Kragujevac J. Math. 24 (2002) 193-205.

Abstract. We consider a batch arrival queueing system M⁽ⁱ⁾/M/1/m/ of varying cluster arrival sizes I. The arrival process thus constitutes an independent compound Poisson stream of rate l_r where

and pr(I³ i) = b_i with b_k=0 for k £ 0.
Acceptance into system is further limited by available space m thus implying a truncation of an otherwise infinite domain.
With the aid of certain combinatoric analysis of partitions and compositions the steady state distributions under various forms of arrival size pattern have been explicitly obtained in terms of system specifications. It is demonstrated that the results can perfectly provide one more class of truncated geometric distribution for a less idealistic modeling of the complex natural process of aggregation, congregation and abundance for such animals as soil microarthropods. A numerical illustration is provided using some copious data from the biological literature.