## Nonidentifiability of the two-state Markovian arrival process.(English)Zbl 1202.60119

Summary: We consider the problem of identifiability for the two-state Markovian arrival process (MAP$$_{2}$$). In particular, we show that the MAP$$_{2}$$ is not identifiable providing the conditions under which two different sets of parameters induce identical stationary laws for the observable process.

### MSC:

 60J05 Discrete-time Markov processes on general state spaces 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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### References:

 [1] Asmussen, S. and Koole, G. (1993). Marked point processes as limits of Markovian arrival streams. J. Appl. Prob. 30, 365–372. JSTOR: · Zbl 0778.60035 [2] Ausín, M. C., Wiper, M. P. and Lillo, R. E. (2008). Bayesian prediction of the transient behaviour and busy period in short- and long-tailed GI/G/1 queueing systems. Comput. Statist. Data Anal. 52, 1615–1635. · Zbl 1452.62215 [3] Bean, N. G. and Green, D. A. (2000). When is a MAP poisson? Math. Comput. Modelling 31, 31–46. · Zbl 1042.60530 [4] Chakravarthy, S. R. (2001). The batch Markovian arrival process: a review and future work. In Advances in Probability Theory and Stochastic Processes , Notable Publications, NJ, pp. 21–49. [5] Fearnhead, P. and Sherlock, C. (2006). An exact Gibbs sampler for the Markov-modulated poisson process. J. R. Statist. Soc. B 68, 767–784. · Zbl 1110.62131 [6] Green, D. (1998). MAP/PH/1 departure processes. Doctoral Thesis, School of Applied Mathematics, University of Adelaide. [7] He, Q.-M. and Zhang, H. (2006). PH-invariant polytopes and Coxian representations of phase type distributions. Stoch. Models 22, 383–409. · Zbl 1159.60340 [8] He, Q.-M. and Zhang, H. (2008). An algorithm for computing minimal Coxian representations. INFORMS J. Computing 20, 179–190. · Zbl 1243.90045 [9] He, Q.-M. and Zhang, H. (2009). Coxian representations of generalized Erlang distributions. Acta Math. Appl. Sinica 25, 489–502. · Zbl 1177.60017 [10] Heffes, H. (1980). A class of data traffic processes-covariance function characterization and related queueing results. Bell Systems Tech. J. 59, 897–929. · Zbl 0439.90032 [11] Heffes, H. and Lucantoni, D. (1986). A Markov-modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE J. Sel. Areas Commun. 4, 856–868. [12] Heyman, D. P. and Lucantoni, D. (2003). Modeling multiple IP traffic streams with rate limits. IEEE/ACM Trans. Networking 11, 948–958. [13] Ito, H., Armari, S.-I. and Kobayashi, K. (1992). Identifiability of hidden Markov information sources and their minimum degrees of freedom. IEEE Trans. Inf. Theory 38, 324–333. · Zbl 0742.60041 [14] Leroux, B. G. (1992). Maximum-likelihood estimation for hidden Markov models. Stoch. Process. Appl. 40, 127–143. · Zbl 0738.62081 [15] Lucantoni, D. M. (1993). The BMAP/G/1 queue: a tutorial. In Performance Evaluation of Computer and Communication Systems , eds L. Donatiello and R. Nelson, Springer, Berlin, pp. 330–358. [16] Lucantoni, D. M., Meier-Hellstern, K. S. and Neuts, M. F. (1990). A single-server queue with server vacations and a class of nonrenewal arrival processes. Adv. Appl. Prob. 22, 676–705. JSTOR: · Zbl 0709.60094 [17] Neuts, M. F. (1979). A versatile Markovian point process. J. Appl. Prob. 16, 764–779. JSTOR: · Zbl 0422.60043 [18] Nielsen, B. F. (2000). Modelling long-range dependent and heavy-tailed phenomena by matrix analytic methods. In Advances in Algorithmic Methods for Stochastic Models , Notable Publications, pp. 265–278. [19] Ramaswami, V. (1980). The N/G/1 queue and its detailed analysis. Adv. Appl. Prob. 12, 222–261. JSTOR: · Zbl 0424.60093 [20] Ramirez, P., Lillo, R. E. and Wiper, M. P. (2008). On identifiability of MAP processes. Technical Report, Statistics and Econometrics Working Papers ws084613, Universidad Carlos III de Madrid. [21] Rydén, T. (1994). Consistent and asymptotically normal parameter estimates for hidden Markov models. Ann. Statist. 22, 1884–1895. · Zbl 0831.62060 [22] Rydén, T. (1996). An EM algorithm for estimation in Markov-modulated Poisson processes. Comput. Statist. Data Anal. 21, 431–447. · Zbl 0875.62405 [23] Rydén, T. (1996). On identifiability and order of continous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions. J. Appl. Prob. 33, 640–653. JSTOR: · Zbl 0865.60064 [24] Scott, S. L. (1999). Bayesian analysis of a two-state Markov modulated Poisson process. J. Comput. Graph. Statist. 8, 662–670. [25] Scott, S. L. and Smyth, P. (2003). The Markov modulated Poisson process and Markov Poisson cascade with applications to web traffic modeling. In Bayesian Statistics , 7 (Tenerife, 2002), Oxford University Press, New York, pp. 671–680.
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