Abstract
We study a discrete-time single-server queue where batches of messages arrive. Each message consists of a geometrically distributed number of packets which do not arrive at the same instant and which require a time unit as service time. We consider the cases of constant spacing and geometrically distributed (random) spacing between consecutive packets of a message. For the probability generating function of the stationary distribution of the embedded Markov chain we derive in both cases a functional equation which involves a boundary function. The stationary mean number of packets in the system can be computed via this boundary function without solving the functional equation. In case of constant (random) spacing the boundary function can be determined by solving a finite-dimensional (an infinite-dimensional) system of linear equations numerically.
For Poisson- and Bernoulli-distributed arrivals of messages numerical results are presented. Further, limiting results are derived.
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References
D. Anick, D. Mitra and M.M. Sondhi, Stochastic theory of a data handling system with multiple sources, Bell System Techn. J. 61 (1982) 1871–1894.
B. van Arem, Discrete time analysis of a slotted transmission system: theoretical solution, Memorandum 624, University of Twente, Enschede, The Netherlands (1987).
B. van Arem, Discrete time analysis of a slotted transmission system: numerical solution; Memorandum 625, University of Twente, Enschede, The Netherlands (1987).
B. van Arem, Discrete time analysis of a slotted transmission system, Comput. Oper. Res. 16 (1989) 101–111.
S. Asmussen and H. Johansen, On a robustness problem for the GI/GI/s queue (in German), Elektron. Informationsverarb. Kybernetik 22 (1986) 565–570.
G.P. Basharin and V.A. Efimushkin, Discrete time one-server queueing system with calls of several types (in Russian), Problemy Peredači Informacii 20 (1984) 95–104.
P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).
A.A. Borovkov,Asymptotic Methods in Queueing Theory (Wiley, Chichester, 1984).
A. Brandt, M. Brandt and H. Sulanke, A single server model for packetwise transmission of messages: Analytical solution for the Poisson case, Preprint Nr. 229, Humboldt-Universität zu Berlin (1989).
A. Brandt, P. Franken and B. Lisek,Stationary Stochastic Models (Akademie-Verlag, Berlin, to appear).
H. Bruneel, Some remarks on discrete-time buffers with correlated arrivals, Comput. Oper. Res. 12 (1985) 445–458.
J.H.A. de Smit, The single server semi-Markov queue, Stoch. Proc. Appl. 22 (1986) 37–50.
K. Fendick, V. Saksena and W. Whitt, Dependence in packet queues: A multi-class batch-Poisson model,Proc. 12th Int. Teletraffic Congress, Torino (1988) vol. 1.
J.-R. Louvion, P. Boyer and G. Gravey, A discrete-time single server queue with Bernoulli arrivals and constant service time, ibid., vol. 2.
M.F. Neuts, Some explicit formulas for the steady state behaviour of the queue with semi-Markovian service times, Adv. Appl. Prob. 9 (1977) 141–157.
L. Takács, A storage process with semi-Markov input, Adv. Appl. Prob. 7 (1975) 830–844.
L. Takács, A Banach space of matrix functions and its application in the theory of queues, Sankhya Ser. A 38 (1976) 201–211.
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Brandt, A., Brandt, M. & Sulanke, H. A single server model for packetwise transmission of messages. Queueing Syst 6, 287–310 (1990). https://doi.org/10.1007/BF02411479
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DOI: https://doi.org/10.1007/BF02411479