Abstract
This paper presents an alternative steady-state solution to the discrete-time Geo/G/1/N + 1 queueing system using roots. The analysis has been carried out for a late-arrival system using the imbedded Markov chain method, and the solutions for the early arrival system have been obtained from those of the late-arrival system. Using roots of the associated characteristic equation, the distributions of the numbers in the system at various epochs are determined. We find a unified approach for solving both finite- and infinite- buffer systems. We investigate the measures of effectiveness and provide numerical illustrations. We establish that, in the limiting case, the results thus obtained converge to the results of the continuous-time counterparts. The applications of discrete-time queues in modeling slotted digital computer and communication systems make the contributions of this paper relevant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akar N (2006) Solving the M E/M E/1 queue with state-space methods and the matrix sign function. Perform Eval 63 (2):131–145
Alfa A (2016) Applied discrete-time queues. Springer, Berlin
Bhat UN (2015) An introduction to queueing theory: modeling and analysis in applications. Birkhäuser, Basel
Bruneel H, Kim BG (2012) Discrete-time models for communication systems including ATM, vol 205. Springer, Berlin
Chaudhry ML (1992) Qroot software package. A & A Publications, Kingston
Chaudhry ML (2000) On numerical computations of some discrete-time queues. In: Grassmann W (ed) Computational probability, vol 10. Kluwer Academic, Boston, pp 365–407
Chaudhry ML, Harris CM, Marchal WG (1990) Robustness of rootfinding in single-server queueing models. ORSA J Comput 2(3):273–286
Chaudhry ML, Gupta UC, Agarwal M (1991) On exact computational analysis of distributions of numbers in systems for M/G/1/N+ 1 and G I/M/1/N + 1 queues using roots. Comput Oper Res 18(8):679–694
Finch PD (1958) The effect of the size of the waiting room on a simple queue. J Royal Stat Soc Ser B (Methodological):182–186
Gravey A, Hébuterne G (1992) Simultaneity in discrete-time single server queues with bernoulli inputs. Perform Eval 14(2):123–131
Kobayashi H, Mark BL (2009), System modeling and analysis: Foundations of system performance evaluation. Pearson Education India
Kobayashi H, Mark BL, Turin W (2011) Probability, random processes and statistical analysis: applications to communications, Signal processing, queueing theory and mathematical finance. Cambridge University Press, Cambridge
Lee HW, Lee SH, Yoon SH, Ahn B, Park NI (1999) A recursive method for bernoulli arrival queues and its application to partial buffer sharing in atm. Comput Oper Res 26(6):559–581
Neuts MF (1981) Matrix-geometric solutions in stochastic models: an algorithmic approach. Courier Corporation
Singh G, Gupta U, Chaudhry ML (2016) Detailed computational analysis of queueing-time distributions of the B M A P/G/1 queue using roots. J Appl Probab 53(4):1078–1097
Takagi H (1993a) Queueing analysis: Discrete-time systems, vol 3. North Holland
Takagi H (1993b) Queueing Analysis: Finite systems, vol 2. North Holland
Woodward ME (1994) Communication and computer networks: modelling with discrete-time queues. Wiley-IEEE Computer Society Press, New York
Acknowledgements
This research was supported (in part) by the Department of National Defense Applied Research Program grant GRC0000B1638.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chaudhry, M.L., Goswami, V. The Queue Geo/G/1/N + 1 Revisited. Methodol Comput Appl Probab 21, 155–168 (2019). https://doi.org/10.1007/s11009-018-9645-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-018-9645-0