An infinite-server queueing system is studied in which units arrive in batches of variable size following a non-homogeneous compound Poisson process and each unit arriving at time t has its own arbitrary service time distribution. Using the standard properties of the Poisson process and elementary probability arguments, we derive the probability generating functions (PGF) of N(t), the number of units in the system at time t and D(t), the number of service completions up to time t or in the interval (0,t] in the transient state. Covariance functions between N(t), D(t); N(t), R(t) and R(t), D(t) are obtained separately where R(t) represents the number of arrivals in the time interval (0,t]. Expressions are obtained for the mean and variance of N(t) and D(t). Finally, some steady state results are obtained under certain conditions.
