Research ArticleOptimization of Joint Power and Bandwidth Allocation inMulti-Spot-Beam Satellite Communication Systems
Heng Wang Aijun Liu and Xiaofei Pan
College of Communications Engineering PLA University of Science amp Technology Nanjing Jiangsu 210007 China
Correspondence should be addressed to Heng Wang wangheng0987654321126com
Received 30 August 2013 Accepted 5 December 2013 Published 12 January 2014
Academic Editor Jun Jiang
Copyright copy 2014 Heng Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Multi-spot-beam technique has been widely applied in modern satellite communication systems However the satellite powerand bandwidth resources in a multi-spot-beam satellite communication system are scarce and expensive it is urgent to utilizethe resources efficiently To this end dynamically allocating the power and bandwidth is an available way This paper initiallyformulates the problem of resource joint allocation as a convex optimization problem taking into account a compromise betweenthe maximum total system capacity and the fairness among the spot beams A joint bandwidth and power allocation iterativealgorithm based on duality theory is then proposed to obtain the optimal solution of this optimization problem Compared with theexisting separate bandwidth or power optimal allocation algorithms it is shown that the joint allocation algorithm improves boththe total system capacity and the fairness among spot beamsMoreover it is easy to be implemented in practice as the computationalcomplexity of the proposed algorithm is linear with the number of spot beams
1 Introduction
In recent years the multi-spot-beam technique has played animportant role in the satellite communication systems as itcan not only supply higher power density to a particular spotbeam but also construct flexible service networks Howeverthe satellite power and bandwidth resources are scarce andexpensive in multi-spot-beam satellite communication sys-tems As a result it is crucial to make effort to enhance theutilization efficiency of power and bandwidth resource
Because the coverage of each spot beam is different thereal traffic demand of each spot beam is different and timevarying In addition the channel condition of each spot beamis also affected by the weather condition Therefore it isimportant to dynamically allocate the bandwidth and powerresource to each spot beam to meet its traffic demand
In the previous work separate optimal power or band-width allocation algorithms were proposed in [1ndash4] and[5] respectively The work in [1] emphasized the mathe-matical formulation and analytic solutions of the optimumpower resource allocation problem and explained the tradeoffbetween the total system capacity and fairness among all spotbeams with different traffic demands and delay constraints
The optimization problem is solved based on the Karush-Kuhn-Tucker (KKT) condition However the works in [1] didnot provide the way to find the optimal Lagrangian multipli-ersTherefore themethods of bisection and subgradient wereapplied to search for the optimal Lagrangianmultipliers in [23] To improve the total system capacity amethod of selectingsmall number of active spot beamswas proposed in [4] whilekeeping the fairness among spot beams The optimal powerallocation algorithms failed to consider one problem thatthe optimal power allocation algorithm required the satellitepower amplifier to operate with high back-off degrading thesystem performance To solve the problem in [5] a dynamicbandwidth allocation algorithm was proposed The aboveworks proved that it was needed to allocate more resourcesto the spot beam with higher traffic demand to get fairnessamong spot beams thus the total system capacity decreaseddue to the concavity of the capacity function with a fixedpower or bandwidth allocation To overcome this drawbackin this paper we propose a joint bandwidth and powerallocation algorithm
For completeness it is noted that the joint bandwidthand power allocation problem has been investigated forterrestrial wireless communication systems [6ndash8] In [6]
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 683604 9 pageshttpdxdoiorg1011552014683604
2 Mathematical Problems in Engineering
the joint bandwidth and power allocation problem in down-link transmission was investigated In [7] an optimal jointbandwidth and power allocation algorithm in wireless mul-tiuser networks with and without relaying was proposedIn [8] a joint power and bandwidth allocation algorithmwith QoS support in heterogeneous wireless networks wasproposed using convex optimization methodology How-ever the conclusions obtained in the above works cannotbe applied to the multi-spot-beam satellite communicationsystem due to the fact that the constraints in the system weinvestigate are different from those in the systems mentionedabove
In this paper we propose solving the problem of jointbandwidth and power allocation for the multi-spot-beamsatellite communication system We initially formulate theproblem of joint bandwidth and power allocation as anonlinear optimization problem and prove the optimizationproblem is convex The object of optimization problem is tomatch the capacity allocated to each spot beam to the trafficdemand as closely as possible taking into account a compro-mise between the maximum total system capacity and theproportional fairness among spot beams A joint bandwidthand power allocation iterative algorithm based on dualitytheory is then proposed to obtain the optimal solution of thisoptimization problem Compared with the separate poweror bandwidth optimal allocation algorithm the proposedjoint bandwidth andpower allocation algorithm improves thetotal system capacity and the fairness among spot beams Inaddition we discuss the impact of traffic demand channelcondition and the delay constraint of each spot beam on theallocation results
The remainder of this paper is organized as follows InSection 2 we formulate the optimization problem of jointbandwidth and power allocation and prove the optimizationproblem is convex Section 3 proposes the optimal jointbandwidth and power allocation algorithm based on dualitytheory and Section 4 presents the simulation results andanalyzes the impact of traffic demand channel conditionsand delay constraints on the allocation result Section 5concludes the paper
2 Mathematical Formulation of JointBandwidth and Power Allocation
21 Modeling of Downlink Multi-Spot-Beam CapacityFigure 1 shows the system configuration of a multi-spot-beam satellite communication system where the trafficdemand of the 119894th spot beam is 119879
119894 the power allocated to the
119894th spot beam is 119875119894 the bandwidth allocated to the 119894th spot
beam is 119882119894 and the signal attenuation factor of the 119894th spot
beam is 1205722119894 It is noted that 1205722
119894mainly consists of the effects of
weather conditions free space loss and antenna gainUsing time sharing for Gaussian broadcast channels [9]
the Shannon bounded capacity 119862119894for the 119894th spot beam is
given as
119862119894= 119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
) (1)
Ci
Ti
Figure 1 System configuration of multi-spot-beam satellite system
where 1198730is the noise power density of each spot beam
It is noted that interbeam interference from the sidelobesof adjacent spot beams degenerates the Shannon capacityHowever in this paper we ignore interbeam interferencebecause we consider very narrow spot beams over a largenumber of spot beams [1] It is observed from (1) that thespot beam capacity119862
119894is increased as the bandwidth or power
allocated to the spot beam increases However the totalbandwidth and power of the satellite are fixed so the capacityof the system is limited
If the total system power and bandwidth resources aresufficient to support the total traffic demand generated byall the spot beams it seems to be meaningless for us tomake efforts to improve the resource utilization efficiencyTherefore we only focus on the scenario where the total trafficdemand exceeds the total available system capacity [1]
22 Delay Constraint In this section we describe the impactof the delay constraint In the practical multi-spot-beamsatellite system there is much real-time traffic such as videoand audio flows for which the delay performance is animportant evaluation criterion According to the analysis in[1] the delay constraint of each spot beam can be transformedinto the traffic demand of each spot beam which is given asfollows
119862119894ge
119879119894
(1 minus 119890119894)119863119894
(2)
where 119890119894is the packet error rate over the link of the 119894th spot
beam 119863119894is the average delay deadline of the 119894th spot beam
and (1 minus 119890119894)119863119894gt 1 The constraint (2) implies that each spot
beam has a minimal traffic demand which is determined by119890119894and 119863
119894 It is seen that both the higher priority traffic with
a smaller119863119894and the worse channel condition with a larger 119890
119894
will lead to a larger fraction of the capacity
23 Optimization Problem Formulation There are manymetrics to evaluate the system performance and different
Mathematical Problems in Engineering 3
metrics may lead to different allocation results Therefore itis very important for us to choose an appropriate metric Inthis paper the metric is to minimize the deficit between thetraffic demand and the capacity allocated taking into accounta compromise between the maximum total system capacityand the proportional fairness among spot beams MotivatedbyChoi andChan [1] the problem can be formulated by usingthe 119899th order objective function as follows
min119875119894119882119894
119873
sum
119894=1
(119879119894minus 119862119894)119899 (3)
st 119862119894= 119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
) le 119879119894
forall119894 (4)
119873
sum
119894=1
119875119894le 119875total (5)
119873
sum
119894=1
119882119894le 119882total (6)
119862119894= 119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
) ge119879119894
(1 minus 119890119894)119863119894
forall119894
(7)
where 119899 is integer and 119899 ge 2 When 119899 = 1 the problem isequivalent to maximizing the total system capacity and thefairness among spot beams is ignored So we ignore the casewith 119899 = 1 The constraint (4) indicates that the allocatedresource should not exceed the traffic demand of each spotbeam Conditions (5) (6) and (7) imply the constraint for thetotal power the total bandwidth and the delay respectively
The above optimization is a nonlinear programmingproblem It is found that the optimal variables 119882
119894and 119875
119894
are coupled in the problem above which makes the globaloptimal solution difficult to be obtained To make the aboveproblem tractable in the following section we propose ajoint power and bandwidth allocation algorithm based onduality optimization theory [10ndash13] It is known that if theoptimization problem is convex the duality gap is zero Asa result the optimal solution obtained by the duality problemis the global optimal solution of the primary problem [14]Fortunately the optimization problem mentioned above isconvex the proof of which is shown in the appendix
As mentioned above the joint bandwidth and power alloca-tion algorithm is based on the duality theory By introducingnonnegative dual variables 120583 120582 and 120588 = [120588
1 1205882 120588
119899] the
Lagrange function is yielded given as
119871 (PW120588 120582 120583) =
119873
sum
119894=1
[119879119894minus119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
)]
119899
minus 120583(119882total minus119873
sum
119894=1
119882119894)
minus
119873
sum
119894=1
120588119894[119862119894minus
119879119894
(1 minus 119890119894)119863119894
]
minus 120582(119875total minus119873
sum
119894=1
119875119894)
(8)
where P = [1198751 1198752 119875
119899] andW = [119882
11198822 119882
119899]
From (8) Lagrange dual function can be obtained by
119863(120588 120582 120583) = minPW
119871 (PW120588 120582 120583) (9)
and the dual problem can be written as
119889lowast= max120582ge0 120583ge0120588ge0
119863(120588 120582 120583) (10)
According to [10] the dual problem in (10) can be furtherdecomposed into the following three sequentially iterativesubproblemsSubproblem 1 Power Allocation Given the dual variables 120583 120582and 120588 for any 119894 = [1 119873] maximizing (8) with respect to119875119894brings the equation
1198991205722
119894119882119894
ln 2 (1198821198941198730+ 1205722
119894119875lowast
119894)
times [119879119894minus119882119894log2(1 +
1205722
119894119875lowast
119894
1198821198941198730
)]
119899minus1
+120588119894
119899 = 120582
(11)
The119875lowast119894can be obtained from (11) by numerical calculation
methods for example the golden section method Theoptimized power allocation of the 119894th spot beam 119875
opt119894
=
max 0 119875lowast
119894
Subproblem 2 Bandwidth Allocation Substituting the 119875opt119894
into (8) maximizing (8) with respect to 119882119894brings the
following equation
[(119879119894minus 119862lowast
119894)119899minus1
+120588119894
119899]
times 119862lowast
119894
119882lowast
119894
minus1
ln 2 [1198730119882lowast
119894 (1205722
119894119875opt119894
) + 1]
=120583
119899
(12)
where 119862lowast119894= 119882lowast
119894log2(1 + (120572
2
119894119875opt119894
119882lowast
1198941198730))
From (12) we can obtain the119882lowast119894by using golden section
method The optimized bandwidth allocated to the 119894th spotbeam119882
opt119894
= max0119882lowast119894
Subproblem 3 Dual Variables Update The optimal dualvariables can be obtained by solving its dual problem
(120588opt 120582
opt 120583
opt) = arg min
120582 120583120588
max 119871 (PoptWopt
120588 120582 120583)
(13)
4 Mathematical Problems in Engineering
Because the dual function is always convex here wecan use a subgradient (a generalization of gradient) updatemethod as follows [11]
120582119899+1
= [120582119899minus Δ119899
120582(119875total minus
119873
sum
119894=1
119875opt119894
)]
+
120583119899+1
= [120583119899minus Δ119899
120583(119882total minus
119873
sum
119894=1
119882opt119894
)]
+
120588119899+1
119894= [120588119899
119894minus Δ119899
120588(119862
opt119894
minus119879119894
(1 minus 119890119894)119863119894
)]
+
(14)
where [119909]+ = max0 119909 119899 is the iteration number and Δ isthe iteration step size
The above dual variable updating algorithm is guaranteedto converge to the optimal solution as long as the iterationstep chosen is sufficiently small [11]
The whole process of the proposed joint bandwidth andpower allocation algorithm can be summarized as followsThe Proposed Joint Bandwidth and Power Allocation Algo-rithm
Step 1 Set appropriate initial values for the dual variables andthe bandwidth of each spot beam
Step 2 Substitute the values of the bandwidth of each spotbeam and the dual variables into (11) and then calculate theoptimized power allocated to each spot beam
Step 3 Substitute into (12) both the power values for eachspot beam obtained from Step 2 and the Lagrangianmultipli-ers calculate the optimized bandwidth allocated to each userand then calculate the optimized bandwidth allocated to eachspot beam
Step 4 Substitute the values of the power and bandwidth ofeach spot beam which are separately obtained from Steps 2and 3 into (14) and then update the dual variables
Step 5 If the conditions of |120582119899+1
(119875total minus sum119894119875119894)| lt 120576
|120583119899+1
(119882total minus sum119894119882
opt119894
)| lt 120576 and |120588119899+1
119894(119862
opt119894
minus 119879119894[(1 minus
119890119894)119863119894)]| lt 120576 forall119894 are satisfied simultaneously then terminate
the algorithm Otherwise jump to Step 2
According to the above process it is seen that thecomputational complexities of Steps 2 3 and 4 are 119874(119878119873)119874(119878119873) and 119874(2 + 119873) where 119873 is the number of thespot beams and 119878 is the computational complexity of thegolden section method Therefore the total computationalcomplexity of the proposed algorithm is 119874(2119870119878119873 + 2119870 +
119870119873) where 119870 is the number of the iterations It is notedthat both 119870 and 119878 are independent of 119873 As a result thecomputational complexity of the proposed algorithm is linearwith the number of the spot beams which indicates that thealgorithm can be implemented in practice
4 Simulation Results and Analysis
For the simulation a Ka bandmulti-spot-beam satellite com-munication system model is set up The system has 10 spotbeams the total power of the satellite is 200W and the totalbandwidth of the satellite is 500MHz The traffic demand ofeach spot beam increases from 80Mbps to 260Mbps by stepsof 20Mbps The minimal traffic demand of each spot beamcaused by the delay constraint is 20 of traffic demand
41 Convergence Behavior of Proposed Joint Allocation Algo-rithm To show the convergence behavior of the proposedjoint allocation algorithm we first assume that the channelcondition of each spot beam is the same and the normalizednoise power spectral density parameter 119873
01205722
119894is 02119890
minus6Moreover the order of the objective function is 2
Figures 2 and 3 show the convergence of the dual variablesand optimal variables As seen in Figure 2 each dual variablehas its own update step size and the dual variables convergeto their optimal value after finite iterations which dependon the choice of the step sizes When the dual variablesconverge it is noted that from Figure 3 the optimal variablesalso converge
42 Efficiency of the Proposed Joint Allocation AlgorithmTo verify the resource efficiency of the proposed optimaljoint allocation algorithm (OBOP) we compare it with thefollowing three algorithms
