In this study, the locally compactness spaces property will be investigated for three topologies, known as tri-locally compact spaces. Additionally, the features of these spaces will be investigated in terms of locally compact spaces. Many theoretical results have been presented and proven, and several well-known theorems concerning locally compact spaces have been extended to apply to three topologies. An illustrative example is provided to support the findings.

Erlang distribution is continuous probability distribution that has application in several field such as stochastic process and mathematical biology, due to its relation with exponential and gamma distribution. In the sense that, the duration of the successive calls follows the Erlang distribution, if individual telephone calls is exponentially distributed to the time period. In this study, Bayesian estimation was employed in the estimation of scale parameter of weighted Erlang distribution. The posterior distribution was derived under two informative priors, which are inverse exponential and inverse chi-square prior. The Bayes estimate and their relative posterior risks were derived under the assumption of Squared Error Loss Function (SELF), and Precautionary Loss Function (PLF). A Monte Carlo simulation was carried out in order to obtain the numerical value of the estimates. It was observed that SELF performs best when inverse exponential prior is used.