In this paper we extend the Shiryaev's quickest change detection formulation
by also accounting for the cost of observations used before the change point.
The observation cost is captured through the average number of observations
used in the detection process before the change occurs.

We study the problem of tracking an object moving through a network of
wireless sensors. In order to conserve energy, the sensors may be put into a
sleep mode with a timer that determines their sleep duration. It is assumed
that an asleep sensor cannot be communicated with or woken up, and hence the
sleep duration needs to be determined at the time the sensor goes to sleep
based on all the information available to the sensor. Having sleeping sensors
in the network could result in degraded tracking performance, therefore, there
is a tradeoff between energy usage and tracking performance.

The two popular criteria of optimality for quickest change detection
procedures are Lorden's criterion and the Bayesian criterion. In this paper a
robust version of these quickest change detection problems is considered when
the pre-change and post-change distributions are not known exactly but belong
to known uncertainty classes of distributions.