Nonidentifiability of the Two-State BMAP

The capability of modeling non-exponentially distributed and dependent inter-arrival times as well as correlated batches makes the Batch Markovian Arrival Processes (BMAP) suitable in different real-life settings as teletraffic, queueing theory or actuarial contexts. An issue to be taken into account for estimation purposes is the identifiability of the process. This paper explores the identifiability of the stationary two-state BMAP noted as BMAP(2)(k), where k is the maximum batch arrival size, under the assumptions that both the interarrival times and batches sizes are observed. It is proven that for k >= 2 the process cannot be identified. The proof is based on the construction of an equivalent BMAP(2)(k) to a given one, and on the decomposition of a BMAP(2)(k) into k BMAP(2)(2)s.