Dependence patterns for modeling simultaneous events

• Study the modeling capabilities of the stationary m  -state BMAPBMAP, with simultaneous events up to size k.
• BMAPsBMAPs allow dependent and identically distributed inter-event times and also correlated batch sizes.
• Correlation patterns are identified for the acfs of the 2-state BMAPBMAP.
• Simulation studies are carried out for m≥3m≥3 cases.
• Influence of the dependence patterns for both acfs for the 2-state BMAPBMAP in the counting process is explored.

In this paper we examine in detail some of the modeling capabilities of the stationary m  -state BMAPBMAP, with simultaneous events up to size k  , noted BMAPm(k)BMAPm(k). Specifically, we study the forms of the auto-correlation functions of the inter-event times and event sizes. We provide a novel characterization of the functions which is suitable for analyzing the dependence patterns. In particular, this allows one to prove a geometrically decrease to zero of the functions and to identify four correlation patterns, when m  =2 . The case m≥3m≥3 is illustrated via an extensive simulation study, from which it can be deduced that, as expected, richer structures can be obtained as m   increases. In addition, the influence of the dependence patterns for both auto-correlation functions for the BMAP2(2)BMAP2(2) in the counting process has been explored through an empirical analysis.