Optimal Power Allocation for Multiple Beam Satellite Systems Yang Hong1, Anand Srinivasan1, Brian Cheng1, Leo Hartman2 and Peter Andreadis3 1

EION Inc., Ottawa, Onatrio, Canada 2Canadian Space Agency, St. Hubert, Quebec, Canada 3 Communications Research Center, Ottawa, Ontario, Canada E-mail: {yhong, anand, brian}@eion.com, [email protected], [email protected] Abstract—Presently, multiple spot-beam satellites are being launched with the ability to allocate power dynamically and to increase throughput, because on-board resources (e.g., power, bandwidth) are scarce and expensive. It is critical to share them efficiently among as many users as possible. In this paper, we propose an innovative power allocation algorithm for the system. We use heuristic method to search the Lagrange multiplier and obtain the optimal power allocation for each spot beam in order to meet the total power constraint and individual SLA constraint together. Scilab simulation demonstrates that our algorithm maximizes the power utilization of the satellite successfully. Index Terms—Satellite Communications, Channel Capacity

I. INTRODUCTION In the past, commercial satellites did not have the ability to support multiple beams on-board. Presently, most of the commercial satellites have the ability to support multiple spot beams enabling services to diverse geographical region. Unfortunately, all the geographical regions do not have the same capacity requirement. Neither does the future demand follow the same progression. This has led to a very interesting problem, namely how do you increase or decrease capacity adaptively based on the increase or decrease in user/service demand? Satellites do have the ability to service large number of users, albeit cumulatively. Satellites, when launched, are engineered to cater to future demands as it is expected to stay in service for at least a few decades. However, many remote areas may not use efficiently the capacity to their maximum due to sparse population, while high demands might come from an urban center. Therefore, it is a real challenge to redistribute the capacity in order to optimize the revenue. Some attempts have been made to adjust power for an optimal resource allocation in on-board processor, e.g., [1]. In addition, satellite service providers (e.g., our industrial partner TeleSat® [2]) are investing time and money to adopt innovative solutions to allocate spot beams and to increase revenue. As part of future missions, multiple spot-beam satellites are being launched with the ability to allocate power dynamically and to increase throughput, because onboard resources (e.g., power, bandwidth, transmitters, receivers and spot-beams) are scarce and expensive. It is critical to share them efficiently among as many users as possible. There are multiple ways to accomplish the end goal of increasing revenue. One of the methods to allocate power is to decrease power in some areas while increasing power in others where there is surge in user demands. Another method is to allocate multiple spot beams over a

geographical area allowing timesharing of active downlink beams. A data satellite network would have to support a broad spectrum of bursty unscheduled users with different constraints over time varying atmospheric satellite channels [3, 4]. The paper [1] proves that there is significant power gain and advantage of fairness with optimized power allocation. In both the cases, the issue of providing efficient resource allocation and channel access should be considered together with system performance guarantees on throughput and delay for individual spot-beams. On the other hand, the extremely long communication propagation distance decreases signal power significantly so that current emerging capacity (or power) allocation schemes for wireless communication systems are not supported by satellite system standard [5]. Currently the conventional schemes provide the excess fixed capacity for multi-spot beam satellite (e.g., TeleSat® ANIK F2) to guarantee any user demands with full resource reservation. However, in addition to guaranteed QoS (Quality of Service) (i.e., Dimin defined later), satellite service providers (e.g., TeleSat®) expect to increase the revenue by introducing more subscribers, therefore maximizing limited resource utilization becomes necessary. We would like to optimize bandwidth allocation among multiple spot beams. To achieve this, we set up the formulation of general power allocation optimization problem for multiple beam satellite systems. Then we use heuristic method to search the Lagrange multiplier and obtain the optimal power allocation for each spot beam in order to meet the total power constraint and individual SLA (Service Level Agreement) constraint together. II. SYSTEM OVERVIEW The DVB-S (Digital Video Broadcasting - Satellite) standard has abstracted a lot of attention in broadband satellite communications. The standards support the resource management by a set of protocols to exchange information between NCC (Network Control Center) and RCSTs (Return Channel via Satellite Terminals). The second generation (DVB-S2) includes the transmission of multimedia traffic and a variety of connection types (from broadcast to unicast or even multicast). It provides a bunch of potential services (e.g., television broadcasting, teleconference, Internet surfing, video and audio over IP (Internet Protocol), etc.) to the end-users through RCSTs [6, 7]. Interactive service with access to the Internet is BoD (Bandwidth on Demand) solution, by which users can

request satellite capacity according to their traffic demands. A centralized algorithm in NCC assigns the bandwidth to satisfy the requirement of each user. How to maximize the utilization of the limited resources remains a great challenge for the satellite service providers. Fig. 1 depicts a typical satellite network with DVB-RCS standard. Many RCSTs provides interface with the endusers. The OBP (On-Board Processor) routes the packets from uplink to downlink in a flexible way. NCC serves the satellite access request from the RCSTs and manages the OBP configuration.