(a) Uniform bandwidth allocation and uniform powerallocation (UBUP) UBUP algorithm divides thebandwidth and power resource equally between eachspot beam
(b) Uniform bandwidth allocation and optimal powerallocation (UBOP) [1] UBOP algorithm divides thebandwidth resource equally between each spot beamand optimizes the power allocation for each spotbeam
(c) Uniform power allocation and optimal bandwidthallocation (OBUP) [5] OBUP algorithm divides thepower resource equally between each spot beam andoptimizes the bandwidth allocation for each spotbeam
When the channel conditions are the same Figure 4shows the capacity distributions of spot beams which areallocated by the four algorithms Table 1 shows the totalsystem capacity of the four algorithms As shown in Figure 4to get the fairness among spot beams the separate optimalallocation algorithms (UBOP and OBUP) will provide morecapacity to higher demand spot beams It is known thatthe capacity does not linearly increase with the power orbandwidth increasing due to the concavity of the capacityfunction with a fixed bandwidth or power allocation Thisresults in a lower total system capacity However in theproposed joint allocation algorithm (OBOP) the capacityfunction is almost linear since the bandwidth and power arejointly dynamically allocated As a result the total systemcapacity is much improved compared to the separate optimal
Mathematical Problems in Engineering 5
0 50 100 1500
100
200
300
400
500
600
700
800
900
1000
Iterations
120588(1) (step size is 04)120582 (step size is 15)120583 (step size is 07)
Figure 2 The convergence behavior of the dual variables
0 50 100 1500
500
1000
1500
Iterations
The sum of the power allocated to each beamThe sum of the bandwidth allocated to each beam
Figure 3 The convergence behavior of the optimal variables
Table 1 The total system capacity of the four algorithms when thechannel conditions are the same
allocation algorithm This conclusion can be also concludedfrom Table 1
When the channel conditions are the same Figure 5shows the deficit between the traffic demand and the capacity
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
Figure 4 Comparison of the four algorithms in terms of thecapacity allocated to each spot beam when channel conditions arethe same
Table 2The total sum of (119879119894minus 119862119894)2 of the four algorithms when the
channel conditions are the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1153611986415
UBOP [1] 1017011986415
OBUP [5] 1067611986415
The proposed OBOP 8336611986415
allocated to each spot beam Table 2 presents the sum of thedeficit between the traffic demand and the capacity allocatedto each spot beam From Figure 5 it is shown that in thejoint allocation algorithm (OBOP) from the spot beam 3to spot beam 10 the deficit between traffic demand andcapacity allocated is almost the same so the fairness amongspot beams is better than the separate optimal allocations(UBOP and OBUP) This conclusion can be also observedfromTable 2 since the deficit between the traffic demand andthe capacity allocated is smaller than the separate optimalallocations Together with the conclusion above regardingtotal system capacity we can conclude that the performanceof the optimal joint allocation algorithm (OBOP) is muchimproved compared with the separate optimal algorithms(UBOP and OBUP)
43 Impact of the Channel Condition and Delay ConstraintsWecompare the allocation result of theOBOP allocationwithdelay constraint with that of the OBOP allocation withoutthe delay constraint (OBOPND) to analyze the impact of thedelay constraints When there is no delay constraint on eachspot beam the allocation result can be obtained by the samealgorithm where we just need to remove the dual variable 120588in (8)
6 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]
OBUP [5]The proposed OBOP
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 5 Comparison of the four algorithms in terms of the deficitbetween the traffic demand and the capacity allocated to each spotbeam when channel conditions are the same
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
OBOPNDUBUPUBOP [1]OBUP [5]
The proposed OBOP
The minimal traffic demand
ith beam
Figure 6 The capacity allocated by five algorithms to each spotbeam when channel conditions are not the same
From spot beam 2 to spot beam 4 we set the normalizednoise power spectral density parameters 119873
01205722
119894to 03119890
minus604119890minus6 and 05119890
minus6 respectively to analyze the impact ofchannel condition on the allocation results Each trafficdemand of the three spot beams is 120Mbsp so the totaltraffic demand is the same as the above simulation scenarioThe order of the objective function also is 2
When the channel conditions are not the same Figure 6shows the capacity distributions of spot beams which areallocated by the 5 algorithms Table 3 shows the total system
Table 3 The total system capacity of the five algorithms when thechannel conditions are not the same
Table 4 The total sum of (119879119894minus 119862119894)2 of the 5 algorithms when the
channel conditions are not the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1239711986415
UBOP [1] 1122511986415
OBUP [5] 1140811986415
The proposed OBOP 9091911986415
OBOPND 8573011986415
capacity of the 5 algorithms From Figure 6 it is shownthat both the separate optimal allocation algorithm and theproposed joint allocation algorithm allocate more capacityto the spot beam with better channel condition than thatwith worse channel condition especially for the OBOPNDalgorithm no capacity is allocated to spot beam 3 and spotbeam 4 Therefore the metric we choose in this paper notonly considers the fairness among spot beams but also triesto maximize the throughput of the system which achieves agood system performance as we expected Due to the delayconstraint we have to allocate aminimal traffic to spot beamswith worse channel condition thus the total system capacitydecreases This conclusion is also shown in Table 3 So weconclude that when the channel condition of each spot beamis not the same the delay constraint of each spot beamdecreases the total system capacity
When the channel conditions are not the same Figure 7shows the deficit between the traffic demand and the capacityallocated to each spot beam Table 4 presents the sum ofthe deficits between the traffic demand and the capacityallocated to each spot beam It is shown that the worsechannel conditions and the delay constraint decrease thefairness among spot beams
44 Impact of the Order of the Objective Function and DelayConstraints As mentioned in Section 23 different metricsmay lead to different allocation results Here we analyze theimpact of the order of the objective function The channelcondition of each spot beam is set to be the same
Figure 8 shows the allocated capacity of each spot beamand Table 5 shows the total system capacity when the order ofthe objective function is different
When there is no delay constraint in each spot beamas we use a higher-order deviation objective function moreresources are provided for higher traffic demand spot beamswhile a lower-order cost function gives relatively moreresource to lower traffic demand spot beams For example
Mathematical Problems in Engineering 7
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 7 The deficit between the traffic demand and the capacityallocated by 5 algorithms to each spot beam when channel condi-tions are not the same
the cubic objective function provides no capacity for the threelowest traffic demand spot beams Although the capacityallocated to each spot beam is different the total systemcapacities of different objective function are almost the same
When there is delay constraint in each spot beam aminimal traffic demand must be allocated to them As aresult for the same cubic objective function the three lowesttraffic demand spot beams are allocated to theminimal trafficdemand It is also seen that the capacity allocated to each spotbeam is the same by using second- and third-order of theobjective function
5 Conclusion
In the multi-spot-beam satellite system due to the scarcenessof the satellite resource it is crucial for us to improve theresource utilization efficiency To this end in this paperwe first formulated the joint bandwidth and power allo-cation problem as a convex optimization problem Thenwe proposed a joint bandwidth and power allocation algo-rithm based on duality theory to get the optimal solutionCompared with the individual optimal power or bandwidthallocation the proposed joint optimal bandwidth and powerallocation algorithm improved the total system capacity andthe fairness among spot beams while the computationalcomplexity of the algorithm was linear with the numberof spot beams Therefore the proposed algorithm can beimplemented in practice
It was shown from the simulation results that the spotbeam with higher traffic demand and better channel con-dition will be allocated more resources to minimize theobjective function Due to the delay constraint in each spotbeam there was a minimal traffic demand in each spot beam
Table 5 The total system capacity of different objective functions
Figure 8 The capacity allocated to each spot beam of differentobjective functions
When the channel condition of each spot beamwas the samethe delay constraint only had an impact on the fairness amongspot beams However when the channel conditions were notthe same both the total system capacity and fairness amongspot beams were influenced by the delay constraint
Appendix
We first prove that constraint (4) can be ignored by contra-diction Assuming that 119862
119894gt 119879119894 it is seen that the constraint
(7) is satisfied obviously As a result the correspondingLagrange multiplier 120588
119894is zero According to (11) the value
of 120582 is negative which contradicts that 120582 is nonnegativeTherefore the constraint (4) is satisfied when the value of 120582 isnonnegative
According to [14] for the optimization problem whichhas the following general form
min 119891 (X)
st 119892119894 (X) le 0
(A1)
when the functions 119892119894(X) and 119891(X) are convex the optimiza-
tion problem is convex
8 Mathematical Problems in Engineering
Taken together with the fact that the functions sum119875119894
and sum119882119894in constraints (5) and (6) are linear to prove
the optimization problem with respect to [11988211198822 119882
119899
1198751 1198752 119875
119899] is convex we only need to prove that the
objective function sum (119879119894minus 119862119894)2 is convex and the function
119862119894= 119882119894log2[1 + 120572
2
119894119875119894(1198821198941198730)] in constraint (7) is concave
Firstly we prove that function 119862119894with respect to119882
119894and 119875
119894is
concaveThe Hessian of the function 119862
119894= 119882119894log2[1 + 120572
2
119894119875119894
(1198821198941198730)] is given as follows
119867119862=
[[[[
[
1205972119862119894
1205971198752
119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
1205971198822
119894
]]]]
]
(A2)
where 1205972119862119894120597119875119894
2= minus120572
2
119894119882119894 ln 2(119873
0119882119894+ 1205722
119894119875119894)2
le 0 and|119867119862| = 0It is seen that the determinant of the119867
119862is zero therefore
119867119862is negative semidefinite and the function 119862
119894is concave
[14] Then we prove that sum (119879119894minus 119862119894)2 is convex
It is known that the sum of convex functions is alsoconvex [14] Therefore to prove that sum (119879
119894minus 119862119894)2 is convex
we just need to prove the following function is convex
119891 (119875119894119882119894) = (119879
119894minus 119862119894)119899 (A3)
where 119862119894= 119882119894log2(1 + 120572
2
1198941198751198941198821198941198730)
The Hessian of 119891(119875119894119882119894) is given as follows
119867119891=
[[[[
[
1205972119891 (119875119894119882119894)
1205971198752
119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
1205971198822
119894
]]]]
]
(A4)
To prove that 119867119891is positive semidefinite we obtain the
following equation
1205972119891 (119875119894119882119894)
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
(120597119862119894
120597119875119894
)
2
minus 119899(119879119894minus 119862119894)119899minus1 1205972119862119894
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
= 1198992(119899 minus 1) (119879119894 minus 119862
119894)2119899minus3
times1198622
119894
119882119894ln 2(119873
01198821198941205722
119894+ 119875119894)2
(A8)
When 119879119894
ge 119862119894 it is obvious that (A5) and (A8)
are nonnegative Therefore 119867119891is positive semidefinite and
sum (119879119894minus 119862119894)2 is convex Taken together with the conclusion
that 119862119894is concave we prove that the optimization problem
is convex
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
The authors would like to thank the National High-TechResearch amp Development Program of China (863 Program)under Grant 2012AA01A508 for the project support
References
[1] J P Choi and V W S Chan ldquoOptimum power and beamallocation based on traffic demands and channel conditionsover satellite downlinksrdquo IEEE Transaction onWireless Commu-nication vol 4 no 6 pp 2983ndash2993 2005
[2] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio and Wireless Symposium pp823ndash826 January 2008
Mathematical Problems in Engineering 9
[3] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the 13thInternational Conference on Communication Technology (ICCTrsquo11) pp 873ndash876 September 2011
[4] U Park H Wook Kim D Sub Oh and B J Ku ldquoOptimumselective beam allocation scheme for satellite network withmulti-spot beamsrdquo in Proceedings of the 4th InternationalConference on Advances in Satellite and Space Communications(SPACOMM rsquo12) pp 78ndash81 April2012
[5] U Park H W Kim D Sub Oh and B J Ku ldquoA dynamicbandwidth allocation scheme for a multi-spot-beam satellitesystemrdquo ETRI Journal vol 34 no 4 pp 613ndash616 2012
[6] K Kumaran and H Viswanathan ldquoJoint power and bandwidthallocation in Downlink transmissionrdquo IEEE Transaction onWireless Communications vol 4 no 3 pp 1008ndash1016 2005
[7] X Gong S A Vorobyov and C Tellambura ldquoJoint bandwidthand power allocation with admission control in wireless multi-user networks with and without relayingrdquo IEEE Transactions onSignal Processing vol 59 no 4 pp 1801ndash1813 2011
[8] J Miao Z Hu K Yang CWang andH Tian ldquoJoint power andbandwidth allocation algorithm with Qos support in heteroge-neous wireless networksrdquo IEEE Communications Letters vol 16no 4 pp 479ndash481 2012
[9] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991
[10] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013
[11] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionon Communications vol 54 no 7 pp 1310ndash1322 2006
[12] R Wang K N Vincent Lau L Lv and B Chen ldquoJoint cross-layer scheduling and spectrum sensing for OFDMA cognitiveradio systemsrdquo IEEE Transaction on Wireless Communicationsvol 8 no 5 pp 2410ndash2416 2009
[13] A G Marques X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009
[14] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
the joint bandwidth and power allocation problem in down-link transmission was investigated In [7] an optimal jointbandwidth and power allocation algorithm in wireless mul-tiuser networks with and without relaying was proposedIn [8] a joint power and bandwidth allocation algorithmwith QoS support in heterogeneous wireless networks wasproposed using convex optimization methodology How-ever the conclusions obtained in the above works cannotbe applied to the multi-spot-beam satellite communicationsystem due to the fact that the constraints in the system weinvestigate are different from those in the systems mentionedabove
In this paper we propose solving the problem of jointbandwidth and power allocation for the multi-spot-beamsatellite communication system We initially formulate theproblem of joint bandwidth and power allocation as anonlinear optimization problem and prove the optimizationproblem is convex The object of optimization problem is tomatch the capacity allocated to each spot beam to the trafficdemand as closely as possible taking into account a compro-mise between the maximum total system capacity and theproportional fairness among spot beams A joint bandwidthand power allocation iterative algorithm based on dualitytheory is then proposed to obtain the optimal solution of thisoptimization problem Compared with the separate poweror bandwidth optimal allocation algorithm the proposedjoint bandwidth andpower allocation algorithm improves thetotal system capacity and the fairness among spot beams Inaddition we discuss the impact of traffic demand channelcondition and the delay constraint of each spot beam on theallocation results
The remainder of this paper is organized as follows InSection 2 we formulate the optimization problem of jointbandwidth and power allocation and prove the optimizationproblem is convex Section 3 proposes the optimal jointbandwidth and power allocation algorithm based on dualitytheory and Section 4 presents the simulation results andanalyzes the impact of traffic demand channel conditionsand delay constraints on the allocation result Section 5concludes the paper
2 Mathematical Formulation of JointBandwidth and Power Allocation
21 Modeling of Downlink Multi-Spot-Beam CapacityFigure 1 shows the system configuration of a multi-spot-beam satellite communication system where the trafficdemand of the 119894th spot beam is 119879
119894 the power allocated to the
119894th spot beam is 119875119894 the bandwidth allocated to the 119894th spot
beam is 119882119894 and the signal attenuation factor of the 119894th spot
