1. A computing system comprising: a database which has a data part storing data, an index part storing an old index which is a complete index of the data and an incomplete new index which is being rebuilt, index information which stores a state of the new index and a state of the old index, and statistical information which stores a number of records registered in the new index and a number of records registered in the old index; and a processing device which receives a query from a client, determines which of the new index and the old index is used for processing the query, processes the query according to the determination, and transmits a response to the query to the client, wherein when determining which of the new index and the old index is used, the processing device performs analysis of a type of the query, analysis of user designation information included in the query, calculation of an index value of the new index, calculation of an index value of the old index, and comparison between the index value of the new index and the index value of the old index, in the calculation of the index value of the new index, the processing device calculates a first cost required for accessing the data part by using the new index, calculates a degree of completion of the new index on a basis of a ratio of the number of records of the data registered in the new index to the number of records of the data stored in the data part, and calculates the index value of the new index from the first cost and the degree of completion of the new index, in the calculation of the index value of the old index, the processing device calculates a second cost required for accessing the data part by using the old index, calculates a degree of completion of the old index on a basis of a ratio of the number of records of the data registered in the old index to the number of records of the data stored in the data part, and calculates the index value of the old index from the second cost and the degree of completion of the old index, and the processing device determines to use the new index in a case where the type of the query is any of aggregate functions MAX (a maximum value), MIN (a minimum value), and AVG (an average value), use of the new index is allowed in the user designation information, and the index value of the new index is larger than the index value of the old index, while the processing device determines to use the old index in other cases.