revenue “R”, at time {t} by finding the optimal allocation of power {P1,…,Pn}, resulting in new capacities {C1,…,Cn}, for all i at time (t), subject to the following constraints: • Total power constraint ∑Pi≤Ptot (Ptot is the total power provided by the satellite), • SLA and QoS (Quality of Service) is maintained within each individual spot beam (i.e., Ci≥1.1 Dimin ,

•

DVB-RCS

where Dimin is minimum traffic demand that is guaranteed by dedicated bandwidth and is not considered in [1]), Pi≥0, Ci≥0, Di≥0, 1.1 Dimin ≤Ci≤Di≤ Dimax for all 1≤i≤n, where demand.

OBP

RCST

RCST

RCST RCST

NCC Fig. 1. DVB-RCS system

The requests generated by all terminals within an individual beam constitute the aggregated traffic demand of that beam. The OBP allocates the bandwidth by adjusting the power based on different traffic demands and different SLAs (Service Level Agreements) of multiple spot beams. In the DVB-RCS standard, there are four major types of capacity request (i.e., traffic demand), from highest to lowest priority: CRA (Continuous Rate Assignment), RBDC (Rate Based Dynamic Capacity), VBDC (Volume Based Dynamic Capacity) and FCA (Free Capacity Assignment). In this paper, we only consider the total capacity request of all the four traffic types instead of that of a specific traffic type within an individual spot beam. III. OPTIMAL POWER ALLOCATION III.1 Problem Formulation

Ci Di

Fig. 2. A satellite system with multiple spot-beams

Fig. 2 depicts a satellite system with multiple spot-beams. Let {C1,C2,…,Cn} be the capacity of the spot beams {S1,S2,…,Sn} respectively. The OBP provides a capacity Ci with a power Pi for the ith beam with an instantaneous traffic demand Di. We want to optimize the cost function, namely

Dimax

is the maximum traffic

III.2 Power allocation The design of any satellite system is based on two major objects: (I) satisfying a minimum SNR (Signal to Noise Ratio) for a specific period; (II) carrying the maximum revenue-earning traffic at a minimum cost [5]. In order to obtain high SNR under all conditions, any satellite link can be designed with very large antennas and high cost. To achieve the best compromise between the system performance and the cost is the target of a good system design. Without loss of generality, we make the following three assumptions that are widely accepted in the satellite system design: (I) each spot beam is equipped with an individual transponder and carries a signal only for that beam [1]; (II) all the RCSTs use the same size of antennae [1]; (III) by taking the transponder antenna gain, the receiver antenna gain and the free space path loss into account, there is a ratio γ between the transmit power Pi and the received power Pri (or Pi=γPri) within the ith spot beam [5]. So the respective SNRi can be approximated as SNRi=(Pri/W)/N0=Pi/(γWN0), where N0 is the noise power density and W is the bandwidth used. Using the time sharing scheme for Gaussian broadcast channels, we can obtain the Shannon bounded capacity [8], Pi Ci = W log 2 1 + W γ 0

.

(1)

This capacity is reached regardless of the number of uncoordinated RCSTs inside the beam when the RCSTs use the same size of antennae [1]. Based on the relationship between the capacity Ci and the power Pi, we can enhance the capacity Ci by increasing the power Pi in order to meet the instantaneous traffic demand Di with a lower limit Dimin and an upper limit Dimax . However, the total power limit of the satellite restricts the total capacity. In order to utilize the total capacity efficiently and satisfy the traffic demands of as many users as possible, we would like to minimize a cost function of the difference between {Ci} and {Di} across all the beams, for i=1,2,…,n. Considering the deviation between actual capacity Ci' and Shannon capacity Ci, we adopt 10% safety margin (i.e., Ci≥1.1 Dimin ) to guarantee the minimum traffic demand Dimin and to meet the

minimum SNR requirement. Therefore, we adopt a square deviation cost function between capacities and demands and formulate our power allocation problem as follows: n

minimize ∑ ( Di − Ci ) 2 ,

(2)

1.1Dimin ≤Ci≤Di≤ Dimax , ∀i ,

(3)

i =1

subject to

n

∑ Pi ≤ Ptot , ∀i .

