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[en] We present simulation results of the build-up of the electron-cloud density ne for the two proposed ILC damping ring lattices, DC04 and DSB3, with particular attention to the potential benefit of an antechamber. We examine a field-free region and a dipole bending magnet, with or without an antechamber. We assume a secondary electronemission model for the chamber surface based on approximate fits to measured data for TiN, except that we let the peak value of the secondary emission yield (SEY), (delta)max, be a variable. We conclude that there is a critical value of (delta)max below which the antechamber provides a substantial benefit, roughly a factor ∼40 reduction in ne relative to the case in which max exceeds the critical value. We estimate the steady-state value of ne as a function of (delta)max, and thereby obtain the critical value of (delta)max for all cases considered. Thus, from the perspective of the electron-cloud effect, the inclusion of an antechamber in the design is justified only if (delta)max is below the critical value. The results presented here constitute a slight extension of those previously presented in March and September, 2010.