  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 time unit, however the optimal
policy remains the same for linear ageing and varies for non-linear ageing.
When it comes to the VoIU the performance difference appears only when the cost
increases non-linearly with time. The study illustrates the importance of the
newly introduced variants of age, furthermore supported in the case of VoIU by
its tractability.