beam is 1205722119894 It is noted that 1205722
119894mainly consists of the effects of
weather conditions free space loss and antenna gainUsing time sharing for Gaussian broadcast channels [9]
the Shannon bounded capacity 119862119894for the 119894th spot beam is
given as
119862119894= 119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
) (1)
Ci
Ti
Figure 1 System configuration of multi-spot-beam satellite system
where 1198730is the noise power density of each spot beam
It is noted that interbeam interference from the sidelobesof adjacent spot beams degenerates the Shannon capacityHowever in this paper we ignore interbeam interferencebecause we consider very narrow spot beams over a largenumber of spot beams [1] It is observed from (1) that thespot beam capacity119862
119894is increased as the bandwidth or power
allocated to the spot beam increases However the totalbandwidth and power of the satellite are fixed so the capacityof the system is limited
If the total system power and bandwidth resources aresufficient to support the total traffic demand generated byall the spot beams it seems to be meaningless for us tomake efforts to improve the resource utilization efficiencyTherefore we only focus on the scenario where the total trafficdemand exceeds the total available system capacity [1]
22 Delay Constraint In this section we describe the impactof the delay constraint In the practical multi-spot-beamsatellite system there is much real-time traffic such as videoand audio flows for which the delay performance is animportant evaluation criterion According to the analysis in[1] the delay constraint of each spot beam can be transformedinto the traffic demand of each spot beam which is given asfollows
119862119894ge
119879119894
(1 minus 119890119894)119863119894
(2)
where 119890119894is the packet error rate over the link of the 119894th spot
beam 119863119894is the average delay deadline of the 119894th spot beam
and (1 minus 119890119894)119863119894gt 1 The constraint (2) implies that each spot
beam has a minimal traffic demand which is determined by119890119894and 119863
119894 It is seen that both the higher priority traffic with
a smaller119863119894and the worse channel condition with a larger 119890
119894
will lead to a larger fraction of the capacity
23 Optimization Problem Formulation There are manymetrics to evaluate the system performance and different
Mathematical Problems in Engineering 3
metrics may lead to different allocation results Therefore itis very important for us to choose an appropriate metric Inthis paper the metric is to minimize the deficit between thetraffic demand and the capacity allocated taking into accounta compromise between the maximum total system capacityand the proportional fairness among spot beams MotivatedbyChoi andChan [1] the problem can be formulated by usingthe 119899th order objective function as follows
min119875119894119882119894
119873
sum
119894=1
(119879119894minus 119862119894)119899 (3)
st 119862119894= 119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
) le 119879119894
forall119894 (4)
119873
sum
119894=1
119875119894le 119875total (5)
119873
sum
119894=1
119882119894le 119882total (6)
119862119894= 119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
) ge119879119894
(1 minus 119890119894)119863119894
forall119894
(7)
where 119899 is integer and 119899 ge 2 When 119899 = 1 the problem isequivalent to maximizing the total system capacity and thefairness among spot beams is ignored So we ignore the casewith 119899 = 1 The constraint (4) indicates that the allocatedresource should not exceed the traffic demand of each spotbeam Conditions (5) (6) and (7) imply the constraint for thetotal power the total bandwidth and the delay respectively
The above optimization is a nonlinear programmingproblem It is found that the optimal variables 119882
119894and 119875
119894
are coupled in the problem above which makes the globaloptimal solution difficult to be obtained To make the aboveproblem tractable in the following section we propose ajoint power and bandwidth allocation algorithm based onduality optimization theory [10ndash13] It is known that if theoptimization problem is convex the duality gap is zero Asa result the optimal solution obtained by the duality problemis the global optimal solution of the primary problem [14]Fortunately the optimization problem mentioned above isconvex the proof of which is shown in the appendix
As mentioned above the joint bandwidth and power alloca-tion algorithm is based on the duality theory By introducingnonnegative dual variables 120583 120582 and 120588 = [120588
1 1205882 120588
119899] the
Lagrange function is yielded given as
119871 (PW120588 120582 120583) =
119873
sum
119894=1
[119879119894minus119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
)]
119899
minus 120583(119882total minus119873
sum
119894=1
119882119894)
minus
119873
sum
119894=1
120588119894[119862119894minus
119879119894
(1 minus 119890119894)119863119894
]
minus 120582(119875total minus119873
sum
119894=1
119875119894)
(8)
where P = [1198751 1198752 119875
119899] andW = [119882
11198822 119882
119899]
From (8) Lagrange dual function can be obtained by
119863(120588 120582 120583) = minPW
119871 (PW120588 120582 120583) (9)
and the dual problem can be written as
119889lowast= max120582ge0 120583ge0120588ge0
119863(120588 120582 120583) (10)
According to [10] the dual problem in (10) can be furtherdecomposed into the following three sequentially iterativesubproblemsSubproblem 1 Power Allocation Given the dual variables 120583 120582and 120588 for any 119894 = [1 119873] maximizing (8) with respect to119875119894brings the equation
1198991205722
119894119882119894
ln 2 (1198821198941198730+ 1205722
119894119875lowast
119894)
times [119879119894minus119882119894log2(1 +
1205722
119894119875lowast
119894
1198821198941198730
)]
119899minus1
+120588119894
119899 = 120582
(11)
The119875lowast119894can be obtained from (11) by numerical calculation
methods for example the golden section method Theoptimized power allocation of the 119894th spot beam 119875
opt119894
=
max 0 119875lowast
119894
Subproblem 2 Bandwidth Allocation Substituting the 119875opt119894
into (8) maximizing (8) with respect to 119882119894brings the
following equation
[(119879119894minus 119862lowast
119894)119899minus1
+120588119894
119899]
times 119862lowast
119894
119882lowast
119894
minus1
ln 2 [1198730119882lowast
119894 (1205722
119894119875opt119894
) + 1]
=120583
119899
(12)
where 119862lowast119894= 119882lowast
119894log2(1 + (120572
2
119894119875opt119894
119882lowast
1198941198730))
From (12) we can obtain the119882lowast119894by using golden section
method The optimized bandwidth allocated to the 119894th spotbeam119882
opt119894
= max0119882lowast119894
Subproblem 3 Dual Variables Update The optimal dualvariables can be obtained by solving its dual problem
(120588opt 120582
opt 120583
opt) = arg min
120582 120583120588
max 119871 (PoptWopt
120588 120582 120583)
(13)
4 Mathematical Problems in Engineering
Because the dual function is always convex here wecan use a subgradient (a generalization of gradient) updatemethod as follows [11]
120582119899+1
= [120582119899minus Δ119899
120582(119875total minus
119873
sum
119894=1
119875opt119894
)]
+
120583119899+1
= [120583119899minus Δ119899
120583(119882total minus
119873
sum
119894=1
119882opt119894
)]
+
120588119899+1
119894= [120588119899
119894minus Δ119899
120588(119862
opt119894
minus119879119894
(1 minus 119890119894)119863119894
)]
+
(14)
where [119909]+ = max0 119909 119899 is the iteration number and Δ isthe iteration step size
The above dual variable updating algorithm is guaranteedto converge to the optimal solution as long as the iterationstep chosen is sufficiently small [11]
The whole process of the proposed joint bandwidth andpower allocation algorithm can be summarized as followsThe Proposed Joint Bandwidth and Power Allocation Algo-rithm
Step 1 Set appropriate initial values for the dual variables andthe bandwidth of each spot beam
Step 2 Substitute the values of the bandwidth of each spotbeam and the dual variables into (11) and then calculate theoptimized power allocated to each spot beam
Step 3 Substitute into (12) both the power values for eachspot beam obtained from Step 2 and the Lagrangianmultipli-ers calculate the optimized bandwidth allocated to each userand then calculate the optimized bandwidth allocated to eachspot beam
Step 4 Substitute the values of the power and bandwidth ofeach spot beam which are separately obtained from Steps 2and 3 into (14) and then update the dual variables
Step 5 If the conditions of |120582119899+1
(119875total minus sum119894119875119894)| lt 120576
|120583119899+1
(119882total minus sum119894119882
opt119894
)| lt 120576 and |120588119899+1
119894(119862
opt119894
minus 119879119894[(1 minus
119890119894)119863119894)]| lt 120576 forall119894 are satisfied simultaneously then terminate
the algorithm Otherwise jump to Step 2
According to the above process it is seen that thecomputational complexities of Steps 2 3 and 4 are 119874(119878119873)119874(119878119873) and 119874(2 + 119873) where 119873 is the number of thespot beams and 119878 is the computational complexity of thegolden section method Therefore the total computationalcomplexity of the proposed algorithm is 119874(2119870119878119873 + 2119870 +
119870119873) where 119870 is the number of the iterations It is notedthat both 119870 and 119878 are independent of 119873 As a result thecomputational complexity of the proposed algorithm is linearwith the number of the spot beams which indicates that thealgorithm can be implemented in practice
4 Simulation Results and Analysis
For the simulation a Ka bandmulti-spot-beam satellite com-munication system model is set up The system has 10 spotbeams the total power of the satellite is 200W and the totalbandwidth of the satellite is 500MHz The traffic demand ofeach spot beam increases from 80Mbps to 260Mbps by stepsof 20Mbps The minimal traffic demand of each spot beamcaused by the delay constraint is 20 of traffic demand
41 Convergence Behavior of Proposed Joint Allocation Algo-rithm To show the convergence behavior of the proposedjoint allocation algorithm we first assume that the channelcondition of each spot beam is the same and the normalizednoise power spectral density parameter 119873
01205722
119894is 02119890
minus6Moreover the order of the objective function is 2
Figures 2 and 3 show the convergence of the dual variablesand optimal variables As seen in Figure 2 each dual variablehas its own update step size and the dual variables convergeto their optimal value after finite iterations which dependon the choice of the step sizes When the dual variablesconverge it is noted that from Figure 3 the optimal variablesalso converge
42 Efficiency of the Proposed Joint Allocation AlgorithmTo verify the resource efficiency of the proposed optimaljoint allocation algorithm (OBOP) we compare it with thefollowing three algorithms
(a) Uniform bandwidth allocation and uniform powerallocation (UBUP) UBUP algorithm divides thebandwidth and power resource equally between eachspot beam
(b) Uniform bandwidth allocation and optimal powerallocation (UBOP) [1] UBOP algorithm divides thebandwidth resource equally between each spot beamand optimizes the power allocation for each spotbeam
(c) Uniform power allocation and optimal bandwidthallocation (OBUP) [5] OBUP algorithm divides thepower resource equally between each spot beam andoptimizes the bandwidth allocation for each spotbeam
When the channel conditions are the same Figure 4shows the capacity distributions of spot beams which areallocated by the four algorithms Table 1 shows the totalsystem capacity of the four algorithms As shown in Figure 4to get the fairness among spot beams the separate optimalallocation algorithms (UBOP and OBUP) will provide morecapacity to higher demand spot beams It is known thatthe capacity does not linearly increase with the power orbandwidth increasing due to the concavity of the capacityfunction with a fixed bandwidth or power allocation Thisresults in a lower total system capacity However in theproposed joint allocation algorithm (OBOP) the capacityfunction is almost linear since the bandwidth and power arejointly dynamically allocated As a result the total systemcapacity is much improved compared to the separate optimal
Mathematical Problems in Engineering 5
0 50 100 1500
100
200
300
400
500
600
700
800
900
1000
Iterations
120588(1) (step size is 04)120582 (step size is 15)120583 (step size is 07)
Figure 2 The convergence behavior of the dual variables
0 50 100 1500
500
1000
1500
Iterations
The sum of the power allocated to each beamThe sum of the bandwidth allocated to each beam
Figure 3 The convergence behavior of the optimal variables
Table 1 The total system capacity of the four algorithms when thechannel conditions are the same
allocation algorithm This conclusion can be also concludedfrom Table 1
When the channel conditions are the same Figure 5shows the deficit between the traffic demand and the capacity
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
Figure 4 Comparison of the four algorithms in terms of thecapacity allocated to each spot beam when channel conditions arethe same
Table 2The total sum of (119879119894minus 119862119894)2 of the four algorithms when the
channel conditions are the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1153611986415
UBOP [1] 1017011986415
OBUP [5] 1067611986415
The proposed OBOP 8336611986415
allocated to each spot beam Table 2 presents the sum of thedeficit between the traffic demand and the capacity allocatedto each spot beam From Figure 5 it is shown that in thejoint allocation algorithm (OBOP) from the spot beam 3to spot beam 10 the deficit between traffic demand andcapacity allocated is almost the same so the fairness amongspot beams is better than the separate optimal allocations(UBOP and OBUP) This conclusion can be also observedfromTable 2 since the deficit between the traffic demand andthe capacity allocated is smaller than the separate optimalallocations Together with the conclusion above regardingtotal system capacity we can conclude that the performanceof the optimal joint allocation algorithm (OBOP) is muchimproved compared with the separate optimal algorithms(UBOP and OBUP)
43 Impact of the Channel Condition and Delay ConstraintsWecompare the allocation result of theOBOP allocationwithdelay constraint with that of the OBOP allocation withoutthe delay constraint (OBOPND) to analyze the impact of thedelay constraints When there is no delay constraint on eachspot beam the allocation result can be obtained by the samealgorithm where we just need to remove the dual variable 120588in (8)
6 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]
OBUP [5]The proposed OBOP
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 5 Comparison of the four algorithms in terms of the deficitbetween the traffic demand and the capacity allocated to each spotbeam when channel conditions are the same
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
OBOPNDUBUPUBOP [1]OBUP [5]
The proposed OBOP
The minimal traffic demand
ith beam
Figure 6 The capacity allocated by five algorithms to each spotbeam when channel conditions are not the same
From spot beam 2 to spot beam 4 we set the normalizednoise power spectral density parameters 119873
01205722
119894to 03119890
minus604119890minus6 and 05119890
minus6 respectively to analyze the impact ofchannel condition on the allocation results Each trafficdemand of the three spot beams is 120Mbsp so the totaltraffic demand is the same as the above simulation scenarioThe order of the objective function also is 2
When the channel conditions are not the same Figure 6shows the capacity distributions of spot beams which areallocated by the 5 algorithms Table 3 shows the total system
Table 3 The total system capacity of the five algorithms when thechannel conditions are not the same
Table 4 The total sum of (119879119894minus 119862119894)2 of the 5 algorithms when the
channel conditions are not the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1239711986415
UBOP [1] 1122511986415
OBUP [5] 1140811986415
The proposed OBOP 9091911986415
OBOPND 8573011986415
capacity of the 5 algorithms From Figure 6 it is shownthat both the separate optimal allocation algorithm and theproposed joint allocation algorithm allocate more capacityto the spot beam with better channel condition than thatwith worse channel condition especially for the OBOPNDalgorithm no capacity is allocated to spot beam 3 and spotbeam 4 Therefore the metric we choose in this paper notonly considers the fairness among spot beams but also triesto maximize the throughput of the system which achieves agood system performance as we expected Due to the delayconstraint we have to allocate aminimal traffic to spot beamswith worse channel condition thus the total system capacitydecreases This conclusion is also shown in Table 3 So weconclude that when the channel condition of each spot beamis not the same the delay constraint of each spot beamdecreases the total system capacity
When the channel conditions are not the same Figure 7shows the deficit between the traffic demand and the capacityallocated to each spot beam Table 4 presents the sum ofthe deficits between the traffic demand and the capacityallocated to each spot beam It is shown that the worsechannel conditions and the delay constraint decrease thefairness among spot beams
44 Impact of the Order of the Objective Function and DelayConstraints As mentioned in Section 23 different metricsmay lead to different allocation results Here we analyze theimpact of the order of the objective function The channelcondition of each spot beam is set to be the same
