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The paper shows that in the analysis of a queuing system with fixed-size batch arrivals, there emerges a set of polynomials which are a generalization of Chebyshev polynomialsof the second kind. The paper uses these polynomials in assessing the transient behaviour of the overflow (equivalently call blocking) probability in the system. A key figure to noteis the proportion of the overflow (or blocking) probability resident in the transient component,which is shown in the results to be more significant at the beginning of the transient and naturally decays to zero in the limit of large t. The results also show that the significanceof transients is more pronounced in cases of lighter loads, but lasts longer for heavier loads.