This paper discusses the sequential estimation of the scale parameter of the Rayleigh distribution using the three-stage sequential sampling procedure proposed by Hall (Ann. Stat.1981, 9, 1229–1238). Both point and confidence interval estimation are considered via a unified optimal decision framework, which enables one to make the maximum use of the available data and, at the same time, reduces the number of sampling operations by using bulk samples. The asymptotic characteristics of the proposed sampling procedure are fully discussed for both point and confidence interval estimation. Since the results are asymptotic, Monte Carlo simulation studies are conducted to provide the feel of small, moderate, and large sample size performance in typical situations using the Microsoft Developer Studio software. The procedure enjoys several interesting asymptotic characteristics illustrated by the asymptotic results and supported by simulation.

In this paper we discuss the multistage sequential estimation of the variance of the Rayleigh distribution using the three-stage procedure that was presented by Hall (Ann. Stat. 9(6):1229–1238, 1981). Since the Rayleigh distribution variance is a linear function of the distribution scale parameter’s square, it suffices to estimate the Rayleigh distribution’s scale parameter’s square. We tackle two estimation problems: first, the minimum risk point estimation problem under a squared-error loss function plus linear sampling cost, and the second is a fixed-width confidence interval estimation, using a unified optimal stopping rule. Such an estimation cannot be performed using fixed-width classical procedures due to the non-existence of a fixed sample size that simultaneously achieves both estimation problems. We find all the asymptotic results that enhanced finding the three-stage regret as well as the three-stage fixed-width confidence interval for the desired parameter. The procedure attains asymptotic second-order efficiency and asymptotic consistency. A series of Monte Carlo simulations were conducted to study the procedure’s performance as the optimal sample size increases. We found that the simulation results agree with the asymptotic results.

This paper is an approach for introducing a mathematical treatment of the problem of finding the optimum location(s) of the emergency service centers concerning the case in which the callers for the service have different degrees of importance (weights) leading to present appropriate algorithm needed to solve it. In the end of the paper we introduce a numerical example to illustrate the steps of the algorithm.

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The rational management of emergency services centers focuses on two important goals, the optimal positioning of these centers to be able to provide services in the fastest possible way, and the second is to ensure that these centers can provide, with no delay, service to every one upon his/her request, especially in emergency cases when demand on these services are at the peak which means that in urban design it is very important to locate the emergency service centers in a position that satisfy both the two above goals. In fact, several research works concerned with determining the optimum location of the service centers but with a little concern of the limited capacity of each center. This research discusses how to achieve these two goals by treating the problem mathematically. Limitation of the capacity of each center is also a matter of concern. Finally, an effective computer algorithm is provided to illustrate the practical implication of the performed mathematical solution.