Distribution of the total message flow in group arrival queueing systems

Queueing systems are considered in which grouped arrival times of messages add up to a simple flow. In addition to the arrival time, every message is described in terms of a certain random length; the simultaneous function of message length and service time is specified. For the case of initial conditions a Laplace-Stiltjes transform $\delta(s,t)$ of the total amount of messages staying in a system, where the number of servers is.infinite, at time $t$ and a Laplace transform for $t$ the function $\delta(s, t)$ for an infinite queue one-server system are obtained.