Quality of Service Radar Resource Management with Task Dependencies

Abstract

This paper presents an exact quality of service radar resource management model which considers pairwise dependencies between tasks and is based on mixed-integer programming. This is in contrast to the common approach, which is not able to model any dependencies and hence has to assume independence among the tasks, which is generally an incorrect simplification. We define and implement substitutabilities between radar tasks with regard to their utility. Numerical results demonstrate that in scenarios with substituabilities, the proposed method can achieve a significantly better operational radar performance than a traditional approach, especially in high load situations. Though, a drawback of the added complexity due to considering dependencies is an increased computational runtime in comparison to the common method. Due to implementing dependencies, the method is especially well suited for complex scenarios with many substituabilities among radar tasks that need to be exploited under high task load.

Full text

Quality of Service Radar
Resource Management with
Task Dependencies
CHRISTOPH VOLLWEITER
Fraunhofer FKIE, Wachtberg, Germany
Abstract— This paper presents an exact quality of service radar
resource management model which considers pairwise dependencies
between tasks and is based on mixed-integer programming. This is
in contrast to the common approach, which is not able to model
any dependencies and hence has to assume independence among
the tasks, which is generally an incorrect simplification. We define
and implement substitutabilities between radar tasks with regard to
their utility. Numerical results demonstrate that in scenarios with
substituabilities, the proposed method can achieve a significantly
better operational radar performance than a traditional approach,
especially in high load situations. Though, a drawback of the
added complexity due to considering dependencies is an increased
computational runtime in comparison to the common method. Due
to implementing dependencies, the method is especially well suited
for complex scenarios with many substituabilities among radar
tasks that need to be exploited under high task load.
Index Terms—Resource management, Quality of service, Cog-
nitive radar, Task analysis, Mixed integer linear programming,
Optimization, Operations research
I. INTRODUCTION
Software based signal- and data processing as well as
the high beam agility provided by electronically steered
array antennas [1]–[3] has enabled multifunction radars
(MFRs) [4] to execute numerous different tasks. Among
others, these include target search, target tracking, com-
munication, synthetic aperture radar (SAR), electronic
support measures (ESM) and electronic counter measures
(ECM). The high beam agility enables the MFR to
dynamically switch between different radar tasks almost
instantaneously [2], [3], which in turn allows a large num-
ber of tasks to be executed by a single antenna. However,
in order to exploit the finite radar sensor resources best,
an efficient radar resource management (RRM) is required
to balance the resources among the tasks.
Manuscript received XXXXX 00, 0000; revised XXXXX 00, 0000;
accepted XXXXX 00, 0000.
Author’s address: Christoph Vollweiter, Department of
Sensor Data and Information Fusion, Fraunhofer Institute
for Communication, Information Processing and Ergonomics
FKIE, Fraunhoferstraße 20, 54434 Wachtberg, Germany. E-mail:
(christoph.vollweiter@fkie.fraunhofer.de).
0018-9251 © 2022 IEEE
One way to handle this problem are rule based
methods [5]. However, they have the drawback that they
generally cannot properly adapt for the mission objectives
in a dynamically changing environment. Quality of ser-
vice (QoS)-based RRM methods are an alternative. Using
task specific performance models, they assign a utility
to the predicted benefit of executing a task with given
operational parameters. In order to achieve the overall
best possible performance for all tasks, the utility sum of
all tasks is maximized.
However, using the simple utility sum of all tasks
is only possible under the assumption that all tasks are
independent, which is generally not the case. For example,
let there be a group of two targets flying closely together
and each target is tracked by a separate tracking task.
Then it might be possible to cover both targets using
only the radar beam of a single task instead of executing
two separate tracking tasks [6]. We define this type of
dependency as substitution, i.e. one or multiple tasks
can be substituted by a single task. In the example, the
substituting task could either be one of the two existing
tracking tasks, a search task or alternatively an entirely
new tracking task could be generated whose beam might
be centered in the area between the two targets. In such
a case, the tracking tasks are obviously not independent.
Hence, it is necessary to model in the RRM that a single
task may not only provide utility for itself, but also for
other tasks. Exploiting substitutions among different tasks
can lead to savings in sensor resources, which are in turn
available for other tasks, thereby increasing the overall
efficiency of a radar system. Given the increase in aerial
radar targets, e.g. due to the affordability of unmanned
aerial vehicles, considering substitutions among tasks
should be viewed as an important part of a cognitive radar
system [7]. Section II B will discuss task substitutabilities
in more detail.
This paper proposes a novel optimization model of a
QoS RRM based on mixed-integer programming (MIP)
that considers substitutabilities among tasks. However,
integrating these dependencies into the optimization prob-
lem increases the complexity considerably and thereby
also the computational runtime. Since RRM algorithms
should be suitable for real time operation, emphasis
was placed on keeping the computational runtime of the
optimization problem short. Hence, some model simplifi-
cations were necessary. One example for this is, that we
only consider pairwise dependencies between the tasks.
However, solving the optimization problem presented in
this paper is even with our simplifications a NP-hard
problem [8]. Though, according to our numerical simu-
lations, considering substituabilities among tasks enables
the proposed QoS RRM to achieve a significantly better
operational performance than the classical QoS RRM
model. This applies especially in scenarios with many
tasks and hence a high load on the radar system and is the
main advantage of the proposed QoS RRM optimization
model.
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Recently some QoS RRM models that also consider
dependencies were published [9]–[12]. Common to all
of them is, that they are extensions of QoS resource
allocation method (Q-RAM) [13]–[15]. Q-RAM is in [9]
adapted for a rotating phased array radar. Hence, in this
case the dependencies are time related, since the tasks
cannot be executed at any time. In [10] task interferences,
e.g. caused by jamming, are considered. That means the
utility of a task is dependent whether it is executed in
an interfered or non interfered time interval. Hence, the
dependencies in [10] are also time dependent and not
dependent on other tasks unlike to the proposed approach.
Both [11] and [12] integrate dependency relations
between one synchronization task and a subset of all other
radar tasks. Depending on the chosen operational param-
eters for the synchronization task, the utility achieved by
the other tasks is influenced. In [11] the dependency exists
due to the synchronization task determining the exact
location of the platform, while in [12] synchronization
is needed in order to limit clock drift in a bistatic radar
setting. In both cases [11] and [12], the dependency is
pairwise between a given single synchronization task and
multiple other tasks, i.e. a single one-to-many relationship
consisting of multiple pairs. This is in contrast to the
method proposed in this article, which supports multiple
pairwise many-to-many dependencies.
In addition to considering dependencies, another dif-
ference of the proposed method in contrast to some
common QoS RRM methods known in the literature like
Q-RAM [13]–[15], methods based on Q-RAM [9]–[12]
or the agent-based approaches presented in [16]–[20] is,
that this method models the possible sensor resources
allocated to each task continuously and not discretely.
Depending on the granularity of the discretization, this
can further improve the total achievable utility when
maximizing the overall utility of all tasks continuously
instead of discretely. Though, the proposed method is
not the first QoS RRM model to optimize the problem
continuously instead of discretely (c.f. [21]).
Worth mentioning are also [22], where integer pro-
gramming is applied to just-in-time optimization of radar
search patterns as well as [23], which presents a schedul-
ing approach based on branch-and-bound. Further, mixed-
integer optimization problems are also used in order to
conduct resource management for radar networks, e.g.
[24]–[28]. However, none of [22]–[28] consider task
dependencies, unlike the model proposed in this article.
The main contributions of this article are threefold.
First, we define pairwise substitutabilities between tasks
with regard to their utility in the context of QoS RRM.
Secondly, we expand the traditional QoS RRM formula-
tion [29] to also consider task substitutabilities and lin-
earize the resulting optimization problem. Thirdly, numer-
ical simulations are conducted and analyzed to verify the
claim, that the operational performance of a radar system
can be improved by considering task substitutabilities in
the RRM.
The rest of this article is structured as follows. Section
II provides an overview of the problem definition. First,
the general QoS RRM problem definition without task
dependencies is described, followed by the definition
of substitutabilities among radar tasks. After that, the
introduced general problem definition is extended in or-
der to support task dependencies and a mixed-integer
nonlinear programming (MINLP) optimization problem
is obtained. Subsequently, in Section III this optimization
problem is linearized in order to be able to solve it as a
mixed-integer linear programming (MILP) problem. Next,
Section IV uses numerical simulations to compare the
proposed method with Q-RAM. First, in Subsection IV A
a more theoretical and abstract qualitative evaluation with
simplifications is performed, which can be understood
and reproduced easily. After that, a more application-
oriented analysis is performed in Subsection IV B. This
includes an evaluation of the optimal subpattern assign-
ment (OSPA) metric [30] achieved by the different QoS
RRM methods as well as an analysis of the required
computational runtime. Finally, Section Vconcludes this
article and provides an outlook for possible future work.
