A single-server queue concatenated with a multilayer channel encoder is considered. The main focus of this work is on minimization of the expected delay of a packet from entering the queue until completion of successful service. Tight bounds are derived for the expected delay for different numbers of coded layers. Numerical optimization is applied to find the optimal resource allocation minimizing the average delay. Similar bounds are also derived for the case of continuous layering. It is demonstrated that code layering may give pronounced performance gains in terms of delay, which are more impressive than those associated with throughput. This makes layering more attractive when communicating under stringent delay constraints.