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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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