II. PROBLEM DEFINITION
The first subsection Aprovides a short overview of
the QoS RRM problem definition, which assumes that
all tasks are independent and hence does not model
dependencies. Next, Subsection Bintroduces and defines
substitutabilities. Finally, in Subsection Cthe problem
definition from Subsection Ais extended in order to
support the task dependencies introduced in Subsection
B.
A. QoS RRM Problem Formulation Without Task
Dependencies
The QoS RRM problem formulation without task
dependencies introduced in this section is mostly based
on [29]. For more details, please refer to [29] or [31].
Let there be independent tasks Ti, i ∈N=
{1, ..., NT}with operational parameter selections υi∈
Υi⊂Rn, i ∈Nand uncontrollable environmental
parameters e∈E. Examples for radar tasks are target
search, target tracking, communication, SAR, ESM or
ECM. Typical operational parameters for a given task
include the number of radar pulses and the revisit time.
Each operational parameter selection υiconsists of a set
of chosen operational parameters for a task Ti. Examples
for environmental parameters include the distance to the
target or the target bearing. For each task Tiwith i∈N
a quality
ˆqi: Υi×E→R(1)
can be defined. The quality of a task characterizes how
well it is executed and depends on the task type (e.g.
expected error for tracking tasks or cumulative detection
probability for a search task). The prediction of the
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content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
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expected quality depending on a given operational param-
eter selection and estimated environmental parameters is
usually implemented via performance models [16], [32]–
[34]. According to the mission requirements, the quality
of each task Tiis mapped to a utility via
ui:R→[0,1].(2)
The utility function uidefines for each task the
mission specific value of achieving a certain quality, e.g.
in the case of tracking tasks some targets may require a
higher tracking accuracy than other targets.
After defining the utility function, the objective func-
tion that maximizes the total utility over all tasks can be
defined:
max X
i∈N
γiωiuiˆqi(υi, e)(3)
The binary variable γi∈ {0,1}defines if task Tiis
executed and the weight ωi∈R+indicates the situation-
and mission-dependent priority of a task. As mentioned
earlier, in this subsection we assume that all tasks are
independent from each other. This means no substitutabil-
ities are considered. Hence, the objective function can be
modeled as simple weighted sum.
Since only a limited amount of radar resources ˆr∈R+
is available, an additional function specifying the required
sensor resources depending on the chosen operational
parameter selection and the environmental parameters is
required for each task Ti:
ˆgi: Υi×E→R+(4)
By using the function ˆgia constraint can be defined that
limits the amount of available radar resources:
X
i∈N
γiˆgi(υi, e)−ˆr≤0(5)
Overall, this leads to the optimization problem
max X
i∈N
γiωiuiˆqi(υi, e)(6a)
X
i∈N
γiˆgi(υi, e)−ˆr≤0(6b)
υ∈Υ(6c)
γi∈ {0,1} ∀i∈N(6d)
with υ= (υ1, ..., υNT),Υ=Υ1×... ×ΥNTand e∈E.
The problem (6) optimizes over the set of all opera-
tional parameter selections υ∈Υfor all tasks. However,
it is only necessary to consider those parameters that lie
on the concave majorant of the resource utility plot [29].
The blue line in Figure 1shows an exemplary concave
majorant of a task Tjwith fixed environmental parameters
e∈Ein the resource utility plot. Each point on the
concave majorant represents one evaluated operational
parameter selection υj∈Υjfor task Tj.
Let ˆ
Υ⊆Υbe the set of operational parameter
selections for all tasks that lie on the concave majorant.
As shown in [29], for operational parameter selections
ˆυ= (ˆυ1, ..., ˆυNT)∈ˆ
Υon the concave majorant a mapping
ϕ:ˆ
Υ→RNT
+(7)
0 0.2 0.4 0.6 0.8 1
0
0.5
1
Sensor resources
Utility
Task Tj
Task Tisubstituted by Tj
Fig. 1: Exemplary resource utility plot for task Tjand the
utility achieved for task Tiby task Tjvia substition.
can be defined. It consists of mappings
ϕi:ˆ
Υi→R+(8)
for each task Tiwith i∈N. Using ϕi(ˆυi) = riwith
ri∈R+and under the assumption that the concave
majorant is continuous, the optimization problem (6) can
be reformulated:
max X
i∈N
γiωiuiqi(ri, e)(9a)
X
i∈N
γiri−ˆr≤0(9b)
r∈RNT
+(9c)
γi∈ {0,1} ∀i∈N(9d)
with qidefined as
qi:R+×E→R,(10)
qi(ϕi(ˆυi), e) = ˆqi(ˆυi, e)∀ˆυi∈ˆ
Υi, e ∈E. (11)
This reformulation allows us to define the optimization
problem dependent on the resources rinstead of the
operational parameters υ∈Υlike in (6). The above
optimization problem (9) is a mixed integer quadratic
optimization problem because both the objective function
as well as the constraint (9b) contain a product of the
binary variable γiand of uiqi(ri, e)as well as ri,
respectively.
Note, that in (9c) the variable ris defined continu-
ously, which allows resource assignments on the complete
concave majorant for each task Ti. This assumes that
each resource selection ri∈R+on the concave majorant
can be mapped to an operational parameter selection.
But in practice the concave majorant is typically only
sampled discretely and linearized in between (e.g. see
the blue line in Figure 1). However, since typically all
radar tasks possess the continuous operational parameter
revisit interval, an easy approximate approach could be to
simply select the operational parameter selection on the
concave majorant nearest to the selected resource amount
and to adjust the revisit interval such that all assigned
resources are used. Though, it is of course also possible to
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develop more sophisticated strategies, e.g. by dynamically
resampling the concave majorant in that area to find a
better matching operational parameter selection.
In the special case that all utiliy functions intersect
the origin, the binary variable γiin optimization problem
(6) or (9) is not required in order to deactivate tasks and
can hence be removed. This can for example be the case
when all tasks Tihave a parameter configuration υ0
i∈Υi
on the concave majorant with
uiˆqi(υ0
i, e)= 0 ∀e∈E , (12)
ˆgi(υ0
i, e)=0 ∀e∈E. (13)
Without γi, the problem formulation (6) is identical to
the classical QoS RRM model found in the literature.
Consequently, since uiqi(ri, e)is concave and the bi-
nary variable γiwas removed, (6) or (9) can be solved
as convex optimization problems using standard methods.
Besides the Karush-Kuhn-Tucker (KKT) constraints used
in convex optimization [35], other methods to solve such
a problem include Q-RAM [13]–[15], continuous double
auction parameter selection (CDAPS) [16]–[19] or an ap-
proach based on an agent trained via deep reinforcement
learning [20].
Although, in general the above special case does
not apply. In theory, it is always possible to insert
an operational parameter selection υ0
iinto the concave
majorant. When optimizing the QoS RRM discretely,
that means only the discrete points on the blue concave
majorant shown in Figure 1would be valid solutions,
inserting a control parameter selection υ0
iwould indeed
be possible and γicould be removed. Though, in case of
the blue concave majorant shown in Figure 1, it would
be necessary to update the concave majorant after adding
υ0
i. This would remove the first point on the blue concave
majorant shown in Figure 1with resource requirements
of 0.1 and zero utility. However, when optimizing the
problem continually, then simply adding a parameter
selection υ0
ito the concave majorant could lead to an
undesired model of the reality. This applies to the above
discussed case of the blue concave majorant pictured in
Figure 1. This is because after removing the point with
resource requirements of 0.1, resource amounts slightly
above zero would already provide some utility, which
does not comply with the original concave majorant.
B. Definition of Substitutabilities
This section introduces additional parameters for the
QoS RRM problem required in order to model substi-
tutabilities among tasks. Substitutabilities among tasks do
not necessarily have to be symmetrical.
A common example for substitutability could be two
tracking tasks or a tracking task and a search task if
both tasks can be executed using a single radar beam.