(4)

i =1

Considering Equation (1), we can rewrite constraint (3) as Pi ≤ Di ≤ Dimax . 1.1Dimin ≤ Ci = W log 2 1 + γ W 0

The constraint (3) hints that we do not generate more power than required by the traffic demand, since supply more than demand (or Ci>Di) is wasteful and unnecessary. To illustrate the resource optimization in this paper, we consider the case that the total traffic demand exceeds the total capacity generated by the total power Ptot. Constraint (4) implies a limit on the total power supply Ptot. With a Lagrange multiplier λ, we can re-write the objective function (2) as: n

n

i =1

i =1

J ( Pi ) = ∑ ( Di − Ci ) 2 + λ ( ∑ Pi − Ptot ) ,

(5)

By differentiating objective function (5) with respect to Pi, we can obtain optimal beam power profile Pi which should satisfy the following equation: Pi λγ 0 ln 2 Pi = 1 + , Di − W log 2 1 + (6) 2 γW 0 γW 0 where λ is a Lagrange multiplier which will be determined from the total power constraint (4). Nonnegative λ means that Equation (6) satisfies the right restriction (i.e., Ci≤Di) in the constraint (3). The paper [1] only provided the solution for Pi in the cases of SNR1. However, there was no solution in the practical range [SNR1]. In addition, the two special cases of SNR1 go far away from the major satellite system design objective [5]. Here we want to find a closed form solution for Pi using an intuitive approximation method to unravel the relationship between the traffic demands and the beam power profiles. We can use the following heuristics method to find the optimal power Pi from nonlinear Equation (6): P (1) Let f1 ( Pi ) = Di − W log 2 1 + i and γW 0 f 2 ( Pi ) =

λγ 0 ln 2 2

Pi 1 + γW . 0

(2) Obtain Dsum = ∑ in=1 Di . We merge Dsum and Ptot to one spot beam to obtain an initial value λ0 for λ, i.e.,

λ = λ0 = Dsum − nW log 2 1 +

Ptot γnW 0

γn 0 ln 2 Ptot 1 + γnW 2 0

(3) Gradually increasing Pi with Pi ∈ [ Ptot Di /(10 Dsum ), Ptot ] until the square error between f1(Pi) and f2(Pi) is negligible enough to get Piopt . (4) We repeat step 3 iteratively for each other Pi. (5) We now have Psum = ∑in=1 Piopt . If Psum≅Ptot and Psum≤Ptot, we obtain the optimal power profile Pi. (5a) If PsumPtot, then set λmin=λ and let λ=(λmin+λmax)/2. Go to step (3). (6) If Ci≥1.1 Dimin for every spot beam i, then we obtain finally the optimal power profile Pi and satisfy all the minimum traffic demands from all the spot beams. (6a) If CjPtot, we update Lagrange multiplier λ and continue searching Pi until Psum≅Ptot and Psum≤Ptot.

.

Using binary search as rule of thumb, we set λmin=λ0/2 and λmax=2λ0. We run many different simulation scenarios and find that the optimal λ locates in the range of λmin and λmax.

i λ i λ i λ

Table 2. Modification process of Lagrange multiplier λ 1 2 3 4 5 6 7 172.4 258.6 215.6 194 183.2 177.8 180.5 8 9 10 11 12 13 14 179.2 179.8 180.2 180 173.4 260.1 216.8 15 16 17 18 19 195.1 184.2 178.8 181.5 180.2

Table 2 shows the modification process of Lagrange multiplier λ with 19 steps. We search Pi at every step until we obtain the final optimal solution for the power allocation. Actually we get the solution for the power allocation after 11th iterative search of λ. However, we find that Cj1.1 D min j further 8 iterative search.

