Radio Resource Management Optimization of Flexible Satellite Payloads for DVB-S2 Systems

Abstract

The increasing demand for high-rate broadcast and multicast services over satellite networks has pushed for the development of High Throughput Satellites characterized by a large number of beams (e.g., more than 100). This, together with the variable distribution of data traffic request across beams and over time, has called for the design of a new generation of satellite payloads, able to flexibly allocate bandwidth and power. In this context, this paper studies the problem of radio
resource allocation in the forward link of multibeam satellite networks adopting the Digital Video Broadcasting - Satellite - Second Generation (DVB-S2) standard. We propose a novel objective function with the aim to meet as close as possible the requested traffic across the beams while taking fairness into account. The resulting non-convex optimization problem is solved using a modified version of the simulated annealing algorithm, for which a detailed complexity analysis is presented. Simulation results obtained under realistic conditions confirm
the effectiveness of the proposed approach and shed some light on possible payload design implications.

Full text

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Radio Resource Management Optimization of
Flexible Satellite Payloads for DVB-S2 Systems
Giuseppe Cocco, IEEE, Member, Tomaso De Cola, IEEE, Member, Martina Angelone, Zoltan Katona, and Stefan
Erl
Abstract
The increasing demand for high-rate broadcast and multicast services over satellite networks has pushed for the development
of High Throughput Satellites characterized by a large number of beams (e.g., more than 100). This, together with the variable
distribution of data traffic request across beams and over time, has called for the design of a new generation of satellite payloads,
able to flexibly allocate bandwidth and power. In this context, this paper studies the problem of radio resource allocation in the
forward link of multibeam satellite networks adopting the Digital Video Broadcasting - Satellite - Second Generation (DVB-
S2) standard. We propose a novel objective function with the aim to meet as close as possible the requested traffic across the
beams while taking fairness into account. The resulting non-convex optimization problem is solved using a modified version
of the simulated annealing algorithm, for which a detailed complexity analysis is presented. Simulation results obtained under
realistic conditions confirm the effectiveness of the proposed approach and shed some light on possible payload design implications.
I. INTRODUCTION
The advent of High Throughput Satellite (HTS) systems [1], [2] has revolutionized the concept of satellite communications
in that new satellite systems operating in the Ka frequency band (and above) are being designed so as to provide geographical
coverage through a large number of beams. Such dramatic change has started upon the ever-increasing user demand for
broadcast/multicast services characterized by high data rates and reliability performance. In order to meet these requirements,
a natural technology candidate is the Digital Video Broadcasting - Satellite - Second Generation (DVB-S2) standard [3], which
is nowadays one of the most widespread and preferred options by satellite broadcasters in the forward link1.
In spite of the attractive performance figures that can be attained by DVB-S2 (e.g., in terms of spectral efficiency), the
problem of optimally allocating bandwidth to beams and optimally operating the payload from a power perspective according
to the amount of requested traffic is still not completely solved. This is because of the large number of variables that play
an important role in the resulting radio resource allocation problem. Such problem has been often addressed from a ground
segment viewpoint, by proposing optimization frameworks able to take into account propagation impairments (e.g., rain) and
interference contribution from other beams (e.g., co-channel interference (CCI)). For instance, reference [5] addresses the
G. Cocco is with the LTS4 Signal Processing Laboratory and the Laboratory of Intelligent Systems, ´
Ecole Polytechnique F´
ed´
erale de Lausanne (EPFL),
Lausanne 1015, Switzerland, (e-mail: giuseppe.cocco@epfl.ch)
T. De Cola, Z. Katona and S. Erl are with the Institute of Communications and Navigation, German Aerospace Center (DLR), Oberpfaffenhofen 82234,
Germany (e-mail: {tomaso.decola, zoltan.katona, stefan.erl}@dlr.de)
M. Angelone is with the Communications and TT&C Systems and Techniques Section, European Space Agency, ESTEC, Noordwijk 2201, The Netherlands
(e-mail: martina.angelone@esa.int)
1The standard has been recently extended in the DVB-S2X [4], which includes new features and yields improved performance with respect to the previous
version.

2
problem from a scheduling viewpoint, allocating different modulation and coding schemes (ModCods) to the different satellite
beams. However, the complex characteristics of data traffic (time- and space-correlation, heavily depending on the specific
geographic area) have always represented a formidable obstacle against the derivation of closed-form solutions, hence requiring
the introduction of approximated models or the use of numerical optimization techniques.
On the other hand, the recent years have also witnessed an evolution of satellite system concepts from a space segment
viewpoint, which also have an important impact on the resource allocation problem [6]. Specifically, more sophisticated payload
designs have been introduced [7], so as to cope with the time and geographic variations of the traffic requested by each beam.
This mainly resulted in two possible design options, namely beam-hopping and flexible payloads. The former makes use of
a time-slotted illumination window so that a sequence of beam illumination and the illumination time assigned to each beam
can be designed according to the traffic demands and the antenna radiation pattern. The latter option makes use of a dual
approach, consisting in allocating bandwidth or power to the beams in relation to the requested traffic. The effectiveness
of both options heavily depend on the specific payload design (e.g., number of traveling-wave tube amplifiers (TWTA) and
structure of the payload connection matrix) and on the constraints (e.g., mass and available power) imposed by state-of-the-art
technology. In [8], [9] the problem of time/beam allocation is studied in the presence of traffic asymmetry. In the papers a
closed form solution for the optimal resource allocation in a simplified setup with no interference is derived for two different
utility functions, aiming at matching the requested bitrate and maximizing the product of the ratios between the offered and
requested capacity across the beams, respectively.
In [9], [10] the advantages of multi-beam with respect to single beam satellite systems are studied considering different
performance metrics. Specifically, the optimal power allocation is derived for two different objective functions, one leading to
throughput maximization and the other related to fairness. Although these two papers offer interesting hints on the problem
of resource allocation, the validity of the results is limited by the assumption of no co-channel interference, which is instead
removed in [11]. In the paper a phased array antenna is assumed at the satellite and call-admission control schemes are
investigated. Differently from the approaches adopted in the papers mentioned above, the studies presented in [12], [13]
explore the benefits of power allocation. In particular, a two-stages sub-optimal algorithm is applied to solve a non-convex
optimization problem, the solution of which gives some insights about the relations between power allocation and offered traffic
on the forward link of satellite networks. Finally, beam-hopping and flexible systems are compared in [14], where the latter
implements a non-uniform bandwidth allocation and makes use of sizable beams. It is worth noting that most optimization
strategies considered in the available literature dedicated to resource allocation make use of genetic algorithms or neural
networks. In [15] the Simulated Annealing (SA) algorithm [16], has been proposed to minimize the co-channel interference in
the uplink of two independent satellite systems. In [17] a reinforcement learning approach against satellite channel impairments
such as orbital perturbations and atmospheric effects is presented. As a side remark, we point out that the problem of radio
resource management (RRM) has been studied also in the context of terrestrial networks [18] [19]. However, the payload
constraints and the different network topologies make the two optimization problems significantly different.
The original contribution of the present paper with respect to the state of the art is the following2:
•We propose a novel resource allocation strategy in which a multi-objective optimization problem is addressed through the
definition of an ad-hoc objective function taking both fairness and absolute capacity mismatch into account.
•Analytical insights are presented for such function and it is shown that its convexity can not be guaranteed for all payload
2Part of the content of the present paper has been presented at the Advanced Satellite Multimedia Systems (ASMS) conference 2016 [20].