This can for example be if both tracking targets are
located at a similar angle from the platform or if the
target is also covered by the search beam. However, this
dependency is not necessarily symmetrical. For example,
if one tracking task T1tracks a target with a low radar
cross section (RCS) and task T2a target with a high RCS,
then T1might be able to substitute T2, however not the
other way around. This is because the number of pulses
required to track a target with a high RCS adequately is in
general much lower than the number of pulses required for
tracking a target with a low RCS. Similarly, substituting
a search task with a tracking task might not be desirable,
because this could create gaps in the search raster [1].
Note, that in addition to executing either T1or T2, it
would also be possible to introduce an additional task T3
that is well suited for substituting both T1and T2. Its beam
could be centered in the area between the two targets and
its operational parameter selections could be chosen such
that both T1and T2could be substituted well. Such a task
T3would be modeled to not create any utility by itself,
but only via substitution for other tasks. By exploiting
substitutabilities among tasks the resource efficiency of a
radar system can be improved.
When considering dependencies among tasks, one has
to decide between how many different tasks dependencies
shall be modeled, since relations among tasks are not
only limited to task pairs. It is also possible to consider
how good a set of tasks can substitute another set of
tasks. However, this increases the number of dependen-
cies exponentially with the number of tasks. Since this
would also increase the complexity of the QoS RRM
optimization problem and hence its computational run-
time, we restrict ourselves to only considering pairwise
dependencies. Apart from the complexity of the QoS
RRM optimization model, radar performance models also
have to be executed in order to provide the necessary
input parameters for the optimization model, e.g. how
good one set of tasks could substitute another set of tasks.
Doing this for dependencies among sets of tasks instead
of just pairwise dependencies would also increase the
computational cost significantly. This is another reason
why we only consider pairwise dependencies.
We model the presence of a substitutability among
two tasks Tiand Tjwith i, j ∈Nvia the parameter
Iij := (−1,if Tican be substituted by Tj
0,if Tiis independent of Tj.(14)
Of course, if i=jthen there cannot be a substitutability
and Iij = 0 holds. If Iij =−1, that means that task Ti
can be substituted by task Tj, but not necessarily the other
way around.
Note, that this restriction of only considering pairwise
dependencies does not limit the ability of a single task to
substitute multiple other tasks. The model approach for
the task dependencies is loosely inspired by the Choquet
integral [36], [37], which is however only able to represent
symmetrical dependencies. Though, it is not limited to
pairwise dependencies unlike our model approach.
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C. Integrating dependencies into a QoS RRM
problem
In this subsection substitutabilities, defined in the
prior Subsection B, are integrated into the mathemati-
cal QoS RRM problem formulations of Subsection A.
As mentioned previously, we restrict ourselves to only
considering pairwise task dependencies in order to limit
the computational complexity and hence also the compu-
tational runtime.
For substitutabilities we assume that a task Tican
only be substituted by a maximum of one other task.
Without this assumption we would either have to model
more than pairwise task dependencies or we would have
to assume that the utility generated by substituting Tiby
Tj1is independent of the utility generated by substituting
Tiby Tj2. In other words, this would mean that the utility
for Tiby substituting with Tj1and the utility for Tiby
substituting with Tj2is additive. However, in general this
is clearly not the case. Hence, it was preferred to only
allow task substitutions by a maximum of one other task.
If task Tican be substituted by a task Tj, i.e. Iij =−1,
then we have to predict the utility that task Tjcan
contribute to task Ti. The utility function of task Tj
is discretely approximated by the evaluated operational
parameter selections ˆυj∈ˆ
Υjon the concave majorant,
as shown in Figure 1. Hence, if Tjis to substitute Ti,
the performance model of Ticould be used to predict the
expected quality for task Tiwhen using the operational
parameter selections ˆυjon the concave majorant of task
Tj. Note, that depending on the current performance
model implementation of Ti, modifications might be
required. For example, it is necessary for the performance
model to support off-beam penalties (see e.g. Appendix
14A in [5]) in order to predict the correct expected quality
due to task substitution.
We define the resulting quality for Tiachieved by a
operational parameter selection ˆυjof task Tjas ˆqij ( ˆυj, e).
Analogous to the procedure described in Section II A, we
can find a mapping ϕij (ˆυj) = rjand define a function
qij (ϕij (ˆυj), e) = ˆqij (ˆυj, e)∀ˆυj∈ˆ
Υj, e ∈E. (15)
The predicted quality generated via substitution can then
be passed to the utility function of Tiin order to calculate
the predicted utility achieved by substituting Tiwith Tj.
That utility function is defined as
uiqij (rj, e).(16)
Based on the evaluated discrete points ˆυj, a continu-
ous concave utility function can be approximated in the
resource utility space (analogously like in Section II A for
uiqi(ri, e)). Figure 1illustrates the concave majorant
of the substituting task Tjin blue as well as the concave
majorant of the utility that Tjcould generate for Tiin red.
This corresponds to the utility functions ujqj(rj, e)and
uiqij (rj, e), respectively. Each marker on the concave
majorants symbolizes one evaluated operational parameter
selection. Note, that since for both concave majorants
the same operational parameter selections were evaluated,
the points are at the identical resource amount. The only
exception are points which are not part of the concave
majorant of uiqij (rj, e), i.e. for the first point on
the blue majorant of Figure 1there is no counterpart
with an identical resource amount on the red concave
majorant. However, while the same operational parameter
selections have been evaluated, they generate different
utility amounts. In this example the utility for Tivia
substitution by Tjis below the utility for Tj. Though,
this must not necessarily be the case and depends on the
tasks Tiand Tj.
Note, that when building the concave majorant of the
substitution utility uiqij (rj, e), we only considered the
operational parameter selections ˆυj∈ˆ
Υjon the concave
majorant of task Tjand not all possible operational
parameter selections υj∈Υj. However, ˆυj∈ˆ
Υjwhich
are optimal for ujqj(rj, e)must not necessarily be
optimal for the concave majorant of uiqij(rj, e). This
means by not restricting ϕij (ˆυj)and in consequence also
not restricting the quality function defined in (15) to
ˆυj∈ˆ
Υj, it might be possible to construct a better concave
majorant for uiqij (rj, e). Though, after having chosen
a resource amount rjfor task Tj, we have to map the
resource selection rjto an operational parameter selection
which should actually be executed by the radar system. If
the concave majorants of ujqj(rj, e)and uiqij(rj, e)
encompass different operational parameter selections than
it becomes unclear which operational parameter selection
to pass to the radar system. This is the main reason why
we only consider ˆυj∈ˆ
Υjwhen determining the concave
majorant for uiqij (rj, e). In addition to that, this also
avoids having to evaluate ˆqij (υj, e)for all υj∈Υjwhich
reduces the computational requirements.
The issue described above could in theory be over-
come by defining concave majorants for each possible
combination of substitutions, i.e. if Tjis executed to
provide utility for itself and also for tasks Ti1, ..., Til,
we could calculate an optimal concave majorant for this
special case using the weighted sum of their individual
utilities. This would result in having to model different
concave majorants for each possible combination of sub-
stitutions, which would ultimately increase the size of the
resulting optimization problem considerably. Hence, this
approach was not pursued further.
Next, we define for each possible substitution
xij ∈ {0,1} ∀i, j ∈N:Iij <0(17)
as the binary variable that decides whether task Tiis
substituted by task Tj. Then we can define the utility for
Tivia substitution provided by other tasks as
Us
i:= X
j∈N:Iij <0
xij γjuiqij (rj, e)(18)
Analogously, we define
Ub
i:= γiuiqi(ri, e)(19)
VOLLWEITER: QoS RRM with Task Dependencies 5
This article has been accepted for publication in IEEE Transactions on Aerospace and Electronic Systems. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
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as the basic utility provided if Tiis executed directly.
Hence, the total utility for task Tibecomes
Ui:= Ub
i+Us
i.(20)
Based on the above definitions, we can update the ob-
jective function from (9) in order to include the utility
provided by substituted tasks:
max X
i∈N
ωiUi(21)
Since we made the assumption that a task Timay only
be substituted by exactly one other task Tjand since
substitutions are only allowed if task Tiis inactive, we
add the following constraint:
X
j∈N:
Iij <0
xij ≤1−γi∀i∈N:∃j∈Nwith Iij <0(22)
In summary, this leads to the following optimization
problem:
max X
i∈N
ωiUi(23a)
X
i∈N
γiri−ˆr≤0(23b)
X
j∈N:Iij <0
xij ≤1−γi
∀i∈N:∃j∈Nwith Iij <0
(23c)
ri∈R+∀i∈N(23d)
γi∈ {0,1} ∀i∈N(23e)
xij ∈ {0,1} ∀i, j ∈N:Iij <0(23f)
III. PROBLEM LINEARIZATION
The optimization problem (23) includes binary vari-
ables γi, xij ∈ {0,1}and continuous variables ri∈R+.
