3312 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 5, MAY 2016

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1 3312 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 5, MAY 2016 L -Problem-Based Transmission Rate Allocation With Packet Loss and Power Metrics Over Satellite Networks Igor Bisio, Senior Member, IEEE, Stefano Delucchi, Student Member, IEEE, Fabio Lavagetto, and Mario Marchese, Senior Member, IEEE Abstract This aer tackles the classical roblem of transmission rate allocation in satellite networks where fading may negatively imact communications. Within the framework of multiobjective rogramming MOP), this aer introduces a transmission rate allocation criterion among Earth stations ESs) called L -roblem-based rate allocation L RA). The allocations roved by L RA are reresentative of a comromise among the need of different erformance metrics such as acket loss and transmission ower TP). This aer determines the condition for the existence and the value of a L RA transmission rate allocation bound R bound, to which the transmission rate globally allocated by L RA converges when the overall available transmission rate R TOT tends to infinity. The erformance analysis, which is carried out through simulations under different satellite channel conditions, is aimed at investigating L RA features, at showing the existence of the rate bound and the advantages concerning the rate allocation given by using L RA, and at comaring L RA with two other schemes in the literature concerning allocated rate, acket loss rate, transmit ower, and execution time. Index Terms L -roblem-based transmission rate allocation, multiobjective rogramming MOP), erformance analysis, satellite communications. I. INTRODUCTION AN emergency network is designed to rove reliable communications during emergency situations and when disasters suddenly strike a certain area. A challenge that arises with disasters is that the telecommunication services both roved by cellular networks e.g., third-generation and Long- Manuscrit received August 20, 2014; revised March 26, 2015; acceted May 14, Date of ublication June 19, 2015; date of current version May 12, The review of this aer was coordinated by Prof. C. Assi. I. Bisio is with the Digital Signal Processing Laboratory and Satellite Communications and Networking Laboratory, Deartment of Naval, Electrical, Electronic, and Telecommunications Engineering, University of Genoa, Genoa, Italy igor.bisio@unige.it). S. Delucchi is with the Satellite Communications and Networking Laboratory and Telecommunications Networks and Telematics Laboratory, Deartment of Naval, Electrical, Electronic, and Telecommunications Engineering, University of Genoa, Genoa, Italy, and also with Aitek S..A, Genoa, Italy stefano.delucchi@unige.it). F. Lavagetto is with the Digital Signal Processing Laboratory, Deartment of Naval, Electrical, Electronic, and Telecommunications Engineering, University of Genoa, Genoa, Italy fabio.lavagetto@unige.it). M. Marchese is with the Satellite Communications and Networking Laboratory, Deartment of Naval, Electrical, Electronic, and Telecommunications Engineering, University of Genoa, Genoa, Italy mario.marchese@ unige.it). Color versions of one or more of the figures in this aer are available online at htt://ieeexlore.ieee.org. Digital Object Identifier /TVT Fig. 1. Reference network. Term Evolution) and Internet infrastructures are usually interruted. In order to deal with this challenge, designing an efficient disaster resilient network has recently gained a significant interest. Satellite communication networks have been consered a leading technology in this domain. Satellites assure the continuous availability of wireless communication channels and can be exloited for the fast deloyment of urgently required communication suorts. Coherently with the state of the art in the field see [1] [5] among many others), we conser a ractical disaster network scenario, which is shown in Fig. 1, comosed of a number of mobile nodes MNs) transmitting traffic flows through access nodes, which are called mobile satellite gateways MSGs). MSGs forward traffic to a shared satellite channel. A grou of MNs and a single MSG define a single Earth station ES). ESs are deloyed in different ones of the disaster area. Z is the overall number of ESs. Traffic flow management and variable satellite channel quality due to several imairments, such as rain fading, limited energy, and bandwth constraints are toical roblems in this field and have a great imact on erformance. Otimiing resource management reresents a key research issue in such environment [6]. In this framework, this aer rooses an algorithm to allocate transmission rate to ESs. The algorithm is entified as L -roblem-based transmission rate allocation L RA) and addresses two aims: maximiing communication erformance by limiting the acket losses and, simultaneously, minimiing the energy consumtion by limiting the transmission ower TP) IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// for more information.

