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Distribution of the final extent of a rumour process. (English) Zbl 0796.60098
Summary: A rumour model due to {\it D. P. Maki} and {\it M. Thompson} [Mathematical models and applications (Prentice-Hall, 1973)] is slightly modified to incorporate a continuous-time random contact process and varying individual behaviours in front of the rumour. Two important measures of the final extent of the rumour are provided by the ultimate number of people who have heard the rumour, and the total personal time units during which the rumour is spread. Our purpose is to derive the exact joint distribution of these two statistics. That will be done by constructing a family of martingales for the rumour process and then using a particular family of Gontcharoff polynomials.

60K30Applications of queueing theory
60J05Discrete-time Markov processes on general state spaces
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