The rapid development of information and network technologies motivates the emergence of various new computing paradigms, such as distributed computing, cloud computing, and edge computing. This also enables more and more network enterprises to provide multiple different services simultaneously. To ensure these services can only be accessed conveniently by authorized users, many password and smart card based authentication schemes for multiserver architecture have been proposed. Recently, Truong et al. introduced an identity based user authentication scheme on elliptic curve cryptography in multiserver environment and claimed that their scheme is secure against popular attacks. However, in this paper, we point out that their scheme suffers from offline password guessing and impersonation attack and fails to achieve security requirements of this kind of authentication scheme. Moreover, we put forward a new scheme to conquer security pitfalls in the above scheme. Security analysis indicates that the proposed scheme can be free from well-known attacks. Performance discussion demonstrates that our scheme has advantages in terms of both security property and computation efficiency and thus is more desirable for practical applications in multiserver environment.