Haight's distribution and busy periods (Q1124222)

\textit{G. Letac} and \textit{V. Seshadri} [ibid. 6, 165-169 (1988; Zbl 0633.62015)] considered the probability density of a compound random variable \[ X(1)=Z(N(1))=Y_ 1+...+Y_{N(1)}, \] where Z(\(\cdot)\) is a gamma process, i.e. \(Y_ 1,Y_ 2,..\). are i.i.d. having an exponential distribution with mean \(\lambda^{-1}\), where \[ P(N(1)=n)=p_ n(\alpha)=e^{-\alpha}(n^{n-2}/(n-1)!)(\alpha e^{-\alpha})^{n- 1}\quad (n=1,2,...). \] The authors show that this distribution studied among others by F. Haight is a busy period distribution for an M/D/1 queueing model assuming that it starts with r present customers. There is also given a generalization to a semigroup of infinitely divisible distributions in which the aforementioned probability density is embedded.