2 BISIO et al.: L RA PROBLEM-BASED TRANSMISSION RATE ALLOCATION 3313 The choice of the secific satellite environment does not affect the general behavior of the allocation scheme roosed in the aer, and as a consequence, it has been left unsecified for the sake of generality. Possible alication environments are: geostationary Earth orbit GEO), medium Earth orbit, and low Earth orbit satellites, and high-altitude latforms HAPs). The main difference among them stands in the roagation delay and consequent round tri time RTT). This aer comares L RA with alternative resource allocation methods through simulations. In summary, this aer contains the following contributions: a survey of the state of the art about resource allocation in satellite and wireless communications, in Section II; the resentation of the L RA, which is the extension of the MOP-based method introduced in [5]; the definition of the existence conditions of a rate bound R bound, which arisen from L RA, to which the overall L RA allocation converges when the system transmission rate availability tends to infinity, in Section IV; the analytical formulations emloyed to model the used erformance metrics: acket loss robability PLP) and TP,inSectionV; the verification of the Transmission Rate Bound existence conditions resented in Section IV for L RA, in Section VI; a simulative erformance evaluation of L RA and a comarison with other aroaches available in the literature, in Section VII. The conclusions are reorted in Section VIII. II. STATE OF THE ART Resource allocation over satellite and wireless channels is a well-investigated toic. The most consered resource by allocation algorithms is either the transmission rate, which is exressed in bits er second, available for traffic flows, or the transmission bandwth, which is exressed in hert, often simly referred to as bandwth. Due to the multile use of the terms transmission rate and bandwth, whose meaning in the literature is also a consequence of the different reference scientific community e.g., communications, comuting, networking, etc.), to avo misunderstanding, we refer secifying the definition of bandwth that we use in this aer and the mathematical relation between bandwth and transmission rate. The transmission bandwth is the measure in hert of the wth of the range of frequency where a given th ES with [1,Z] or the overall system comosed of Z ES oerates. Let W be the bandwth of the th station, the transmission rate R available for the th station is given as done in [2], [7] [9]) by the Hartley Shannon law for a white Gaussian channel as follows: R = W log1 + h TP ) [1,Z] 1) where TP is the TP in watts, and h is the channel gain of the th station. h is defined in 27) of this aer, which is shown below. In the ast, the aim of allocation algorithms was the maximiation of the transmission rate, which is emloyed by a single entity, to imrove the quality of communications [1], [7], [8]. The action was carried out without consering ower, or energy, consumtion. More recently, algorithms are for joint transmission rate and ower allocation, e.g., in [2], [3], and [9] [15]. L RA may be included in this category. Most joint transmission rate and ower allocation schemes in the state of the art may be classified in one of the following formulations. A. Sum Caacity Maximiation With Constrained Power The total amount of ower is not art of the cost function, but it is constrained under a given threshold. This aroach maximies the weighted sum of the transmission rates R = R 1,...,R,...,R Z ) assigned to each entity through two sets of variables: bandwth W =W 1,...,W,...,W Z ) and ower P =P 1,...,P,...,P Z ), The roblem is formalied in the following: max α R W, P) W,P 2) W W TOT ; P P TOT α =α 1,...,α,...,α Z ) is a vector of weights. Formulation in 2) reresents a generalied version of a grou of methods alied in different scenarios and reorted in the following. In more detail, the algorithm roosed in [2] is aimed at maximiing the weighted sum of downlink transmission rates by allocating downlink bandwth and ower to a given number of users in a wireless acket data system. In [9], an allocation roblem was formulated to maximie the information rate in a multiho network based on an orthogonal frequency-division multilexing OFDM) technique with ower and subcarrier constraints. In [10], an adative radio resource allocation algorithm for different traffic flows was described. Consered resources are ower and bandwth and are assigned first to real-time users to guarantee their requirements. Remaining resources are allocated to non-real-time users. The aim of the algorithm is to maximie the weighted sum of nonreal-time user throughut. In [11], the goal is to maximie a quantity defined as average energy efficiency EE. In [11], the acket loss effect in wireless transmissions is taken into account, articularly the ratio between successfully delivered bits and total consumed ower. The sum of allocated transmission rates is not exlicitly taken into account; nevertheless, the mentioned average energy efficiency is a function of the bandwth allocated to each entity. B. Power Minimiation With Constrained Caacity A second set of algorithms, such as in [3] and [12] [14], which are groued here as ower minimiation PM), is aimed at minimiing the TP by the variable bandwths W = W 1,...,W,...,W Z ), i.e., min β P W, R) W 3) W W TOT ; R R th [1,Z]

