Cloud computing is a model for enabling convenient, on-demand network access to a shared pool of configurable computing resources. To provide cloud computing services economically, it is important to optimize resource allocation under the assumption that the required resource can be taken from a shared resource pool. In addition, to be able to provide processing ability and storage capacity, it is necessary to allocate bandwidth to access them at the same time. This paper proposes an optimal resource allocation method for cloud computing environments. First, this paper develops a resource allocation model of cloud computing environments, assuming both processing ability and bandwidth are allocated simultaneously to each service request and rented out on an hourly basis. The allocated resources are dedicated to each service request. Next, this paper proposes an optimal joint multiple resource allocation method, based on the above resource allocation model. It is demonstrated by simulation evaluation that the proposed method can reduce the request loss probability and as a result, reduce the total resource required, compared with the conventional allocation method. Then, this paper defines basic principles and a measure for achieving fair resource allocation among multiple users in a cloud computing environment, and proposes a fair joint multiple resource allocation method. It is demonstrated by simulation evaluations that the proposed method enables the fair resource allocation among multiple users without a large decline in resource efficiency. Keywords: Cloud computing, joint multiple resource allocation, fairness