Stationary Analysis of Geo/Geo/1 Queue with Two-Speed Service and the Optimal Switching Threshold for the Service Rate


This paper considers a Geo/Geo/1 queueing system with infinite capacity, in which the service rate changes depending on the workload. Initially, when the number of customers in the system is less than a certain threshold L, low service rate is provided for cost saving. On the other hand, the high service rate is activated as soon as L customers accumulate in the system and such service rate is preserved until the system becomes completely empty even if the number of customers falls below L. The steady-state probability distribution and the expected number of customers in the system are derived. Through the first-step argument, a recursive algorithm for computing the first moment of the conditional sojourn time is obtained. Furthermore, employing the results of regeneration cycle analysis, the direct search method is also implemented to determine the optimal value of L for minimizing the long-run average cost rate function.

Share and Cite:

Lin, X. (2015) Stationary Analysis of Geo/Geo/1 Queue with Two-Speed Service and the Optimal Switching Threshold for the Service Rate. Applied Mathematics, 6, 908-921. doi: 10.4236/am.2015.66083.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Bekker, R., Borst, S., Boxma, O. and Kella, O. (2004) Queues with Workload-Dependent Arrival and Service Rates. Queueing Systems, 46, 537-556.
[2] Chaudhry, M.L. and Gupta, U.C. (1996) On the Analysis of the Discrete-Time Geom(n)/G(n)/1/N Queue. Probability in the Engineering and Informational Sciences, 10, 415-428.
[3] Chaudhry, M.L., Templeton, J.G.C. and Gupta, U.C. (1996) Analysis of the Discrete-Time GI(n)/Geom(n)/1/N Queue. Computers & Mathematics with Applications, 31, 59-68.
[4] Garg, R.L. and Singh, P. (1993) Queue-Dependent servers Queueing System. Microelectronics Reliability, 33, 2289-2295.
[5] Gebhard, R.F. (1967) A Queueing Process with Bilevel Hysteretic Service-Rate Control. Naval Research Logistics Quarterly, 14, 55-67.
[6] Gross, D. and Harris, C.M. (1985) Fundamentals of Queueing Theory. 2nd Edition, John Wiley, New York.
[7] Harris, C.M. and Marchal, W.G. (1988) State Dependence in M/G/1 Server-Vacation Models. Operations Research, 36, 560-565.
[8] Hunter, J.J. (1983) Mathematical Techniques of Applied Probability, Discrete-Time Models: Techniques and Applications. Vol. II, Academic Press, New York.
[9] Jain, M. (2005) Finite Capacity M/M/r Queueing System with Queue Dependent Servers. Computers & Mathematics with Applications, 50, 187-199.
[10] Lin, C.H. and Ke, J.C. (2011) Optimization Analysis for an Infinite Capacity Queueing System with Multiple Queue-Dependent Servers: Genetic Algorithm. International Journal of Computer Mathematics, 88, 1430-1442.
[11] Parthasarathy, P.R. and Lenin, R.B. (1999) Exact Busy Period Distribution of a Discrete Queue with Quadratic Rates. International Journal of Computer Mathematics, 71, 427-436.
[12] Saaty, T.L. (1961) Elementary of Queueing Theory with Applications. McGraw-Hill, New York.
[13] Singh, V.P. (1973) Queue-Dependent Servers. Journal of Engineering Mathematics, 7, 123-126.
[14] Takagi, H. (1993) Queueing Analysis: A Foundation of Performance Evaluation. Vol. 3, North-Holland, New York.
[15] Wang, K.H. and Tai, K.Y. (2000) A Queueing System with Queue-Dependent Servers and Finite Capacity. Applied Mathematical Modelling, 24, 807-814.
[16] William, J.G. and Wang, P. (1992) An M/G/1-Type Queueing Model with Service Times Depending on Queue Length. Applied Mathematical Modelling, 16, 652-658.
[17] Zhernovyi, Y.V. (2012) Stationary Characteristics of MX/M/1 Systems with Two-Speed Service. Journal of Communications Technology and Electronics, 57, 920-931.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.