3
constraints.
•Unlike [15], the present paper applies a variant of SA to the forward link of a multibeam satellite system equipped with
a payload which is flexible in terms of power allocation, bandwidth allocation or both.
•We perform a detailed analysis of the computational complexity of the proposed optimization algorithm showing that it
grows linearly with relevant system parameters such as the number of TWTAs in the payload.
•We show the potential of the proposed resource allocation scheme considering a realistic requested traffic profile, realistic
operative conditions (TWTA characteristics, intermodulation interference, beam pattern) and considering the DVB-S2
standard as reference for the physical layer.
•We compare the performance of the new objective function with others previously proposed in literature and show that it
achieves comparatively good performance in both low traffic and high traffic regimes.
We stress the fact that the proposed solution is tested under realistic conditions and taking into account effects such as
intermodulation interference, ModCod constraints, dependency between symbol constellation and nonlinear effects in the
transponder, antenna radiation pattern and consequent inter-beam interference as well as realistic traffic request time variation.
Up to our knowledge all such effects have not been jointly considered so far in previous works on multibeam satellite RRM
problems.
The remainder of this paper is structured as follows. Section II presents the system model and the formulation of the resource
allocation problem. In Section III we introduce and analyze the properties of the objective function. In Section IV our resource
allocation strategy is presented and its complexity analyzed. Section V contains the simulation results, while the conclusions
are presented in Section VI.
II. RA DI O RESOURCE ALL OC ATIO N PROB LE M
A. System Model
A multi-beam geostationary satellite system is taken as reference. The satellite generates a geographical footprint subdivided
into Nbbeams, where each beam i,i= 1, . . . , Nb, serves Ni
ufixed satellite terminals. The population of users active on beam
igenerates an aggregate traffic request which we denote as Ti
r. Let us denote with hj
i(j),i0the gain experienced by the signal
transmitted in beam i0and received by user jlocated in the footprint of beam i(j), where jcan take values between 1and
Ni
uand Ntot
u=PNb
i=1 Ni
u,Ntot
ubeing the total number of users in the system. Such gains account for the receiving and
transmitting antennas gains and the propagation impairments (e.g., free space loss and atmospheric attenuation). Ideally, each
satellite terminal jis expected to receive only the signal transmitted by its reference beam, which we denote as i(j). However,
due to the secondary lobes of the satellite antennas, user terminals suffer from interference generated by beams others than the
reference one operating in the same frequency band (co-channel beams), leading to hj
i(j),i06= 0 for some i06=i(j). For ease
of notation in the following we will indicate the reference beam with iand the interference beams as i0. As far as the payload
model is concerned, a single feed per beam (SFPB) architecture is considered. It is assumed that a number of TWTAs equal
to NTWTA is available on-board the satellite and that each of them amplifies the same amount of bandwidth. Each tube has a
maximum available power equal to Ptot. Depending on the operating point of the tube a higher or lower power efficiency3can
be obtained. Each TWTA reuses the whole bandwidth and serves a subset of beams. The association between beams and the
TWTAs is specified by a so-called connection matrix. We consider as a reference, or conventional system, one in which the
3The power efficiency is intended as the ratio of power transmitted in RF with respect to the DC power provided to the tube.

4
total useful bandwidth4Bof each TWTA is shared uniformly among the subset of amplified beams, so that the bandwidth per
beam depends only on the specific coloring scheme adopted (e.g., B,B/2, or B/4for 1,2, or 4colours, respectively) and can
not be modified in order to follow the traffic request variations. The TWTA bandwidth is shared among the connected beams
in a way that such beams cannot have overlapping portions of bandwidth. Data is transmitted through a beam making use of
multiple carriers, each being assigned a fraction of the bandwidth allocated to the beam. The portions of data traffic addressed
to the users served by a given beam are multiplexed in time according to a time division multiplexing (TDM) framing.
B. Problem Formulation
Our goal is to allocate resources such that each beam receives an offered capacity Ti
o,i∈ {1, . . . , Nb}, that is as close as
possible to the requested capacity Ti
rwhile taking fairness into account. Ti
odepends on the bandwidth allocated to beam i,
the power settings of the TWTA to which beam iis connected, as well as on the co-channel interference generated by other
beams and on the channel gains (relative to both reference signal and interferers) of each single user. The resource allocation
takes place at the gateway. We assume that the gateway has knowledge of the gains Gi,j ,i∈ {1, . . . , Nb},j∈ {1, . . . , N i
u},
Ni
ubeing the number of users served by beam i. This assumption is a realistic one since fixed terminals are considered and
thus the rate of channel variation can be assumed to be relatively slow. The channel gains are assumed to be periodically
estimated by the gateway through a return channel. This is actually the case in real systems, in which knowledge of such gains
are used by the ground station to choose the ModCod which is best suited to each terminal’s current channel condition, i.e.,
the ModCod with the highest spectral efficiency that can be supported by the channel.
We consider three different payloads. The first payload can be optimized both in terms of bandwidth and power allocation.
The second one has only bandwidth flexibility while in the third payload only the TWTA operating conditions in terms of
power can be adjusted. For a fair comparison, both the number and the characteristics of the TWTAs are the same for all
payloads. We also assume that the connection matrix, which determines the subset of beams that are connected to each TWTA,
is the same in all payloads. The subset of beams connected to different TWTAs are disjoint, i.e., one beam cannot be connected
to more than one TWTA. The flexibility in terms of bandwidth allows to modify, in each TWTA, the spectrum assignment to
the subset of beams connected to it, under the constraint that the same portion of spectrum can not be assigned to more than
one beam connected to the same TWTA. Note that the same portion of spectrum can be assigned from different TWTAs to
one of the beams in their relative subset, so that from a resource allocation point of view each TWTA acts as an independent
unit. The bandwidth can be adjusted with a granularity of Bch MHz, i.e., the bandwidth allocated to a beam must be a multiple
of Bch MHz. In the following we will refer to such elementary unit of bandwidth as chunk. Thus, the amount of bandwidth
that can be assigned to a certain beam can be expressed as NchBch , where Nch belongs to the set {1,2, . . . , N tot
ch },Ntot
ch being
the maximum number of chunks available in the system. Transmission takes place on different carriers, each corresponding to
a chunk.
The flexibility in terms of power allocation consists in the possibility to change operating point (i.e., input back-off (IBO))
and power setting of each TWTA independently of the others. The power setting defines the attenuation (in dB) of the radio
frequency saturated power delivered by the TWTA with respect to the peak in the reference operation mode (no attenuation).
A larger power profile indicates larger output back-off (OBO) for a given IBO, which reduces the non-linear effects of the
tube at the cost of a reduced power efficiency. The power settings can be chosen from the set {0,1, . . . , Sp−1, Sp}, where
4by useful bandwidth we mean the overall bandwidth once the guard bands between adjacent channels has been subtracted.

