The Correlation Function of Multiple Dependent Poisson Processes Generated by the Alternating Renewal Process Method

We derive conditions under which alternating renewal processes can be used to construct correlated Poisson processes. The pairwise correlation function is also derived, showing that the resulting correlations can be negative. The technique and the analysis can be extended to the generation of two or more dependent renewal processes.

💡 Research Summary

The paper investigates how alternating renewal processes (ARPs) can be employed to generate multiple Poisson processes that are statistically dependent. A Poisson process is defined by exponentially distributed inter‑event times with a constant rate λ. An ARP, in contrast, alternates between two waiting‑time distributions, (F_{1}(t)) and (F_{2}(t)), producing a sequence of intervals (X_{1}, X_{2}, …) where each odd interval follows (F_{1}) and each even interval follows (F_{2}). The authors first ask under what conditions the superposition of these alternating intervals yields a renewal process whose inter‑event distribution is still exponential, i.e., a genuine Poisson process. By applying Laplace transforms they derive the necessary and sufficient condition

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