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On the thrashing cusp in slotted Aloha systems. (English) Zbl 0582.94002
We develop a new tool for performance evaluation of a multiaccess communication system. We give an explicit detailed analytic description of a cusp catastrophe in a computer communication system. An approximate model is formulated for a slotted Aloha system assuming a single buffer, based on the Markovian model studied by S. S. Lam [”Packet switching in a multiaccess broadcast channel with application to satellite communication in a computer network”, Ph. D. Thesis, Dep. Comput. Sci., Univ. California (1974)] and A. B. Carleial and M. E. Hellman [IEEE Trans. Commun. COM-23, 401-410 (1975; Zbl 0347.94002)].
Next, we outline the research on quantitative processing of thrashing and its results. Sudden changes, which can be observed in the throughput rate, the average delay, and the average number of backlogged users, are induced by smooth alterations of control parameters in the behavior of a slotted Aloha system. These catastrophic phenomena, such as long lasting periods of vanishing throughput rate and very high delays, are analyzed in line with catastrophe theory. The system behavior in a slotted Aloha system is characterized by the cusp catastrophe. In particular, introducing a new parameter $$\sigma$$ in our model, we make it clear why and in what cases these catastrophic phenomena occur. The amount of change is also estimated.
##### MSC:
 94A05 Communication theory 68N99 Theory of software 58K35 Catastrophe theory 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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