Delay analysis of two batch-service queueing models with batch arrivals: Geo(X)/Geo(c)/1

In this paper, we compute the probability generating functions (PGF's) of the customer delay for two batch-service queueing models with batch arrivals. In the first model, the available server starts a new service whenever the system is not empty (without waiting to fill the capacity), while the server waits until he can serve at full capacity in the second model. Moments can then be obtained from these PGF's, through which we study and compare both systems. We pay special attention to the influence of the distribution of the arrival batch sizes. The main observation is that the difference between the two policies depends highly on this distribution. Another conclusion is that the results are considerably different as compared to Bernoulli (single) arrivals, which are frequently considered in the literature. This demonstrates the necessity of modeling the arrivals as batches