Figure 8 shows the allocated capacity of each spot beamand Table 5 shows the total system capacity when the order ofthe objective function is different
When there is no delay constraint in each spot beamas we use a higher-order deviation objective function moreresources are provided for higher traffic demand spot beamswhile a lower-order cost function gives relatively moreresource to lower traffic demand spot beams For example
Mathematical Problems in Engineering 7
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 7 The deficit between the traffic demand and the capacityallocated by 5 algorithms to each spot beam when channel condi-tions are not the same
the cubic objective function provides no capacity for the threelowest traffic demand spot beams Although the capacityallocated to each spot beam is different the total systemcapacities of different objective function are almost the same
When there is delay constraint in each spot beam aminimal traffic demand must be allocated to them As aresult for the same cubic objective function the three lowesttraffic demand spot beams are allocated to theminimal trafficdemand It is also seen that the capacity allocated to each spotbeam is the same by using second- and third-order of theobjective function
5 Conclusion
In the multi-spot-beam satellite system due to the scarcenessof the satellite resource it is crucial for us to improve theresource utilization efficiency To this end in this paperwe first formulated the joint bandwidth and power allo-cation problem as a convex optimization problem Thenwe proposed a joint bandwidth and power allocation algo-rithm based on duality theory to get the optimal solutionCompared with the individual optimal power or bandwidthallocation the proposed joint optimal bandwidth and powerallocation algorithm improved the total system capacity andthe fairness among spot beams while the computationalcomplexity of the algorithm was linear with the numberof spot beams Therefore the proposed algorithm can beimplemented in practice
It was shown from the simulation results that the spotbeam with higher traffic demand and better channel con-dition will be allocated more resources to minimize theobjective function Due to the delay constraint in each spotbeam there was a minimal traffic demand in each spot beam
Table 5 The total system capacity of different objective functions
Figure 8 The capacity allocated to each spot beam of differentobjective functions
When the channel condition of each spot beamwas the samethe delay constraint only had an impact on the fairness amongspot beams However when the channel conditions were notthe same both the total system capacity and fairness amongspot beams were influenced by the delay constraint
Appendix
We first prove that constraint (4) can be ignored by contra-diction Assuming that 119862
119894gt 119879119894 it is seen that the constraint
(7) is satisfied obviously As a result the correspondingLagrange multiplier 120588
119894is zero According to (11) the value
of 120582 is negative which contradicts that 120582 is nonnegativeTherefore the constraint (4) is satisfied when the value of 120582 isnonnegative
According to [14] for the optimization problem whichhas the following general form
min 119891 (X)
st 119892119894 (X) le 0
(A1)
when the functions 119892119894(X) and 119891(X) are convex the optimiza-
tion problem is convex
8 Mathematical Problems in Engineering
Taken together with the fact that the functions sum119875119894
and sum119882119894in constraints (5) and (6) are linear to prove
the optimization problem with respect to [11988211198822 119882
119899
1198751 1198752 119875
119899] is convex we only need to prove that the
objective function sum (119879119894minus 119862119894)2 is convex and the function
119862119894= 119882119894log2[1 + 120572
2
119894119875119894(1198821198941198730)] in constraint (7) is concave
Firstly we prove that function 119862119894with respect to119882
119894and 119875
119894is
concaveThe Hessian of the function 119862
119894= 119882119894log2[1 + 120572
2
119894119875119894
(1198821198941198730)] is given as follows
119867119862=
[[[[
[
1205972119862119894
1205971198752
119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
1205971198822
119894
]]]]
]
(A2)
where 1205972119862119894120597119875119894
2= minus120572
2
119894119882119894 ln 2(119873
0119882119894+ 1205722
119894119875119894)2
le 0 and|119867119862| = 0It is seen that the determinant of the119867
119862is zero therefore
119867119862is negative semidefinite and the function 119862
119894is concave
[14] Then we prove that sum (119879119894minus 119862119894)2 is convex
It is known that the sum of convex functions is alsoconvex [14] Therefore to prove that sum (119879
119894minus 119862119894)2 is convex
we just need to prove the following function is convex
119891 (119875119894119882119894) = (119879
119894minus 119862119894)119899 (A3)
where 119862119894= 119882119894log2(1 + 120572
2
1198941198751198941198821198941198730)
The Hessian of 119891(119875119894119882119894) is given as follows
119867119891=
[[[[
[
1205972119891 (119875119894119882119894)
1205971198752
119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
1205971198822
119894
]]]]
]
(A4)
To prove that 119867119891is positive semidefinite we obtain the
following equation
1205972119891 (119875119894119882119894)
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
(120597119862119894
120597119875119894
)
2
minus 119899(119879119894minus 119862119894)119899minus1 1205972119862119894
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
= 1198992(119899 minus 1) (119879119894 minus 119862
119894)2119899minus3
times1198622
119894
119882119894ln 2(119873
01198821198941205722
119894+ 119875119894)2
(A8)
When 119879119894
ge 119862119894 it is obvious that (A5) and (A8)
are nonnegative Therefore 119867119891is positive semidefinite and
sum (119879119894minus 119862119894)2 is convex Taken together with the conclusion
that 119862119894is concave we prove that the optimization problem
is convex
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
The authors would like to thank the National High-TechResearch amp Development Program of China (863 Program)under Grant 2012AA01A508 for the project support
References
[1] J P Choi and V W S Chan ldquoOptimum power and beamallocation based on traffic demands and channel conditionsover satellite downlinksrdquo IEEE Transaction onWireless Commu-nication vol 4 no 6 pp 2983ndash2993 2005
[2] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio and Wireless Symposium pp823ndash826 January 2008
Mathematical Problems in Engineering 9
[3] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the 13thInternational Conference on Communication Technology (ICCTrsquo11) pp 873ndash876 September 2011
[4] U Park H Wook Kim D Sub Oh and B J Ku ldquoOptimumselective beam allocation scheme for satellite network withmulti-spot beamsrdquo in Proceedings of the 4th InternationalConference on Advances in Satellite and Space Communications(SPACOMM rsquo12) pp 78ndash81 April2012
[5] U Park H W Kim D Sub Oh and B J Ku ldquoA dynamicbandwidth allocation scheme for a multi-spot-beam satellitesystemrdquo ETRI Journal vol 34 no 4 pp 613ndash616 2012
[6] K Kumaran and H Viswanathan ldquoJoint power and bandwidthallocation in Downlink transmissionrdquo IEEE Transaction onWireless Communications vol 4 no 3 pp 1008ndash1016 2005
[7] X Gong S A Vorobyov and C Tellambura ldquoJoint bandwidthand power allocation with admission control in wireless multi-user networks with and without relayingrdquo IEEE Transactions onSignal Processing vol 59 no 4 pp 1801ndash1813 2011
[8] J Miao Z Hu K Yang CWang andH Tian ldquoJoint power andbandwidth allocation algorithm with Qos support in heteroge-neous wireless networksrdquo IEEE Communications Letters vol 16no 4 pp 479ndash481 2012
[9] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991
[10] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013
[11] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionon Communications vol 54 no 7 pp 1310ndash1322 2006
[12] R Wang K N Vincent Lau L Lv and B Chen ldquoJoint cross-layer scheduling and spectrum sensing for OFDMA cognitiveradio systemsrdquo IEEE Transaction on Wireless Communicationsvol 8 no 5 pp 2410ndash2416 2009
[13] A G Marques X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009
[14] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
metrics may lead to different allocation results Therefore itis very important for us to choose an appropriate metric Inthis paper the metric is to minimize the deficit between thetraffic demand and the capacity allocated taking into accounta compromise between the maximum total system capacityand the proportional fairness among spot beams MotivatedbyChoi andChan [1] the problem can be formulated by usingthe 119899th order objective function as follows
min119875119894119882119894
119873
sum
119894=1
(119879119894minus 119862119894)119899 (3)
st 119862119894= 119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
) le 119879119894
forall119894 (4)
119873
sum
119894=1
119875119894le 119875total (5)
119873
sum
119894=1
119882119894le 119882total (6)
119862119894= 119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
) ge119879119894
(1 minus 119890119894)119863119894
forall119894
(7)
where 119899 is integer and 119899 ge 2 When 119899 = 1 the problem isequivalent to maximizing the total system capacity and thefairness among spot beams is ignored So we ignore the casewith 119899 = 1 The constraint (4) indicates that the allocatedresource should not exceed the traffic demand of each spotbeam Conditions (5) (6) and (7) imply the constraint for thetotal power the total bandwidth and the delay respectively
The above optimization is a nonlinear programmingproblem It is found that the optimal variables 119882
119894and 119875
119894
are coupled in the problem above which makes the globaloptimal solution difficult to be obtained To make the aboveproblem tractable in the following section we propose ajoint power and bandwidth allocation algorithm based onduality optimization theory [10ndash13] It is known that if theoptimization problem is convex the duality gap is zero Asa result the optimal solution obtained by the duality problemis the global optimal solution of the primary problem [14]Fortunately the optimization problem mentioned above isconvex the proof of which is shown in the appendix
As mentioned above the joint bandwidth and power alloca-tion algorithm is based on the duality theory By introducingnonnegative dual variables 120583 120582 and 120588 = [120588
1 1205882 120588
119899] the
Lagrange function is yielded given as
119871 (PW120588 120582 120583) =
119873
sum
119894=1
[119879119894minus119882119894log2(1 +
1205722
119894119875119894
1198821198941198730
)]
119899
minus 120583(119882total minus119873
sum
119894=1
119882119894)
minus
119873
sum
119894=1
120588119894[119862119894minus
119879119894
(1 minus 119890119894)119863119894
]
minus 120582(119875total minus119873
sum
119894=1
119875119894)
(8)
where P = [1198751 1198752 119875
119899] andW = [119882
11198822 119882
119899]
From (8) Lagrange dual function can be obtained by
119863(120588 120582 120583) = minPW
119871 (PW120588 120582 120583) (9)
and the dual problem can be written as
119889lowast= max120582ge0 120583ge0120588ge0
119863(120588 120582 120583) (10)
According to [10] the dual problem in (10) can be furtherdecomposed into the following three sequentially iterativesubproblemsSubproblem 1 Power Allocation Given the dual variables 120583 120582and 120588 for any 119894 = [1 119873] maximizing (8) with respect to119875119894brings the equation
1198991205722
119894119882119894
ln 2 (1198821198941198730+ 1205722
119894119875lowast
119894)
times [119879119894minus119882119894log2(1 +
1205722
119894119875lowast
119894
1198821198941198730
)]
119899minus1
+120588119894
119899 = 120582
(11)
The119875lowast119894can be obtained from (11) by numerical calculation
methods for example the golden section method Theoptimized power allocation of the 119894th spot beam 119875
opt119894
=
max 0 119875lowast
119894
Subproblem 2 Bandwidth Allocation Substituting the 119875opt119894
into (8) maximizing (8) with respect to 119882119894brings the
following equation
[(119879119894minus 119862lowast
119894)119899minus1
+120588119894
119899]
times 119862lowast
119894
119882lowast
119894
minus1
ln 2 [1198730119882lowast
119894 (1205722
119894119875opt119894
) + 1]
=120583
119899
(12)
where 119862lowast119894= 119882lowast
119894log2(1 + (120572
2
119894119875opt119894
119882lowast
1198941198730))
From (12) we can obtain the119882lowast119894by using golden section
method The optimized bandwidth allocated to the 119894th spotbeam119882
opt119894
= max0119882lowast119894
Subproblem 3 Dual Variables Update The optimal dualvariables can be obtained by solving its dual problem
(120588opt 120582
opt 120583
opt) = arg min
120582 120583120588
max 119871 (PoptWopt
120588 120582 120583)
(13)
4 Mathematical Problems in Engineering
Because the dual function is always convex here wecan use a subgradient (a generalization of gradient) updatemethod as follows [11]
120582119899+1
= [120582119899minus Δ119899
120582(119875total minus
119873
sum
119894=1
119875opt119894
)]
+
120583119899+1
= [120583119899minus Δ119899
120583(119882total minus
119873
sum
119894=1
119882opt119894
)]
+
120588119899+1
119894= [120588119899
119894minus Δ119899
120588(119862
opt119894
minus119879119894
(1 minus 119890119894)119863119894
)]
+
(14)
where [119909]+ = max0 119909 119899 is the iteration number and Δ isthe iteration step size
The above dual variable updating algorithm is guaranteedto converge to the optimal solution as long as the iterationstep chosen is sufficiently small [11]
The whole process of the proposed joint bandwidth andpower allocation algorithm can be summarized as followsThe Proposed Joint Bandwidth and Power Allocation Algo-rithm
Step 1 Set appropriate initial values for the dual variables andthe bandwidth of each spot beam
Step 2 Substitute the values of the bandwidth of each spotbeam and the dual variables into (11) and then calculate theoptimized power allocated to each spot beam
Step 3 Substitute into (12) both the power values for eachspot beam obtained from Step 2 and the Lagrangianmultipli-ers calculate the optimized bandwidth allocated to each userand then calculate the optimized bandwidth allocated to eachspot beam
Step 4 Substitute the values of the power and bandwidth ofeach spot beam which are separately obtained from Steps 2and 3 into (14) and then update the dual variables
Step 5 If the conditions of |120582119899+1
(119875total minus sum119894119875119894)| lt 120576
|120583119899+1
(119882total minus sum119894119882
opt119894
)| lt 120576 and |120588119899+1
119894(119862
opt119894
minus 119879119894[(1 minus
119890119894)119863119894)]| lt 120576 forall119894 are satisfied simultaneously then terminate
the algorithm Otherwise jump to Step 2
According to the above process it is seen that thecomputational complexities of Steps 2 3 and 4 are 119874(119878119873)119874(119878119873) and 119874(2 + 119873) where 119873 is the number of thespot beams and 119878 is the computational complexity of thegolden section method Therefore the total computationalcomplexity of the proposed algorithm is 119874(2119870119878119873 + 2119870 +
119870119873) where 119870 is the number of the iterations It is notedthat both 119870 and 119878 are independent of 119873 As a result thecomputational complexity of the proposed algorithm is linearwith the number of the spot beams which indicates that thealgorithm can be implemented in practice
4 Simulation Results and Analysis
For the simulation a Ka bandmulti-spot-beam satellite com-munication system model is set up The system has 10 spotbeams the total power of the satellite is 200W and the totalbandwidth of the satellite is 500MHz The traffic demand ofeach spot beam increases from 80Mbps to 260Mbps by stepsof 20Mbps The minimal traffic demand of each spot beamcaused by the delay constraint is 20 of traffic demand
41 Convergence Behavior of Proposed Joint Allocation Algo-rithm To show the convergence behavior of the proposedjoint allocation algorithm we first assume that the channelcondition of each spot beam is the same and the normalizednoise power spectral density parameter 119873
01205722
119894is 02119890
minus6Moreover the order of the objective function is 2
Figures 2 and 3 show the convergence of the dual variablesand optimal variables As seen in Figure 2 each dual variablehas its own update step size and the dual variables convergeto their optimal value after finite iterations which dependon the choice of the step sizes When the dual variablesconverge it is noted that from Figure 3 the optimal variablesalso converge
42 Efficiency of the Proposed Joint Allocation AlgorithmTo verify the resource efficiency of the proposed optimaljoint allocation algorithm (OBOP) we compare it with thefollowing three algorithms
(a) Uniform bandwidth allocation and uniform powerallocation (UBUP) UBUP algorithm divides thebandwidth and power resource equally between eachspot beam
(b) Uniform bandwidth allocation and optimal powerallocation (UBOP) [1] UBOP algorithm divides thebandwidth resource equally between each spot beamand optimizes the power allocation for each spotbeam
(c) Uniform power allocation and optimal bandwidthallocation (OBUP) [5] OBUP algorithm divides thepower resource equally between each spot beam andoptimizes the bandwidth allocation for each spotbeam