SNRi=Pi/10>1

∑

n (D i =1 i

[5]. The initial minimum value of − Ci ) without satisfying Ci≥1.1 Dimin is 14621.56, 2

while the final minimum value of ∑in=1 ( Di − Ci ) 2 with Ci≥1.1 Dimin is 14629.24, so the deviation is 0.05%. As a whole, the optimal power allocation is completed successfully. Fig. 4 shows the capacity allocation, the minimum traffic demand Dimin and the instantaneous traffic demand Di of the multiple spot beams.

300 f1 250

f2

V. CONCLUSIONS

solution 200

We have set up a problem formulation to optimize the power allocation for multiple spot beam satellite systems. We employ a heuristic search method to find the appropriate Lagrange multiplier so that the optimal power allocation is achieved based on the different traffic demands from multiple spot beams. At the same time, the individual SLA is also guaranteed by minimum resource reservation. Scilab® simulation demonstrates that our algorithm maximizes the power utilization of the satellite effectively.

150

f(Pi)

100

50

0

-50

-100

-150 0

20

40

60

80

100

120

140

160

180

200

Pi(w)

Fig. 3. Heuristic search optimal power profile

REFERENCES

Fig. 3 depicts the plot of the functions f1(P1) and f2(P1) for the first time search of P1. The solution of the optimal power profile P1 (i.e., P1=14.04w) is pegged in the intersection as we expected. Table 3. Optimal power allocation for the multi-beam satellite system i 1 2 3 4 5 6 min 51 58 54 59 51 57 D i

1.1 Dimin

56

64

59

65

56

63

Di Ci Pi i

92 62.37 13.74 7 53

108 73.44 17.68 8 58

87 59 12.66 9 52

150 100.02 30.01 10 52

106 72.07 17.16 Sum 545

125 84.62 22.32

1.1 Dimin

58

64

57

57

599

Di Ci Pi

146 97.65 28.72

141 94.63 27.13

108 73.44 17.68

88 59.51 12.82

1151 776.75 199.92

Dimin

150 Di Ci

[1]

J. P. Choi, V. W. S. Chan, “Optimum multi-beam satellite downlink power allocation based on traffic demands”, Proceedings of IEEE Globecom, 2002, pp.2875-2881.

[2]

Telesat Inc., Telesat Annual Report, 2003-2004.

[3]

M. Neely, E. Modiano, and C. Rohrs, “Power Allocation and Routing in Multi-Beam Satellites with Time Varying Channels,” IEEE/ACM Transactions on Networking, February 2003.

[4]

A. Morell, G. Seco-Granados and M.A. Vazquez-Castro, “Joint Time Slot Optimization and Fair Bandwidth Allocation for DVB-RCS Systems”, IEEE Globecom, 2006.

[5]

Timothy Pratt, Charles Bostian and Jeremy Allnutt, Satellite Communications, 2nd Edition, John Wiley & Sons, New York, 2003.

[6]

ETSI, “Digital Video Broadcasting (DVB); Interaction Channel for Satellite Distribution Systems,” ETSI EN 301 790, April 2005.

[7]

ETSI, “Satellite Earth Stations and Systems (SES); Broadband Satellite Multimedia (BSM) Services and Architectures: QoS Functional Architecture,” ETSI TS 102 462, December 2005.

[8]

T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, 1991.

[9]

W.K. Ho, Yang Hong, A. Hansson, H. Hjalmarsson and J.W. Deng, “Relay Auto-Tuning of PID Controllers using Iterative Feedback Tuning”, Automatica, Volume 39, Issue 1, January 2003, pp.149-157.

Di_min

[10] Yang Hong, O. W.W. Yang, “Design of an Adaptive PI Rate Controller for Streaming Media Traffic Based on Gain and Phase Margins”, IEE Proceedings on Communications, Volume 153, Issue 1, February 2006, pp. 5-14.

Capacity (Mb/s)

100

[11] INRIA, Introduction to Scilab, Scilab 4.1, December 2006. 50

0 1

2

3

4

5

6

7

8

9

10

Spot Beam #

Fig. 4. Capacity allocation of multiple spot-beam satellite

Table 3 shows the result of the optimal power allocation for the multi-beam satellite system. Each individual power Pi satisfies the minimum permitted overall SNR, that is,

Citation of this paper: Y. Hong, A. Srinivasan, B. Cheng, L. Hartman, and P. Andreadis, “Optimal Power Allocation for Multiple Beam Satellite Systems,” In Proceedings of IEEE Radio and Wireless Symposium (IEEE RWS), Orlando, FL, January 2008, pp.823-826.

