot problem arising in modeling blood flow. Communications on Pure & Applied Analysis, 2010, 9 (4) : 839-865. doi: 10.3934/cpaa.2010.9.839
[20]

Biswajit Basu. On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4783-4796. doi: 10.3934/dcds.2019195

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences