Age Minimization in Energy Harvesting
Communications: Energy-Controlled Delays
Ahmed Arafa

Sennur Ulukus

arXiv:1712.03945v1 [] 11 Dec 2017

Department of Electrical and Computer Engineering
University of Maryland, College Park, MD 20742
arafa@umd.edu ulukus@umd.edu

Abstract— We consider an energy harvesting source that is collecting measurements from a physical phenomenon and sending
updates to a destination within a communication session time.
Updates incur transmission delays that are function of the energy
used in their transmission. The more transmission energy used
per update, the faster it reaches the destination. The goal is to
transmit updates in a timely manner, namely, such that the total
age of information is minimized by the end of the communication
session, subject to energy causality constraints. We consider
two variations of this problem. In the first setting, the source
controls the number of measurement updates, their transmission
times, and the amounts of energy used in their transmission
(which govern their delays, or service times, incurred). In the
second setting, measurement updates externally arrive over time,
and therefore the number of updates becomes fixed, at the
expense of adding data causality constraints to the problem. We
characterize age-minimal policies in the two settings, and discuss
the relationship of the age of information metric to other metrics
used in the energy harvesting literature.

I. I NTRODUCTION
A source collects measurements from a physical phenomenon and sends information updates to a destination.
The source relies solely on energy harvested from nature to
communicate, and the goal is to send these updates in a timely
manner during a given communication session time, namely,
such that the total age of information is minimized by the end
of the session time. The age of information is the time elapsed
since the freshest update has reached the destination.
Power scheduling in energy harvesting communication systems has been extensively studied in the recent literature.
Earlier works [1]–[4] consider the single-user setting under different battery capacity assumptions, with and without
fading. References [5]–[8] extend this to multiuser settings:
broadcast, multiple access, and interference channels; and [9]–
[13] consider two-hop, relay, and two-way channels.
Minimizing the age of information metric has been studied mostly in a queuing-theoretic framework; [14] studies a
source-destination link under random and deterministic service
times. This is extended to multiple sources in [15]. References
[16]–[18] consider variations of the single source system, such
as randomly arriving updates, update management and control,
and nonlinear age metrics, while [19] shows that last-comefirst-serve policies are optimal in multi-hop networks.
This work was supported by NSF Grants CNS 13-14733, CCF 14-22111,
CCF 14-22129, and CNS 15-26608.

Our work is most closely related to [20], [21], where age
minimization in single-user energy harvesting systems is considered; the difference of these works from energy harvesting
literature in [1]–[13] is that the objective is age of information
as opposed to throughput or transmission completion time,
and the difference of them from age minimization literature
in [14]–[19], [22] is that sending updates incurs energy expenditure where energy becomes available intermittently. [20]
considers random service time (time for the update to take
effect) and [21] considers zero service time. Recently in [23],
we considered a fixed non-zero service time in two-hop and
single hop settings. In our work here, we consider an energycontrolled (variable) service time in a single-user setting.
We consider a source-destination pair where the source
relies on energy harvested from nature to send information
updates to the destination. Different from [20], [21], updates’
service times depend on the amounts of energy used to send
them; the higher the energy used to send an update, the faster
it reaches the destination. Hence, a tradeoff arises; given an
amount of energy available at the source, it can either send a
few number of updates with relatively small service times, or
it can send a larger number of updates with relatively higher
service times. In this paper, we investigate this tradeoff and
characterize the optimal solution in the offline setting. We
formulate the most general setting of this problem where the
source decides on the number of updates to be sent, when
to send them, and the amounts of energy consumed in their
transmission (and therefore the amounts of service times or
delays they incur), such that the total age of information is
minimized by the end of the session time, subject to energy
causality constraints. We present some structural insights of
the optimal solution in this general setting, and propose an
iterative solution. Our results show that the optimal number
of updates depends on the parameters of the problem: the
amounts and times of the harvested energy, delay-energy
consumption relationship, and the session time.
We also consider the scenario where update arrival times
at the source (measurement times) cannot be controlled; they
arrive during the communication session. Thus, two main
changes occur to the previously mentioned model. First, the
total number of updates gets fixed; and second, data causality
constraints are enforced, since the source cannot transmit an
update before receiving it. We formulate the problem in this
setting and characterize its optimal solution.

