Abstract
In this paper, a discrete-time single-server queueing system with an infinite waiting room, referred to as theG (G)/Geo/1 model, i.e., a system with general interarrival-time distribution, general arrival bulk-size distribution and geometrical service times, is studied. A method of analysis based on integration along contours in the complex plane is presented. Using this technique, analytical expressions are obtained for the probability generating functions of the system contents at various observation epochs and of the delay and waiting time of an arbitrary customer, assuming a first-come-first-served queueing discipline, under the single restriction that the probability generating function for the interarrival-time distribution be rational. Furthermore, treating several special cases we rediscover a number of well-known results, such as Hunter's result for theG/Geo/1 model. Finally, as an illustration of the generality of the analysis, it is applied to the derivation of the waiting time and the delay of the more generalG (G)/G/1 model and the system contents of a multi-server buffer-system with independent arrivals and random output interruptions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A.O. Allen,Probability, Statistics, and Queueing Theory (Academic Press, New York, 1978).
H. Bruneel and B.G. Kim,Discrete-Time Models for Communication Systems Including ATM (Kluwer Academic, Boston, 1993).
H. Bruneel, A general model for the behaviour of infinite buffers with periodic service opportunities, Eur. J. Oper. Res. 16 (1984) 98–106.
H. Bruneel, Method of analysis for discrete-time buffers with randomly interrupted arrival stream, IEE Proc. part E 131 (1984) 187–193.
H. Bruneel, On buffers with stochastic input and output interruptions, AEU (Electronics and Communication) 38 (1984) 265–271.
J.-F. Chang and R.-F. Chang, The application of the residue theorem to the study of a finite queue with batch Poisson arrivals and synchronous servers, SIAM J. App1. Math. 44 (1984) 626–656.
M.L. Chaudhry, Alternative numerical solutions of stationary queueing-time distributions in discrete-time queues:GI/G/1, J. Oper. Res. Soc. 44 (1993) 1035–1051.
M.L. Chaudhry and J.G.C. Templeton,A First Course in Bulk Queues (Wiley, 1983).
M.O. González,Classical Complex Analysis (Marcel Dekker, New York, 1992).
D. Gross and C.M. Harris,Fundamentals of Queueing Theory (Wiley, New York, 1974).
J.F. Hayes,Modeling and Analysis of Computer Communications Networks (Plenum Press, New York, 1984).
J.J. Hunter,Mathematical Techniques of Applied Probability, Vol. 2:Discrete Time Models: Techniques and Applications (Academic Press, New York, 1983).
L. Kleinrock,Queueing Systems, Vol. I:Theory (Wiley, New York, 1975).
M. Murata and H. Miyahara, An analytic solution of the waiting time distribution for the discrete-timeGI/G/1 queue, Perf. Eval. 13 (1991) 87–95.
B.A. Powell and B. Avi-Itzhak, Queueing systems with enforced idle time, Oper. Res. 15 (1967) 1145–1156.
N.U. Prabhu,Stochastic Storage Processes: Queues, Insurance Risk and Dams, Applications of Mathematics 15 (Springer, New York, 1980).
L. Takács,Introduction to the Theory of Queues (Wiley, New York, 1975).
Author information
Authors and Affiliations
Additional information
Both authors wish to thank the Belgian National Fund for Scientific Research (NFWO) for support of this work.
Rights and permissions
About this article
Cite this article
Vinck, B., Bruneel, H. Analyzing the discrete-timeG (G)/Geo/1 queue using complex contour integration. Queueing Syst 18, 47–67 (1994). https://doi.org/10.1007/BF01158774
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01158774