## 1 Introduction  
The number of IoT (Internet of Things) devices [1; 2] is growing rapidly, and also the published data obtain a huge number and great variety. As an example of the published data, sensor data are expected such as temperature, power consumption, and camera images. The sensor data are temporal and spatial data which relate to the specific information about geographical positions, areas, and time. In addition, those users and systems search and utilize sensor data based on such related data. To accommodate a huge number of devices (nodes), data, and users, therefore, the high-scalable mechanisms are necessary to search the required data efficiently. As the techniques to realize high scalability, overlay network construction techniques have been proposed such as distributed hash tables (DHTs), skip graphs, and geographical-based ones [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. Those techniques construct logical networks among nodes based on the specific information of each node such as a one-dimensional ID and multi-dimensional attribute values. Nodes are discovered by sending queries with the specific value or range called \"key\". Each node forwards the queries to the destination node based on its own routing table. In addition, the existing techniques assign territories to the nodes for distributed management and search of sensor data. On the other hand, the actual users and systems are expected to request sensor data with the specific time intervals such as \"I want to know the data for each one month from the specific day\", \"I want to show the data in the same day of week\". In addition, the specified time intervals are expected to have several patterns at the same time. Figure 1 shows the example of such queries as sensor data collection. In this paper, we call those queries \"interval queries\". In previous work, the author has proposed an overlay network construction method which can efficiently process the queries based on multiple different time intervals [21]. The proposed method assumes the ring-shaped topology and assigns the nodes to the key space based on one-dimensional information about time. In addition, to reduce the forwarded messages for the assumed queries, each node constructs shortcut links with the specific intervals which the actual users tend to request. However, the previous method causes unfairness and a concentration of the loads to the specific node because the density of data or those generators/providers is different from those related key values. Therefore, we proposed the enhanced scheme to construct structured overlay networks based on multiple different time intervals [22]. The proposed scheme uses node virtualization to equalize the loads of each real (physical) node. By the proposed scheme, each physical node manages multiple virtual nodes with those assigned keys (time). Load concentration to the specific physical node can be reduced because a large number of virtual nodes are managed by various physical nodes and widely placed on the whole of the key space. As an example of the applications run by virtual nodes, distributed content management systems are expectable for time-related contents such as a photograph, video, and sensor data. This paper is an extension of [22] to show new experiment results obtained in another simulation environment. In addition, comparisons with related work and discussion are also described in Sections 2.1 and 4, respectively. The contributions of this paper are: (1) clarification of \"interval queries\" with problems in the existing techniques, (2) establishment of a node virtualization scheme for structured overlay networks based on multiple different time intervals, and (3) the experiment results in the viewpoints of fairness and a communication load among nodes. In the following, Section 2 describes the materials and methods which contain the related work and the proposed scheme. The evaluation of the proposed scheme and discussion is summarized in Sections 3 and 4, respectively.