In general, this means that it has to be solved using
MIP methods. In addition to that, there are non linear
functions both in the objective function as well as in
some constraints (e.g. products γiuiqi(ri, e)). Hence,
the problem at hand is a MINLP. Since MIP problems can
usually be solved much more efficiently if they are linear,
we linearize the optimization problem in this section in
multiple steps.
In the first step we linearize the product γiuiqi(ri, e)
by substituting it with fi∈[0,1]. The product equals Ub
i
defined in (19), which is part of Ui. Since γi≥0holds
as well as uiqi(ri, e)≥0and because the product is
part of the objective function, which is maximized, fi
only has to be limited upwards. Due to γi∈ {0,1}and
uiqi(ri, e)∈[0,1], we can do this using the constraints
fi≤γi∀i∈N(24)
fi≤uiqi(ri, e)∀i∈N. (25)
Analogously, we substitute the term
xij γjuiqij (rj, e), which is part of Us
i, with aij
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Sensor resources
Utility
Fig. 2: Example of a piecewise linear approximation of
a concave function (black) using approximating tangents
(red). The feasible solution space for ˜uiis shown in blue.
and add the constraints
aij ≤xij ∀i, j ∈N:Iij <0(26)
aij ≤γj∀i, j ∈N:Iij <0(27)
aij ≤uiqij (rj, e)∀i, j ∈N:Iij <0.(28)
Additionally, the quadratic resource constraint (23b)
is replaced with
X
i∈N
ri−ˆr≤0(29)
ri≤γi∀i∈N. (30)
Next, we have to linearize the utility functions. For
that, we will substitute uiqi(ri, e)and uiqij (rj, e)
with variables ˜ui∈(−∞,1] and ˜uij ∈(−∞,1], respec-
tively. In order to limit ˜uiand ˜uij upwards, we approx-
imate the utility function from above using piecewise
linear tangents as shown in Figure 2. This is possible,
because the function uiqi(ri, e)was in Section II A
assumed to be concave (and analogus for uiqij(rj, e)).
The accuracy of this approximation can be increased
arbitrarily by increasing the number of approximating
tangents.
The next steps will only be described for ˜ui, however
the procedure required for ˜uij is analogous. Limiting ˜uiis
achieved by selecting NLpoints r1
i, ..., rNL
iin the interval
(r0
i, r1
i)(for example, but not mandatory, equidistant). We
define r0
ito be the highest resource amount that achieves
a utility of zero and r1
ito be the lowest resource amount
that reaches the maximal utility (which is often the utility
one). Then for each point rl
iwith l∈L={1, ..., NL},
the corresponding tangent tl
i(ri)that touches the utility
function uiqi(ri, e)at ri=rl
iis determined. After
that, uiqi(ri, e)can be approximated in the interval
ri∈(r0
i, r1
i)by the variable ˜ui∈Rwith the constraints
˜ui≤tl
i(ri)∀i∈N, l ∈L. (31)
If there is already a concave majorant of the utility
function available that consists of discrete points (as
required by Q-RAM) and linearizations in between (e.g.
as shown in Figure 1), than it is not necessary to calculate
a set of tangents as approximation. In that case it is
6 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. XX, No. XX XXXXX 2022
This article has been accepted for publication in IEEE Transactions on Aerospace and Electronic Systems. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
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sufficient to simply add the equations of all lines that
connect adjacent points on the concave majorant.
However, a problem with this substitution of the utility
function is that in some cases the substituting variable
˜uimay become negative for ri∈[0, r0
i]due to some
approximating tangents. This can be observed in Figure 2
where resource amounts smaller than 0.1 would cause
negative utility due to the dashed tangent. This problem
always occurs if one of the tangents becomes negative for
positive resource amounts.
Due to constraint (25), fimust always be smaller
than uiqi(ri, e), which we are about to substitute with
˜ui∈R. After that substitution, (25) becomes fi≤˜ui.
Since we defined fi∈[0,1],˜uican only assume non-
negative values. This leads to the problem that under
some circumstances positive resource amounts riwould
be required even though the task bears no utility. However,
this would obviously not be the correct model for the
application.
Since fiis part of the objective function, it may not
become negative. The relation ˜ui≥0is not strictly
necessary, but fi≥0must hold, because only fiis
used directly in the objective function and the constraints.
However, the positivity of the utility was an important
assumption in the prior linearization of Ub
iby fi. Thus,
we need to update the introduced substitution of the term
γiuiqi(ri, e)by fi. We want fito be defined as follows:
fi:= (0if ˜ui<0
γi˜uiotherwise (32)
Since fi∈[0,1] and the constraint (25) holds, ˜ui<0
is infeasible. As described above, this however implicates
that in order to achieve zero utility for a task, strictly
positive values rimight be necessary under some circum-
stances. Hence, we relax (25) by replacing it with
fi≤˜ui+ (1 −γi)M∀i∈N. (33)
The introduced constant M∈R+must be sufficiently
large such that tl
i(ri)+M≥0holds for all approximating
tangents tl
i(ri)with l∈L,i∈Nfor all ri∈[0,ˆr].
However, since all tangents have a positive slope, their
minimum in the interval [0,ˆr]is at ri= 0 and it is there-
fore sufficient to only check their value for this ri. Further,
due to the concavity of the utility function approximated
by the set of tangents, only the first approximating tangent
of each task Tineeds to be considered since that is the
tangent with the steepest slope.
By adding the term (1−γi)Mthe restrictiveness of the
above constraint can be activated or deactivated based on
the binary variable γi. Hence, values ˜ui<0are possible,
however only if γi= 0, because then the constraint is
relaxed by adding Mon the right side. In the case of
γi= 0,ri= 0 is implicated by constraint (30) and fi= 0
by constraint (24). This means γi= 0 disables the task
Ti. Therefore, a utility of zero and thereby a deactivation
of the task is always achievable with a resource amount
of zero, as desired.
Finally, we also need to handle the problem of ˜uij
becoming negative. This is achieved almost analogously
by replacing (28) with the constraints
aij ≤˜uij + (1 −xij )M∀i, j ∈N:Iij <0.(34)
This concludes the necessary steps of linearizing the
optimization problem defined in (23). However, we fur-
ther add the following constraints to the problem:
˜ui≤γi∀i∈N(35)
˜uij ≤γj∀i, j ∈N:Iij <0(36)
We can add these, because if task Tiis deactivated, i.e.
γi= 0, than it can of course not achieve any utility and
˜ui= 0 must hold. Analogously for ˜uij : If task Tjis
deactivated due to γj= 0, than task Tjcan not provide
any utility for task Tiand hence ˜uij = 0.
MILPs are typically solved using a technique called
Branch and Cut [38]. In the first step the linear program-
ming (LP) relaxation is solved, i.e. the integer require-
ments are relaxed and integer variables are treated as
continuous variables. After that, the integer variables that
assumed non-integer values are considered. On the one
hand, additional cuts can be added to the LP relaxation
so that the non-integer solutions become infeasible. Care
must be taken to not remove feasible integer solutions
from the solution space. Then, the updated LP relaxation
is solved again. On the other hand, Branch and Bound can
be applied, i.e. fixing integer variables to discrete values
and searching a decision tree.
By adding the constraints (35) and (36), additional
cuts are introduced which results in a tighter feasible
solution space of the LP relaxation. While introducing
additional constrains also enlargens the optimization prob-
lem, tests have shown that in our case these additional
constraints help to reduce the runtime. In summary, we
get the following new optimization problem:
max X
i∈N
ωifi+X
j∈N:Iij <0
aij (37a)
X
i∈N
ri−ˆr≤0(37b)
X
j∈N:Iij <0
xij ≤1−γi
∀i∈N:∃j∈Nwith Iij <0
(37c)
fi≤˜ui+ (1 −γi)M∀i∈N(37d)
aij ≤˜uij + (1 −xij )M∀i, j ∈N:Iij <0(37e)
ri≤γi∀i∈N(37f)
fi≤γi∀i∈N(37g)
aij ≤xij ∀i, j ∈N:Iij <0(37h)
aij ≤γj∀i, j ∈N:Iij <0(37i)
˜ui≤tl
i(ri)∀i∈N, l ∈L(37j)
˜ui≤γi∀i∈N(37k)
˜uij ≤tl
ij (rj)∀i, j ∈N:Iij <0, l ∈L(37l)
˜uij ≤γj∀i, j ∈N:Iij <0(37m)
VOLLWEITER: QoS RRM with Task Dependencies 7
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content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
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ri∈R+∀i∈N(37n)
fi∈[0,1] ∀i∈N(37o)
aij ∈[0,1] ∀i, j ∈N:Iij <0(37p)
˜ui∈(−∞,1] ∀i∈N(37q)
˜uij ∈(−∞,1] ∀i, j ∈N:Iij <0, l ∈L(37r)
γi∈ {0,1} ∀i∈N(37s)
xij ∈ {0,1} ∀i, j ∈N:Iij <0(37t)
This linearized optimization problem can be solved using
standard MILP solvers like Gurobi [39].