EION Inc., Ottawa, Onatrio, Canada 2Canadian Space Agency, St. Hubert, Quebec, Canada 3 Communications Research Center, Ottawa, Ontario, Canada E-mail: {yhong, anand, brian}@eion.com, [email protected], [email protected] Abstract—Presently, multiple spot-beam satellites are being launched with the ability to allocate power dynamically and to increase throughput, because on-board resources (e.g., power, bandwidth) are scarce and expensive. It is critical to share them efficiently among as many users as possible. In this paper, we propose an innovative power allocation algorithm for the system. We use heuristic method to search the Lagrange multiplier and obtain the optimal power allocation for each spot beam in order to meet the total power constraint and individual SLA constraint together. Scilab simulation demonstrates that our algorithm maximizes the power utilization of the satellite successfully. Index Terms—Satellite Communications, Channel Capacity

I. INTRODUCTION In the past, commercial satellites did not have the ability to support multiple beams on-board. Presently, most of the commercial satellites have the ability to support multiple spot beams enabling services to diverse geographical region. Unfortunately, all the geographical regions do not have the same capacity requirement. Neither does the future demand follow the same progression. This has led to a very interesting problem, namely how do you increase or decrease capacity adaptively based on the increase or decrease in user/service demand? Satellites do have the ability to service large number of users, albeit cumulatively. Satellites, when launched, are engineered to cater to future demands as it is expected to stay in service for at least a few decades. However, many remote areas may not use efficiently the capacity to their maximum due to sparse population, while high demands might come from an urban center. Therefore, it is a real challenge to redistribute the capacity in order to optimize the revenue. Some attempts have been made to adjust power for an optimal resource allocation in on-board processor, e.g., [1]. In addition, satellite service providers (e.g., our industrial partner TeleSat® [2]) are investing time and money to adopt innovative solutions to allocate spot beams and to increase revenue. As part of future missions, multiple spot-beam satellites are being launched with the ability to allocate power dynamically and to increase throughput, because onboard resources (e.g., power, bandwidth, transmitters, receivers and spot-beams) are scarce and expensive. It is critical to share them efficiently among as many users as possible. There are multiple ways to accomplish the end goal of increasing revenue. One of the methods to allocate power is to decrease power in some areas while increasing power in others where there is surge in user demands. Another method is to allocate multiple spot beams over a

geographical area allowing timesharing of active downlink beams. A data satellite network would have to support a broad spectrum of bursty unscheduled users with different constraints over time varying atmospheric satellite channels [3, 4]. The paper [1] proves that there is significant power gain and advantage of fairness with optimized power allocation. In both the cases, the issue of providing efficient resource allocation and channel access should be considered together with system performance guarantees on throughput and delay for individual spot-beams. On the other hand, the extremely long communication propagation distance decreases signal power significantly so that current emerging capacity (or power) allocation schemes for wireless communication systems are not supported by satellite system standard [5]. Currently the conventional schemes provide the excess fixed capacity for multi-spot beam satellite (e.g., TeleSat® ANIK F2) to guarantee any user demands with full resource reservation. However, in addition to guaranteed QoS (Quality of Service) (i.e., Dimin defined later), satellite service providers (e.g., TeleSat®) expect to increase the revenue by introducing more subscribers, therefore maximizing limited resource utilization becomes necessary. We would like to optimize bandwidth allocation among multiple spot beams. To achieve this, we set up the formulation of general power allocation optimization problem for multiple beam satellite systems. Then we use heuristic method to search the Lagrange multiplier and obtain the optimal power allocation for each spot beam in order to meet the total power constraint and individual SLA (Service Level Agreement) constraint together. II. SYSTEM OVERVIEW The DVB-S (Digital Video Broadcasting - Satellite) standard has abstracted a lot of attention in broadband satellite communications. The standards support the resource management by a set of protocols to exchange information between NCC (Network Control Center) and RCSTs (Return Channel via Satellite Terminals). The second generation (DVB-S2) includes the transmission of multimedia traffic and a variety of connection types (from broadcast to unicast or even multicast). It provides a bunch of potential services (e.g., television broadcasting, teleconference, Internet surfing, video and audio over IP (Internet Protocol), etc.) to the end-users through RCSTs [6, 7]. Interactive service with access to the Internet is BoD (Bandwidth on Demand) solution, by which users can

request satellite capacity according to their traffic demands. A centralized algorithm in NCC assigns the bandwidth to satisfy the requirement of each user. How to maximize the utilization of the limited resources remains a great challenge for the satellite service providers. Fig. 1 depicts a typical satellite network with DVB-RCS standard. Many RCSTs provides interface with the endusers. The OBP (On-Board Processor) routes the packets from uplink to downlink in a flexible way. NCC serves the satellite access request from the RCSTs and manages the OBP configuration.

