An application of the correlation structure of a Markov chain for the estimation of shift parameters in queueing systems. (Ukrainian, English) Zbl 1101.62069

Voĭna, O. A.; Chaplya, E.

Teor. Jmovirn. Mat. Stat. 71, 49-56 (2004); translation in Theory Probab. Math. Stat. 71, 53-61 (2005).

This authors deal with the queueing system \(M/M/1/0\), which consists of one service device with elementary flow of demands on input. There is no information about the state of the system at the arrival/leaving moments \(s_1,s_2,\ldots, s_{k},\ldots\). Let \(\xi_{k}=\tau_{k}+\theta_{x(k-1)}\), \(k=1,2,\ldots,K(T)\), \(\xi'_{K(T)}=T-s_{K(T)}-\theta_{x(K(T))}\) be observations, where \(\tau_{k}=s_{k}-s_{k-1}\), \(s_0=0\), \(\theta_{x(k)}\) is a delay of observations depending on state \(x(k)\) of the system at the moment \(s_{k}\). The states \(x(k), k=0,1,\ldots\), are not observable. The problem is to estimate the unknown shift parameter \(\theta\). The authors use the correlation structure of Markov chains to obtain consistent estimates of \(\theta\). The asymptotic properties of the proposed estimates are investigated.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

62M05	Markov processes: estimation; hidden Markov models
60J27	Continuous-time Markov processes on discrete state spaces
60K25	Queueing theory (aspects of probability theory)

Keywords:

correlation structure; Markov chain; shift parameter; queuing system; delay of observations