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In a cloud computing environment, it is necessary to simultaneously allocate both processing ability and network bandwidth needed to access it. The authors proposed the congestion control method for a cloud computing environment which reduces the size of required resource for congested resource type, instead of restricting all service requests as in the existing networks. Although this method can achieve an efficient use of resources in congested situation, it may result in an `unfair' use of resources. Therefore, it is required to enhance the proposed congestion control method in order to enable the fair resource allocation even in congested situation. First, this paper proposes a definition of fairness in congested situation, assuming that multiple resource types are allocated simultaneously to each service request. Next, this paper identifies a measure for evaluating fair resource allocation. On the basis of the above concepts, this paper proposes to enhance the previous congestion control method, so as to enable the fair resource allocation among users in congested situation. It is demonstrated by simulation evaluations that the proposed method enables fair resource allocation, compared with the previous method which does not consider the fair allocation.