revenue “R”, at time {t} by finding the optimal allocation of power {P1,…,Pn}, resulting in new capacities {C1,…,Cn}, for all i at time (t), subject to the following constraints: • Total power constraint ∑Pi≤Ptot (Ptot is the total power provided by the satellite), • SLA and QoS (Quality of Service) is maintained within each individual spot beam (i.e., Ci≥1.1 Dimin ,

•

DVB-RCS

where Dimin is minimum traffic demand that is guaranteed by dedicated bandwidth and is not considered in [1]), Pi≥0, Ci≥0, Di≥0, 1.1 Dimin ≤Ci≤Di≤ Dimax for all 1≤i≤n, where demand.

OBP

RCST

RCST

RCST RCST

NCC Fig. 1. DVB-RCS system

The requests generated by all terminals within an individual beam constitute the aggregated traffic demand of that beam. The OBP allocates the bandwidth by adjusting the power based on different traffic demands and different SLAs (Service Level Agreements) of multiple spot beams. In the DVB-RCS standard, there are four major types of capacity request (i.e., traffic demand), from highest to lowest priority: CRA (Continuous Rate Assignment), RBDC (Rate Based Dynamic Capacity), VBDC (Volume Based Dynamic Capacity) and FCA (Free Capacity Assignment). In this paper, we only consider the total capacity request of all the four traffic types instead of that of a specific traffic type within an individual spot beam. III. OPTIMAL POWER ALLOCATION III.1 Problem Formulation

Ci Di

Fig. 2. A satellite system with multiple spot-beams

Fig. 2 depicts a satellite system with multiple spot-beams. Let {C1,C2,…,Cn} be the capacity of the spot beams {S1,S2,…,Sn} respectively. The OBP provides a capacity Ci with a power Pi for the ith beam with an instantaneous traffic demand Di. We want to optimize the cost function, namely

Dimax

is the maximum traffic

III.2 Power allocation The design of any satellite system is based on two major objects: (I) satisfying a minimum SNR (Signal to Noise Ratio) for a specific period; (II) carrying the maximum revenue-earning traffic at a minimum cost [5]. In order to obtain high SNR under all conditions, any satellite link can be designed with very large antennas and high cost. To achieve the best compromise between the system performance and the cost is the target of a good system design. Without loss of generality, we make the following three assumptions that are widely accepted in the satellite system design: (I) each spot beam is equipped with an individual transponder and carries a signal only for that beam [1]; (II) all the RCSTs use the same size of antennae [1]; (III) by taking the transponder antenna gain, the receiver antenna gain and the free space path loss into account, there is a ratio γ between the transmit power Pi and the received power Pri (or Pi=γPri) within the ith spot beam [5]. So the respective SNRi can be approximated as SNRi=(Pri/W)/N0=Pi/(γWN0), where N0 is the noise power density and W is the bandwidth used. Using the time sharing scheme for Gaussian broadcast channels, we can obtain the Shannon bounded capacity [8], Pi Ci = W log 2 1 + W γ 0

.

(1)

This capacity is reached regardless of the number of uncoordinated RCSTs inside the beam when the RCSTs use the same size of antennae [1]. Based on the relationship between the capacity Ci and the power Pi, we can enhance the capacity Ci by increasing the power Pi in order to meet the instantaneous traffic demand Di with a lower limit Dimin and an upper limit Dimax . However, the total power limit of the satellite restricts the total capacity. In order to utilize the total capacity efficiently and satisfy the traffic demands of as many users as possible, we would like to minimize a cost function of the difference between {Ci} and {Di} across all the beams, for i=1,2,…,n. Considering the deviation between actual capacity Ci' and Shannon capacity Ci, we adopt 10% safety margin (i.e., Ci≥1.1 Dimin ) to guarantee the minimum traffic demand Dimin and to meet the

minimum SNR requirement. Therefore, we adopt a square deviation cost function between capacities and demands and formulate our power allocation problem as follows: n

minimize ∑ ( Di − Ci ) 2 ,

(2)

1.1Dimin ≤Ci≤Di≤ Dimax , ∀i ,

(3)

i =1

subject to

n

∑ Pi ≤ Ptot , ∀i .