age

age

Q2

Q2

0

t1

L

Q3

Q1

L
t 1 + d1

t 2 + d2

t2

x1

t 3 t 3 + d3

T

0

time

t1

t 1 + d1

a1

x3
x2

t 2 + d2

t2

a2

t 3 t 3 + d3

T

time

a3

Fig. 2.
Age evolution versus time in a system where N = 3 update
measurements arriving during communication.

x4

Fig. 1. Age evolution versus time in a controlled measurement times system,
with N = 3 updates.

harvested. We also have the service time constraints
ti + di ≤ ti+1 ,

II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
A source node acquires measurement updates from some
physical phenomenon and sends them to a destination during a
communication session of duration T time units. Updates need
to be sent as timely as possible, i.e., such that the total age of
information is minimized by time T . The age of information
metric is defined as
a(t) , t − U (t),

∀t

(1)

where U (t) is the time stamp of the latest received information
(measurement) update, i.e., the time at which it was acquired
at the source. Without loss of generality, we assume a(0) = 0.
The objective is to minimize the following quantity
Z T
AT ,
a(t)dt
(2)
0

The source powers itself using energy harvested from
nature, and is equipped with an infinite battery to store its
incoming energy. Energy is harvested in packets of sizes Ej
at times sj , 1 ≤ j ≤ M . Without loss of generality, we assume
s1 = 0. The total energy harvested by time t is
X
E(t) =
Ej
(3)
j: sj ≤t

We denote by ei , the energy used in transmitting update i,
and denote by di , its transmission delay (service time) until it
reaches the destination. These are related as follows
ei = f (di )

(4)

where f is a deceasing convex function1. Let ti denote the
transmission time of update i. The following then holds
k
X
i=1

f (di ) ≤

k
X

E(ti ),

∀k

(5)

i=1

which represent the energy causality constraints [1], which
mean that energy cannot be used in transmission prior to being
1 This relationship is valid, for instance, if the channel is AWGN. With

normalized bandwidth and noise variance, we have f (d) = d 22B/d − 1 ,
with B denoting the size of the update packet in bits [24].

∀i

(6)

which ensure that there can be only one transmission at a time.
A. Controlled Measurements
In this setting, the source controls when to take a new
measurement update, and the goal is to choose total number
N
of updates N , transmission times {ti }N
i=1 , and delays {di }i=1 ,
such that AT is minimized, subject to energy causality constraints in (5) and service time constraints in (6). We note
that the source should start the transmission of an update
measurement whenever it is acquired. Otherwise, its age can
only increase. In Fig. 1, an example run of the age evolution
versus time is presented in a system with N = 3 updates.
The area under the age curve is given by the sum of the
areas of the three trapezoids Q1 , Q2 , and Q3 , plus the area
of the triangle L. The area of Q2 for instance is given by
2
1
1 2
2 (t2 + d2 − t1 ) − 2 d2 . Computing the area for a general N
updates, we formulate the problem as follows
min

N,t,d

s.t.

N
X

(ti + di − ti−1 )2 − d2i + (T − tN )2

i=1

ti + di ≤ ti+1 ,
k
X
i=1

f (di ) ≤

k
X

1≤i≤N
E(ti ),

1≤k≤N

(7)

i=1

with t0 , 0 and tN +1 , T .
B. Externally Arriving Measurements
In this setting, measurement updates arrive during the communication session at times {ai }N
i=1 , where N is now fixed.
We now have the following constraints
ti ≥ a i ,

∀i

(8)

representing the data causality constraints [1], which mean
that updates cannot be transmitted prior to being received at
the source. In Fig. 2, we show an example of the age evolution
in a system with N = 3 arriving updates. The area of Q2 in
this case is given by 21 (t2 +d2 −a1 )2 − 12 (t2 +d2 −a2 )2 and the
area of L is the constant term 21 (T − a3 )2 . Computing the area

for P
general N update arrivals, we write the objective function
N
2
2
as
i=1 (ti + di − ai−1 ) − (ti + di − ai ) , with a0 , 0.
This can be further simplified after some algebra to get the
following problem formulation2
min
t,d