IV. EXPERIMENTAL RESULTS
This section presents the experimental results using
the proposed QoS RRM method. First, in Subsection A
we compare the proposed method with Q-RAM qualita-
tively on a rather theoretical level. For that, we generate
abstract input data required by the RRM algorithms and
solve the RRM problem for a single time step. This means
no full scenario over multiple timesteps with executed
radar tasks, received measurements, track updates etc. is
simulated. The tasks are modeled abstract, i.e. random
concave majorants are generated without using complex
and task-specific performance models and we compare the
achieved overall utility as well as the number of planned
tasks. The reason for these simplifications is to provide an
evaluation that can be understood and reproduced easily.
After that, a full scenario simulation is analyzed in
Subsection B. This includes an evaluation of the OSPA
metric over the scenario time as well as a runtime
evaluation of the proposed RRM method. The goal of this
analysis is to demonstrate the advantage of the proposed
RRM method compared to Q-RAM in an application-
oriented way.
A difference between Q-RAM and the proposed RRM
approach is, that the proposed method can select the
resources continuously unlike Q-RAM, which can only
select the discrete resource utility points defined for each
utility function. This can enable the proposed method to
achieve higher utility values and to hold more tasks active.
However, its impact depends on the discretization of the
resource utility points. In order to be able to evaluate
the qualitative impact of modeling substitutabilities, it
is important to distinguish the benefit achieved due to
considering substitutabilities from the advantage of being
able to assign resources continuously instead of discretely.
We ensure this by also executing our proposed approach
without modeling any substitutabilities.
The mapping of the continuous resource selection
rito the respective discrete operational parameters on
the concave majorant is implemented according to the
easy approximate approach outlined in Section II A: the
operational parameter selection on the concave majorant
nearest to the selected resource amount is chosen and the
revisit interval is adjusted such that all assigned resources
are used.
TABLE I: Tuned Gurobi parameters used for the evalua-
tions.
Parameter Value
Method 0
MIPFocus 1
SimplexPricing 3
Cuts 1
MIRCuts 0
AggFill 100
PreDual 0
In all simulations seeds are used to initialize random
generators in order to guarantee comparability over the
conducted Monte Carlo (MC) runs of the proposed RRM
method and Q-RAM. The MILP optimization problems
are solved using the solver Gurobi (version 10.0.0) [39].
The tuned Gurobi parameters listed in Table Iare used in
order to shorten the optimization runtime for our specific
MIP problem. All evaluations were performed on an Intel
Xeon Gold 6126 CPU.
A. Theoretical RRM Performance Comparison
In this subsection we present a qualitative comparison
of the proposed RRM method with Q-RAM on a rather
theoretical level. This allows for an evaluation that can
be understood and reproduced easily.
1. Evaluation Setup
For the evaluation we execute the proposed QoS RRM
approach in multiple MC runs with different input data.
Each MC simulation is independent. Hence, we evaluate
the execution of the QoS RRM problem for a single time
step and not a full scenario with moving targets over
multiple time steps. For each MC run we need input
parameters for the optimization problem. These are
•utility functions uiqi(ri, e)for all tasks Ti,i∈N,
•definition of dependency between task pairs formal-
ized by Iij for all i, j ∈N,
•substitution utility functions uiqij (rj, e)for i, j ∈
N:Iij <0.
In order to be able to execute a large number of MC
simulations, the above mentioned input parameters are
generated randomly. We model the radar tasks abstract
and do not distinguish among different task types. That
means for the tasks Tiwe generate the corresponding
utility functions as well as their task dependencies ran-
domly without executing complex performance models
specific to a task type. The utility functions uiqi(ri, e)
of the tasks Tirequired in (19) are generated based on
the function
ui(ri) = (1 −ˆα)−exp(−βri).(38)
The parameter ˆαlimits the maximum achievable utility,
while βcontrols the slope of the utility function. Both
8 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. XX, No. XX XXXXX 2022
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content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
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0 0.05 0.1 0.15 0.2
0
0.2
0.4
0.6
0.8
1
Sensor resources
Utility
Fig. 3: 30 randomly generated utility functions based on
utility function (38).
parameters are selected randomly in the simulations for
each task Ti. The parameter ˆαis chosen based on the
normally distributed random variable αusing
ˆα= min max(α, 0),0.99(39)
with α∼ N (0,0.25). The parameter βis chosen uni-
formly distributed based on the distribution U(βmin, βmax)
with βmin = 20 and βmax = 200. Figure 3visualizes 30
randomly generated utility functions based on the above
parameters.
After having generated utility functions for all tasks
Ti, task dependencies are chosen randomly. In detail, that
means selecting random pairs i, j ∈Nof task indexes
until the given number of substitutabilities is achieved
and setting Iij =−1. Additionally, we need to define the
substitution utility for task Tiachieved by executing task
Tjdefined by uiqij (rj, e)required in (18). Since the
tasks are modeled abstract, we assume that uiqij (rj, e)
is calculated by scaling the utility of task Tjlinearly using
a randomly selected factor Is
ij while ensuring a maximum
utility of 1, i.e.
uiqij (rj, e):= min Is
ij ujqj(rj, e),1.(40)
The linear factor Is
ij is chosen uniformly distributed based
on U(0,1.25). However, as stated previously in Section
II C, in practice task dependent performance models
need to be used in order to predict the utility function
uiqij (rj, e)instead of the simplification used here. In
Section IV B such a task dependent performance model
will be used.
Further, we also assume due to simplicity that all tasks
are weighted equally, i.e. ωi=ωjholds for all tasks
i, j ∈N. As discussed in Section III when linearizing the
utility functions and visualized in Figure 2, each utility
function has to be approximated using discrete points.
This is true for both the proposed approach of solving the
QoS RRM problem as well as for the classical approach
Q-RAM. For that, we select 30 resource utility points per
utility function. These points are distributed equidistantly
in the resource interval at which the utility is positive.
In case of the proposed new approach, the first point
is at the last resource amount for which the utility is
still zero. Formally, for the task Tithis is the resource
amount ˜r0
iwith u(˜r0
i) = 0 and u(˜r0
i)>0for all r > ˜r0
i.
However, for Q-RAM the first discrete point always needs
to be at the resource amount of zero, because otherwise
it would not be possible to deactivate the task. Hence,
if ˜r0
i>0holds, than the proposed new approach and
Q-RAM use different first points. Points after that are
identical in both approaches. The last of the discrete
resource utility points is at the resource value ˜r1
ifor which
u(˜r1
i)=1−ˆα−0.0001 holds. We subtract the constant
0.0001 because the utility function can become very flat at
the end (see Figure 3for some examples). In those cases
not subtracting a constant would lead to many discrete
points with very similar utility values, which we want
to avoid. The other resource utility points are distributed
equidistantly between the first and the last point. In
the evaluation we compare the solution quality of the
proposed new method with Q-RAM. This includes a
comparison of the overall achieved utility and the number
of tasks that can be held active, i.e. tasks that achieve a
utility higher than zero.
We evaluate scenarios with |N|=
10,20,30,40,50,75,100,200, ..., 1000 tasks and generate
a number of pairwise substituability relationships equal
to 0%, 10%, 20% and 30% of the total number of
tasks in the scenario. These scenarios are then solved
using the proposed new method and Q-RAM. For each
combination of tasks and substitutabilities 1000 MC runs
are performed. The accepted MIP gap after which Gurobi
terminates is set to 0.0001. The MIP gap is defined as
|ObjBound −ObjVal|
|ObjVal|,(41)
where ObjBound is an upper bound for the optimization
problem found by the solver and ObjVal is the current
objective value of the best solution found [39].
