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Estimation for renewal processes with unobservable gamma or Erlang interarrival times. (English) Zbl 0872.62081

Summary: An estimator based on the number of renewals \(N^t\) in a fixed time \(t\) is proposed to estimate the parameter \(k\) in a renewal process where the interarrival times have gamma \((\mu,k)\) distribution. Consideration is also given to estimation in the case when \(k\) is an integer, the Erlang distribution. Actual interarrival times are assumed unobservable and only periodic inspection of renewal process is possible. The estimator is improved by recording the excess life times, \(Y(t)\), occurring between the end of one inspection period and the time the next begins. The joint distribution of \(N^t\) and \(Y(t)\) assuming gamma interarrival times is derived. For the gamma case \((k\in\mathbb{R})\), Fisher’s information measure is used to define an efficiency criterion to show that when the cost of fully observing the renewal process is large compared to the cost of periodic inspection, the proposed estimator is superior. For the case of Erlang interarrivals, the probability of correctly estimating the parameter is used as the criterion for comparing estimators, and a similar conclusion is reached. Application to the estimation of \(k\) in the \(\text{M}/\text{E}_k/1\) queue is considered by making use of the embedded renewal process consisting of departure points while the queue is busy.

MSC:

62M05 Markov processes: estimation; hidden Markov models
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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