N
X

(ai − ai−1 ) (ti + di )

k
X

k
X

i=1

s.t.

ti + di ≤ ti+1 ,
f (di ) ≤

1≤i≤N
E(ti ),

1≤k≤N

i=1

i=1

ti ≥ a i

1≤i≤N

(9)

We note that both problems (7) and (9) are non-convex.
One main reason is that the total energy arriving up to time
t, E(t), is not concave in t. Henceforth, in the next sections,
we solve the two problems when all the energy packets arrive
at the beginning of communication, i.e., when M = 1 energy
arrival. In this case E(t) = E, ∀t. The solutions in the case
of multiple energy arrivals follow similar structures.
III. C ONTROLLED M EASUREMENTS

propose an iterative algorithm to find the optimal inter-update
times {x∗i } given the optimal number of updates N ∗ and the
optimal service times {d∗i }. This is described as follows.
+1
Let {x̄i }N
and let
i=1 denote the output of this algorithm,
PN ∗ +1
us define the stopping condition to be when
x̄i =
i=1
PN ∗ ∗
∗
∗
T + i=1 di . We initialize by setting x̄1 = d1 ; x̄i = di +d∗i−1 ,
2 ≤ i ≤ N ; and x̄N +1 = d∗N . We then check the stopping
condition. If it is not satisfied, we compute m1 , arg min x̄i ,
and increase x̄m1 until either the stopping condition is satisfied, or x̄m1 is equal to mini6=m1 x̄i . In the latter case,
we compute m2 , arg mini6=m1 x̄i , and increase both x̄m1
and x̄m2 simultaneously until either the stopping condition is
satisfied, or they are both equal to mini∈{m
x̄i . In the
/
1 ,m2 }
latter case, we compute m3 and proceed similarly as above
until the stopping condition is satisfied. Note that if mk is
not unique at some stage k of the algorithm, we increase the
whole set {x̄i , i ∈ mk } simultaneously.
The above algorithm has a water-filling flavor; it evens out
the xi ’s to the extent allowed by the service times di ’s and the
session time T , while keeping them as low as possible. The
next lemma shows its optimality.

In this section, we focus on problem (7) with a single energy
arrival. We first have the following lemma.

Lemma 2 In problem (10), given N ∗ and {d∗i }N
i=1 , the optimal x∗i = x̄i , 1 ≤ i ≤ N + 1.

Lemma 1 In problem (7), all energy is consumed by the end
of communication.

Proof: First, note that the algorithm initializes xi ’s by their
least possible values. If this satisfies the stopping (feasibility)
condition, then it is optimal. Otherwise, since we need to
increase at least one of the xi ’s, the algorithm chooses the least
one; this gives the least objective function since y < z implies
(y + ǫ)2 < (z + ǫ)2 for y, z ≥ 0 and ǫ > 0. Next, observe that
while increasing one of the xi ’s, if the stopping condition is
satisfied, then we have reached the minimal feasible solution.
Otherwise, if two xi ’s become equal, then by convexity of
the square function, it is optimal to increase both of them
simultaneously [25]. This shows that each step of the algorithm
is optimal, and hence it achieves the age-minimal solution. 
We note that the above algorithm is essentially a variation
of the solution of the single-hop problem in [23]. There, all the
inter-update delays are fixed, while here they can be different.
Next, we present an example to show how the choice of the
number of updates and inter-update delays affect the solution,
in a specific scenario. In particular, we focus on the case where
the inter-update delays are fixed for all update packets, i.e.,
di = d, ∀i. In this case, by Lemma 1, for a given N , the
optimal inter-update delay is given by d = f −1 (E/N ). We
can then use the algorithm above to find the optimal xi ’s, as
shown in Lemma 2. For example, we consider a system with
energy E = 20 energy units, with f (d) = d 22/d − 1 , and
T = 10 time units. We plot the optimal age in this case versus
N in Fig. 3. We see that the optimal number of updates is equal
to 5; it is not optimal to send too few or too many updates
(the maximum feasible is 7 in this example). This echoes the
early results in [14], where the optimal rate of updating is not
the maximum (throughput-wise) or the minimum (delay-wise),
but rather lies in between.