2. Evaluation Results
The evaluation results based on the mean over 1000
MC runs are presented in the subfigures of Figure 4. We
distinguish the results based on the number of substi-
tutabilities present in the simulated scenario. Data plotted
in blue, orange, green and red corresponds to scenarios
with substitutabilties equal to 0%, 10%, 20% and 30% of
the total number of tasks, respectively.
Figure 4a illustrates for different numbers of tasks
the achieved relative utility difference compared to the
Q-RAM solution, i.e. the utility achieved by Q-RAM
equals one. The different solid lines visualize the mean
utility of the proposed method for different scenario
parameter settings. For example, the utility achieved by
our proposed method in case of a scenario with 300
tasks and substitutabilities equal to 20% of all tasks is
on average 12.9% better than the utility of the solution
found by Q-RAM. First, we will focus exclusively on the
solid lines in Figure 4a, before we introduce later in the
evaluation the meaning of the dashed lines.
Figure 4a exhibits that the utility increases with
an increasing percentage of generated substitutabilities.
VOLLWEITER: QoS RRM with Task Dependencies 9
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content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
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0 200 400 600 800 1,000
1
1.1
1.2
1.3
Number of tasks
Relative utility difference
(a) Relative utility difference of the proposed method (solid lines)
and the adjusted utility value of Q-RAM (dashed lines) compared
to the unadjusted utility value achieved by Q-RAM for different
substitutability amounts.
0 200 400 600 800 1,000
0
100
200
300
Number of tasks in scenario
Number of substituted/
substitutable tasks
(b) Number of substituted tasks (solid lines) and substitutable tasks
(dashed lines) depending on the total number of tasks for different
substitutability amounts.
0 200 400 600 800 1,000
1
1.1
1.2
Number of tasks
Relative planned active
tasks difference
(c) Relative difference of the number of planned active tasks
(including substituted tasks) for different substitutability amounts
compared to the number of planned active tasks when using
Q-RAM.
0 200 400 600 800 1,000
0
100
200
300
400
Number of tasks
Number of planned
active tasks
QRAM
(d) Absolute number of planned active tasks depending on the
total number of tasks for different substitutability amounts. Solid
lines also consider substituted tasks active, while dashed lines only
show tasks planned for actual execution.
Fig. 4: Evaluation results of theoretical RRM performance comparison for scenarios without substitutabilities (blue),
substitutabilities equal to 10% (orange), 20% (green) and 30% (red) of the total number of tasks (results based on
the mean over 1000 MC runs for each parameter set).
Further, for each of the solid lines the positive utility
difference increases monotonically with an increasing
number of tasks (with the exception of the case without
substitutabilities). A reason for that is because the number
of substitutabilities is defined as percentage of the total
number of tasks, hence the number of substitutable tasks
increases linearly with the number of tasks, as displayed
by the dashed lines in Figure 4b. This in turn provides
on average better possibilities to exploit substitutabilities
and thereby increases the overall utility.
However, the slope of the orange, green and red solid
lines in Figure 4a is decreasing with an increasing number
of tasks. The reason for that is, that at a high number
of tasks the radar resources become more exhausted and
only a fraction of the tasks are active (see the solid lines in
Figure 4d, which show the number of tasks with a positive
utility). This in turn results that the number of conducted
substitutions does not increase linearly with the number of
available substitutions, as illustrated in Figure 4b. Hence,
the utility difference does for higher task numbers not
increase at the same high rate as it does for lower task
numbers.
The solid blue line in Figure 4a plots the utility
difference of the proposed method applied to a scenario
with no substitutabilities. It shows that there is hardly
any difference in the utility achieved when compared
with Q-RAM, even though the proposed new algorithm
is able to choose continuous resource values in contrast
to Q-RAM. This shows, that in the case of the chosen
discretization of the resource utility points, described in
Subsection 1, the utility improvement indicated by the
other solid lines is almost exclusively due to considering
substitutabilities.
Finally, we introduce the meaning of the dashed lines
in Figure 4a. We refer to them as the adjusted utility
value of Q-RAM. These lines show the total utility of
the Q-RAM solution including the utility that would have
been achieved via substitution. This means that even
though Q-RAM does not consider substitutabilities when
calculating the solution, it can still happen by chance that
in the Q-RAM solution a task is executed which could
10 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. XX, No. XX XXXXX 2022
This article has been accepted for publication in IEEE Transactions on Aerospace and Electronic Systems. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/

substitute another task that is not executed. Since that
utility achieved via substitution would materialize when
the tasks are actually executed, even though this was not
planned by Q-RAM and was by pure chance, it should
still be analyzed in a fair comparison. That utility value is
calculated by inserting the solution provided by Q-RAM
into the optimization problem (37) described in Section
III. That means fixing all resource assignments made by
Q-RAM via appropriate constraints and then optimizing
the MIP optimization problem.
Comparing in Figure 4a the dashed lines based on
the Q-RAM solution with the respective solid lines based
on the solution of the proposed method displays a very
significant utility difference. For example, the utility
achieved by a scenario with substitutabilities equal to 10%
and 300 tasks is on average 6.7% better than the unad-
justed Q-RAM objective, while the adjusted Q-RAM ob-
jective is only 2.8% better. This shows that explicitly mod-
eling the substitutabilities offers a considerable advantage
in terms of achieved utility, especially in scenarios with
many tasks. Further, it should be noted that in real world
scenarios the gap between the utility achieved by the
proposed method and the utility indicated by the dashed
lines would probably be bigger. On the one hand, this is
because in practice the tasks are usually not weighted
equally. On the other hand, in many real applications
substitutabilites are not as evenly distributed over the tasks
as in our simulations. Typically substitutabilites are more
clustered, e.g. in the case of a target group there can be
substitutabilites expected among the set of tracking tasks
and the search tasks covering that area and nearby areas.
Hence, it is assumed that if the tasks are not weighted
equally or if substitutabilites are not distributed evenly,
than it becomes more difficult to randomly select a task
that can substitute another task of high utility weight and
thus high influence on the overall utility.
Next, we will evaluate the number of planned active
tasks. The word planned takes into account that the QoS
RRM can only plan the execution of a task. The actual
task execution depends on the task scheduler, which is
separate from the QoS RRM. Figure 4c and 4d depict the
mean amount of planned active tasks depending on the
number of substitutions and total tasks in the scenario. In
Figure 4d the planned active tasks are recorded in absolute
numbers, while Figure 4c shows the relative difference
compared to the mean number of planned active tasks in
Q-RAM for the conducted MC runs.
Figure 4d distinguishes between two different defini-
tions of planned active tasks: in the first definition, which
we will use per default, a planned active task is a task
for which a positive utility is achieved in the QoS RRM
problem. For that it does not matter whether that is due to
executing the task itself or due to another task providing
utility for that task by substitution. This definition is used
by the solid lines in Figure 4d. In the other definition used
by the dashed lines, only tasks that are actually planned
to be executed are counted. This means that tasks which
only achieve a positive utility due to being substituted by
another task fall not into this definition.
Both Figure 4c and 4d reveal that the number of
planned active tasks increases with an increasing percent-
age of generated substitutabilities. This is not surpris-
ing and coincides with the utility observation in Figure
4a. However, Figure 4d also shows that the number of
planned active tasks without considering substitutions
is for all scenarios with substitutabilities lower than in
Q-RAM, as indicated by the dashed lines in that figure.
This means that the proposed method abstained from
executing certain tasks which could be held active by
substituting that task with another task.
Since in Figure 4d the blue line is below the red,
green and orange solid lines and only slightly above
Q-RAM, this increase in planned active tasks cannot be
attributed due to solving the problem continuously instead
of discretely. However, solving the problem continuously
has the effect that more tasks are active as evidenced by
the blue line in Figure 4c, albeit only to a very limited
extend. This effect applies especially for lower number of
tasks and decreases as the number of tasks increases. It
is assumed that the decrease with an increasing number
of tasks is because the increase in task numbers gives
Q-RAM more possibilities to select tasks that possess
discrete resource utility points with low resource require-
ments. Since for high task numbers only a fraction of
the tasks can actually be executed due to radar resource
limitations, more choices are available. Further, due to
the utility functions being concave, i.e. their slope is
biggest at the beginning of the curve for low resource
amounts, and because of equal weighting of the tasks,
this is consistent with Q-RAM’s optimization objective
of maximizing the overall utility.
In summary, it can be noted that considering substi-
tutabilities provides a significant advantage in terms of
achieved utility. Further, it also allows considerably more
tasks to be held active, which could be very beneficial in
target-rich environments with many tracking tasks.
