Fast algorithms for generating discrete random variates with changing distributions

One of the most fundamental operations when simulating a stochastic discrete-event dynamic system is the generation of a nonuniform discrete random variate. The simplest form of this operation can be stated as follows: Generate a random variable X that is distributed over the integers 1,2,…,<italic>n</italic> such that P(X=<italic>i</italic>) = <italic>a<subscrpt>i</subscrpt></italic>/(<italic>a</italic><subscrpt>1</subscrpt> +…+<italic>a<subscrpt>n</subscrpt></italic>), where <italic>a<subscrpt>i</subscrpt></italic>'s are fixed nonnegative numbers. The well-known “alias algorithm” is available to accomplish this task in O(1) time. A more difficult problem is to generate variates for X when the <italic>a<subscrpt>i</subscrpt></italic>'s are changing with time. We present three rejection-based algorithms for this task, and for each algorithm we characterize the performance in terms of acceptance probability and the expected effort to generate a variate. We show that, under fairly unrestrictive conditions, the long-run expected effort is O(1). Applications to Markovian queuing networks are discussed. We also compare the three algorithms with competing schemes appearing in the literature.

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