Proof: By direct first derivatives, we observe that the objective
function is increasing in {di }. Thus, if not all energy is
consumed, then one can simply use the remaining amount to
decrease the last service time and achieve lower age. 
Next, we apply the change of variables x1 , t1 + d1 , xi ,
ti + di − ti−1 P
for 2 ≤ i ≤ N , and
PNxN +1 , T − tN . Then,
N +1
we must have i=1 xi = T + i=1 di , which reflects the
dependence relationship between the variables. This can also
be seen geometrically in Fig. 1. Then, the problem becomes
min

N,x,d

s.t.

N
+1
X
i=1

x2i −

N
X

d2i

i=1

x1 ≥ d1
xi ≥ di + di−1 ,

2≤i≤N

xN +1 ≥ dN
N
+1
X
i=1
N
X

xi = T +

f (di ) ≤ E

N
X

di

i=1

(10)

i=1

+1
The variables {xi }N
i=1 control the inter-update times, which
are lower bounded by the service times {di }N
i=1 , which are in
turn controlled by the amount of harvested energy, E. We
2 An inherent assumption in this model is that 0 < a < a < · · · < a .
1
2
N
Otherwise the areas of the trapezoids become 0 and the problem becomes
degenerate. Also, the parameters of the problem are such that it is feasible.

Hence, the optimal λ∗ is given by the unique solution of

28

N
X

26

(15)

i=1

24

Age of Information

h (−ci /λ∗ ) = E

22

20

18

16

14

12
1

2

3

4

5

6

7

Number of updates
Fig. 3.

Age of information versus number of updates.

IV. E XTERNALLY A RRIVING M EASUREMENTS
In this section, we solve problem (9) with a single energy
arrival. We observe that the problem in this case is convex and
can be solved by standard techniques [25]. We first have the
following lemma.
Lemma 3 In problem (9), the optimal update times satisfy
t∗1 = a1
t∗i = max

(11)



ai , ai−1 + d∗i−1 , . . . , a1 +



i−1
X
j=1

d∗j




, i ≥ 2 (12)



Proof: This follows directly from the constraints of problem
(9); the optimal update times should always be equal to their
lower bounds. Hence, we have t∗1 = a1 , t∗2 = max{a2 , t∗1 +
d∗1 } = max{a2 , a1 + d∗1 }, t∗3 = max{a3 , t∗2 + d∗2 } =
max{a3 , a2 + d∗2 , a1 + d∗1 + d∗2 }, and so on. 
By the previous lemma, the problem now reduces to finding
the optimal inter-update delays {d∗i }. We note that starting
from t∗1 = a1 , we have two choices for t∗2 ; either a2 or a1 +d∗1 .
Once t∗2 is fixed, t∗3 in turn has two choices; either a3 or t∗2 +d∗2 .
Now observe that once a choice pattern is fixed,
PN the objective
function of problem (9) will be given by
i=1 ci di where
ci > 0 is a constant that depends on the choice pattern. For
instance, for N = 3, choosing the pattern t∗2 = a1 + d∗1 and
t∗3 = a3 gives c1 = a2 , c2 = a2 − a1 , and c3 = a3 − a2 . We
introduce the following Lagrangian for this problem [25]
!
N
N
X
X
f (di ) − E
(13)
ci di + λ
L=
i=1

i=1

where λ is a non-negative Lagrange multiplier. The KKT
conditions are
ci = −λf ′ (di )

(14)

−1

where h , f ◦ g and g , (f ′ ) . To see this, note that since
f is convex, it follows that g exists and is increasing. By (14),
we then have d∗i = g (−ci /λ∗ ). Substituting in the energy
constraint, which has to be satisfied with equality, gives (15).
By monotonicity of f and g, h is also monotone, and therefore
(15) has a unique solution in λ∗ .
Therefore, we solve problem (9) by first fixing a choice
pattern for the update times, which gives us a set of constants
{ci } allowing us to solve for λ∗ using (15). We go through
all possible choice patterns and choose the one that is feasible
and gives minimal age.
We finally note that the measurements’ arrival times can be
so close to each other that the optimal solution is such that
t∗i > ai+l for some i and l ≥ 1. That is, there would be l + 1
measurements waiting in the data queue before t∗i . If the total
number of updates can be changed, then this solution can be
made better by transmitting only the freshest, i.e., the (i+l)th,
measurement packet at t∗i and ignoring all the rest. This strictly
improves the age and saves some energy as well. The solution
can be further optimized by re-solving the problem with Ñ =
N − l arriving measurements at times ã1 = a1 , . . . , ãi−1 =
ai−1 , ãi = ai+l , . . . , ãÑ = aN .
V. D ISCUSSION : R ELATIONSHIP

