Descriptors of arrival-process burstiness with application to the discrete Markovian arrival process

We use the concept of burstiness to propose new descriptors of arrival processes or, more generally, point or traffic processes. We say that the arrival process is in a burst when its interarrival times are less than or equal to some threshold value; during periods in which interarrival times are greater than the threshold value, the process is in a gap. We propose to describe the arrival process in terms of the number of arrivals during a burst (gap) and the duration of a burst (gap). For the case of discrete-time Markovian arrival processes we derive the distribution of the number of arrivals during a burst (gap) and the mean duration of a burst (gap). We present numerical results to illustrate how our descriptors can be used to understand the behavior of an arrival process and the congestion it induces in a queueing system. We also report the results of an experiment which shows a statistical relationship between two of our descriptors and a measure of queueing congestion.