5
IBO [dB]
OBO [dB]
Maximum saturated RF power ↓
Power setting 0
Power setting 1
Power setting 2
Power setting 3
Power setting 4
Fig. 1: Example of TWTA characteristics for five different power settings (Sp= 4). The operation point corresponding to the maximum saturated RF
power is also indicated.
0corresponds to the reference setting. An example of TWTA characteristics for five different power settings (i.e., Sp= 4) is
shown in Fig. 1. Payloads with no power flexibility keep a fixed IBO (equal to 3dB) and a power profile equal to 2.
The optimization problem that we aim to solve is:
minimize
v,p,Bf(v,p,B)
subject to vt∈ {−IBOmax,−IBOmax + ∆IBO,...,−IBOmin }
pt∈ {0,1, . . . , Sp}
∀t∈ {1,2, . . . , NTWTA}
B∈ B
where NTWTA is the number of TWTAs in the payload, f(v,p,B)is the objective function to be minimized, which will be
defined Section III, v= (v1, . . . , vNTWTA)and p= (p1, . . . , pNTWTA )are vectors containing the IBO’s and the power profiles
for all TWTAs, respectively, while B∈ B ⊂ ZNb×Nch is the bandwidth allocation matrix, which belongs to the set of feasible
bandwidth allocation matrices B.Bis a subset of the set of binary matrices whose structure depends on the specific payload
bandwidth constraints. ∆IBO is the granularity in the IBO setting and is a payload specific parameter. As in real systems, the
power optimization of each TWTA is done by selecting the IBO which, in turn, gives a certain OBO.
As a common practice in optimization problems, we aim at minimizing an objective function. The objective function reflects
the system key performance indicators (KPI). In order to define the KPIs we start with some general considerations. Our goal
is to efficiently allocate the satellite resources with the aim of satisfying the requested traffic in all beams. The capacity request
satisfaction can be looked at from a system (or global) perspective as well as from a beam (or user) perspective. From a global
perspective, a valid choice would be to take as objective function a measure of the error in matching the requested capacity
across the beams, i.e.,
E=
Nb
X
i=1
(Ti
o−Ti
r)2.
Although this may be a valid indicator to measure the error with respect to an ideal resource allocation condition (by ideal we
mean one in which the offered capacity exactly matches the requested capacity in each beam), the offered capacity exceeding
the requested one is treated in the same way as the missing capacity, which is not desirable. Moreover, the measure E is
potentially unbounded (i.e., it can assume arbitrary positive values) and this makes it difficult to evaluate the goodness of a

6
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5x 108
Beam Id
Capacity [bps]
Offered (Conventional payload)
Requested
Fig. 2: Example of requested capacity and offered capacity in bits per second (bps) for the conventional payload plotted versus beam Id.
given solution. Furthermore, even if relatively good results are obtained in terms of matching error E, it can still happen that
traffic requests are largely unmatched for a non-negligible number of beams. This may be the case mainly for beams that
present relatively low requests and for which too little resources are allocated. As a matter of fact, beams with relatively little
capacity request may not have great impact on E and thus an optimization solution that performs well at a global scale may
neglect such beams. Although beams with higher capacity requests are likely to be the most profitable ones from the satellite
operator perspective, low traffic beams should be taken into account in the optimization process since other considerations
may make such beams appealing (e.g., presence in the territory, reputation of the operator, etc.). Fairness is indeed a relevant
parameter to be accounted for in the RRM optimization. Several different measures of fairness have been proposed in literature,
such as the Jain Index and the normalized entropy. Using one of these measures as (negative) objective function would have
the disadvantage of not accounting for the absolute value of the mismatch in terms of capacity, either missing or in excess.
Let us now define the satisfaction index of beam ias SIi=Ti
o
Ti
r. SI is a non-negative number which gives a measure of
the extent up to which the requested capacity is satisfied. If SI<1the beam has been allocated insufficient resources for its
capacity needs, while SI>1indicates that the beam is being over-provisioned. Ideally it would be desirable to keep track of
both request satisfaction and absolute gap between the requested and the offered capacity. A way to visualize the system state
in such terms is to represent of all beams as points of a scatter plot in a plane having as axes the satisfaction index and the
difference ∆i=Ti
o−Ti
r, which gives a measure of the missing (if negative) or exceeded (if positive) capacity. We refer to
such plane as the satisfaction/gap (SG) plane. In Fig. 2 an example of requested and offered capacity for the conventional
payload plotted versus the beam Id is shown. The corresponding SG representation is depicted in Fig. 3.
III. OBJECTIVE FUNCTION
The representation in the SG plane of the system state in terms of offered/requested capacity gives a qualitative idea of the
goodness of a resource allocation solution in terms of both satisfaction and gap distribution. In order to have also a quantitative
measure, we define in the following an objective function which is derived from the SG plane. We will refer to it as the
satisfaction-gap measure (SGM).
A. SGM Definition
Our aim is to find an objective function that satisfies the following:

7
0 10 20 30 40 50 60
−3
−2
−1
0
1
2
3x 108
∆[bps]
SI
Fig. 3: Representation of beams in the SG plane. Each of the 200 beams is represented as a point (blue circle) in the plane having as xcoordinate the
satisfaction index and as ycoordinate the capacity gap ∆as defined in this section.
1) Provides a measure of the mismatch with respect to the ideal case accounting for both capacity gap and satisfaction in
all beams.
2) A beam with satisfaction 1−δ, with 0≤δ < 1, should weight more than a beam with satisfaction 1 + δ(which is also
undesired but not as bad as having beams with missing capacity).
3) Assumes values in the interval [0,1], where 1corresponds to the ideal state in which there is perfect match between
offered and requested capacity through all beams5.
We apply a transformation to the SG plane in such a way that the measure we look for satisfies the three conditions above.
Let us start with point 1). In order to take both SI and ∆into account, we treat the SG plane as a complex plane, in which
SI represents the real axis and ∆the imaginary axis. Beam i,i∈ {1, . . . , Nb}, is represented as a complex number in such
plane, with coordinates:
ci=SIi+j∆i
where j=√−1. In the following we drop the index ito simplify the notation. In order to satisfy point 2) we apply the
following transformation to the beams having real part lower than 1:
Re{c} → 1−1
Re{c},∀c:Re{c} ≤ 1.(1)
This transformation translates a smaller SI into a larger distance from the origin. In order to satisfy point 3) we shift the points
with real part (satisfaction) larger than or equal to 1 towards the origin by applying the transformation c→c−1. In this way
the point representing the optimal solution becomes (0,0). In Fig. 3 it can be seen how, depending on the unit of measure
adopted to measure the excess/missing capacity (e.g., kbps, Mbps, Gbps) the range of the y axis can be quite wide with respect
to the x axis. This can be easily fixed with the following scaling operation:
Im{c} → Im{c}
β(2)
with β > 0. The value of βcan be chosen, for instance, equal to (or a function of) the system throughput of the conventional
payload. In this way it is possible to make a comparison in terms of the goodness in the resource allocation solution between
systems with different total capacities.
5Note that this can not be achieved in systems that are under-dimensioned in terms of total system bandwidth and TWTA number.

8
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x(SI)
y(∆)
Fig. 4: Representation of beams in the modified SG plane. Each beam is represented as a point (blue circle) in the plane having as xcoordinate a function
of the satisfaction index and as ycoordinate a function of the capacity gap ∆. The unitary circle is also shown (red line).
In order to get a measure which takes values between 0and 1, we apply one last transformation to the plane which confines
all the points within the circle of radius 1 around the origin. This is done applying the following transformation to the absolute
value of each point, without modifying its phase:
|c|→ 1−e−|c|.(3)
According to this transformation, a point at infinite distance from the ideal condition (origin) will lie on the unitary circle,
while a point that has |c| 1before applying (3) will have a distance from the origin approximately equal to |c|. The
modified plot corresponding to the example in Fig. 3 is shown in Fig. 4.
Starting from the transformed plot, we define the SGM as a measure of the average distance from the optimal condition:
SGM = 1 −1
Nb
Nb
X
i=1 |ci|3.(4)
The third power rise in the sum on the right hand side of expression (4) is included in order to give more weight to beams
that are farther apart from the ideal condition, i.e., have SI which is either close to zero or much larger than 1or have a large
mismatch in terms of absolute capacity. Larger values have been tested but lead no significant advantage.
The SGM is the complement to 1of the average (cube of the) distance from the origin of the points in the transformed
scatter plot. It can be easily seen that such measure takes values in [0,1] and is close to 1when all points are gathered around
the origin, which corresponds to the case in which the offered capacity matches almost exactly the requested capacity in each
of the beams and the ∆’s are relatively small.