When the channel conditions are the same Figure 4shows the capacity distributions of spot beams which areallocated by the four algorithms Table 1 shows the totalsystem capacity of the four algorithms As shown in Figure 4to get the fairness among spot beams the separate optimalallocation algorithms (UBOP and OBUP) will provide morecapacity to higher demand spot beams It is known thatthe capacity does not linearly increase with the power orbandwidth increasing due to the concavity of the capacityfunction with a fixed bandwidth or power allocation Thisresults in a lower total system capacity However in theproposed joint allocation algorithm (OBOP) the capacityfunction is almost linear since the bandwidth and power arejointly dynamically allocated As a result the total systemcapacity is much improved compared to the separate optimal
Mathematical Problems in Engineering 5
0 50 100 1500
100
200
300
400
500
600
700
800
900
1000
Iterations
120588(1) (step size is 04)120582 (step size is 15)120583 (step size is 07)
Figure 2 The convergence behavior of the dual variables
0 50 100 1500
500
1000
1500
Iterations
The sum of the power allocated to each beamThe sum of the bandwidth allocated to each beam
Figure 3 The convergence behavior of the optimal variables
Table 1 The total system capacity of the four algorithms when thechannel conditions are the same
allocation algorithm This conclusion can be also concludedfrom Table 1
When the channel conditions are the same Figure 5shows the deficit between the traffic demand and the capacity
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
Figure 4 Comparison of the four algorithms in terms of thecapacity allocated to each spot beam when channel conditions arethe same
Table 2The total sum of (119879119894minus 119862119894)2 of the four algorithms when the
channel conditions are the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1153611986415
UBOP [1] 1017011986415
OBUP [5] 1067611986415
The proposed OBOP 8336611986415
allocated to each spot beam Table 2 presents the sum of thedeficit between the traffic demand and the capacity allocatedto each spot beam From Figure 5 it is shown that in thejoint allocation algorithm (OBOP) from the spot beam 3to spot beam 10 the deficit between traffic demand andcapacity allocated is almost the same so the fairness amongspot beams is better than the separate optimal allocations(UBOP and OBUP) This conclusion can be also observedfromTable 2 since the deficit between the traffic demand andthe capacity allocated is smaller than the separate optimalallocations Together with the conclusion above regardingtotal system capacity we can conclude that the performanceof the optimal joint allocation algorithm (OBOP) is muchimproved compared with the separate optimal algorithms(UBOP and OBUP)
43 Impact of the Channel Condition and Delay ConstraintsWecompare the allocation result of theOBOP allocationwithdelay constraint with that of the OBOP allocation withoutthe delay constraint (OBOPND) to analyze the impact of thedelay constraints When there is no delay constraint on eachspot beam the allocation result can be obtained by the samealgorithm where we just need to remove the dual variable 120588in (8)
6 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]
OBUP [5]The proposed OBOP
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 5 Comparison of the four algorithms in terms of the deficitbetween the traffic demand and the capacity allocated to each spotbeam when channel conditions are the same
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
OBOPNDUBUPUBOP [1]OBUP [5]
The proposed OBOP
The minimal traffic demand
ith beam
Figure 6 The capacity allocated by five algorithms to each spotbeam when channel conditions are not the same
From spot beam 2 to spot beam 4 we set the normalizednoise power spectral density parameters 119873
01205722
119894to 03119890
minus604119890minus6 and 05119890
minus6 respectively to analyze the impact ofchannel condition on the allocation results Each trafficdemand of the three spot beams is 120Mbsp so the totaltraffic demand is the same as the above simulation scenarioThe order of the objective function also is 2
When the channel conditions are not the same Figure 6shows the capacity distributions of spot beams which areallocated by the 5 algorithms Table 3 shows the total system
Table 3 The total system capacity of the five algorithms when thechannel conditions are not the same
Table 4 The total sum of (119879119894minus 119862119894)2 of the 5 algorithms when the
channel conditions are not the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1239711986415
UBOP [1] 1122511986415
OBUP [5] 1140811986415
The proposed OBOP 9091911986415
OBOPND 8573011986415
capacity of the 5 algorithms From Figure 6 it is shownthat both the separate optimal allocation algorithm and theproposed joint allocation algorithm allocate more capacityto the spot beam with better channel condition than thatwith worse channel condition especially for the OBOPNDalgorithm no capacity is allocated to spot beam 3 and spotbeam 4 Therefore the metric we choose in this paper notonly considers the fairness among spot beams but also triesto maximize the throughput of the system which achieves agood system performance as we expected Due to the delayconstraint we have to allocate aminimal traffic to spot beamswith worse channel condition thus the total system capacitydecreases This conclusion is also shown in Table 3 So weconclude that when the channel condition of each spot beamis not the same the delay constraint of each spot beamdecreases the total system capacity
When the channel conditions are not the same Figure 7shows the deficit between the traffic demand and the capacityallocated to each spot beam Table 4 presents the sum ofthe deficits between the traffic demand and the capacityallocated to each spot beam It is shown that the worsechannel conditions and the delay constraint decrease thefairness among spot beams
44 Impact of the Order of the Objective Function and DelayConstraints As mentioned in Section 23 different metricsmay lead to different allocation results Here we analyze theimpact of the order of the objective function The channelcondition of each spot beam is set to be the same
Figure 8 shows the allocated capacity of each spot beamand Table 5 shows the total system capacity when the order ofthe objective function is different
When there is no delay constraint in each spot beamas we use a higher-order deviation objective function moreresources are provided for higher traffic demand spot beamswhile a lower-order cost function gives relatively moreresource to lower traffic demand spot beams For example
Mathematical Problems in Engineering 7
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 7 The deficit between the traffic demand and the capacityallocated by 5 algorithms to each spot beam when channel condi-tions are not the same
the cubic objective function provides no capacity for the threelowest traffic demand spot beams Although the capacityallocated to each spot beam is different the total systemcapacities of different objective function are almost the same
When there is delay constraint in each spot beam aminimal traffic demand must be allocated to them As aresult for the same cubic objective function the three lowesttraffic demand spot beams are allocated to theminimal trafficdemand It is also seen that the capacity allocated to each spotbeam is the same by using second- and third-order of theobjective function
5 Conclusion
In the multi-spot-beam satellite system due to the scarcenessof the satellite resource it is crucial for us to improve theresource utilization efficiency To this end in this paperwe first formulated the joint bandwidth and power allo-cation problem as a convex optimization problem Thenwe proposed a joint bandwidth and power allocation algo-rithm based on duality theory to get the optimal solutionCompared with the individual optimal power or bandwidthallocation the proposed joint optimal bandwidth and powerallocation algorithm improved the total system capacity andthe fairness among spot beams while the computationalcomplexity of the algorithm was linear with the numberof spot beams Therefore the proposed algorithm can beimplemented in practice
It was shown from the simulation results that the spotbeam with higher traffic demand and better channel con-dition will be allocated more resources to minimize theobjective function Due to the delay constraint in each spotbeam there was a minimal traffic demand in each spot beam
Table 5 The total system capacity of different objective functions
Figure 8 The capacity allocated to each spot beam of differentobjective functions
When the channel condition of each spot beamwas the samethe delay constraint only had an impact on the fairness amongspot beams However when the channel conditions were notthe same both the total system capacity and fairness amongspot beams were influenced by the delay constraint
Appendix
We first prove that constraint (4) can be ignored by contra-diction Assuming that 119862
119894gt 119879119894 it is seen that the constraint
(7) is satisfied obviously As a result the correspondingLagrange multiplier 120588
119894is zero According to (11) the value
of 120582 is negative which contradicts that 120582 is nonnegativeTherefore the constraint (4) is satisfied when the value of 120582 isnonnegative
According to [14] for the optimization problem whichhas the following general form
min 119891 (X)
st 119892119894 (X) le 0
(A1)
when the functions 119892119894(X) and 119891(X) are convex the optimiza-
tion problem is convex
8 Mathematical Problems in Engineering
Taken together with the fact that the functions sum119875119894
and sum119882119894in constraints (5) and (6) are linear to prove
the optimization problem with respect to [11988211198822 119882
119899
1198751 1198752 119875
119899] is convex we only need to prove that the
objective function sum (119879119894minus 119862119894)2 is convex and the function
119862119894= 119882119894log2[1 + 120572
2
119894119875119894(1198821198941198730)] in constraint (7) is concave
Firstly we prove that function 119862119894with respect to119882
119894and 119875
119894is
concaveThe Hessian of the function 119862
119894= 119882119894log2[1 + 120572
2
119894119875119894
(1198821198941198730)] is given as follows
119867119862=
[[[[
[
1205972119862119894
1205971198752
119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
1205971198822
119894
]]]]
]
(A2)
where 1205972119862119894120597119875119894
2= minus120572
2
119894119882119894 ln 2(119873
0119882119894+ 1205722
119894119875119894)2
le 0 and|119867119862| = 0It is seen that the determinant of the119867
119862is zero therefore
119867119862is negative semidefinite and the function 119862
119894is concave
[14] Then we prove that sum (119879119894minus 119862119894)2 is convex
It is known that the sum of convex functions is alsoconvex [14] Therefore to prove that sum (119879
119894minus 119862119894)2 is convex
we just need to prove the following function is convex
119891 (119875119894119882119894) = (119879
119894minus 119862119894)119899 (A3)
where 119862119894= 119882119894log2(1 + 120572
2
1198941198751198941198821198941198730)
The Hessian of 119891(119875119894119882119894) is given as follows
119867119891=
[[[[
[
1205972119891 (119875119894119882119894)
1205971198752
119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
1205971198822
119894
]]]]
]
(A4)
To prove that 119867119891is positive semidefinite we obtain the
following equation
1205972119891 (119875119894119882119894)
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
(120597119862119894
120597119875119894
)
2
minus 119899(119879119894minus 119862119894)119899minus1 1205972119862119894
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
= 1198992(119899 minus 1) (119879119894 minus 119862
119894)2119899minus3
times1198622
119894
119882119894ln 2(119873
01198821198941205722
119894+ 119875119894)2
(A8)
When 119879119894
ge 119862119894 it is obvious that (A5) and (A8)
are nonnegative Therefore 119867119891is positive semidefinite and
sum (119879119894minus 119862119894)2 is convex Taken together with the conclusion
that 119862119894is concave we prove that the optimization problem
is convex
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
The authors would like to thank the National High-TechResearch amp Development Program of China (863 Program)under Grant 2012AA01A508 for the project support
References
[1] J P Choi and V W S Chan ldquoOptimum power and beamallocation based on traffic demands and channel conditionsover satellite downlinksrdquo IEEE Transaction onWireless Commu-nication vol 4 no 6 pp 2983ndash2993 2005
[2] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio and Wireless Symposium pp823ndash826 January 2008
Mathematical Problems in Engineering 9
[3] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the 13thInternational Conference on Communication Technology (ICCTrsquo11) pp 873ndash876 September 2011
[4] U Park H Wook Kim D Sub Oh and B J Ku ldquoOptimumselective beam allocation scheme for satellite network withmulti-spot beamsrdquo in Proceedings of the 4th InternationalConference on Advances in Satellite and Space Communications(SPACOMM rsquo12) pp 78ndash81 April2012
[5] U Park H W Kim D Sub Oh and B J Ku ldquoA dynamicbandwidth allocation scheme for a multi-spot-beam satellitesystemrdquo ETRI Journal vol 34 no 4 pp 613ndash616 2012
[6] K Kumaran and H Viswanathan ldquoJoint power and bandwidthallocation in Downlink transmissionrdquo IEEE Transaction onWireless Communications vol 4 no 3 pp 1008ndash1016 2005
[7] X Gong S A Vorobyov and C Tellambura ldquoJoint bandwidthand power allocation with admission control in wireless multi-user networks with and without relayingrdquo IEEE Transactions onSignal Processing vol 59 no 4 pp 1801ndash1813 2011
[8] J Miao Z Hu K Yang CWang andH Tian ldquoJoint power andbandwidth allocation algorithm with Qos support in heteroge-neous wireless networksrdquo IEEE Communications Letters vol 16no 4 pp 479ndash481 2012
[9] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991
[10] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013
[11] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionon Communications vol 54 no 7 pp 1310ndash1322 2006
[12] R Wang K N Vincent Lau L Lv and B Chen ldquoJoint cross-layer scheduling and spectrum sensing for OFDMA cognitiveradio systemsrdquo IEEE Transaction on Wireless Communicationsvol 8 no 5 pp 2410ndash2416 2009
[13] A G Marques X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009
[14] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
Because the dual function is always convex here wecan use a subgradient (a generalization of gradient) updatemethod as follows [11]
120582119899+1
= [120582119899minus Δ119899
120582(119875total minus
119873
sum
119894=1
119875opt119894
)]
+
120583119899+1
= [120583119899minus Δ119899
120583(119882total minus
119873
sum
119894=1
119882opt119894
)]
+
120588119899+1
119894= [120588119899
119894minus Δ119899
120588(119862
opt119894
minus119879119894
(1 minus 119890119894)119863119894
)]
+
(14)
where [119909]+ = max0 119909 119899 is the iteration number and Δ isthe iteration step size
The above dual variable updating algorithm is guaranteedto converge to the optimal solution as long as the iterationstep chosen is sufficiently small [11]
The whole process of the proposed joint bandwidth andpower allocation algorithm can be summarized as followsThe Proposed Joint Bandwidth and Power Allocation Algo-rithm
Step 1 Set appropriate initial values for the dual variables andthe bandwidth of each spot beam
Step 2 Substitute the values of the bandwidth of each spotbeam and the dual variables into (11) and then calculate theoptimized power allocated to each spot beam
Step 3 Substitute into (12) both the power values for eachspot beam obtained from Step 2 and the Lagrangianmultipli-ers calculate the optimized bandwidth allocated to each userand then calculate the optimized bandwidth allocated to eachspot beam
Step 4 Substitute the values of the power and bandwidth ofeach spot beam which are separately obtained from Steps 2and 3 into (14) and then update the dual variables
Step 5 If the conditions of |120582119899+1
(119875total minus sum119894119875119894)| lt 120576
|120583119899+1
(119882total minus sum119894119882
opt119894
)| lt 120576 and |120588119899+1
119894(119862
opt119894
minus 119879119894[(1 minus
119890119894)119863119894)]| lt 120576 forall119894 are satisfied simultaneously then terminate
the algorithm Otherwise jump to Step 2
According to the above process it is seen that thecomputational complexities of Steps 2 3 and 4 are 119874(119878119873)119874(119878119873) and 119874(2 + 119873) where 119873 is the number of thespot beams and 119878 is the computational complexity of thegolden section method Therefore the total computationalcomplexity of the proposed algorithm is 119874(2119870119878119873 + 2119870 +
119870119873) where 119870 is the number of the iterations It is notedthat both 119870 and 119878 are independent of 119873 As a result thecomputational complexity of the proposed algorithm is linearwith the number of the spot beams which indicates that thealgorithm can be implemented in practice
4 Simulation Results and Analysis
For the simulation a Ka bandmulti-spot-beam satellite com-munication system model is set up The system has 10 spotbeams the total power of the satellite is 200W and the totalbandwidth of the satellite is 500MHz The traffic demand ofeach spot beam increases from 80Mbps to 260Mbps by stepsof 20Mbps The minimal traffic demand of each spot beamcaused by the delay constraint is 20 of traffic demand
41 Convergence Behavior of Proposed Joint Allocation Algo-rithm To show the convergence behavior of the proposedjoint allocation algorithm we first assume that the channelcondition of each spot beam is the same and the normalizednoise power spectral density parameter 119873
