This paper concerns a discrete time Geo/G/1 retrial queue with general retrial time in which all the arriving customers require first essential service with probability ${\alpha }_0$ while only some of them demand one of other optional services: − ( = 1, 2, 3,...) service with probability ${\alpha }_r$. The system state distribution, the orbit size and the system size distributions are obtained in terms of generating functions. The stochastic decomposition law holds for the proposed model. Performance...

This paper concerns a discrete time Geo/G/1 retrial queue with general retrial time in which all the arriving customers require first essential service with probability ${\alpha }_0$ while only some of them demand one of other optional services: − ( = 1, 2, 3,...) service with probability ${\alpha }_r$. The system state distribution, the orbit size and the system size distributions are obtained in terms of generating functions. The stochastic decomposition law holds for the proposed model. Performance...

We consider an estimate of the mode of a multivariate probability density with support in ${ℝ}^d$ using a kernel estimate drawn from a sample . The estimate is defined as any in {} such that $f_n\left(x\right)={max}_{i=1,\cdots ,n}{f}_n\left(X_i\right)$. It is shown that behaves asymptotically as any maximizer ${\widehat{\theta }}_n$ of . More precisely, we prove that for any sequence ${\left(r_n\right)}_{n\ge 1}$ of positive real numbers such that $r_n\to \infty$ and $r_n^{d}logn/n\to 0$, one has $r_n\parallel {\theta }_n-{\widehat{\theta }}_n\parallel \to 0$ in probability. The asymptotic normality of follows without further work.

Let be a distribution function (d.f) in the domain of attraction of an extreme value distribution $H_{\gamma }$; it is well-known that , where is the d.f of the excesses over , converges, when tends to , the end-point of , to $G_{\gamma }\left(\frac{x}{\sigma \left(u\right)}\right)$, where $G_{\gamma }$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma >-1$, a function which verifies ${lim}_{u\to s_{+}\left(F\right)}\Lambda \left(u\right)=\gamma$ and is such that $\Delta \left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\Lambda \left(u\right)}(x/\sigma \left(u\right))|$ converges to faster than $d\left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\gamma }(x/\sigma \left(u\right))|$.