9
The optimization problem to be solved is, finally:
minimize
v,p,B−SGM(v,p,B)
subject to vt∈ {−IBOmax,−IBOmax + 1,...,−IBOmin }
pt∈ {0,1, . . . , Sp}
∀t∈ {1,2, . . . , NTWTA}
B∈ B.
B. SGM Analysis
In the following we provide some analytical insight on the SGM.
We start by considering the implications of the transformation undergone by the points in the SG plane according to
expressions (1)-(3). Following transformation (1), the points in the SG plane corresponding to beams with SI≤1are modified
as follows:
c→1−1
SI +j∆.(5)
Transformation (2) is a simple re-scaling which is applied to all points. The scaling factor βcan be chosen at will. One option
is to choose βequal to the average requested traffic across all beams, i.e.,
β=1
Nb
Nb
X
i=1
Ti
r.
After applying transformation (3) the absolute value of each point in the SG plane is transformed as follows:
|c| → 




1−exp −q1−1
SI 2+∆2
β2,if SI ≤1
1−exp −qSI2+∆2
β2,if SI >1.
(6)
Plugging Eqn. (6) into Eqn. (4) we have:
SGM = 1 −1
NbX
i|SIi≤1|ci|3−1
NbX
i|SIi>1|ci|3
= 1 −1
NbX
i|SIi≤1

1−e−θu
i


3−1
NbX
i|SIi>1

1−e−θo
i


3
=3
NbX
i|SIi≤1
e−θu
i1−e−θu
i+e−2θu
i
3
+3
NbX
i|SIi>1
e−θo
i1−e−θo
i+e−2θo
i
3(7)
where we defined θu
i=r1−1
SIi2+∆2
i
β2and θo
i=qSI2
i+∆2
i
β2as the argument of the exponential term for underprovisioned
and overprovisioned beams, respectively. If the system under consideration is characterized by a significant mismatch in terms
of SI or ∆in all the beams, expression (7) can be simplified by neglecting the terms of higher order. In this case we have:
SGM '3
NbX
i|SIi≤1
e−θu
i+3
NbX
i|SIi>1
e−θo
i.(8)
From Eqn. (8) some interesting conclusions can be drawn about the way the two performance measures combined in the
SGM affect the objective function. Let us now focus on beams in the following situations: one in which there is a significant

10
mismatch in terms of absolute capacity while having a satisfaction index close to 1(∆-mismatch) and a second situation in
which the absolute capacity mismatch is small while the satisfaction index mismatch is much larger than one (SI-mismatch).
1) ∆-Mismatch: In this case the mismatch in terms of SI is much smaller than the capacity mismatch, specifically ∆2
i
β2
1−1
SI 2for the underprovisioned beams and ∆2
i
β2SI2for the overprovisioned beams. For underprovisioned beams, this
represents a situation in which a beam with a capacity request much larger than the average has an offered capacity which is
slightly smaller in percentage to the requested one 6. Similar considerations can be done for overprovisioned beams. In this
case the terms in the sums of Eqn. (7) corresponding to beams with ∆-mismatch can be approximated as:





e−θu
i=e−r(1−1
SI )2+∆2
β2'e−|∆|
β,for SI ≤1
e−θo
i=e−qSI2+∆2
β2'e−|∆|
βfor SI >1.
(9)
2) SI-Mismatch: In case of SI-mismatch, a beam is not in an ideal condition mostly due to the SI axis in the SG plane.
Formally, we say that a beam has an SI-mismatch if 1−1
SI 2∆2
i
β2in case of underprovisioned beam, or SI ∆2
i
β2for an
overprovisioned beam. The terms in the sums of Eqn. (7) corresponding to beams with SI-mismatch can be approximated as:





e−θu
i=e−r(1−1
SI )2+∆2
β2'e−|1−1
SI |for SI ≤1
e−θo
i=e−qSI2+∆2
β2'e−SI for SI >1.
(10)
In both mismatch cases we see that, in a first order approximation, the weight of a beam on the overall SGM calculation
depends exponentially on the metric for which the mismatch is larger. In intermediate cases the weight of a beam in the SGM
depends on a combination of the two metrics.
Now we show that for some payload constraints the objective function is non-convex. Since the objective function is defined
as f=−SGM, this is equivalent to prove the non-concavity of SGM. Note that for some specific systems characterized by
a given channel matrix and traffic request and for certain power and bandwidth allocation constraints the function may be
convex. However, since the user population as well as the traffic requests may change over time, the convexity of −SGM may
not always hold. This is shown in the following proposition.
Proposition 3.1: There exist some system configurations for which the SGM function is non concave.
Proof 3.2: See Appendix.
Since an algorithm designed to work well for a convex function may perform poorly if the function is not convex, we opt for
a stochastic optimization solution, which is presented in the next section. Considerations on the performance of such algorithm
in the convex case are also presented.
IV. RESOURCE ALL OC ATION STR ATEGY
Even assuming full channel state information at the transmitter, finding the optimal resource allocation is not trivial. This is
due, on one side, to the non-convexity of the objective function and on the other side to the large number of possible solutions,
which makes exhaustive search not viable 7.
6For instance, consider a system with average requested capacity equal to 100 Mbps. Assuming a beam with a requested capacity equal to 900 Mbps and
an SI equal to 0.93, we have ∆2
i
β2'0.397 while 1−1
SI 2= 0.0057.
7If we consider a flexible payload with 50 TWTAs, a number of bandwidth chunks equal to 8and a number of allowed IBO levels equal to 10, the number
of possible allocations (feasible points) is equal to (256 ×10)50 which is on the order of 10170.

