Parameter estimation for a discretely observed population process under Markov-modulation

Staffing for many-server systems facing non-standard arrival processes

“…In addition, when using a MAP, one needs to estimates parameters pertaining to the unobservable background process, which is typically rather involved (see e.g. de Gunst, Knapik, Mandjes, & Sollie, 2019;Okamura, Dohi, & Trivedi, 2009 ); our approach circumvents this complication.…”

Parameter estimation for multivariate population processes: a saddlepoint approach

“…We now concentrate on the first factor in (12), which can be computed using a saddlepoint approximation. To this end, we first observe that the occurrence of the event Eðm 0 Þ (i.e., M k ð'Þ ¼ 0 for all ' 2 Nðm 0 Þ) changes the distribution of the random vectors A k and D k ; in the sequel we denote the random vectors under this condition byà k andD k : To describe the distribution ofà k andD k , we use the following 'renormalized' probabilities, for ' 00 6 2 Nðm 0 Þ :…”

“…Observe in particular the similarity with the result stated in Lemma 1. Using this mgf, we can use a saddlepoint technique to approximate PðM k ¼ m 0 j Eðm 0 Þ, M kÀ1 ¼ m, X kÀ1 ¼ iÞ in (12) by following the same argument as in Section 4.1, evidently only including the non-zero elements of m 0 : We observe that the dimension of this saddlepoint approximation is now LÀfNðm 0 Þg, which is smaller than L as a consequence of m 0 2 S i ðmÞ n S i ðmÞ : In summary, according to (12) the probability t i ðm 0 j mÞ can be factorized into two probabilities. The probability corresponding to the nodes included in Nðm 0 Þ can be computed explicitly according to (13), whereas the probability corresponding to the remaining nodes can be evaluated relying on the saddlepoint approximation of reduced dimension.…”

“…In addition, when focusing on a Markovian arrival process only, rather than the resulting population process, in [10,11] estimation procedures based on discrete-time observations are presented. We finally mention, [12] in which a parameter estimation procedure for a univariate population process under Markov modulation is proposed and assessed, based on discrete-time observations of the population size.…”