(4)

i =1

Considering Equation (1), we can rewrite constraint (3) as Pi ≤ Di ≤ Dimax . 1.1Dimin ≤ Ci = W log 2 1 + γ W 0

The constraint (3) hints that we do not generate more power than required by the traffic demand, since supply more than demand (or Ci>Di) is wasteful and unnecessary. To illustrate the resource optimization in this paper, we consider the case that the total traffic demand exceeds the total capacity generated by the total power Ptot. Constraint (4) implies a limit on the total power supply Ptot. With a Lagrange multiplier λ, we can re-write the objective function (2) as: n

n

i =1

i =1

J ( Pi ) = ∑ ( Di − Ci ) 2 + λ ( ∑ Pi − Ptot ) ,

(5)

By differentiating objective function (5) with respect to Pi, we can obtain optimal beam power profile Pi which should satisfy the following equation: Pi λγ 0 ln 2 Pi = 1 + , Di − W log 2 1 + (6) 2 γW 0 γW 0 where λ is a Lagrange multiplier which will be determined from the total power constraint (4). Nonnegative λ means that Equation (6) satisfies the right restriction (i.e., Ci≤Di) in the constraint (3). The paper [1] only provided the solution for Pi in the cases of SNR1. However, there was no solution in the practical range [SNR1]. In addition, the two special cases of SNR1 go far away from the major satellite system design objective [5]. Here we want to find a closed form solution for Pi using an intuitive approximation method to unravel the relationship between the traffic demands and the beam power profiles. We can use the following heuristics method to find the optimal power Pi from nonlinear Equation (6): P (1) Let f1 ( Pi ) = Di − W log 2 1 + i and γW 0 f 2 ( Pi ) =

λγ 0 ln 2 2

Pi 1 + γW . 0

(2) Obtain Dsum = ∑ in=1 Di . We merge Dsum and Ptot to one spot beam to obtain an initial value λ0 for λ, i.e.,

λ = λ0 = Dsum − nW log 2 1 +

Ptot γnW 0

γn 0 ln 2 Ptot 1 + γnW 2 0

(3) Gradually increasing Pi with Pi ∈ [ Ptot Di /(10 Dsum ), Ptot ] until the square error between f1(Pi) and f2(Pi) is negligible enough to get Piopt . (4) We repeat step 3 iteratively for each other Pi. (5) We now have Psum = ∑in=1 Piopt . If Psum≅Ptot and Psum≤Ptot, we obtain the optimal power profile Pi. (5a) If PsumPtot, then set λmin=λ and let λ=(λmin+λmax)/2. Go to step (3). (6) If Ci≥1.1 Dimin for every spot beam i, then we obtain finally the optimal power profile Pi and satisfy all the minimum traffic demands from all the spot beams. (6a) If CjPtot, we update Lagrange multiplier λ and continue searching Pi until Psum≅Ptot and Psum≤Ptot.

.

Using binary search as rule of thumb, we set λmin=λ0/2 and λmax=2λ0. We run many different simulation scenarios and find that the optimal λ locates in the range of λmin and λmax.

i λ i λ i λ

Table 2. Modification process of Lagrange multiplier λ 1 2 3 4 5 6 7 172.4 258.6 215.6 194 183.2 177.8 180.5 8 9 10 11 12 13 14 179.2 179.8 180.2 180 173.4 260.1 216.8 15 16 17 18 19 195.1 184.2 178.8 181.5 180.2

Table 2 shows the modification process of Lagrange multiplier λ with 19 steps. We search Pi at every step until we obtain the final optimal solution for the power allocation. Actually we get the solution for the power allocation after 11th iterative search of λ. However, we find that Cj1.1 D min j further 8 iterative search.