11
We propose a suboptimal algorithm based on a slightly modified version of the Simulated Annealing algorithm [16]. The
algorithm tries to minimize the objective function defined in Section II-B. This is done by running iteratively the SA algorithm,
each time using lower starting and stopping temperatures. The way the SA algorithm is applied at each run is described in the
following.
A. Perturbation of the Feasible Point
The SA algorithm uses as starting point the same bandwidth and power allocation as the conventional payload.
At each iteration the algorithm perturbates the feasible point. Depending on the payload to which the algorithm is applied,
either the bandwidth, the power or both can be modified. For a payload with full flexibility the algorithm chooses randomly
at each iteration whether to modify one or the other.
The perturbation of the feasible point is done as follows. A beam is selected at random, then:
•If the bandwidth is to be modified, the number of bandwidth chunks Nch currently allocated to the beam is modified
by adding to such number a random variable u∈ {−1,0,+1}while keeping the number of allocated beams within the
set {1,2, . . . , N tot
ch −1, N tot
ch }. Once the new number of chunks Nch is fixed, their spectral location within the available
bandwidth is selected at random among the Ntot
ch
Nch possible dispositions. Afterwards, the algorithms switches off the
chunks allocated to the selected beam from the other beams connected to the same TWTA (if necessary).
•If the power is to be modified, the algorithm selects the TWTA to which the selected beam is connected and modifies
either its IBO or its power profile. Modifying the operating conditions of the TWTA induces a modification in the amount
of power delivered by the TWTA, its power efficiency (i.e., the ratio of the delivered RF power to the absorbed DC power)
and the intermodulation interference associated with the TWTA nonlinearity. All these effects are taken into account by
the algorithm through a realistic payload model. Note also that all the beams connected to the same TWTA are affected
by the same attenuation/amplification of the signal on the selected beam. This is done in order to avoid the so-called
capture effect, which takes place in TWTAs when carriers of different power are fed to the amplifier [21].
B. SGM Evaluation
Once the resources of the TWTA corresponding to the selected beam have been modified, the resulting signal-to-interference-
plus-noise ratio (SINR) of all users are calculated for each bandwidth chunk8. The new SINR’s are used to determine the
ModCod with highest spectral efficiency supported by the channel of each terminal in each chunk according to the system
specifications. Once the spectral efficiencies for all terminals are obtained, they are averaged out across users and chunks.
We call ηj
i,c(v,p,B)the spectral efficiency achievable by terminal jin chunk cof beam iat the feasible point (v,p,B). In
most works dedicated to RRM optimization the spectral efficiency is evaluated using the Shannon capacity formula. Although
such approach is useful to get an insight on the theoretical performance limits of the system, the transmission rate of real
systems is constrained by the symbol constellation size as well as by the finiteness of the channel codeword lengths. In current
satellite systems it is common practice to use look-up tables that contain the highest ModCod (and thus the spectral efficiency)
achievable at a given SINR. The SINR seen by terminal jin chunk cof beam iat the feasible point (v,p,B)is:
SINRj
i,c(v,p,B) = P(vt(i), pt(i))|hj
i,i|2bi,c
Pi0,i06=iP(vt(i0), pt(i0))|hj
i,i0|2bi0,c +N(11)
8The SINR is calculated on a chunk-by-chunk basis since each chunk is assumed to be a single carrier.

12
where we indicated with hj
i,i0the channel coefficient from beam i0to user jin beam i.hj
i,i0accounts for the propagation loss
as well as the satellite and the terminal antenna gains, t(i)identifies the TWTA feeding beam iwhile Nrepresents the AWGN
noise power. bi0,c represents element (i0, c)in the binary matrix B, and takes value 1 or 0 depending on whether chunk cis
allocated to beam i0or not, respectively. According to this notation, P(vt(i), pt(i))represents the power allocated to beam i,
which is delivered by TWTA t(i), in correspondence of the IBO vt(i)and the TWTA power profile pt(i). Finally, Nrepresents
the AWGN noise power which is assumed to be the same for all the terminals. The average spectral efficiency in beam iis
then:
ηi=1
Ni
uNi
ch
Ni
u
X
j=1
Ni
ch
X
c=1
ηj
i,c(v,p,B)(12)
where Ni
uis the number of users in beam iwhile Ni
ch is the number of chunks allocated to beam i. Expression (12) follows
from the assumption that all users within a beam access their content in a TDM fashion such that each user is allocated the
whole chunk of bandwidth during the assigned reception slot. Using the average spectral efficiency together with the number
of bandwidth chunks allocated to each beam and taking the roll-off into account, the new value of the offered capacity is
calculated as follows:
Ti
o=Ni
chBch
1 + αηi(13)
αbeing the roll-off factor. Finally, using Eqn. (13), the SGM is calculated as described in Section III.
C. Feasible Point Update
Once the new SGM is calculated, there are two possibilities:
•The SGM obtained in the new point is larger than the old one. In this case the new point is kept and a new iteration
starts.
•The SGM obtained in the new point is smaller than the old one. In this case the new point is kept with a certain probability.
The probability of keeping the new point depends on a simulation parameter that is updated periodically. In the literature
related to simulation annealing, such parameter is called temperature and is indicated in the following as Tsa. The new
point is accepted with probability:
exp −SGMold −SGMnew
SGMoldTsa .
Since we are considering the case in which the new point is worse than the previous one, in the expression above we have
SGMold −SGMnew >0. Note also that such probability decreases as the temperature Tsa decreases. Similar expressions are
commonly found in literature, except for division by SGMold, which we use to increase the sensitivity to SGM variations.
The temperature is decreased once a predefined number of iterations is reached. The cooling law at iteration nis:
Tsa(n)=∆T×Tsa(n−1) (14)
where 0<∆T < 1is a design parameter. The block diagram describing one call of the algorithm is shown in Fig. 5.
As mentioned earlier in this section, the proposed algorithm is a modified version of the SA. The modification consists in
that the SA algorithm is run iteratively, each time using lower starting and stopping temperatures. Specifically, if we indicate
with T_start[l] and T_stop[l] the starting and stopping temperatures at the l-th algorithm call, respectively, the following
holds:
T_start[l] = T_stop[l−1].

13
Initialization: allocate bandwidth
and power as in conventional
payload;
set ܶ
௦௔= T_start
Select a beam
Perturbate feasible point
ܵܩܯ௢௟ௗ <ܵܩܯ௡௘௪?
Keep new
point
With probability
݁ିௌீெ೚೗೏ିௌீெ೙೐ೢ
ௌீெ೚೗೏்
ೞೌ
With probability
1െ݁ିௌீெ೚೗೏ିௌீெ೙೐ೢ
ௌீெ೚೗೏்
ೞೌ
Decrease
ܶ
௦௔
Yes
No
Yes
No
Return point with
largest SGM and exit
Yes
No
Number of iterations
per temperature
reached?
ܶ
௦௔ > T_stop?
Fig. 5: Flow diagram for one call of the SA algorithm. At each call the initial and starting temperatures T_start and T_stop are decreased such that
T_start[l] = T_stop[l−1], lbeing the call index.
The reason behind such modification is that, for the specific problem considered, we observed a tendency of the SA to converge
to local minima. This is a well known behavior of stochastic optimization algorithm with non convex objective functions. The
solution usually adopted is to run the algorithm more than once, each time starting from a different starting point. In the setup
considered in the present paper, such solution showed limited advantages. For this reason we introduced a variant of such
approach in which i) the starting point of the new run is the feasible point output of the previous one, ii) rather than running
the SA anew, we decrease the starting and stopping temperature at each call. The overall effect is to break the path of the
algorithm in the feasible set, avoiding that it gets stuck in regions characterized by values of SGM that are lower than the last
accepted one.
As mentioned in Section III, the choice of a stochastic algorithm is due to the fact that the objective function can be non-
convex depending on the specific system. There is a large body of literature indicating that SA has good performance when
applied to non-convex problems. Even if the problem is not convex, however, the SA algorithm has been shown to lead to
good results. See [22] and references therein for further details.
D. Analysis of Computational Complexity
The SA algorithm can be seen as a hybrid between a random walk and a greedy algorithm. The larger the temperature Tsa,
the closer the behavior of the algorithm is to that of a random walk. As shown in Fig. 5, at each iteration of the algorithm
a random perturbation of the feasible point is performed followed by the evaluation of the new value of the SGM and a
comparison with the old one. Within an iteration, the most demanding part in terms of number of elementary operations is
the evaluation of the feasible point. This requires the calculation of the SINR for each user in the network on each chunk of
bandwidth as shown in Eqn. (11). The sum in the denominator of (11) has at most NTWTA terms since each bandwidth chunk
can be used at most once by each TWTA. For each term in the sum, a multiplication between real numbers is performed.

