Asymptotic of maxima in the infinite server queue with bounded batch sizes

This paper considers the infinite server queue with the batch input $M^X|G|\infty$. Let all servers be free at time zero and $M(t)$ denote the maximum number of customers simultaneously present in the queue during $[0,t]$. The following theorem is proved.
Theorem 1. If $L$ is the maximum number of customers in a batch, then almost sure
$$ M(t)\frac{\ln\ln t}{\ln t}\to L\quadas $t\to\infty$.\eqno (*) $$

Some generalizations are discussed: nonstationary queues (with time-dependent parameters) and queues with heterogeneous customers. For these monotony theorems are proved. Conditions under which the asymptotic $(*)$ stays correct are obtained.