01205722
119894is 02119890
minus6Moreover the order of the objective function is 2
Figures 2 and 3 show the convergence of the dual variablesand optimal variables As seen in Figure 2 each dual variablehas its own update step size and the dual variables convergeto their optimal value after finite iterations which dependon the choice of the step sizes When the dual variablesconverge it is noted that from Figure 3 the optimal variablesalso converge
42 Efficiency of the Proposed Joint Allocation AlgorithmTo verify the resource efficiency of the proposed optimaljoint allocation algorithm (OBOP) we compare it with thefollowing three algorithms
(a) Uniform bandwidth allocation and uniform powerallocation (UBUP) UBUP algorithm divides thebandwidth and power resource equally between eachspot beam
(b) Uniform bandwidth allocation and optimal powerallocation (UBOP) [1] UBOP algorithm divides thebandwidth resource equally between each spot beamand optimizes the power allocation for each spotbeam
(c) Uniform power allocation and optimal bandwidthallocation (OBUP) [5] OBUP algorithm divides thepower resource equally between each spot beam andoptimizes the bandwidth allocation for each spotbeam
When the channel conditions are the same Figure 4shows the capacity distributions of spot beams which areallocated by the four algorithms Table 1 shows the totalsystem capacity of the four algorithms As shown in Figure 4to get the fairness among spot beams the separate optimalallocation algorithms (UBOP and OBUP) will provide morecapacity to higher demand spot beams It is known thatthe capacity does not linearly increase with the power orbandwidth increasing due to the concavity of the capacityfunction with a fixed bandwidth or power allocation Thisresults in a lower total system capacity However in theproposed joint allocation algorithm (OBOP) the capacityfunction is almost linear since the bandwidth and power arejointly dynamically allocated As a result the total systemcapacity is much improved compared to the separate optimal
Mathematical Problems in Engineering 5
0 50 100 1500
100
200
300
400
500
600
700
800
900
1000
Iterations
120588(1) (step size is 04)120582 (step size is 15)120583 (step size is 07)
Figure 2 The convergence behavior of the dual variables
0 50 100 1500
500
1000
1500
Iterations
The sum of the power allocated to each beamThe sum of the bandwidth allocated to each beam
Figure 3 The convergence behavior of the optimal variables
Table 1 The total system capacity of the four algorithms when thechannel conditions are the same
allocation algorithm This conclusion can be also concludedfrom Table 1
When the channel conditions are the same Figure 5shows the deficit between the traffic demand and the capacity
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
Figure 4 Comparison of the four algorithms in terms of thecapacity allocated to each spot beam when channel conditions arethe same
Table 2The total sum of (119879119894minus 119862119894)2 of the four algorithms when the
channel conditions are the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1153611986415
UBOP [1] 1017011986415
OBUP [5] 1067611986415
The proposed OBOP 8336611986415
allocated to each spot beam Table 2 presents the sum of thedeficit between the traffic demand and the capacity allocatedto each spot beam From Figure 5 it is shown that in thejoint allocation algorithm (OBOP) from the spot beam 3to spot beam 10 the deficit between traffic demand andcapacity allocated is almost the same so the fairness amongspot beams is better than the separate optimal allocations(UBOP and OBUP) This conclusion can be also observedfromTable 2 since the deficit between the traffic demand andthe capacity allocated is smaller than the separate optimalallocations Together with the conclusion above regardingtotal system capacity we can conclude that the performanceof the optimal joint allocation algorithm (OBOP) is muchimproved compared with the separate optimal algorithms(UBOP and OBUP)
43 Impact of the Channel Condition and Delay ConstraintsWecompare the allocation result of theOBOP allocationwithdelay constraint with that of the OBOP allocation withoutthe delay constraint (OBOPND) to analyze the impact of thedelay constraints When there is no delay constraint on eachspot beam the allocation result can be obtained by the samealgorithm where we just need to remove the dual variable 120588in (8)
6 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]
OBUP [5]The proposed OBOP
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 5 Comparison of the four algorithms in terms of the deficitbetween the traffic demand and the capacity allocated to each spotbeam when channel conditions are the same
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
OBOPNDUBUPUBOP [1]OBUP [5]
The proposed OBOP
The minimal traffic demand
ith beam
Figure 6 The capacity allocated by five algorithms to each spotbeam when channel conditions are not the same
From spot beam 2 to spot beam 4 we set the normalizednoise power spectral density parameters 119873
01205722
119894to 03119890
minus604119890minus6 and 05119890
minus6 respectively to analyze the impact ofchannel condition on the allocation results Each trafficdemand of the three spot beams is 120Mbsp so the totaltraffic demand is the same as the above simulation scenarioThe order of the objective function also is 2
When the channel conditions are not the same Figure 6shows the capacity distributions of spot beams which areallocated by the 5 algorithms Table 3 shows the total system
Table 3 The total system capacity of the five algorithms when thechannel conditions are not the same
Table 4 The total sum of (119879119894minus 119862119894)2 of the 5 algorithms when the
channel conditions are not the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1239711986415
UBOP [1] 1122511986415
OBUP [5] 1140811986415
The proposed OBOP 9091911986415
OBOPND 8573011986415
capacity of the 5 algorithms From Figure 6 it is shownthat both the separate optimal allocation algorithm and theproposed joint allocation algorithm allocate more capacityto the spot beam with better channel condition than thatwith worse channel condition especially for the OBOPNDalgorithm no capacity is allocated to spot beam 3 and spotbeam 4 Therefore the metric we choose in this paper notonly considers the fairness among spot beams but also triesto maximize the throughput of the system which achieves agood system performance as we expected Due to the delayconstraint we have to allocate aminimal traffic to spot beamswith worse channel condition thus the total system capacitydecreases This conclusion is also shown in Table 3 So weconclude that when the channel condition of each spot beamis not the same the delay constraint of each spot beamdecreases the total system capacity
When the channel conditions are not the same Figure 7shows the deficit between the traffic demand and the capacityallocated to each spot beam Table 4 presents the sum ofthe deficits between the traffic demand and the capacityallocated to each spot beam It is shown that the worsechannel conditions and the delay constraint decrease thefairness among spot beams
44 Impact of the Order of the Objective Function and DelayConstraints As mentioned in Section 23 different metricsmay lead to different allocation results Here we analyze theimpact of the order of the objective function The channelcondition of each spot beam is set to be the same
Figure 8 shows the allocated capacity of each spot beamand Table 5 shows the total system capacity when the order ofthe objective function is different
When there is no delay constraint in each spot beamas we use a higher-order deviation objective function moreresources are provided for higher traffic demand spot beamswhile a lower-order cost function gives relatively moreresource to lower traffic demand spot beams For example
Mathematical Problems in Engineering 7
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 7 The deficit between the traffic demand and the capacityallocated by 5 algorithms to each spot beam when channel condi-tions are not the same
the cubic objective function provides no capacity for the threelowest traffic demand spot beams Although the capacityallocated to each spot beam is different the total systemcapacities of different objective function are almost the same
When there is delay constraint in each spot beam aminimal traffic demand must be allocated to them As aresult for the same cubic objective function the three lowesttraffic demand spot beams are allocated to theminimal trafficdemand It is also seen that the capacity allocated to each spotbeam is the same by using second- and third-order of theobjective function
5 Conclusion
In the multi-spot-beam satellite system due to the scarcenessof the satellite resource it is crucial for us to improve theresource utilization efficiency To this end in this paperwe first formulated the joint bandwidth and power allo-cation problem as a convex optimization problem Thenwe proposed a joint bandwidth and power allocation algo-rithm based on duality theory to get the optimal solutionCompared with the individual optimal power or bandwidthallocation the proposed joint optimal bandwidth and powerallocation algorithm improved the total system capacity andthe fairness among spot beams while the computationalcomplexity of the algorithm was linear with the numberof spot beams Therefore the proposed algorithm can beimplemented in practice
It was shown from the simulation results that the spotbeam with higher traffic demand and better channel con-dition will be allocated more resources to minimize theobjective function Due to the delay constraint in each spotbeam there was a minimal traffic demand in each spot beam
Table 5 The total system capacity of different objective functions
Figure 8 The capacity allocated to each spot beam of differentobjective functions
When the channel condition of each spot beamwas the samethe delay constraint only had an impact on the fairness amongspot beams However when the channel conditions were notthe same both the total system capacity and fairness amongspot beams were influenced by the delay constraint
Appendix
We first prove that constraint (4) can be ignored by contra-diction Assuming that 119862
119894gt 119879119894 it is seen that the constraint
(7) is satisfied obviously As a result the correspondingLagrange multiplier 120588
119894is zero According to (11) the value
of 120582 is negative which contradicts that 120582 is nonnegativeTherefore the constraint (4) is satisfied when the value of 120582 isnonnegative
According to [14] for the optimization problem whichhas the following general form
min 119891 (X)
st 119892119894 (X) le 0
(A1)
when the functions 119892119894(X) and 119891(X) are convex the optimiza-
tion problem is convex
8 Mathematical Problems in Engineering
Taken together with the fact that the functions sum119875119894
and sum119882119894in constraints (5) and (6) are linear to prove
the optimization problem with respect to [11988211198822 119882
119899
1198751 1198752 119875
119899] is convex we only need to prove that the
objective function sum (119879119894minus 119862119894)2 is convex and the function
119862119894= 119882119894log2[1 + 120572
2
119894119875119894(1198821198941198730)] in constraint (7) is concave
Firstly we prove that function 119862119894with respect to119882
119894and 119875
119894is
concaveThe Hessian of the function 119862
119894= 119882119894log2[1 + 120572
2
119894119875119894
(1198821198941198730)] is given as follows
119867119862=
[[[[
[
1205972119862119894
1205971198752
119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
1205971198822
119894
]]]]
]
(A2)
where 1205972119862119894120597119875119894
2= minus120572
2
119894119882119894 ln 2(119873
0119882119894+ 1205722
119894119875119894)2
le 0 and|119867119862| = 0It is seen that the determinant of the119867
119862is zero therefore
119867119862is negative semidefinite and the function 119862
119894is concave
[14] Then we prove that sum (119879119894minus 119862119894)2 is convex
It is known that the sum of convex functions is alsoconvex [14] Therefore to prove that sum (119879
119894minus 119862119894)2 is convex
we just need to prove the following function is convex
119891 (119875119894119882119894) = (119879
119894minus 119862119894)119899 (A3)
where 119862119894= 119882119894log2(1 + 120572
2
1198941198751198941198821198941198730)
The Hessian of 119891(119875119894119882119894) is given as follows
119867119891=
[[[[
[
1205972119891 (119875119894119882119894)
1205971198752
119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
1205971198822
119894
]]]]
]
(A4)
To prove that 119867119891is positive semidefinite we obtain the
following equation
1205972119891 (119875119894119882119894)
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
(120597119862119894
120597119875119894
)
2
minus 119899(119879119894minus 119862119894)119899minus1 1205972119862119894
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
= 1198992(119899 minus 1) (119879119894 minus 119862
119894)2119899minus3
times1198622
119894
119882119894ln 2(119873
01198821198941205722
119894+ 119875119894)2
(A8)
When 119879119894
ge 119862119894 it is obvious that (A5) and (A8)
are nonnegative Therefore 119867119891is positive semidefinite and
sum (119879119894minus 119862119894)2 is convex Taken together with the conclusion
that 119862119894is concave we prove that the optimization problem
is convex
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
The authors would like to thank the National High-TechResearch amp Development Program of China (863 Program)under Grant 2012AA01A508 for the project support
References
[1] J P Choi and V W S Chan ldquoOptimum power and beamallocation based on traffic demands and channel conditionsover satellite downlinksrdquo IEEE Transaction onWireless Commu-nication vol 4 no 6 pp 2983ndash2993 2005
[2] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio and Wireless Symposium pp823ndash826 January 2008
Mathematical Problems in Engineering 9
[3] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the 13thInternational Conference on Communication Technology (ICCTrsquo11) pp 873ndash876 September 2011
[4] U Park H Wook Kim D Sub Oh and B J Ku ldquoOptimumselective beam allocation scheme for satellite network withmulti-spot beamsrdquo in Proceedings of the 4th InternationalConference on Advances in Satellite and Space Communications(SPACOMM rsquo12) pp 78ndash81 April2012
[5] U Park H W Kim D Sub Oh and B J Ku ldquoA dynamicbandwidth allocation scheme for a multi-spot-beam satellitesystemrdquo ETRI Journal vol 34 no 4 pp 613ndash616 2012
[6] K Kumaran and H Viswanathan ldquoJoint power and bandwidthallocation in Downlink transmissionrdquo IEEE Transaction onWireless Communications vol 4 no 3 pp 1008ndash1016 2005
[7] X Gong S A Vorobyov and C Tellambura ldquoJoint bandwidthand power allocation with admission control in wireless multi-user networks with and without relayingrdquo IEEE Transactions onSignal Processing vol 59 no 4 pp 1801ndash1813 2011
[8] J Miao Z Hu K Yang CWang andH Tian ldquoJoint power andbandwidth allocation algorithm with Qos support in heteroge-neous wireless networksrdquo IEEE Communications Letters vol 16no 4 pp 479ndash481 2012
[9] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991
[10] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013
[11] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionon Communications vol 54 no 7 pp 1310ndash1322 2006
[12] R Wang K N Vincent Lau L Lv and B Chen ldquoJoint cross-layer scheduling and spectrum sensing for OFDMA cognitiveradio systemsrdquo IEEE Transaction on Wireless Communicationsvol 8 no 5 pp 2410ndash2416 2009
[13] A G Marques X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009
[14] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
allocation algorithm This conclusion can be also concludedfrom Table 1
When the channel conditions are the same Figure 5shows the deficit between the traffic demand and the capacity
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
Figure 4 Comparison of the four algorithms in terms of thecapacity allocated to each spot beam when channel conditions arethe same
Table 2The total sum of (119879119894minus 119862119894)2 of the four algorithms when the
channel conditions are the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1153611986415
UBOP [1] 1017011986415
OBUP [5] 1067611986415
The proposed OBOP 8336611986415
allocated to each spot beam Table 2 presents the sum of thedeficit between the traffic demand and the capacity allocatedto each spot beam From Figure 5 it is shown that in thejoint allocation algorithm (OBOP) from the spot beam 3to spot beam 10 the deficit between traffic demand andcapacity allocated is almost the same so the fairness amongspot beams is better than the separate optimal allocations(UBOP and OBUP) This conclusion can be also observedfromTable 2 since the deficit between the traffic demand andthe capacity allocated is smaller than the separate optimalallocations Together with the conclusion above regardingtotal system capacity we can conclude that the performanceof the optimal joint allocation algorithm (OBOP) is muchimproved compared with the separate optimal algorithms(UBOP and OBUP)
43 Impact of the Channel Condition and Delay ConstraintsWecompare the allocation result of theOBOP allocationwithdelay constraint with that of the OBOP allocation withoutthe delay constraint (OBOPND) to analyze the impact of thedelay constraints When there is no delay constraint on eachspot beam the allocation result can be obtained by the samealgorithm where we just need to remove the dual variable 120588in (8)
6 Mathematical Problems in Engineering
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]
