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A broad array of data series have embedded discrete event?occurrences, cumulative counts of which can be viewed as superpositions $$N(t) = \sum {_{i = 1}^n } N_i (t)$$ of independent identically distributed (iid) counting processes with intensities of the form $$\lambda _i (t) = h(Z_i ,V(t),\vartheta )$$ , where ? is unknown, n may be unknown, Z i are unobservable iid vectors, and V(t) is an observable "environmental" vector process, but the function h is known or hypothesized. It is shown how such data?structures and models arise naturally within the application areas of environmental epidemiology and software reliability, and lead to statistical analyses using novel forms of Poisson regression models with random effects. Such models are potentially very useful in analysis and characterization of teletraffic patterns.