Abstract Linear multistate consecutively-connected systems (LMCCS) are systems that consist of a set of linearly ordered nodes with some of them containing statistically independent multistate connection elements (MCEs). Each MCE can provide a connection between its host node and a random number of next nodes along the sequence based on a known probability mass function. In traditional LMCCS models, the disconnection of any node causes the failure of the entire system. These models are too strict and thus not appropriate for some real-world applications such as those in sensor detection systems and flow transfer systems, which can tolerate a certain number of disconnected nodes referred to as gaps. In this work, we generalize the traditional LMCCS models by introducing a limited number of allowable gaps. The system fails if the number of gaps exceeds a specified limit. To analyze the reliability of the generalized LMCCS subject to a constrained total number of gaps, a universal generating function based method is first suggested. An optimal element sequencing problem is then solved considering that the system reliability can strongly depend on the sequence of different MCEs along the line. Examples are provided to demonstrate the proposed methodology.