14
In order to get an estimation of the complexity growth as a function of the main system and algorithm parameters, in the
following we assume that a multiplication or a division between two mbits numbers requires m2elementary operations
while moperations are required for an addition or a subtraction 9. Assuming that each number is represented with an m-bits
precision, NTWTA ×(m2+m)operations are required to calculate the denominator and 2m2more are needed to calculate the
numerator and perform the division. The complexity deriving from the calculation of expressions (12) and (13) can be upper
bounded as (Ni
u×Nch)×[NTWTA ×(m+m2) + 2m2] + (Ni
u×Nch −1) ×m+ 5m2. The total number of operations Ψoc
required to calculate the offered capacity in all the Nbbeams of the system is:
Ψoc =
Nb
X
i=1
Ni
u×Nch ×(NTWTA ×(m2+m)+2m2)
+(Ni
u×Nch −1)m+ 5m2
=Ntot
u×Nch ×(NTWTA ×(m2+m)+2m2)
+(Ntot
u×Nch −Nb)m+ 5Nbm2.(15)
The evaluation of the SGM requires the calculation of Nbterms of the kind shown in expression (6). Lookup tables can be
used to evaluate the exponential function, while the calculation of its argument involves two divisions (calculation of SI and
division by β2) three additions (calculation of ∆, sum inside the square root, sum of 1 and the inverse of SI) two multiplications
(squares calculation) and a square root, for an overall number of operations that is on the order of 5m2+3m. The evaluation
of Eqn. (4) requires the calculation of Nbcubes, Nbadditions and one division, for a total of (2Nb+ 1)m2+Nbmoperations.
The overall number of operations Ψiter required for one iteration of the algorithm can be approximated as:
Ψiter =Ntot
u×Nch ×(NTWTA ×(m2+m)+2m2)
+(Ntot
u×Nch −Nb)m+Nb(5m2+m)+6m2+ 3m
≈Ntot
u×Nch ×NTWTA ×(m2+m)(16)
where the last approximation is tight if NTWTA 2and Ntot
uNb, which is often the case in practical systems. Thus, the
evaluation of the SGM requires a complexity which grows as the product of the total number of users in the system, the
number of bandwidth chunks and the number of TWTAs. Among such terms, the largest one is usually the number of users
in the system.
The SGM is evaluated at each iteration of the algorithm. The total number of iterations within a run depends on the cooling
factor (∆T) and on the value of the initial and final temperature. Specifically, from Eqn. (14) it follows that:
Tsa(n) = (∆T)nT_start
which implies that the total number ntot of cooling steps to be performed before the algorithm stops is:
ntot =log∆TT_stop
T_start 
9This is of course an upper bound, since much more efficient methods allow to reduce significantly the number of elementary operations to perform
multiplications or additions [23], but the aim of this subsection is to get an estimation of the computational effort needed.

15
where dxeis the smallest integer larger than or equal to x. For each temperature a certain number of iterations is to be
performed. Let us refer to such number as Nit10 . The overall complexity Ψtot of one run of the algorithm is thus:
Ψtot ≈Ntot
u×Nch ×NTWTA ×(m2+m)×ntot ×Nit.(17)
The time needed to perform the Ψtot operations depend on the specific hardware considered. The simulations in the next section
took roughly four hours on a commercial laptop with two 2.7 GHz cores and 8 Gigabytes of RAM. Note that one of the
most computational demanding parts is the calculation of the SINR for each of the users. Its impact on the complexity can be
drastically reduced by implementing the calculation on graphics cards, that can achieve high degree of parallelization and thus
notably decrease the computation time. Performance can be further enhanced by using a dedicated hardware. Furthemore, notice
that the algorithm does not need to run in real-time. More specifically, it should be run either when some users experience a
significant change in their SINR or when the requested traffic changes significantly. Since in the present paper we consider
fixed satellite services in the Ka frequency band and above, SINR variations are mostly due to atmospheric events such as rain
ant thus the rate of variation is in the order of the tens of minutes or even hours. Furthermore, the rate of traffic variation is
usually slow, as shown in the next section, with the requested traffic showing little variations on “coherence” periods of about
one hour. Such time correlation can be exploited as follows. Let us assume that a solution to the RRM problem is provided
by the algorithm, the next solution can then be calculated using the previous one as a starting point. We observed in our
simulations that this can be exploited to speed-up the convergence of the algorithm.
V. PE RF OR MA NC E ANALYSI S
In the following we present the results obtained by applying the proposed solution to three different payloads. The first
payload we consider has full flexibility, in the sense that it can be optimized both in terms of bandwidth and power allocation.
The second payload has only bandwidth flexibility, while in the third one only the power settings can be modified. The starting
point of the algorithm is the resource allocation used in the conventional payload. In all simulations we fixed NTWTA = 50, an
overall useful bandwidth B= 500 MHz, bandwidth chunks of Bch = 31.25 MHz (and thus Ntot
ch = 16), an IBO granularity
∆IBO = 1 dB, Sp= 4 for the power profiles, Nb= 200 beams and Ntot
u= 2000 user (with Ni
u= 10,∀i= 1, . . . , Nb). The
TWTA characteristics shown in Fig. 1 have been used and the corresponding nonlinear effects have been taken into account.
Specifically, an increase in the noise level due to intermodulation has been included in the simulator. Such increase in the noise
level depends on the modulation order (and specifically on the amplitude levels in the constellation) and has been accurately
taken into account according to the approach described in [24]. The gains of the reference beam and of the interference beams
as seen by each user (coefficients hj
i,i and hj
i,i0in Eqn. (11), respectively) have been generated using GRASP, a commercial
software for satellite antenna design and analysis [25]. The antenna gain in dB scale as a function of the geographical location of
the user terminals is shown in Fig. 6. The average SNR (including the intermodulation interference in the noise calculation but
not the interbeam interference) across the users for the conventional payload is 11.6dB. The bandwidth and power allocation
of the conventional payload are used as a starting point for the optimization algorithm.
We compare the results for the different payloads in terms of both the effectiveness of the algorithm to meet the requested
capacity and the fairness with which the different beams are treated. Specifically, the Jain Index of the ceiled satisfaction index
10It is common practice to set a maximum number of iterations per temperature together with additional stopping conditions that trigger the cooling event
before Nit iterations are performed. Nit can thus be regarded to as an upper bound on the number of iterations per temperature.

16
Fig. 6: Antenna gain in dB scale as a function of the geographical location of the user terminals.
is used to measure the fairness in the system. The ceiled satisfaction index is defined as SI = min{SI,1}and is a measure of
the satisfaction level of a beam which focuses on the missing capacity. The JI is calculated as:
JI =PNb
i=1 SIi2
NbPNb
i=1 SI2
i
.(18)
Another relevant figure of merit for satellite communications systems is the unmet capacity (UC), which is the overall amount
of requested capacity that can not be provided. UC is defined as:
UC =
Nb
X
i=1 Ti
r−Ti
o+(19)
where (x)+= max(x, 0). Similarly as for UC, we define the excess capacity (EC) as:
EC =
Nb
X
i=1 Ti
o−Ti
r+(20)
which corresponds to the sum across the beams of the offered capacity which exceeds the requested capacity. The UC and
the EC give an indication of the effectiveness of the resources allocation in the system. Finally, we define the total offered
capacity (TOC) as:
TOC =
Nb
X
i=1
Ti
o.(21)
The requested capacity per beam considered in the following has been generated according to a traffic model developed by
the German Aerospace Center (DLR), accounting for the geographical and time variations of traffic requests as well as the
availability of the satellite network. The model provides good matches with real requested traffic statistics, as discussed in [6,
section III-E], to which the interested reader can refer for more details. The average across the beams of the traffic pattern
time evolution obtained with such model is shown in Fig. 7.
The proposed solution has been tested with both static and time varying capacity requests. The results are presented in
subsections V-A and V-B, respectively.

