Discrete-time Queueing models are frequently utilized to analyze the performance of computing and communication systems. The length of busy period is one of important performance measures for such systems. In this paper, we consider the busy period of the Geo/Geo/1/K queue with a single vacation. We derive the moments of the length of the busy (idle) period, the number of customers who arrive and enter the system during the busy (idle) period and the number of customers who arrive but are lost due to no vacancies in the system for both early arrival system (EAS) and late arrival system (LAS). In order to do this, recursive equations for the joint probability generating function of the busy period of the Geo/Geo/1/K queue starting with n, 1≤n≤K, customers, the number of customers who arrive and enter the system, and arrive but are lost during that busy period are constructed. Using the result of the busy period analysis, we also numerically study differences of various performance measures between EAS and LAS. This numerical study shows that the performance gap between EAS and LAS increases as the system capacity K decrease, and the arrival rate (probability) approaches the service rate (probability). This performance gap also decreases as the vacation rate (probability) decrease, but it does not shrink to zero.

This study is a basic investigation of the contents and services relevant to the domestic vacation house business. In which, the trade scale, types of housing, and environmental conditions of various property locations were analyzed. The characteristics of properties listed on the Japanese website that conducts the greatest volume of vacation house trade in Japan were examined, and the following results were obtained: Villa areas, villas, and resort condos (resort mansions) are the three basic types of properties handled in the vacation house trade. In this market, sales per unit in villa areas and per spaces in resort condos accounted for the highest volume of trade, followed by that of villas (individual houses). In terms of land area, floor area, and sales price per house type, the relatively cheaper small and medium-sized vacation houses are more frequently traded, than expensive large-scale villas. In particular, small multi-family type villas (such as in resort condos) are the most popular. Land and floor area, and sales prices all show considerable variation depending on the type of property considered. Therefore, a business initiative to provide a more detailed classification of properties is required. In terms of the environment of vacation properties, most are located on coasts, plateaus, or inland mountains, and are generally within three-hours’ traveling distance of large cities.