  In this paper, Bayesian quickest change detection problems with sampling
right constraints are considered. Specifically, there is a sequence of random
variables whose probability density function will change at an unknown time.
The goal is to detect this change in a way such that a linear combination of
the average detection delay and the false alarm probability is minimized. Two
types of sampling right constrains are discussed. The first one is a limited
sampling right constraint, in which the observer can take at most $N$
observations from this random sequence. Under this setup, we show that the cost
function can be written as a set of iterative functions, which can be solved by
Markov optimal stopping theory. The optimal stopping rule is shown to be a
threshold rule. An asymptotic upper bound of the average detection delay is
developed as the false alarm probability goes to zero. This upper bound
indicates that the performance of the limited sampling right problem is close
to that of the classic Bayesian quickest detection for several scenarios of
practical interest. The second constraint discussed in this paper is a
stochastic sampling right constraint, in which sampling rights are consumed by
taking observations and are replenished randomly. The observer cannot take
observations if there are no sampling rights left. We characterize the optimal
solution, which has a very complex structure. For practical applications, we
propose a low complexity algorithm, in which the sampling rule is to take
observations as long as the observer has sampling rights left and the detection
scheme is a threshold rule. We show that this low complexity scheme is first
order asymptotically optimal as the false alarm probability goes to zero.
