Abstract
This paper studies a discrete-time batch arrival GI/Geo/1 queue where the server may take multiple vacations depending on the state of the queue/system. However, during the vacation period, the server does not remain idle and serves the customers with a rate lower than the usual service rate. The vacation time and the service time during working vacations are geometrically distributed. Keeping note of the specific nature of the arrivals and departures in a discrete-time queue, we study the model under late arrival system with delayed access and early arrival system independently. We formulate the system using supplementary variable technique and apply the theory of difference equation to obtain closed-form expressions of steady-state system content distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the associated characteristic equations. We discuss the stability conditions of the system and develop few performance measures as well. Through some numerical examples, we illustrate the feasibility of our theoretical work and highlight the asymptotic behavior of the probability distributions at pre-arrival epochs. We further discuss the impact of various parameters on the performance of the system. The model considered in this paper covers a wide class of vacation and non-vacation queueing models which have been studied in the literature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abolnikov L, Dukhovny A (1991) Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications. Int Jof Stoch Anal 4(4):333–355
Baba Y (2005) Analysis of a GI/M/1 queue with multiple working vacations. Oper Res Lett 33(2):201– 209
Banik A, Gupta U, Pathak S (2007) On the GI/M/1/N queue with multiple working vacations — analytic analysis and computation. Appl Math Model 31(9):1701–1710
Chaudhry ML (2000) On numerical computations of some discrete-time queues. computational probability. In: Grassmann, WK (ed). Springer Science & Business Media
Chaudhry ML, Gupta UC (1997) Queue-length and waiting-time distributions of discrete-time GIX/Geom/1 queueing systems with early and late arrivals. Queue Syst 25(1–4):307–324
Chaudhry M, Gupta U, Templeton JG (1996) On the relations among the distributions at different epochs for discrete-time GI/Geom/1 queues. Oper Res Lett 18(5):247–255
Chaudhry ML, Samanta S, Pacheco A (2012) Analytically explicit results for the \({{GI}}/C-{{MSP}}/1/\infty \) queueing system using roots. Probab Eng Inform Sci 26(2):221–244
Chaudhry M, Banik AD, Pacheco A, Ghosh S (2016) A simple analysis of system characteristics in the batch service queue with infinite-buffer and markovian service process using the roots method: \({{GI}}/C-{{MSP}}^{(a, b)}/1/\infty \). RAIRO-Oper Res 50(3):519–551
Cheng C, Li J, Wang Y (2015) An energy-saving task scheduling strategy based on vacation queuing theory in cloud computing. Tsinghua Sci Technol 20(1):28–39
Doshi BT (1986) Queueing systems with vacations — a survey. Queue Syst 1(1):29–66
Elaydi S (2005) An introduction to difference equations. Springer, New York
Fiems D, Bruneel H (2002) Analysis of a discrete-time queueing system with timed vacations. Queue Syst 42(3):243–254
Gao S, Wang J, Zhang D (2013) Discrete-time GIX/Geo/1/N queue with negative customers and multiple working vacations. J Korean Statist Soc 42(4):515–528
Goswami V, Mund G (2010) Analysis of a discrete-time GI/Geo/1/N queue with multiple working vacations. J Syst Sci Syst Eng 19(3):367–384
Goswami V, Mund G (2011) Analysis of discrete-time batch service renewal input queue with multiple working vacations. Comput Indus Eng 61(3):629–636
Gravey A, Hebuterne G (1992) Simultaneity in discrete-time single server queues with bernoulli inputs. Perform Eval 14(2):123–131
Guha D, Banik AD (2013) On the renewal input batch-arrival queue under single and multiple working vacation policy with application to epon. INFOR: Inf Syst Oper Res 51(4):175– 191
Hunter JJ (1983) Mathematical techniques of applied probability: discrete time models, techniques and applications. Academic Press
Ke JC, Wu CH, Zhang ZG (2010) Recent developments in vacation queueing models: a short survey. Int J Oper Res 7(4):3–8
Li JH, Tian NS, Liu WY (2007) Discrete-time GI/Geo/1 queue with multiple working vacations. Queue Syst 56(1):53–63
Li J h, Wq Liu, Ns Tian (2010) Steady-state analysis of a discrete-time batch arrival queue with working vacations. Perform Eval 67(10):897–912
Neuts MF (1994) Matrix-geometric solutions in stochastic models: an algorithmic approach. Courier Corporation
Samanta SK, Chaudhry ML, Gupta UC (2007a) Discrete-time GeoX/G(a,b)/1/N queues with single and multiple vacations. Math Comput Model 45(1–2):93–108
Samanta SK, Gupta U, Sharma R (2007b) Analysis of finite capacity discrete-time GI/Geo/1 queueing system with multiple vacations. J Oper Res Soc 58(3):368–377
Servi LD, Finn SG (2002) M/M/1 queues with working vacations (M/M/1/WV). Perform Eval 50(1):41–52
Takagi H (1993) Queuing analysis: a foundation of performance evaluation. Discrete time systems, vol 3. North-Holland
Tian N, Zhang ZG (2002) The discrete-time GI/Geo/1 queue with multiple vacations. Queue Syst 40(3):283–294
Tian N, Zhang ZG (2006) Vacation queueing models: theory and applications, vol 93. Springer Science & Business Media
Vilaplana J, Solsona F, Teixidó I, Mateo J, Abella F, Rius J (2014) A queuing theory model for cloud computing. J Supercomput 69(1):492–507
Ye Q, Liu L (2016) Performance analysis of the GI/M/1 queue with single working vacation and vacations. Methodol Comput Appl Probab 19(3):685–714
Yu MM, Tang YH, Fu YH (2009) Steady state analysis and computation of the GIX/Mb/1/L queue with multiple working vacations and partial batch rejection. Comput Indus Eng 56 (4):1243– 1253
Zhang ZG, Tian N (2001) Discrete time Geo/G/1 queue with multiple adaptive vacations. Queue Syst 38(4):419–429
Acknowledgements
The author F. P. Barbhuiya is grateful to the Indian Institute of Technology Kharagpur, India for the financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Barbhuiya, F.P., Gupta, U.C. A Discrete-Time GIX/Geo/1 Queue with Multiple Working Vacations Under Late and Early Arrival System. Methodol Comput Appl Probab 22, 599–624 (2020). https://doi.org/10.1007/s11009-019-09724-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-019-09724-6
Keywords
- Bulk arrival
- Difference equation method
- Discrete-time
- GI/Geo/1 queue
- Multiple working vacations
- Supplementary variable technique