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The substability and the ergodicity of various queueing models are discussed. This paper considers the vector-valued queueing process Xrn = (xrn,1,xrn,2, · ··) with non-negative components and a constant initial value Xrr= a. For this, the substability is derived under simple conditions by showing the finiteness of 
With respect to the ergodicity, in order to make use of Borovkov's theorem, we additionally assume that the distribution of the interarrival of customers has non-bounded tail for any given past sequence.