OBUP [5]The proposed OBOP
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 5 Comparison of the four algorithms in terms of the deficitbetween the traffic demand and the capacity allocated to each spotbeam when channel conditions are the same
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
OBOPNDUBUPUBOP [1]OBUP [5]
The proposed OBOP
The minimal traffic demand
ith beam
Figure 6 The capacity allocated by five algorithms to each spotbeam when channel conditions are not the same
From spot beam 2 to spot beam 4 we set the normalizednoise power spectral density parameters 119873
01205722
119894to 03119890
minus604119890minus6 and 05119890
minus6 respectively to analyze the impact ofchannel condition on the allocation results Each trafficdemand of the three spot beams is 120Mbsp so the totaltraffic demand is the same as the above simulation scenarioThe order of the objective function also is 2
When the channel conditions are not the same Figure 6shows the capacity distributions of spot beams which areallocated by the 5 algorithms Table 3 shows the total system
Table 3 The total system capacity of the five algorithms when thechannel conditions are not the same
Table 4 The total sum of (119879119894minus 119862119894)2 of the 5 algorithms when the
channel conditions are not the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1239711986415
UBOP [1] 1122511986415
OBUP [5] 1140811986415
The proposed OBOP 9091911986415
OBOPND 8573011986415
capacity of the 5 algorithms From Figure 6 it is shownthat both the separate optimal allocation algorithm and theproposed joint allocation algorithm allocate more capacityto the spot beam with better channel condition than thatwith worse channel condition especially for the OBOPNDalgorithm no capacity is allocated to spot beam 3 and spotbeam 4 Therefore the metric we choose in this paper notonly considers the fairness among spot beams but also triesto maximize the throughput of the system which achieves agood system performance as we expected Due to the delayconstraint we have to allocate aminimal traffic to spot beamswith worse channel condition thus the total system capacitydecreases This conclusion is also shown in Table 3 So weconclude that when the channel condition of each spot beamis not the same the delay constraint of each spot beamdecreases the total system capacity
When the channel conditions are not the same Figure 7shows the deficit between the traffic demand and the capacityallocated to each spot beam Table 4 presents the sum ofthe deficits between the traffic demand and the capacityallocated to each spot beam It is shown that the worsechannel conditions and the delay constraint decrease thefairness among spot beams
44 Impact of the Order of the Objective Function and DelayConstraints As mentioned in Section 23 different metricsmay lead to different allocation results Here we analyze theimpact of the order of the objective function The channelcondition of each spot beam is set to be the same
Figure 8 shows the allocated capacity of each spot beamand Table 5 shows the total system capacity when the order ofthe objective function is different
When there is no delay constraint in each spot beamas we use a higher-order deviation objective function moreresources are provided for higher traffic demand spot beamswhile a lower-order cost function gives relatively moreresource to lower traffic demand spot beams For example
Mathematical Problems in Engineering 7
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 7 The deficit between the traffic demand and the capacityallocated by 5 algorithms to each spot beam when channel condi-tions are not the same
the cubic objective function provides no capacity for the threelowest traffic demand spot beams Although the capacityallocated to each spot beam is different the total systemcapacities of different objective function are almost the same
When there is delay constraint in each spot beam aminimal traffic demand must be allocated to them As aresult for the same cubic objective function the three lowesttraffic demand spot beams are allocated to theminimal trafficdemand It is also seen that the capacity allocated to each spotbeam is the same by using second- and third-order of theobjective function
5 Conclusion
In the multi-spot-beam satellite system due to the scarcenessof the satellite resource it is crucial for us to improve theresource utilization efficiency To this end in this paperwe first formulated the joint bandwidth and power allo-cation problem as a convex optimization problem Thenwe proposed a joint bandwidth and power allocation algo-rithm based on duality theory to get the optimal solutionCompared with the individual optimal power or bandwidthallocation the proposed joint optimal bandwidth and powerallocation algorithm improved the total system capacity andthe fairness among spot beams while the computationalcomplexity of the algorithm was linear with the numberof spot beams Therefore the proposed algorithm can beimplemented in practice
It was shown from the simulation results that the spotbeam with higher traffic demand and better channel con-dition will be allocated more resources to minimize theobjective function Due to the delay constraint in each spotbeam there was a minimal traffic demand in each spot beam
Table 5 The total system capacity of different objective functions
Figure 8 The capacity allocated to each spot beam of differentobjective functions
When the channel condition of each spot beamwas the samethe delay constraint only had an impact on the fairness amongspot beams However when the channel conditions were notthe same both the total system capacity and fairness amongspot beams were influenced by the delay constraint
Appendix
We first prove that constraint (4) can be ignored by contra-diction Assuming that 119862
119894gt 119879119894 it is seen that the constraint
(7) is satisfied obviously As a result the correspondingLagrange multiplier 120588
119894is zero According to (11) the value
of 120582 is negative which contradicts that 120582 is nonnegativeTherefore the constraint (4) is satisfied when the value of 120582 isnonnegative
According to [14] for the optimization problem whichhas the following general form
min 119891 (X)
st 119892119894 (X) le 0
(A1)
when the functions 119892119894(X) and 119891(X) are convex the optimiza-
tion problem is convex
8 Mathematical Problems in Engineering
Taken together with the fact that the functions sum119875119894
and sum119882119894in constraints (5) and (6) are linear to prove
the optimization problem with respect to [11988211198822 119882
119899
1198751 1198752 119875
119899] is convex we only need to prove that the
objective function sum (119879119894minus 119862119894)2 is convex and the function
119862119894= 119882119894log2[1 + 120572
2
119894119875119894(1198821198941198730)] in constraint (7) is concave
Firstly we prove that function 119862119894with respect to119882
119894and 119875
119894is
concaveThe Hessian of the function 119862
119894= 119882119894log2[1 + 120572
2
119894119875119894
(1198821198941198730)] is given as follows
119867119862=
[[[[
[
1205972119862119894
1205971198752
119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
1205971198822
119894
]]]]
]
(A2)
where 1205972119862119894120597119875119894
2= minus120572
2
119894119882119894 ln 2(119873
0119882119894+ 1205722
119894119875119894)2
le 0 and|119867119862| = 0It is seen that the determinant of the119867
119862is zero therefore
119867119862is negative semidefinite and the function 119862
119894is concave
[14] Then we prove that sum (119879119894minus 119862119894)2 is convex
It is known that the sum of convex functions is alsoconvex [14] Therefore to prove that sum (119879
119894minus 119862119894)2 is convex
we just need to prove the following function is convex
119891 (119875119894119882119894) = (119879
119894minus 119862119894)119899 (A3)
where 119862119894= 119882119894log2(1 + 120572
2
1198941198751198941198821198941198730)
The Hessian of 119891(119875119894119882119894) is given as follows
119867119891=
[[[[
[
1205972119891 (119875119894119882119894)
1205971198752
119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
1205971198822
119894
]]]]
]
(A4)
To prove that 119867119891is positive semidefinite we obtain the
following equation
1205972119891 (119875119894119882119894)
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
(120597119862119894
120597119875119894
)
2
minus 119899(119879119894minus 119862119894)119899minus1 1205972119862119894
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
= 1198992(119899 minus 1) (119879119894 minus 119862
119894)2119899minus3
times1198622
119894
119882119894ln 2(119873
01198821198941205722
119894+ 119875119894)2
(A8)
When 119879119894
ge 119862119894 it is obvious that (A5) and (A8)
are nonnegative Therefore 119867119891is positive semidefinite and
sum (119879119894minus 119862119894)2 is convex Taken together with the conclusion
that 119862119894is concave we prove that the optimization problem
is convex
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
The authors would like to thank the National High-TechResearch amp Development Program of China (863 Program)under Grant 2012AA01A508 for the project support
References
[1] J P Choi and V W S Chan ldquoOptimum power and beamallocation based on traffic demands and channel conditionsover satellite downlinksrdquo IEEE Transaction onWireless Commu-nication vol 4 no 6 pp 2983ndash2993 2005
[2] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio and Wireless Symposium pp823ndash826 January 2008
Mathematical Problems in Engineering 9
[3] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the 13thInternational Conference on Communication Technology (ICCTrsquo11) pp 873ndash876 September 2011
[4] U Park H Wook Kim D Sub Oh and B J Ku ldquoOptimumselective beam allocation scheme for satellite network withmulti-spot beamsrdquo in Proceedings of the 4th InternationalConference on Advances in Satellite and Space Communications(SPACOMM rsquo12) pp 78ndash81 April2012
[5] U Park H W Kim D Sub Oh and B J Ku ldquoA dynamicbandwidth allocation scheme for a multi-spot-beam satellitesystemrdquo ETRI Journal vol 34 no 4 pp 613ndash616 2012
[6] K Kumaran and H Viswanathan ldquoJoint power and bandwidthallocation in Downlink transmissionrdquo IEEE Transaction onWireless Communications vol 4 no 3 pp 1008ndash1016 2005
[7] X Gong S A Vorobyov and C Tellambura ldquoJoint bandwidthand power allocation with admission control in wireless multi-user networks with and without relayingrdquo IEEE Transactions onSignal Processing vol 59 no 4 pp 1801ndash1813 2011
[8] J Miao Z Hu K Yang CWang andH Tian ldquoJoint power andbandwidth allocation algorithm with Qos support in heteroge-neous wireless networksrdquo IEEE Communications Letters vol 16no 4 pp 479ndash481 2012
[9] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991
[10] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013
[11] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionon Communications vol 54 no 7 pp 1310ndash1322 2006
[12] R Wang K N Vincent Lau L Lv and B Chen ldquoJoint cross-layer scheduling and spectrum sensing for OFDMA cognitiveradio systemsrdquo IEEE Transaction on Wireless Communicationsvol 8 no 5 pp 2410ndash2416 2009
[13] A G Marques X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009
[14] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
Figure 5 Comparison of the four algorithms in terms of the deficitbetween the traffic demand and the capacity allocated to each spotbeam when channel conditions are the same
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
160
180
Capa
city
allo
cate
d (M
bps)
OBOPNDUBUPUBOP [1]OBUP [5]
The proposed OBOP
The minimal traffic demand
ith beam
Figure 6 The capacity allocated by five algorithms to each spotbeam when channel conditions are not the same
From spot beam 2 to spot beam 4 we set the normalizednoise power spectral density parameters 119873
01205722
119894to 03119890
minus604119890minus6 and 05119890
minus6 respectively to analyze the impact ofchannel condition on the allocation results Each trafficdemand of the three spot beams is 120Mbsp so the totaltraffic demand is the same as the above simulation scenarioThe order of the objective function also is 2
When the channel conditions are not the same Figure 6shows the capacity distributions of spot beams which areallocated by the 5 algorithms Table 3 shows the total system
Table 3 The total system capacity of the five algorithms when thechannel conditions are not the same
Table 4 The total sum of (119879119894minus 119862119894)2 of the 5 algorithms when the
channel conditions are not the same
Algorithms sum (119879119894minus 119862119894)2
UBUP 1239711986415
UBOP [1] 1122511986415
OBUP [5] 1140811986415
The proposed OBOP 9091911986415
OBOPND 8573011986415
capacity of the 5 algorithms From Figure 6 it is shownthat both the separate optimal allocation algorithm and theproposed joint allocation algorithm allocate more capacityto the spot beam with better channel condition than thatwith worse channel condition especially for the OBOPNDalgorithm no capacity is allocated to spot beam 3 and spotbeam 4 Therefore the metric we choose in this paper notonly considers the fairness among spot beams but also triesto maximize the throughput of the system which achieves agood system performance as we expected Due to the delayconstraint we have to allocate aminimal traffic to spot beamswith worse channel condition thus the total system capacitydecreases This conclusion is also shown in Table 3 So weconclude that when the channel condition of each spot beamis not the same the delay constraint of each spot beamdecreases the total system capacity
When the channel conditions are not the same Figure 7shows the deficit between the traffic demand and the capacityallocated to each spot beam Table 4 presents the sum ofthe deficits between the traffic demand and the capacityallocated to each spot beam It is shown that the worsechannel conditions and the delay constraint decrease thefairness among spot beams
44 Impact of the Order of the Objective Function and DelayConstraints As mentioned in Section 23 different metricsmay lead to different allocation results Here we analyze theimpact of the order of the objective function The channelcondition of each spot beam is set to be the same
Figure 8 shows the allocated capacity of each spot beamand Table 5 shows the total system capacity when the order ofthe objective function is different
When there is no delay constraint in each spot beamas we use a higher-order deviation objective function moreresources are provided for higher traffic demand spot beamswhile a lower-order cost function gives relatively moreresource to lower traffic demand spot beams For example
Mathematical Problems in Engineering 7
1 2 3 4 5 6 7 8 9 10
UBUPUBOP [1]OBUP [5]
The proposed OBOPThe minimal traffic demand
ith beam
0
05
1
15
2
25
3
35times1016
(TiminusCi)2
Figure 7 The deficit between the traffic demand and the capacityallocated by 5 algorithms to each spot beam when channel condi-tions are not the same
the cubic objective function provides no capacity for the threelowest traffic demand spot beams Although the capacityallocated to each spot beam is different the total systemcapacities of different objective function are almost the same
When there is delay constraint in each spot beam aminimal traffic demand must be allocated to them As aresult for the same cubic objective function the three lowesttraffic demand spot beams are allocated to theminimal trafficdemand It is also seen that the capacity allocated to each spotbeam is the same by using second- and third-order of theobjective function
5 Conclusion
In the multi-spot-beam satellite system due to the scarcenessof the satellite resource it is crucial for us to improve theresource utilization efficiency To this end in this paperwe first formulated the joint bandwidth and power allo-cation problem as a convex optimization problem Thenwe proposed a joint bandwidth and power allocation algo-rithm based on duality theory to get the optimal solutionCompared with the individual optimal power or bandwidthallocation the proposed joint optimal bandwidth and powerallocation algorithm improved the total system capacity andthe fairness among spot beams while the computationalcomplexity of the algorithm was linear with the numberof spot beams Therefore the proposed algorithm can beimplemented in practice
It was shown from the simulation results that the spotbeam with higher traffic demand and better channel con-dition will be allocated more resources to minimize theobjective function Due to the delay constraint in each spotbeam there was a minimal traffic demand in each spot beam
Table 5 The total system capacity of different objective functions
Figure 8 The capacity allocated to each spot beam of differentobjective functions
When the channel condition of each spot beamwas the samethe delay constraint only had an impact on the fairness amongspot beams However when the channel conditions were notthe same both the total system capacity and fairness amongspot beams were influenced by the delay constraint
Appendix
We first prove that constraint (4) can be ignored by contra-diction Assuming that 119862
119894gt 119879119894 it is seen that the constraint
(7) is satisfied obviously As a result the correspondingLagrange multiplier 120588
119894is zero According to (11) the value
of 120582 is negative which contradicts that 120582 is nonnegativeTherefore the constraint (4) is satisfied when the value of 120582 isnonnegative
According to [14] for the optimization problem whichhas the following general form
min 119891 (X)
st 119892119894 (X) le 0
(A1)
when the functions 119892119894(X) and 119891(X) are convex the optimiza-
