  We consider the problem of quickest event detection with sleep-wake
scheduling in small extent wireless sensor networks in which, at each time
slot, each sensor node in the awake state observes a sample and communicates
the information to the fusion centre. The sensor nodes in the sleep state do
not sample or communicate any information to the fusion centre (FC), thereby
conserving energy. At each time slot, the FC, after having received the samples
from the sensor nodes in the wake state, makes a decision to stop (and thus
declare that the event has occurred) or to continue observing. If it decides to
continue, the FC also makes the decision of choosing the number of sensor nodes
to be in the wake state in the next time slot. We consider three alternative
approaches to the problem of choosing the number of sensor nodes to be in the
wake state in time slot k+1, based on the information available at time slot k,
namely, 1. optimal control of M_{k+1}, the number of sensor nodes to be in the
awake state in time slot k+1, 2. optimal control of q_{k+1}, the probability of
a sensor node to be in the awake state in time slot k+1, and 3. optimal
probability q that a sensor node is in the awake state in any time slot. In
each case, we formulate the problem as a sequential decision process. We show
that a sufficient statistic for the decision at time k is the a posteriori
probability of change Pi_k. Also, we show that the optimal stopping rule is a
threshold rule on Pi_k. The optimal policy for M_{k+1} can keep very few
sensors wake during the prechange phase and then quickly increase the number of
sensors in the wake state when a change is "suspected". Among the three
sleep-wake algorithms described, we observe that the total cost is minimum for
the optimum control of M_{k+1} and is maximum for the optimum control on q.
