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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 BMAP(m)(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 BMAP(2) (2) in the counting process has been explored through an empirical analysis. (C) 2016 Elsevier Ltd. All rights reserved.