Dependence patterns for modeling simultaneous events

• Published in: Reliability engineering & system safety 2016-10, Vol.154, p.19-30
• Main Authors: Rodríguez, Joanna, Lillo, Rosa E., Ramírez-Cobo, Pepa
• Format: Article
• Language: English
• Subjects: Autocorrelation function; Batch Markovian arrival process (BMAP); Dependent event arrivals size; Dependent inter-event times
• ISSN: 0951-8320; 1879-0836
• DOI: 10.1016/j.ress.2016.05.008

In this paper we examine in detail some of the modeling capabilities of the stationary m-state BMAP, with simultaneous events up to size k, noted 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≥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) in the counting process has been explored through an empirical analysis. •Study the modeling capabilities of the stationary m-state BMAP, with simultaneous events up to size k.•BMAPs allow dependent and identically distributed inter-event times and also correlated batch sizes.•Correlation patterns are identified for the acfs of the 2-state BMAP.•Simulation studies are carried out for m≥3 cases.•Influence of the dependence patterns for both acfs for the 2-state BMAP in the counting process is explored.