TO

OTHER M ETRICS

In this section we discuss the relationship between the proposed problems in this work and other well-known problems
in the energy harvesting literature: transmission completion
time minimization, and delay minimization. Reference [1]
introduced the transmission completion time minimization
problem. In this problem, given some amounts of data arriving
during the communication session, the objective is to minimize
the time by which all the data is delivered to the destination,
subject to energy and data causality constraints.
Reference [26] studies this problem from a different perspective. Instead of minimizing the completion time of all the
data, the objective is to minimize the delay experienced by
each bit, which is equal to the difference between the time
of its reception at the receiver and the time of its arrival
at the transmitter. Delay-minimal policies are fundamentally
different than those minimizing completion time. For instance,
in [1], due to the concave rate-power relationship, transmitting
with constant powers in between energy harvests is optimal.
While in [26], the optimal delay-minimal powers are decreasing over time in between energy harvests, since earlier arriving
bits contribute more to the cumulative delay and are thus given
higher priorities (transmission powers and rates).
We note that minimizing the age of information problem
is similar to the delay minimization problem formulated in
[26]. In both problems, there is a time counter that counts
time between data transmissions and receptions. In the age
of information problem, the time counter starts increasing

delay

affected by changing its arrival time as it represents the time
stamp of when the packet arrived, and hence transmission and
service times need to change if arrival times change in order
to achieve the same age.

3B
2B

R EFERENCES

B

0

t1

t 1 + d1

a1

t 2 + d2

t2

a2

t 3 t 3 + d3

T

time

a3

Fig. 4. Cumulative data packets arriving (blue) and departing (black) versus
time with N = 3 data packets. The shaded area in yellow between the two
curves represents the total delay of the system.

from the beginning of the communication session. While in
the delay problem, the time counter is bit-dependent; it starts
increasing only from the moment a new bit enters the system
and stops when it reaches the destination.
The delay minimization problem was previously formulated
in [27] for the case where the delay is computed per packet, as
opposed to per bit in [26] (note that the age is also computed
per packet and not per bit). The transmitter in [27] was energy
constrained but not harvesting energy over time, which models
the case where all energy packets arrive at the beginning of
communication. For the sake of comparison, we extend the
delay minimization problem in [27] to the energy harvesting
case as in [26] and relate it to the age minimization problem
considered in this work.
Following the model in Section II-B, the ith arriving data
packet waits for ti − ai time in queue, and then gets served
in di time units. Following [26], the total delay is defined
as the area in between the cumulative departing data curve,
and the cumulative arriving data curve. In Fig. 4, we show an
example realization using the same transmission, arrival, and
service times used in Fig. 2. The solid blue curve represents the
cumulative received data packets over time; the dotted black
curve represents cumulative departed (served) data packets
over time; and the shaded area in yellow represents the total
delay DT . The delay of the first data packet for instance is
given by B(t1 − a1 ) + 12 Bd1 , where B is the length of the
data packet in bits. Computing the area for general N arrivals,
the delay minimization problem is given by
min
t,d

s.t.

N
X

2ti + di

i=1

problem (9) constraints

(16)

We see that minimizing delay in problem (16) is almost the
same as minimizing age in problem (9). The main difference
is that to minimize age, transmission and service times are
weighted by arrival times, while this is not the case when
minimizing delay. The reason lies in the definitions of age and
delay; the delay of a packet arriving at time a stays the same
if it arrives at time a + δ, provided that its service time is the
same, and that its transmission time relative to its arrival time
is the same. The age of a packet on the other hand is directly

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