On the GI/M/∞ queue with batch arrivals of constant size

In this note, the GI/M/∞ queue with batch arrivals of constant sizek is investigated. It is shown that the stationary probabilities that an arriving batch findsi customers in the system can be computed in terms of the corresponding binomial moments (Jordan's formula), which are determined by a recursive relation. This generalizes well-known results by Takács [12] for GI/M/∞. Furthermore, relations between batch arrival- and time-stationary probabilities are given.