ing the spatial and temporal correlations of sensor
data. Santini and Romer (2006) use a LeastMean
Square (LMS) adaptive algorithm for making predic
tions given a data stream. Han, et al. (2007) consider
sensor transmissions that are either sourcetriggered
or consumertriggered (the consumers are the queries
that come into the system, asking for parameter infor
mation). In all these cases, the sensor transmission is
suppressed if it would not add sufﬁcient value.
Ahmadi and Abdelzaher (2009) take reliability
into consideration. In particular, they take into ac
count the fact that wireless networks are often noisy
and drop packets.
2.2 Energy Harvesting
Energy harvesting has been the focus of increasing
interest. Surveys of energy harvesting techniques can
be found in Chalasani and Conrad (2008) and Park,
et al. (2007). Related power management techniques
are studied in Kansal, et al. (2007) and Sharma, et
al. (2010). These include migrating tasks to nodes
depending on their energy levels. Task scheduling in
energy harvesting realtime systems is considered in
Moser, et al. (2007); their algorithms take into ac
count both the prevailing energy and time constraints,
rather than simply the task timing constraints.
3 MODELS
3.1 Environmental Model
The sensed environment has a behaviour as projected
by the user. For example, if we have a sensor mea
suring outdoor temperature, there could be a formula
that uses the current timeofday and the last few re
ported observations to estimate the current tempera
ture at the sensor. Since there are stochastic aspects
to the sensed environment (if the environment were
not stochastic there would be no need to use a sen
sor), the actual parameter value can vary from that es
timated. The amount of variation from the estimated
value obviously depends on the age of the information
that is used to make the estimate: the temperature at
10:05 AM is likely to be estimated very accurately if
the temperature at 10:00 AM was reported; by con
trast, if the last temperature report was at 8:00 AM,
the estimate is likely to be of poorer accuracy.
In our environmental model, we do not model the
actual value of the sensed variable. Instead, we model
the difference, δ, between the actual value and the
projected value. It is in this difference that all the
stochastic nature of the environment is captured. For
example, if the sensed variable is treated as falling
in some set of discrete quantities (which is the case
in ﬁnite wordlength machines even if the underlying
sensed variable is continuous), we use the probability
mass function (pmf), π
δ
(∆), of the additional devia
tion, ∆, of the actual, from the projected (or modelled)
value, arising from the passage of one sampling pe
riod. That is, if the state variable was last reported at
sampling point n, the divergence from the projected
value would be ∆
1
+···+∆
m
at sampling point n+m,
where the ∆
i
follow the pmf π
δ
(·).
It is important to recognise that our algorithm does
not require a prior model of the environment to be
available. If one such is available, it can be used to
project into the future, the value of the next parameter
sample. A simple case would be where, for instance,
the ∆
i
can be modeled as i.i.d. random variables. For
example, in our numerical examples, we assume for
concreteness that π
δ
(·) is a geometrically distributed
random variable truncated at some maximum devia
tion: π
δ
(i) = Kα
i
if −D
max
≤ i ≤ D
max
and 0 oth
erwise, where α is a constant characterizing the ran
dom process, K is a normalization constant and D
max
is some given truncation point.
However, if such a prior model is not available, we
can simply use some extrapolation techniques based
on recent observationsto project what the next sample
value will be.
Regardless of whether we use an environmental
model, extrapolation, or a combination of the two, the
sensor is able to replicate, without communication,
the value that the base station would project (based
on its prior transmissions) in the absence of a report
of its current value. In other words, the sensor can
calculate what the base station would estimate for the
current value of the parameter based on the prior sen
sor reports. Since it has the actual measured value of
the current parameter value, it knows the divergence
between these two quantities.
3.2 Cost Measure
Our cost measure is the sum of the absolute errors as
a result of not reporting the value of every sample that
is measured. That is, let B
i
be the broadcast indicator
function,
B
i
=
1 if sampling point i is reported
0 otherwise
Deﬁne L
i
as the last sampling point prior to i whose
value was reported. Then, the aggregate cost incurred
up to (and including) some sampling point i is given
by
Θ(i) =
i
∑
j=1
(1− B
i
)
"
L
i
−1
∑
k=1
∆
k

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