SNRi=Pi/10>1

∑

n (D i =1 i

[5]. The initial minimum value of − Ci ) without satisfying Ci≥1.1 Dimin is 14621.56, 2

while the final minimum value of ∑in=1 ( Di − Ci ) 2 with Ci≥1.1 Dimin is 14629.24, so the deviation is 0.05%. As a whole, the optimal power allocation is completed successfully. Fig. 4 shows the capacity allocation, the minimum traffic demand Dimin and the instantaneous traffic demand Di of the multiple spot beams.

300 f1 250

f2

V. CONCLUSIONS

solution 200

We have set up a problem formulation to optimize the power allocation for multiple spot beam satellite systems. We employ a heuristic search method to find the appropriate Lagrange multiplier so that the optimal power allocation is achieved based on the different traffic demands from multiple spot beams. At the same time, the individual SLA is also guaranteed by minimum resource reservation. Scilab® simulation demonstrates that our algorithm maximizes the power utilization of the satellite effectively.

150

f(Pi)

100

50

0

-50

-100

-150 0

20

40

60

80

100

120

140

160

180

200

Pi(w)

Fig. 3. Heuristic search optimal power profile

REFERENCES

Fig. 3 depicts the plot of the functions f1(P1) and f2(P1) for the first time search of P1. The solution of the optimal power profile P1 (i.e., P1=14.04w) is pegged in the intersection as we expected. Table 3. Optimal power allocation for the multi-beam satellite system i 1 2 3 4 5 6 min 51 58 54 59 51 57 D i

1.1 Dimin

56

64

59

65

56

63

Di Ci Pi i

92 62.37 13.74 7 53

108 73.44 17.68 8 58

87 59 12.66 9 52

150 100.02 30.01 10 52

106 72.07 17.16 Sum 545

125 84.62 22.32

1.1 Dimin

58

64

57

57

599

Di Ci Pi

146 97.65 28.72

141 94.63 27.13

108 73.44 17.68

88 59.51 12.82

1151 776.75 199.92

Dimin

150 Di Ci

[1]

J. P. Choi, V. W. S. Chan, “Optimum multi-beam satellite downlink power allocation based on traffic demands”, Proceedings of IEEE Globecom, 2002, pp.2875-2881.

[2]

Telesat Inc., Telesat Annual Report, 2003-2004.

[3]

M. Neely, E. Modiano, and C. Rohrs, “Power Allocation and Routing in Multi-Beam Satellites with Time Varying Channels,” IEEE/ACM Transactions on Networking, February 2003.

[4]

A. Morell, G. Seco-Granados and M.A. Vazquez-Castro, “Joint Time Slot Optimization and Fair Bandwidth Allocation for DVB-RCS Systems”, IEEE Globecom, 2006.

[5]

Timothy Pratt, Charles Bostian and Jeremy Allnutt, Satellite Communications, 2nd Edition, John Wiley & Sons, New York, 2003.

[6]

ETSI, “Digital Video Broadcasting (DVB); Interaction Channel for Satellite Distribution Systems,” ETSI EN 301 790, April 2005.

[7]

ETSI, “Satellite Earth Stations and Systems (SES); Broadband Satellite Multimedia (BSM) Services and Architectures: QoS Functional Architecture,” ETSI TS 102 462, December 2005.

[8]

T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, 1991.

[9]

W.K. Ho, Yang Hong, A. Hansson, H. Hjalmarsson and J.W. Deng, “Relay Auto-Tuning of PID Controllers using Iterative Feedback Tuning”, Automatica, Volume 39, Issue 1, January 2003, pp.149-157.

Di_min

[10] Yang Hong, O. W.W. Yang, “Design of an Adaptive PI Rate Controller for Streaming Media Traffic Based on Gain and Phase Margins”, IEE Proceedings on Communications, Volume 153, Issue 1, February 2006, pp. 5-14.

Capacity (Mb/s)

100

[11] INRIA, Introduction to Scilab, Scilab 4.1, December 2006. 50

0 1

2

3

4

5

6

7

8

9

10

Spot Beam #

Fig. 4. Capacity allocation of multiple spot-beam satellite

Table 3 shows the result of the optimal power allocation for the multi-beam satellite system. Each individual power Pi satisfies the minimum permitted overall SNR, that is,

Citation of this paper: Y. Hong, A. Srinivasan, B. Cheng, L. Hartman, and P. Andreadis, “Optimal Power Allocation for Multiple Beam Satellite Systems,” In Proceedings of IEEE Radio and Wireless Symposium (IEEE RWS), Orlando, FL, January 2008, pp.823-826.