B. Applied RRM Performance Evaluation
This subsection presents a more application-oriented
evaluation of the proposed RRM method and Q-RAM.
In contrast to the analysis performed in Subsection Ain
which the scenario and the tasks were modeled abstract
and only for a single time step, this evaluation considers
a concrete scenario fully simulated over a time period of
five minutes. This includes the simulated execution of the
by the RRM planned tasks as well as the received radar
measurements and the corresponding track updates.
1. Evaluation Setup
We evaluate the scenario displayed in Figure 5with a
duration of five minutes. It consists of a single airborne
radar platform and 23 targets flying towards the platform.
Both the platform as well as the targets travel at a speed
VOLLWEITER: QoS RRM with Task Dependencies 11
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content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/

0.0 50.0 100.0 150.0 200.0 250.0
-60
-40
-20
0
20
40
x pos (km)
y pos (km)
Fig. 5: Overview of the evaluated scenario. The radar
platform is visualized in black, while the targets are
plotted in green color.
of 200 m s−1and at an altitude of 6000 m. The targets’
RCS equals 16 m2or 18 m2.
In the scenario the task optimization by the QoS RRM
is performed every second. We execute for this scenario
100 MC runs with the following RRM approaches:
•the RRM proposed in this paper with substituabilies
modeled (MipRrm),
•the RRM proposed in this paper without any substi-
tutabilities modeled (MipRrmNoSubs),
•Q-RAM.
In each of the 100 MC runs only the measurement noise
differs. For RRM approaches based on MIP the accepted
MIP gap is set to 0.01 and the tuned Gurobi parameters
from Table Iare used. The runtime of Gurobi for solving
the MIP-based optimization problem is limited to 100 ms.
If the time limit is reached, the currently best found
solution is applied. As starting solution the xij ,rias well
as the γivectors of the previous solution found by Gurobi
are used (the task indices of the previous solution are
mapped accordingly to match the currently active tasks).
The target search covers a region of ±60 degrees in
azimuth and ±40 degrees in elevation and consists of 221
partly overlapping individual search beams. The search
performance model uses the number of looks required in
order to achieve a cumulative detection probability of 90%
as quality metric. It considers different combinations of
target velocities and target pop-up ranges and assumes a
RCS of 10 m2for all targets. The worst case combination
of target velocity and target pop-up range is used to
calculate the cumulative detection probability as described
in [32].
The targets are tracked using an independent
interacting multiple model integrated probabilistic data
association (IMM-IPDA) tracker [40]. In the simulations
we allow tracking tasks to be substituted by search tasks
or other tracking tasks. However, substituting search tasks
by tracking tasks is not allowed since it could create gaps
in the search raster. A substitutability between a pair of
tasks is modeled if the beam center offset in both azimuth
and elevation is less than 10 degrees and if the expected
signal-to-noise ratio (SNR) loss caused due to the beam
offset is less than 50%. Depending on the application and
user preference, other thresholds are also possible. The
expected SNR loss is calculated as described in Appendix
14A of [5].
In contrast to the previous section, in this section we
calculate the terms uiqi(ri, e)required in (19) as well
as uiqij (rj, e)required in (18) using the tracking per-
formance model described in the following. The tracking
performance model is very similar to the one presented
in [34]. It predicts the utility over a time window of 12 s
using the revisit interval and the number of pulses as
operational parameters, i.e. these parameters are assumed
to be fixed for the whole time window.
First, the number of dwells is calculated that fit into
the time window given the chosen revisit time. Based on
that, the time window is split into multiple smaller periods
with the length of the chosen revisit time and, if required,
one additional period with the residual time at the end of
the time window.
Then, at the start of the first period, the current track
sharpness [5] [41] is calculated based on the IMM-IPDA
tracker’s current track state. The track sharpness is the
fraction of the angular estimation error with respect to
the radar beamwidth along the major axis of the track
angular uncertainty ellipse, as described in [42]. After
that, the IMM-IPDA tracker predicts the track state at the
end of the period and we calculate the track sharpness
again. For this predicted target position at the end of
the period, we determine the expected radar measurement
with covariance and update the track using that predicted
measurement. Note, that if the tracking performance
model is used to calculate the utility of tracking task Ti
substituted by task Tj, then off-beam penalties need to be
considered when determining the SNR of the expected
radar measurement. This was implemented as described
in Appendix 14A of [5].
This procedure is iterated for all periods, however
in case of the last period containing only the residual
time, no track update is performed. Assuming that the
track sharpness increases linearly over time, we calculate
the average track sharpness for each time period by
using the track sharpness calculated at the beginning
and at the end of the respective period. Based on that,
we determine for each period’s average track sharpness
the utility. A track sharpness value of 0.2 and below is
mapped to a utility of 1, while values above 0.7 result
in no utility. Values in between are linearly interpolated.
Using the utility achieved in each individual period, we
calculate the predicted overall utility by weighting the
individual period’s utility values according to their time.
For example, in the case of a revisit interval of 2.5 s,
the utility of the first four periods would be weighted
by 2.5/12, while the last period’s utility would only be
weighted by 2/12. This concludes the utility prediction
of the tracking performance model.
Note, that at no time the tracking performance model
uses any ground truth. Only the currently known track
states of the IMM-IPDA tracker are used. We would
also like to mention, that at the start of each parameter
12 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. XX, No. XX XXXXX 2022
This article has been accepted for publication in IEEE Transactions on Aerospace and Electronic Systems. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/

evaluation, the performance model uses a fresh copy of
the current state of the independent IMM-IPDA tracker.
2. Evaluation Results
In this section, the simulation results of the scenario
visualized in Figure 5are evaluated. First, we shortly
compare the utility achieved by the different RRM ap-
proaches displayed in Figure 6a. The utility of MipRrm,
which exploits substitutabilities, is plotted as solid orange
line. Its utility is significantly higher than the utility
achieved by both other approaches. A considerable share
of that utility is attained by exploiting substitutabilities,
as indicated by the dotted orange line. The value of the
dotted orange line equates to the mean value of Pi∈NUs
i
over all conducted MC runs at the corresponding time
steps. The second highest utility is most of the time
obtained by MipRrmNoSubs (solid green line), which
optimizes the RRM problem continuously without con-
sidering substitutabilities. The worst utility is generated
by the classical Q-RAM approach, however the difference
to MipRrmNoSubs is very small.
Next, we verify if the higher utility achieved by the
proposed new approach is reflected by a better OSPA
metric. Figure 6b compares the OSPA metric of the
different RRM approaches. It can be seen, that MipRrm
obtains indeed a significantly better OSPA performance
than the approaches not exploiting substitutabilities. This
is mainly because it detects the targets sooner and con-
sequently tracks a significantly higher number of targets,
as illustrated in Figure 6c. The earlier target detection is
because MipRrm needs a significantly smaller share of
the total radar time for tracking tasks, as shown in Figure
6d, leaving more time for search tasks. Most of the time,
less than 20% of the radar time is spent on tracking tasks,
while the other RRM approaches roughly spent 50% of
the radar time on tracking tasks. The decrease in time
spent on tracking tasks by the approach MipRrm is due
to exploiting substitutabilities. This further becomes clear
when comparing the tracking usage peaks of MipRrm
in Figure 6d at around 3:45 minutes and 4:30 minutes
with the relevant time periods in Figure 6a: the tracking
usage peaks coincide with a slight decrease in utility
achieved via substitutions. This is probably because it was
no longer possible to exploit some substitutabilities due
to changes in the target geometry, hence more resources
for tracking tasks were required.
Note, that the decrease in tracking usage in MipRrm
only results in a slightly worse mean tracking error despite
tracking a higher number of targets (see Figure 6e). The
tracking error is defined as the distance between the
ground truth target position and the target track position
provided by the independent IMM-IPDA tracker. Further,
despite tracking a significantly higher number of targets,
MipRrm dropped in our MC runs on average only 8.57
tracks, while MipRrmNoSubs and Q-RAM dropped 8.72
and 10.66 tracks, respectively.
We can conclude, that the performance of MipRrm-
NoSubs is, with the exception of the average number of
track drops, very similar to Q-RAM. Hence, the observed
performance improvement of MipRrm cannot be caused
solely by optimizing the QoS RRM continuously instead
of discretely. Thus, the performance improvement must
be caused by exploiting substitutabilities.
