M/G/1 Vacation Queueing Systems with Server Timeout


We consider a single-server vacation queueing system that operates in the following manner. When the server returns from a vacation, it observes the following rule. If there is at least one customer in the system, the server commences service and serves exhaustively before taking another vacation. If the server finds the system empty, it waits a fixed time c. At the expiration of this time, the server commences another vacation if no customer has arrived; otherwise, it serves exhaustively before commencing another vacation. Analytical results are derived for the mean waiting time in the system. The timeout scheme is shown to be a generalized scheme of which both the single vacation and multiple vacations schemes are special cases, with c= and c=0, respectively. The model is extended to the N-policy vacation queueing system.

Share and Cite:

Ibe, O. (2015) M/G/1 Vacation Queueing Systems with Server Timeout. American Journal of Operations Research, 5, 77-88. doi: 10.4236/ajor.2015.52007.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Levy, Y. and Yechiali, U. (1975) Utilization of Idle Time in an M/G/1 Queueing System. Management Science, 22, 202-211.
[2] Doshi, B.T. (1986) Queueing Systems with Vacations, a Survey. Queueing Systems, 1, 29-66.
[3] Doshi, B.T. (1990) Single-Server Queues with Vacations. In: Takagi, H., Ed., Stochastic Analysis of Computer and Communications Systems, Elsevier, Amsterdam.
[4] Takagi, H. (1991) Queueing Analysis: A Foundation of Performance Analysis, Volume 1: Vacation and Priority Systems, Part 1. Elsevier Science Publishers B.V., Amsterdam.
[5] Tian, N. and Zhang, G. (2006) Vacation Queueing Models: Theory and Applications. Springer-Verlag, New York.
[6] Fuhrmann, S.W. and Cooper, R.B. (1985) Stochastic Decomposition in M/G/1 Queue with Generalized Vacations. Operations Research, 33, 1117-1129.
[7] Servi, L.D. and Finn, S.G. (2002) M/M/1 Queue with Working Vacations (M/M/1/WV). Performance Evaluation, 50, 41-52.
[8] Wu, D. and Takagi, H. (2006) M/G/1 Queue with Multiple Working Vacation. Performance Evaluation, 63, 654-681.
[9] Baba, Y. (2005) Analysis of a GI/M/1 Queue with Multiple Working Vacations. Operations Research Letters, 33, 201- 209.
[10] Banik, A., Gupta, U. and Pathak, S. (2007) On the GI/M/1/N Queue with Multiple Working Vacations-Analytic Analysis and Computation. Applied Mathematical Modelling, 31, 1701-1710.
[11] Liu, W., Xu, X. and Tian, N. (2007) Stochastic Decompositions in the M/M/1 Queue with Working Vacations. Operations Research Letters, 35, 595-600.
[12] Krishnamoorthy, A. and Sreenivasan, C. (2012) An M/M/2 Queueing System with Heterogeneous Servers including One with Working Vacation. International Journal of Stochastic Analysis, 2012, Article ID: 145867.
[13] Ibe, O.C. and Isijola, O.A. (2014) M/M/1 Multiple Vacation Queueing Systems with Differentiated Vacations. Modeling and Simulation in Engineering, 2014, Article 158247.
[14] Li, J. and Tian, N. (2007) The M/M/1 Queue with Working Vacation and Vacation Interruption. Journal of Systems Science and Systems Engineering, 16, 121-127.
[15] Li, J. and Tian, N. (2007) The Discrete-Time GI/Geo/1 Queue with Working Vacations and Vacation Interruption. Applied Mathematics and Computation, 185, 1-10.
[16] Li, J., Tian, N. and Ma, Z. (2008) Performance Analysis of GI/M/1 Queue with Working Vacations and Vacation Interruption. Applied Mathematical Modelling, 32, 2715-2730.
[17] Zhang, M. and Hou, Z. (2010) Performance Analysis of M/G/1 Queue with Working Vacations and Vacation Interruption. Journal of Computational and Applied Mathematics, 234, 2977-2985.
[18] Sreenivasan, C., Chakravarthy, S.R. and Krishnamoorthy, A. (2013) MAP/PH/1 Queue with Working Vacations, Vacation Interruptions and N-Policy. Applied Mathematical Modeling, 37, 3879-3893.
[19] Isijola-Adakeja, O.A. and Ibe, O.C. (2014) M/M/1 Multiple Vacation Queueing Systems with Differentiated Vacations and Vacation Interruptions. IEEE Access, 2, 1384-1395.
[20] Ibe, O.C. and Trivedi, K.S. (1991) Stochastic Petri Net Analysis of Finite-Population Vacation Queueing Systems. Queueing Systems, 8, 111-128.
[21] Kella, O. (1990) Optimal Control of the Vacation Scheme in an M/G/1 Queue. Operations Research Letters, 38, 724- 728.
[22] Ibe, O.C. (2014) Fundamentals of Applied Probability and Random Processes. 2nd Edition, Elsevier Academic Press, Waltham.
[23] Yadin, M. and Naor, P. (1963) Queueing Systems with a Removable Service Station. Operational Research Quarterly, 14, 395-405.
[24] Kleinrock, L. (1975) Queueing Systems, Volume 1: Theory. John Wiley & Sons, New York.
[25] Little, J.D.C. (1961) A Proof of the Formula: L = λW. Operations Research, 9, 383-387.

Copyright © 2020 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.