Burstiness descriptors for markov renewal processes and markovian arrival processes

Quantitative descriptors of the burstiness of an arrival process are derived for Markov renewal processes (MRP's) and Markovian arrival processes (MAP's). Our burstiness descriptors are based on simple definitions of a burst and a gap in an arrival process. Briefly, for threshold v, we define a burst to be a (maximal) interval during which all interarrival times are less than or equal to v and a gap to be a (maximal) interval during which all interarrival times are greater than v. Thus, an arrival process alternates between bursts and gaps. For the case of a MRP, we derive the distribution of the number of arrivals in a burst (gap) and the mean of the duration of a burst (gap). These results are then specialized for the case of a MAP. For four example MAP z.repos s, six complementary descriptors are plotted as a function of threshold parameter i. These plots illustrate how our burstiness descriptors can be used to gain insight into the behavior of the arrival process and into the behavior of a queueing system to which the arrivals are fed