Analysis of the transient delay in a discrete-time buffer with batch arrivals

We perform a discrete-time analysis of the delay of customers in a FIFO buffer with batch arrivals. The numbers of arrivals per slot are independent and identically distributed variables. Since the arrivals come in batches, the delays of the subsequent customers do not constitute a Markov chain, which complicates the analysis. By using generating functions and the supplementary variable technique, moments of the delay of the k-th customer are calculated