Finally, we examine the solve runtimes of the dif-
ferent RRM methods. The solve runtimes of the two
MIP-based RRM methods over the scenario time are
shown as solid lines in Figure 6f. Note, that the solve
runtime only encompasses the time the solver spent on
the optimization. This means that the runtime required to
evaluate the performance models in order to construct the
required concave majorants, which are input parameters
for the optimization model, is not included. Depending
on how the concave majorant is determined, the required
computational runtime can vary significantly. For ways of
how to identify the concave majorants of tasks efficiently,
see e.g. [34] or [43].
The runtimes of Q-RAM were generally less than
1 ms, hence these runtimes are not plotted. Almost all
MC runs were solved optimally within the time limit at
the root node. Only at a small number of timesteps the
optimization time limit was hit and the solve aborted
before achieving the desired MIP gap. Though, at the
affected timesteps, the time limit was never hit by more
than 6% of the conduced MC runs (mostly only by 1%
or 2%). As mentioned previously, in those cases the best
currently found solution was used, even if it did not satisfy
the desired MIP gap. It can be seen, that the runtime of
MipRrm is in the second half of the scenario much longer
than the runtime of MipRrmNoSubs. The secondary y-
axis shows the number of constraints and reveals a
correlation between the solve runtime and the number
of constraints in the optimization problem. This shows
that the increased runtime of MipRrm in this scenario
is caused by the additionally introduced complexity of
considering substitutabilities in the RRM optimization
problem. However, with a maximum mean runtime of
under 65 ms and a time limit of 100 ms, the proposed
method MipRrm could still be a able to handle this dense
scenario in a real-time setting, depending on the exact
requirements. Additionally, depending on the application
and user preferences, it is possible to reduce the com-
plexity of the optimization problem by further restricting
the threshold at which we consider substitutabilities worth
modeling.
V. CONCLUSION AND FUTURE WORK
In this article substitutabilities between pairwise radar
tasks have been explained and integrated into a QoS RRM
model based on a MINLP problem. Subsequently, this
model has been linearized in order to solve it as a MILP.
In numerical simulations the MILP problem formulation
was evaluated using different scenarios with a varying
number of tasks and substitutabilities and compared with
the traditional RRM approach Q-RAM. In scenarios with
substitutabilities present, the model has been demon-
VOLLWEITER: QoS RRM with Task Dependencies 13
This article has been accepted for publication in IEEE Transactions on Aerospace and Electronic Systems. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/

0:00 1:00 2:00 3:00 4:00 5:00
0
0.5
1
Time [m:ss]
Utility
QRAM
MipRrm
Subs. utility
MipRrmNoSubs
(a) Achieved utility (dotted line symbolizes utility achieved
by exploiting substitutabilities).
0:00 1:00 2:00 3:00 4:00 5:00
400
600
800
1,000
Time [m:ss]
OSPA value
QRAM
MipRrm
MipRrmNoSubs
(b) OSPA metric.
0:00 1:00 2:00 3:00 4:00 5:00
0
10
20
Time [m:ss]
Number of tracks
QRAM
MipRrm
MipRrmNoSubs
(c) Number of tracks.
0:00 1:00 2:00 3:00 4:00 5:00
0
25
50
75
100
Time [m:ss]
Tracking radar usage (%)
QRAM
MipRrm
MipRrmNoSubs
(d) Tracking radar usage (% of total radar time).
0:00 1:00 2:00 3:00 4:00 5:00
2,000
4,000
Time [m:ss]
Mean tracking error [m]
QRAM
MipRrm
MipRrmNoSubs
(e) Mean tracking error.
0:00 1:00 2:00 3:00 4:00 5:00
20
40
60
Time [m:ss]
Solve runtime [ms]
MipRrm (runt.)
MipRrmNoSubs (runt.)
MipRrm (constr.)
MipRrmNoSubs (constr.)
0:00 1:00 2:00 3:00 4:00 5:00
2.5
3.0
3.5
4.0
Time [m:ss]
Number of constraints
(in thousands)
(f) Solve runtime (solid lines in lighter color shade) and number of
constraints (dashed lines in darker color shade). The runtime of Q-RAM
is not included.
Fig. 6: Evaluation results of the scenario shown in Figure 5averaged over 100 MC runs. The results of Q-RAM are
plotted in blue, of MipRrm in orange and of MipRrmNoSubs in green.
strated to achieve a significantly better overall utility
than traditional approaches not exploiting dependencies.
Especially in high load scenarios, the ability to consider
substituabilities among tasks has proven to be valuable
in increasing the overall resource efficiency of the radar
system. Further, the increased utility also coincided with
a higher number of active tasks that the radar system
was able to support. Additionally, in the realistic scenario
shown in Figure 5, the superiority of the proposed QoS
RRM model was also verified by comparing the OSPA
metric. It has been demonstrated that by considering
substitutabilities, the accomplished OSPA metric can be
improved significantly compared to approaches that do
not consider task dependencies. This is especially due
to a decreased resource need for tracking achieved by
exploiting substitutabilities. This in turn frees up precious
radar time for other tasks like radar search in case of the
examined scenario.
A disadvantage of considering substitutabilities is
however, that it is necessary to evaluate more task per-
formance models in order to be able to determine the
utility one task could provide by substituting another
14 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. XX, No. XX XXXXX 2022
This article has been accepted for publication in IEEE Transactions on Aerospace and Electronic Systems. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/

task. Another drawback is the added complexity due to
considering dependencies which results in a significantly
increased computational runtime when solving the QoS
RRM optimization problem compared to the common
method Q-RAM. Hence, the solve runtime of the MILP
model formulation was analyzed. It was found out, that
the proposed QoS RRM approach was able to provide
a solution for the complex scenario shown in Figure 5
in less than 100 ms. Though, this is in stark contrast to
Q-RAM, which was able to provide a solution after less
than 1 ms. However, Q-RAM’s solution was shown to be
of significantly lower quality due to not considering task
dependencies.
While the proposed QoS RRM method was in our
complex scenario able to provide solutions of significantly
higher quality than Q-RAM within the defined time limit
of 100 ms, which could be sufficient for many real-world
applications, we still want to emphasize that solving a
MIP optimization problem is NP-hard. Hence, in a real-
world application we would suggest to have a fallback
solution (e.g. Q-RAM’s solution) readily available in case
that no suitable solution is found within the required time
limit. Even though the proposed method might be too
slow for some applications that require rapid reactions,
the algorithm would still be suitable for the planning
of time periods further in the future. Additionally, it
could also be used to generate optimal solutions for the
QoS RRM problem with dependencies. On the one hand,
such solutions would be valuable as benchmark for the
evaluation of future heuristics that take task dependencies
into account. On the other hand, similarly to the outlook
in the conclusion of [23], optimal solutions could also
be used as training data for a neural network developed
to conduct RRM. Hence, the proposed method is an
important step towards the development of a cognitive
radar system.
In future work the proposed algorithm suitable for
single platform QoS RRM could be expanded to the
multiplatform domain. Further, in order to allow for rapid
reactions, a fast heuristic could be developed which read-
justs the solution provided by the slower MILP approach
proposed in this article if rapid reactions are necessary.
This heuristic could for example be based on Q-RAM
and refrain from considering new substituabilities due
to performance reasons, however it could still keep task
consolidations based on the substitutions made in the
MILP solution of the previous planning interval. Hence,
the single stage approach proposed in this article would be
transferred to a two stage approach with the slower MILP
model in the first stage exploiting dependencies, while a
fast heuristic in the second stage readjusts that solution
and allows for rapid reactions. This would combine the
desirable fast performance of heuristics like Q-RAM with
the ability to exploit task substitutabilities and thereby
remove the main drawback of the proposed method.
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content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/

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Christoph Vollweiter obtained his B.Sc. and
M.Sc. degree in Economics and Mathematics
from the University of Erlangen-Nuremberg,
Erlangen, Germany in 2015 and 2018, respec-
tively. In 2019 he joined the Sensor Data and
Information Fusion Department at the Fraun-
hofer Institute for Communication, Information
Processing and Ergonomics (FKIE), Wacht-
berg, Germany, where he is a member of the
Sensor and Resource Management team. In this
role he conducts research for adaptive and optimized management
of multi-sensor systems, multifunctional sensors and mobile sensor
platforms interlinking methods derived from mathematical optimization
theory with data fusion at the signal and data level.
16 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. XX, No. XX XXXXX 2022
This article has been accepted for publication in IEEE Transactions on Aerospace and Electronic Systems. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TAES.2024.3506497
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