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The paper deals with asymptotic stationarity of the process 
where 
is a vector in 
with non-negative coordinates, 
is an 
-valued process, S is a separable metric space and all operations in 
are meant in the coordinate-wise sense. It is shown that a type of asymptotic stationarity of (X, Y), together with some conditions, implies the same type of asymptotic stationarity of (w, X, Y). This result is applied to analyze asymptotic stationarity of multichannel queues. It may also be used to analyze asymptotic stationarity of series of multichannel queues.