17
0 5 10 15 20
0
0.5
1
1.5
2
x 108
Hour
Mean requested capacity [bps]
Fig. 7: Spatial average of the requested capacity during a day.
A. Static Capacity Request
In figures 8 and 9 the requested capacity is plotted against the beam Id together with the offered capacity obtained by
applying the proposed algorithm to different payloads. Specifically, the capacity offered by the payload with both bandwidth
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
6
7
8x 108
Beam Id
Capacity [bps]
Full
Bandwidth
Power
Conventional
Requested
Fig. 8: Capacity versus beam number. The requested capacity at 00:00 h is shown together with the capacity offered by the fully flexible payload, the
bandwidth flexible payload, the power flexible payload and the conventional payload.
and power flexibility (Full), the one with only bandwidth flexibility (Bandwidth) and the one with power flexibility only (Power)
obtained with the proposed algorithm are shown. The offered capacity of the conventional payload is shown as a benchmark.
In Table I and Table II the comparison among the four different payloads is presented for the requested traffic at off peak
(00:00) and peak (19:00) hours, respectively. In order to have a deeper understanding of the SGM as performance metric and
the implications of using it as objective function, in the tables SGM, Jain Index, unmet capacity and excess capacity are shown
for each payload. All values are rounded to the third decimal. Since the last three parameters have been previously used in
literature or have an intuitive interpretation, they help to understand the SGM more in depth.
With reference to Fig. 8, the payload with bandwidth flexibility and the payload with full flexibility are able to provide an
offered capacity which closely follows the requested one in most of the beams. In beams with very low requested capacity,
such as the beam with Id 194, the offered capacity is relatively larger than the requested one. This is in part due to the limited

18
TABLE I: SGM, Jain Index, unmet capacity and excess capacity at off-peak hour (00:00) for the four payloads. Values are rounded to the third decimal.
SGM JI UC [Gbps] EC [Gbps] TOC [Gbps]
Full 0.923 0.995 1.34 1.664 27.222
Bandw. 0.905 0.995 1.33 1.984 27.552
Power 0.62 0.978 4.263 13.26 35.895
Conv. 0.567 0.982 3.237 17.509 39.529
20 40 60 80 100 120 140 160 180 200
0
1
2
3
4
5
6
7
8x 108
Beam Id
Capacity [bps]
Full
Bandwidth
Power
Conventional
Requested
Fig. 9: Capacity versus beam number. The requested capacity at 19:00 h is shown together with the capacity offered by the fully flexible payload, the
bandwidth flexible payload, the power flexible payload and the conventional payload.
granularity in terms of bandwidth. The payload with power flexibility is not able to follow the requested traffic as closely as
the other two flexible payloads. One of the reasons for this is the fact that the power can be optimized only at TWTA level,
so that all beams experience the same power increase or decrease, while this is not the case for the payload with bandwidth
flexibility, for which the number of bandwidth chunks assigned to a beam can be different from that of other beams connected
to the same TWTA (provided the same chunk of bandwidth is not allocated to more than one of the beams connected to it).
As a last remark, we notice that the offered capacity of the conventional payload shows some fluctuations across the beams.
These are due to the slight differences in channel gains and interference levels experienced by the different users, that, on turn,
depend on the realistic satellite antenna radiation pattern used. From Table I we can see that the qualitative considerations
presented above are backed up by the numerical values in the table. As a matter of fact, the SGM is higher in the payloads
with full and bandwidth flexibility. This corresponds to higher JI, lower UC and lower EC with respect to the other two
payloads. Comparing the Full and the Bandwidth payloads we see that the UC in the two payloads are almost the same while
the smallest EC is achieved by the Full payload, which leads to a larger SGM. This shows how the SGM jointly accounts for
UC, EC and fairness, although the mapping from SGM to the three measures is not straightforward. It is interesting to note
that, as shown in Table I, the Power and the Conventional payloads provide a larger total offered capacity than the other two
schemes. Despite of this, the UC is higher in the two systems, which indicates that the Bandwidth and Full payloads make a
more efficient use of the satellite resources.
In Fig. 9 we show the results relative to a more demanding pattern of requested capacity. The setup is much more challenging
than the one presented in Fig. 8 from an optimization perspective. This can be inferred from the peaks of requested traffic
reaching more than three times the capacity offered by the conventional payload, and from the fact that the overall system
bandwidth and TWTA number in all the advanced payloads is the same as for the conventional one in both simulations. Also
in this case the payloads with full and bandwidth flexibility can best follow the requested traffic profile, while the payload

19
with power flexibility has only a limited adaptation capability. Interestingly, even though the power flexibility alone is not able
to follow the requested traffic, it provides an advantage to the payload with full flexibility. This can be seen from the fact that
the Full payload can better approximate the highest peaks with respect to the bandwidth flexible one. This is further confirmed
by the results shown in the Table II. In the table we see that the payload with full flexibility achieves the best performance in
TABLE II: SGM, Jain Index, unmet capacity and excess capacity at peak hour (19:00) for the four payloads. Values are rounded to the third decimal.
SGM JI UC [Gbps] EC [Gbps] TOC [Gbps]
Full 0.912 0.978 8.514 1.161 37.415
Bandw. 0.884 0.966 10.692 1.081 35.157
Power 0.638 0.93 14.808 10.779 40.738
Conv. 0.603 0.914 15.765 12.166 39.529
all the four figures of merit considered, reducing the unmet and the excess capacity and increasing the system fairness with
respect to any of the other payloads. In this case we see how a higher SGM corresponds to a better performance through the
whole spectrum of figures of merits considered.
In order to have a better understanding of the benefits deriving from the adoption of the SGM rather than other related
objective functions, we applied the proposed optimization algorithm to two different objective functions. One is −JI (the “−”
sign is used so that the optimization problem can be cast as a minimization) while the other is the average absolute value of
the capacity gap ∆,1/NbPNb
i=1 |∆i|. We chose such functions because they are the performance metrics at the basis of the
SGM definition. The comparison in terms of JI, UC and EC is presented in Table III and Table IV for the off-peak and peak
hours, respectively. From Table III we see that applying the proposed algorithm to maximize the Jain index leads to relatively
TABLE III: SGM, Jain Index, unmet capacity and excess capacity at off-peak hour (00:00) for the payload with full flexibility for three different
objective functions. Values are rounded to the third decimal.
JI UC [Gbps] EC [Gbps]
SGM, Full 0.995 1.34 1.664
JI, Full 1 0.264 5.048
∆, Full 0.865 1.144 0.738
TABLE IV: SGM, Jain Index, unmet capacity and excess capacity at peak hour (19:00) for the four payloads. Values are rounded to the third decimal.
JI UC [Gbps] EC [Gbps]
SGM, Full 0.978 8.514 1.161
JI, Full 0.978 9.884 1.572
∆, Full 0.893 7.087 0.624
good results in terms of both JI and UC. Similar results in terms of JI and UC are obtained using the proposed SGM function,
while the SGM outperforms the JI in terms of excess capacity. The system which minimizes ∆achieves relatively good results
in terms of EC and UC, while performing poorly in terms of fairness. This is due to the fact that some of the beams, especially
those with lower capacity request, are in some case left in starvation with no capacity allocated. Such results are in line with
the expectations, since the ∆system tries to minimize the capacity mismatch without taking fairness into account, while the
JI system maximizes fairness disregarding the capacity mismatch. However, since the system is well dimensioned to meet the
capacity request for the off-peak traffic, also the result in terms of UC for the JI system is relatively good. In the off-peak

