Analysis of a discrete-time GI/Geo/1/N queue with multiple working vacations

Abstract

This paper analyzes a finite-buffer renewal input single server discrete-time queueing system with multiple working vacations. The server works at a different rate rather than completely stopping working during the multiple working vacations. The service times during a service period, service time during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. The analysis of actual waiting-time distribution and some performance measures are carried out. We present some numerical results and discuss special cases of the model.

Keywords

Discrete-time / finite-buffer / working vacations / supplementary variable / waiting-time

References

[1] Alfa A.S.. Vacation models in discrete time. Queueing Systems, 2003, 55: 5-30.

[2] Baba Y.. Analysis of GI/M/1 queue with multiple working vacations. Operations Research Letters, 2005, 33: 201-209.

[3] Banik A.D., Gupta U.C., Pathak S.S.. On the GI/M/1/N queue with multiple working vacations - analytic analysis and computation. Applied Mathematical Modelling, 2007, 31: 1701-1710.

[4] Bruneel H., Kim B.G.. Discrete-Time Models for Communication Systems Including ATM, 1983, Boston: Kluwer Academic Publishers

[5] Chae K.C., Lee S.M., Lee H.W.. On stochastic decomposition in the GI/M/1 queue with exponential vacations. Operations Research Letters, 2007, 35: 114-118.

[6] Chae K.C., Lim D.E., Yang W.S.. The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation. Performance Evaluation, 2009, 66: 356-367.

[7] Chatterjee U., Mukherjee S.P.. GI/M/1 queue with server vacations. Journal of the Operational Research Society, 1994, 37: 171-181.

[8] Chaudhry M.L., Gupta U.C.. Performance analysis of the discrete-time GI/Geom/1/N queue. Journal of Applied Probability, 1996, 33: 239-255.

[9] Doshi B.T.. Queueing systems with vacations - a survey. Queueing Systems, 1986, 1: 29-66.

[10] Gravey A., Hebuterne G.. Simultaneity in discrete time single server queues with Bernoulli inputs. Performance Evaluation, 1992, 14: 123-131.

[11] Hunter J.J.. Mathematical Techniques of Applied Probability, vol-II, Discrete Time Models: Techniques and Applications, 1983, New York: Academic Press

[12] Karaesmen F., Gupta S. M.. The finite capacity GI/M/1 queue with server vacations. Journal of the Operational Research Society, 1996, 47: 817-828.

[13] Latouche G., Ramaswami V.. Introduction to Matrix Analytic Methods in Stochastic Modelling, 1999, Philadelphia: SIAM & ASA

[14] Li J., Tian N.. Analysis of the discrete time Geo/Geo/1 queue with single working vacation. Quality Technology & Quantitative Management, 2008, 5: 77-89.

[15] Li J., Tian N., Liu W.. Discrete-time GI/Geo/1 queue with multiple working vacations. Queueing Systems, 2007, 56: 53-63.

[16] Li J., Tian N.. The discrete-time GI/Geo/1 queue with working vacations and vacation interruption. Applied Mathematics and Computation, 2007, 185: 1-10.

[17] Samanta S.K., Gupta U.C., Sharma R.K.. Analysis of finite capacity discrete-time GI/Geo/1 queueing system with multiple vacations. Journal of the Operational Research Society, 2007, 58: 368-377.

[18] Servi L.D., Finn S.G.. M/M/1 queue with working vacations (M/M/1/WV). Performance Evaluation, 2002, 50: 41-52.

[19] Takagi H.. Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority Systems, Part I, 1991, Amsterdam: North Holland Elsevier

[20] Takagi H.. Queueing Analysis - Discrete time Systems, 1993, Amsterdam: North Holland Elsevier

[21] Tian N., Zhang D., Cao C.. The GI/M/1 queue with exponential vacations. Queueing Systems, 1989, 5: 331-144.

[22] Tian N., Zhang Z.G.. The discrete-time GI/Geo/1 queue with multiple vacations. Queueing Systems, 2002, 40: 283-294.

[23] Tian N., Zhang Z.G.. Vacation Queueing Models: Theory and Applications, 2006, New York: Springer-Verlag.

[24] Tian N., Ma M., Liu M.. The discrete time Geo/Geo/1 queue with multiple working vacations. Applied Mathematical Modelling, 2008, 32: 2941-2953.

[25] Woodward M.E.. Communication and Computer Networks: Modelling with Discrete-Time Queues, 1994, Los Alamitos: CA: IEEE Computer Society Press

[26] Wu D., Takagi H.. M/G/1 queue with multiple working vacations. Performance Evaluation, 2006, 63: 654-681.