Abstract
This paper introduces a unified framework for the analysis of a class of random allocation processes that include: (i) the birthday paradox; (ii) the coupon collector problem; (iii) least-recently-used (LRU) caching in memory management systems under the independent reference model; (iv) the move-to-front heuristic of self-organizing search. All analyses are relative to general (non uniform) probability distributions.
Our approach to these problems comprises two stages. First, the probabilistic phenomena of interest are described by means of regular languages extended by addition of the shuffle product. Next, systematic translation mechanisms from languages to generating functions are used to derive integral representations of expectations and probability distributions for allocation processes.
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Flajolet, P., Gardy, D., Thimonier, L. (1988). Random allocations and probabilistic languages. In: Lepistö, T., Salomaa, A. (eds) Automata, Languages and Programming. ICALP 1988. Lecture Notes in Computer Science, vol 317. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-19488-6_120
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DOI: https://doi.org/10.1007/3-540-19488-6_120
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