We consider a status update communication system consisting of a source-destination link. A stochastic process is observed at the source, where samples are extracted at random time instances, and delivered to the destination, thus, providing status updates for the source. In this paper, we expand the concept of information ageing by introducing the cost of update delay (CoUD) metric to characterize the cost of having stale information at the destination. The CoUD captures the freshness of the information at the destination and can be used to reflect the information structure of the source. Moreover, we introduce the value of information of update (VoIU) metric that captures the reduction of CoUD upon reception of an update. Using the CoUD, its by-product metric called peak cost of update delay (PCoUD), and the VoIU, we evaluate the performance of an M/M/1 system in various settings that consider exact expressions and bounds. Our results indicate that the performance of CoUD differs depending on the cost assigned per This work extends the preliminary study in [1]. A. Kosta, N. Pappas, A. Ephremides, and V. Angelakis are with the