tion problem is convex
8 Mathematical Problems in Engineering
Taken together with the fact that the functions sum119875119894
and sum119882119894in constraints (5) and (6) are linear to prove
the optimization problem with respect to [11988211198822 119882
119899
1198751 1198752 119875
119899] is convex we only need to prove that the
objective function sum (119879119894minus 119862119894)2 is convex and the function
119862119894= 119882119894log2[1 + 120572
2
119894119875119894(1198821198941198730)] in constraint (7) is concave
Firstly we prove that function 119862119894with respect to119882
119894and 119875
119894is
concaveThe Hessian of the function 119862
119894= 119882119894log2[1 + 120572
2
119894119875119894
(1198821198941198730)] is given as follows
119867119862=
[[[[
[
1205972119862119894
1205971198752
119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
1205971198822
119894
]]]]
]
(A2)
where 1205972119862119894120597119875119894
2= minus120572
2
119894119882119894 ln 2(119873
0119882119894+ 1205722
119894119875119894)2
le 0 and|119867119862| = 0It is seen that the determinant of the119867
119862is zero therefore
119867119862is negative semidefinite and the function 119862
119894is concave
[14] Then we prove that sum (119879119894minus 119862119894)2 is convex
It is known that the sum of convex functions is alsoconvex [14] Therefore to prove that sum (119879
119894minus 119862119894)2 is convex
we just need to prove the following function is convex
119891 (119875119894119882119894) = (119879
119894minus 119862119894)119899 (A3)
where 119862119894= 119882119894log2(1 + 120572
2
1198941198751198941198821198941198730)
The Hessian of 119891(119875119894119882119894) is given as follows
119867119891=
[[[[
[
1205972119891 (119875119894119882119894)
1205971198752
119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
1205971198822
119894
]]]]
]
(A4)
To prove that 119867119891is positive semidefinite we obtain the
following equation
1205972119891 (119875119894119882119894)
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
(120597119862119894
120597119875119894
)
2
minus 119899(119879119894minus 119862119894)119899minus1 1205972119862119894
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
= 1198992(119899 minus 1) (119879119894 minus 119862
119894)2119899minus3
times1198622
119894
119882119894ln 2(119873
01198821198941205722
119894+ 119875119894)2
(A8)
When 119879119894
ge 119862119894 it is obvious that (A5) and (A8)
are nonnegative Therefore 119867119891is positive semidefinite and
sum (119879119894minus 119862119894)2 is convex Taken together with the conclusion
that 119862119894is concave we prove that the optimization problem
is convex
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
The authors would like to thank the National High-TechResearch amp Development Program of China (863 Program)under Grant 2012AA01A508 for the project support
References
[1] J P Choi and V W S Chan ldquoOptimum power and beamallocation based on traffic demands and channel conditionsover satellite downlinksrdquo IEEE Transaction onWireless Commu-nication vol 4 no 6 pp 2983ndash2993 2005
[2] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio and Wireless Symposium pp823ndash826 January 2008
Mathematical Problems in Engineering 9
[3] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the 13thInternational Conference on Communication Technology (ICCTrsquo11) pp 873ndash876 September 2011
[4] U Park H Wook Kim D Sub Oh and B J Ku ldquoOptimumselective beam allocation scheme for satellite network withmulti-spot beamsrdquo in Proceedings of the 4th InternationalConference on Advances in Satellite and Space Communications(SPACOMM rsquo12) pp 78ndash81 April2012
[5] U Park H W Kim D Sub Oh and B J Ku ldquoA dynamicbandwidth allocation scheme for a multi-spot-beam satellitesystemrdquo ETRI Journal vol 34 no 4 pp 613ndash616 2012
[6] K Kumaran and H Viswanathan ldquoJoint power and bandwidthallocation in Downlink transmissionrdquo IEEE Transaction onWireless Communications vol 4 no 3 pp 1008ndash1016 2005
[7] X Gong S A Vorobyov and C Tellambura ldquoJoint bandwidthand power allocation with admission control in wireless multi-user networks with and without relayingrdquo IEEE Transactions onSignal Processing vol 59 no 4 pp 1801ndash1813 2011
[8] J Miao Z Hu K Yang CWang andH Tian ldquoJoint power andbandwidth allocation algorithm with Qos support in heteroge-neous wireless networksrdquo IEEE Communications Letters vol 16no 4 pp 479ndash481 2012
[9] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991
[10] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013
[11] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionon Communications vol 54 no 7 pp 1310ndash1322 2006
[12] R Wang K N Vincent Lau L Lv and B Chen ldquoJoint cross-layer scheduling and spectrum sensing for OFDMA cognitiveradio systemsrdquo IEEE Transaction on Wireless Communicationsvol 8 no 5 pp 2410ndash2416 2009
[13] A G Marques X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009
[14] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
Figure 7 The deficit between the traffic demand and the capacityallocated by 5 algorithms to each spot beam when channel condi-tions are not the same
the cubic objective function provides no capacity for the threelowest traffic demand spot beams Although the capacityallocated to each spot beam is different the total systemcapacities of different objective function are almost the same
When there is delay constraint in each spot beam aminimal traffic demand must be allocated to them As aresult for the same cubic objective function the three lowesttraffic demand spot beams are allocated to theminimal trafficdemand It is also seen that the capacity allocated to each spotbeam is the same by using second- and third-order of theobjective function
5 Conclusion
In the multi-spot-beam satellite system due to the scarcenessof the satellite resource it is crucial for us to improve theresource utilization efficiency To this end in this paperwe first formulated the joint bandwidth and power allo-cation problem as a convex optimization problem Thenwe proposed a joint bandwidth and power allocation algo-rithm based on duality theory to get the optimal solutionCompared with the individual optimal power or bandwidthallocation the proposed joint optimal bandwidth and powerallocation algorithm improved the total system capacity andthe fairness among spot beams while the computationalcomplexity of the algorithm was linear with the numberof spot beams Therefore the proposed algorithm can beimplemented in practice
It was shown from the simulation results that the spotbeam with higher traffic demand and better channel con-dition will be allocated more resources to minimize theobjective function Due to the delay constraint in each spotbeam there was a minimal traffic demand in each spot beam
Table 5 The total system capacity of different objective functions
Figure 8 The capacity allocated to each spot beam of differentobjective functions
When the channel condition of each spot beamwas the samethe delay constraint only had an impact on the fairness amongspot beams However when the channel conditions were notthe same both the total system capacity and fairness amongspot beams were influenced by the delay constraint
Appendix
We first prove that constraint (4) can be ignored by contra-diction Assuming that 119862
119894gt 119879119894 it is seen that the constraint
(7) is satisfied obviously As a result the correspondingLagrange multiplier 120588
119894is zero According to (11) the value
of 120582 is negative which contradicts that 120582 is nonnegativeTherefore the constraint (4) is satisfied when the value of 120582 isnonnegative
According to [14] for the optimization problem whichhas the following general form
min 119891 (X)
st 119892119894 (X) le 0
(A1)
when the functions 119892119894(X) and 119891(X) are convex the optimiza-
tion problem is convex
8 Mathematical Problems in Engineering
Taken together with the fact that the functions sum119875119894
and sum119882119894in constraints (5) and (6) are linear to prove
the optimization problem with respect to [11988211198822 119882
119899
1198751 1198752 119875
119899] is convex we only need to prove that the
objective function sum (119879119894minus 119862119894)2 is convex and the function
119862119894= 119882119894log2[1 + 120572
2
119894119875119894(1198821198941198730)] in constraint (7) is concave
Firstly we prove that function 119862119894with respect to119882
119894and 119875
119894is
concaveThe Hessian of the function 119862
119894= 119882119894log2[1 + 120572
2
119894119875119894
(1198821198941198730)] is given as follows
119867119862=
[[[[
[
1205972119862119894
1205971198752
119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
120597119875119894119882119894
1205972119862119894
1205971198822
119894
]]]]
]
(A2)
where 1205972119862119894120597119875119894
2= minus120572
2
119894119882119894 ln 2(119873
0119882119894+ 1205722
119894119875119894)2
le 0 and|119867119862| = 0It is seen that the determinant of the119867
119862is zero therefore
119867119862is negative semidefinite and the function 119862
119894is concave
[14] Then we prove that sum (119879119894minus 119862119894)2 is convex
It is known that the sum of convex functions is alsoconvex [14] Therefore to prove that sum (119879
119894minus 119862119894)2 is convex
we just need to prove the following function is convex
119891 (119875119894119882119894) = (119879
119894minus 119862119894)119899 (A3)
where 119862119894= 119882119894log2(1 + 120572
2
1198941198751198941198821198941198730)
The Hessian of 119891(119875119894119882119894) is given as follows
119867119891=
[[[[
[
1205972119891 (119875119894119882119894)
1205971198752
119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
120597119875119894120597119882119894
1205972119891 (119875119894119882119894)
1205971198822
119894
]]]]
]
(A4)
To prove that 119867119891is positive semidefinite we obtain the
following equation
1205972119891 (119875119894119882119894)
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
(120597119862119894
120597119875119894
)
2
minus 119899(119879119894minus 119862119894)119899minus1 1205972119862119894
1205971198752
119894
= 119899 (119899 minus 1) (119879119894 minus 119862119894)119899minus2
= 1198992(119899 minus 1) (119879119894 minus 119862
119894)2119899minus3
times1198622
119894
119882119894ln 2(119873
01198821198941205722
119894+ 119875119894)2
(A8)
When 119879119894
ge 119862119894 it is obvious that (A5) and (A8)
are nonnegative Therefore 119867119891is positive semidefinite and
sum (119879119894minus 119862119894)2 is convex Taken together with the conclusion
that 119862119894is concave we prove that the optimization problem
is convex
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
The authors would like to thank the National High-TechResearch amp Development Program of China (863 Program)under Grant 2012AA01A508 for the project support
References
[1] J P Choi and V W S Chan ldquoOptimum power and beamallocation based on traffic demands and channel conditionsover satellite downlinksrdquo IEEE Transaction onWireless Commu-nication vol 4 no 6 pp 2983ndash2993 2005
[2] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio and Wireless Symposium pp823ndash826 January 2008
Mathematical Problems in Engineering 9
[3] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the 13thInternational Conference on Communication Technology (ICCTrsquo11) pp 873ndash876 September 2011
[4] U Park H Wook Kim D Sub Oh and B J Ku ldquoOptimumselective beam allocation scheme for satellite network withmulti-spot beamsrdquo in Proceedings of the 4th InternationalConference on Advances in Satellite and Space Communications(SPACOMM rsquo12) pp 78ndash81 April2012
[5] U Park H W Kim D Sub Oh and B J Ku ldquoA dynamicbandwidth allocation scheme for a multi-spot-beam satellitesystemrdquo ETRI Journal vol 34 no 4 pp 613ndash616 2012
[6] K Kumaran and H Viswanathan ldquoJoint power and bandwidthallocation in Downlink transmissionrdquo IEEE Transaction onWireless Communications vol 4 no 3 pp 1008ndash1016 2005
[7] X Gong S A Vorobyov and C Tellambura ldquoJoint bandwidthand power allocation with admission control in wireless multi-user networks with and without relayingrdquo IEEE Transactions onSignal Processing vol 59 no 4 pp 1801ndash1813 2011
[8] J Miao Z Hu K Yang CWang andH Tian ldquoJoint power andbandwidth allocation algorithm with Qos support in heteroge-neous wireless networksrdquo IEEE Communications Letters vol 16no 4 pp 479ndash481 2012
[9] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991
[10] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013
[11] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionon Communications vol 54 no 7 pp 1310ndash1322 2006
[12] R Wang K N Vincent Lau L Lv and B Chen ldquoJoint cross-layer scheduling and spectrum sensing for OFDMA cognitiveradio systemsrdquo IEEE Transaction on Wireless Communicationsvol 8 no 5 pp 2410ndash2416 2009
[13] A G Marques X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009
[14] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
= 1198992(119899 minus 1) (119879119894 minus 119862
119894)2119899minus3
times1198622
119894
119882119894ln 2(119873
01198821198941205722
119894+ 119875119894)2
(A8)
When 119879119894
ge 119862119894 it is obvious that (A5) and (A8)
are nonnegative Therefore 119867119891is positive semidefinite and
sum (119879119894minus 119862119894)2 is convex Taken together with the conclusion
that 119862119894is concave we prove that the optimization problem
is convex
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
The authors would like to thank the National High-TechResearch amp Development Program of China (863 Program)under Grant 2012AA01A508 for the project support
References
[1] J P Choi and V W S Chan ldquoOptimum power and beamallocation based on traffic demands and channel conditionsover satellite downlinksrdquo IEEE Transaction onWireless Commu-nication vol 4 no 6 pp 2983ndash2993 2005
[2] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio and Wireless Symposium pp823ndash826 January 2008
Mathematical Problems in Engineering 9
[3] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the 13thInternational Conference on Communication Technology (ICCTrsquo11) pp 873ndash876 September 2011
[4] U Park H Wook Kim D Sub Oh and B J Ku ldquoOptimumselective beam allocation scheme for satellite network withmulti-spot beamsrdquo in Proceedings of the 4th InternationalConference on Advances in Satellite and Space Communications(SPACOMM rsquo12) pp 78ndash81 April2012
[5] U Park H W Kim D Sub Oh and B J Ku ldquoA dynamicbandwidth allocation scheme for a multi-spot-beam satellitesystemrdquo ETRI Journal vol 34 no 4 pp 613ndash616 2012
[6] K Kumaran and H Viswanathan ldquoJoint power and bandwidthallocation in Downlink transmissionrdquo IEEE Transaction onWireless Communications vol 4 no 3 pp 1008ndash1016 2005
[7] X Gong S A Vorobyov and C Tellambura ldquoJoint bandwidthand power allocation with admission control in wireless multi-user networks with and without relayingrdquo IEEE Transactions onSignal Processing vol 59 no 4 pp 1801ndash1813 2011
[8] J Miao Z Hu K Yang CWang andH Tian ldquoJoint power andbandwidth allocation algorithm with Qos support in heteroge-neous wireless networksrdquo IEEE Communications Letters vol 16no 4 pp 479ndash481 2012
[9] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991
[10] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013
[11] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionon Communications vol 54 no 7 pp 1310ndash1322 2006
[12] R Wang K N Vincent Lau L Lv and B Chen ldquoJoint cross-layer scheduling and spectrum sensing for OFDMA cognitiveradio systemsrdquo IEEE Transaction on Wireless Communicationsvol 8 no 5 pp 2410ndash2416 2009
[13] A G Marques X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009
[14] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[3] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the 13thInternational Conference on Communication Technology (ICCTrsquo11) pp 873ndash876 September 2011
[4] U Park H Wook Kim D Sub Oh and B J Ku ldquoOptimumselective beam allocation scheme for satellite network withmulti-spot beamsrdquo in Proceedings of the 4th InternationalConference on Advances in Satellite and Space Communications(SPACOMM rsquo12) pp 78ndash81 April2012
[5] U Park H W Kim D Sub Oh and B J Ku ldquoA dynamicbandwidth allocation scheme for a multi-spot-beam satellitesystemrdquo ETRI Journal vol 34 no 4 pp 613ndash616 2012
[6] K Kumaran and H Viswanathan ldquoJoint power and bandwidthallocation in Downlink transmissionrdquo IEEE Transaction onWireless Communications vol 4 no 3 pp 1008ndash1016 2005
[7] X Gong S A Vorobyov and C Tellambura ldquoJoint bandwidthand power allocation with admission control in wireless multi-user networks with and without relayingrdquo IEEE Transactions onSignal Processing vol 59 no 4 pp 1801ndash1813 2011
[8] J Miao Z Hu K Yang CWang andH Tian ldquoJoint power andbandwidth allocation algorithm with Qos support in heteroge-neous wireless networksrdquo IEEE Communications Letters vol 16no 4 pp 479ndash481 2012
[9] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991
[10] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013
[11] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionon Communications vol 54 no 7 pp 1310ndash1322 2006
[12] R Wang K N Vincent Lau L Lv and B Chen ldquoJoint cross-layer scheduling and spectrum sensing for OFDMA cognitiveradio systemsrdquo IEEE Transaction on Wireless Communicationsvol 8 no 5 pp 2410ndash2416 2009
[13] A G Marques X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009
[14] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