20
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Day hour
SGM
Full
Bandwidth
Conventional
Fig. 10: SGM plotted against the hour of the day obtained applying the proposed algorithm to two payloads each having a different degree of flexibility
(bandwdth and power, bandwidth only). The performance of a conventional (i.e., not optimized) payload is also shown.
hour the proposed SGM function achieves a result which is a compromise between the two systems and achieves relatively
good results in terms of both fairness and capacity mismatch. When the traffic request increases, as shown in Table IV, the
usefulness of the proposed SGM function becomes more evident. As a matter of facts, it achieves the same fairness as the JI
system while having lower UC and lower EC. The ∆system shows good results in terms of EC and UC, but has much worse
results in terms of fairness.
B. Time Varying Capacity Request
In order to evaluate the performance of the proposed solution in case of time-varying requested capacity, we performed
extensive simulations increasing the number of user terminals to 104(50 terminals per beam) and using the requested capacity
model presented in [6, section III-E]. The results of the simulations are shown in Fig. 10, where the SGM is plotted against the
hour of the day. The results obtained applying the proposed algorithm to the two most promising payloads, i.e. the one with
full flexibility and the one with only bandwidth flexibility, are shown. The performance of a conventional (i.e., not optimized)
payload is also shown for comparison. The optimization algorithm is run once per hour so as to follow the traffic request
variation. In the initial state of the optimization algorithm for the time 00:00 the bandwidth and power allocation of all payloads
are set as in the conventional payload. The initial state for each subsequent hour has been set equal to the solution of the
previous hour. This was done in order to save computational resources. In fact, since the traffic distribution across beams
shows only limited changes within an hour, using the solution obtained in the previous hour as a starting point is likely to lead
to a better results for a given amount of computational resources with respect to the case in which the conventional payload
allocation is used as a starting point. From the figure it can be seen how the proposed algorithm achieves a significant gain
in terms of SGM for both flexible payloads with respect to the conventional case. The advantage of having both power and
bandwidth flexibility with respect to having bandwidth flexibility only appears to be relatively limited throughout the whole
day. The largest gain, around 10%, can be observed at around 5:00, in correspondence to a period of low traffic request. Such
gain is due mostly to the lower excess capacity of the fully flexible payload, which confirms what presented in Table I. The
low SGM values achieved by the conventional payload in the same period are also due mostly to excess capacity. During
periods of high traffic demand (around 19:00) the flexible payloads show a minimum in the SGM, which is caused by the high
capacity request and possibly by an under-dimensioning of the system. Despite of this, a gain of around 40%with respect

21
to the conventional payload can be observed. The fact that the performance of the fully flexible payload are at least as good
as those obtained by the payload with only bandwidth flexibility, both evaluated in the respective optimal configurations, is
intuitive, since the Full payload can, at worst, reproduce the optimal allocation of the Bandwidth one. However, this is not
guaranteed if non-optimal feasible points are chosen. The fact that, as shown in Fig. 10, the Full always upper-bounds the
Bandwidth one is an indication of the goodness of the algorithm.
VI. CONCLUSIONS
We studied the problem of radio resource management in multibeam satellite systems under realistic conditions. A novel
objective function for the optimization problem has been introduced with the aim to match the requested capacity across the
beams as close as possible while taking fairness into account. After showing that, at least for some constraints imposed by the
payload, the proposed function is not convex, we proposed a stochastic optimization algorithm to minimize such function. The
algorithm is based on a modified version of the simulated annealing algorithm. A detailed complexity analysis of the proposed
algorithm has been presented. The algorithm has been applied to three payloads having different degrees of flexibility, namely
flexibility both in bandwidth and power, in bandwidth only and in power only. Realistic payload models, antenna pattern,
co-channel interference and requested traffic distribution were used in the simulations. Our results show that the proposed
approach is more efficient than the traditional conventional payload in matching the requested capacity across the beams and
leads to interesting results both under low and high traffic demand. From the results it also emerges that power flexibility
shows limited performance enhancement with respect to a conventional payload design, while bandwidth flexibility definitely
leads to better results. If the two kinds of flexibility are combined, a limited improvement in performance with respect to the
bandwidth-flexible case is observed. The goodness of the proposed optimization approach has been supported by measuring
different figures of merit traditionally used in this kind of analysis, namely missing capacity, excess capacity and Jain index.
Furthermore, we showed that adopting the proposed objective function leads to good results compared to other relevant objective
functions (namely Jain Index and capacity mismatch) in low traffic conditions, while outperforms such functions when higher
traffic requests are considered. Finally, we performed extensive simulations to test the proposed solution in the case of time-
varying capacity request. The results show significant gains with respect to the conventional payload in the whole considered
time span.
APPENDIX
Proof of Proposition 3.1
In order to prove that the SGM is in general not concave we look for a setup in which this is not verified. We recall that
a function g(x) : Rk→R, is concave if and only if its domain Dis a convex set and ∀λ∈[0,1],∀x1,x2∈ D the following
holds [26]:
λg(x1) + (1 −λ)g(x2)≤g(λx1+ (1 −λ)x2).
Let us consider a system with 4beams, two unitary-bandwidth bandwidth chunks and two TWTAs. Each TWTA is connected
to two beams. Bandwidth and power can be allocated by a TWTA independently of the other TWTA. Assume that one user
per beam is present. Now consider the following channel matrix:
H=2
3I+1
314×4

22
where 1k×lis a k×lall-one matrix while Irepresents the identity matrix. Element hi,j of matrix Hrepresents the channel
coefficient from beam jto user in beam i. Element hi,i represents the gain of the reference beam while hi,j,i6=jrepresents
the channel gain of the interfering beam jas observed by the user in beam i. Finally, consider the following vector of requested
capacities: (T1
rT2
rT3
rT4
r) = (1 2 0.5 0). The spectral efficiency is evaluated using the Shannon capacity formula for the
AWGN channel and calculating the SINR using expression (11) 11. Consider now a family of feasible points characterized by
the following bandwidth matrix:
B=








1 0
0 1
1 1
0 0








.
Let us now consider two different feasible points defined by Band the following power allocation vectors (in linear scale)
each containing the OBO of the two TWTA:
v1=

0.2
0.1429

v2=

0.3333
0.5
.
Let SGM(v1,B)and SGM(v2,B)be the SGM values obtained with the two configurations. Let us now define λ1= 0.3and
λ2= 1 −λ1. Finally we have:
λ1SGM(v1,B) + λ2SGM(v2,B)=0.5975
>0.5964 = SGM(λ1v1+λ2v2,B)(22)
which concludes the proof.
ACK NOW LE DG ME NT
The research activity reported in the present paper has been partly funded by the ESA/ESTEC ARTES 5.1 project “MultiBeam-
RRM Efficient load-aware dynamic global radio resource management in multi-spotbeam broadband satellite networks” (contract
#4000109718/2014/NL/WE).
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