Problem of the sequence synchronization in multiuser systems, which are designed using chaotic sequences, is very well investigated for the perfect sequence synchronization. However, mathematical models and expressions for the probability of error in closed form in these systems having imperfect time synchronization, are not developed. The imperfect synchronization can be expressed by a random delay between the received and the locally generated spreading sequence inside the receiver. A comprehensive analysis of the multi-user CDMA system is presented in this article for the case when the random delay is distributed according to the uniform distribution, which is expressed using Dirac's delta functions. The delays in the system are of finite values, thus the truncated uniform density function is applied, which simplifies the derivation of expressions for the probability of error that are confirmed by simulations.