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It is commonly assumed that the arrival process of customers to a service system is a nonhomogeneous Poisson process. Call center data often refute this assumption, and several authors have postulated a doubly-stochastic Poisson process for arrivals instead. We develop approximations for both the long-run fraction of calls answered quickly, and the distribution of the fraction of calls answered quickly within a short period. We also perform a computational study to evaluate the approximations and improve our understanding of such systems.