This paper describes discrete-time models for a system that consists of subsystems connected by time-delay elements representing the moving time of a material from one subsystem to another. When the moving speed is variable and given as prior information, it leads to an imprecise model to simply replace the time-invariant parameters with the time-variant ones. Therefore, we propose a model consisting of a sufficient number of time-invariant delay operators and an output expressed by a linear combination of the outputs of the delay operators. In this model, the update formula for the state variable is time-invariant and the time-varying coefficients of the output equation are determined by the position of the material in the time-delay element. Observer-based state estimation for the proposed model is also discussed. It is shown that the response of the observer with uniform gains results in a time-invariant first-order lag even when the moving speed changes.<br/>