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We estimate the temperature dependence of the bulk viscosity in a relativistic hadron gas. Employing the Green-Kubo formalism in the SMASH (Simulating Many Accelerated Strongly-interacting Hadrons) transport approach, we study different hadronic systems in increasing order of complexity. We analyze the (in)validity of the single exponential relaxation ansatz for the bulk-channel correlation function and the strong influence of the resonances and their lifetimes. We discuss the difference between the inclusive bulk viscosity of an equilibrated, long-lived system, and the effective bulk viscosity of a short-lived mixture like the hadronic phase of relativistic heavy-ion collisions, where the processes whose inverse relaxation rate are larger than the fireball duration are excluded from the analysis. This clarifies the differences between previous approaches which computed the bulk viscosity including/excluding the very slow processes in the hadron gas. We compare our final results with previous hadron gas calculations and confirm a decreasing trend of the inclusive bulk viscosity over entropy density as temperature increases, whereas the effective bulk viscosity to entropy ratio, while being lower than the inclusive one, shows no strong dependence to temperature.