3 3314 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 5, MAY 2016 Fig. 2. Proosed model for a hysical entity. β =β 1,...,β,...,β Z ) is a vector of weights. The transmission rate emloyed by each entity must be greater or equal to R th to assure a certain level of quality for the communication. III. L -PROBLEM-BASED ALLOCATION A. Model of the ES The model roosed in this aer cannot be included in the two grous resented in Section II; it uses MOP theory, which is detailed in Section III-D, and is based on three main comonents: hysical entities, virtual entities, and objective functions. A hysical entity is a system such as a satellite ES. A virtual entity is a comonent within a hysical system such as a single-air buffer server. Each virtual entity is reresented by a grou of objective functions that model erformance arameters such as, in this aer, PLP and TP. Fig. 2 schematically reresents the roosed model for a single hysical entity. B. Aim and General Structure of the Proosed Allocation Algorithm Assuming overall transmission rate R TOT and bandwth W TOT, L RA distributes R TOT among the Z stations obviously, it is true that Z R R TOT and Z W W TOT ). The transmission rate allocation is carried out by a centralied decision maker, which slits R TOT among all virtual entities. The rate allocated to each hysical entity is the sum of the rates allocated to each related virtual entities. The centralied decision-maker is localied within an ES or in the satellite/hap itself, if switching on board is allowed) that reresents the master station MS). Each satellite channel can be corruted by ath loss, noise, and fading. We suose that each ES alies a forward-error-correcting code as a countermeasure and may adatively change the amount of redundancy bits e.g., the correction ower of the code) deending on the channel status. It imlies, in ractice, a reduction of the rate to transmit the information. Consequently, not only a change of the offered load but also different channel conditions imly a modification of the transmission rate that needs to kee a given level of quality of service for the communication. Different from other aroaches in the literature, which focus on the otimiation through a single metric and satisfy the other metrics by using constraints, this aer consers all the metrics at the same time through a multiobjective otimiation and finds a comromise solution among the needs of the different metrics. As roosed in revious works of the same authors see Section II), this aer also consers the transmission rate, which is exressed in bits er second, as the shared resource and the allocation rocess as a cometitive roblem where each entity i.e., ES) accessing the shared available transmission rate is reresented by a grou of functions that need to be otimied. These functions model the used metrics PLP and TP) used, in this aer, as a function of the transmission rate allocated to the entity, because the metrics are ossibly in contrast with each other, the allocation must necessarily reresent a comromise. Multiobjective rogramming MOP) theory defines the multiobjective otimiation roblem and reresents the theoretical reference of the roosed allocation algorithm. C. Previous Scientific Work of the Same Authors This aer originated with [4], where MOP was used to allocate resources over satellite communications by the authors for the first time. In [4], only acket loss is used as a erformance metric. It is extended in [16] by also including acket delay. The first time that an MOP-based transmission rate allocation was resented to minimie, jointly, acket loss and ower, was in [5], which also contained the original ea, the formulation, and the reliminary results. The existence of a transmission rate bound was observed through simulation results in [17], whereas the consequent ossible bandwth saving was exerimentally shown in [18]. This aer starts from the original ea in [5], resents the overall transmission rate allocation scheme, formally checks the existence of the rate bound, and resents a dee erformance evaluation, which is aimed at showing the ractical imact of the resented algorithm. D. Multiobjective Programming Theoretical Framework As in [19], a MOP roblem is defined as min {f 1 x),...,f i x),...,f k x)} subject to x S S = {x R n gx) =g 1 x),g 2 x),...,g m x))},k 2 4) where f i : R n R i [1,k] are the objective functions that comose the objective function vector fx)= {f 1 x),f 2 x),...,f k x)}. x is the decision vector that belongs to a feasible region S, which is a subset of the sace R n. In this aer, the objective functions reresent erformance metrics i.e., acket loss and RP) that need to be otimied. The feasible region is the set of all available resources i.e., the overall available transmission rate R TOT ) shared among the virtual entities. The set of solutions of the roblem described in 4) is called the Pareto otimal oint POP) set, which contains all the accetable solutions of the MOP roblem. According to [19], a formal definition of Pareto otimality is the following: A decision vector x S is Pareto Otimal if another decision vector x S does not exist such that f i x) f i x ) for all i = 1,...,k,andiff j x) <f j x ) for at least one index j.the definition ractically means that any other decision vector that

4 BISIO et al.: L RA PROBLEM-BASED TRANSMISSION RATE ALLOCATION 3315 imroves the value of an object function without worsening, at least, another one does not exist. E. Transmission Rate Allocation Model As sa earlier, the transmission rate allocation roblem is modeled as a MOP roblem in this aer. The system is comosed of Z hysical entities; each hysical entity is entified by the index [1,Z]. Y is the number of virtual entities of the th hysical entity. Each virtual entity is entified by y [1,Y ]. M y is the number of objective functions for each virtual entity y. Each objective function, of a given y th virtual entity, is entified by the index m [1,M y ]. R y is the rate allocated to the virtual entity y of the hysical entity, i.e., R =R 11,R 21,R 31,...,R Y1,...,R 1Z,R 2Z,R 3Z,...,R YZ ) 5) is the vector that contains the rate allocated to each virtual entity, and Y R = R y 6) y=1 is the rate allocated to the hysical entity. F m,y R) is the mth objective function, which is analytically defined in Section V, of the yth virtual entity of the th hysical entity. The full set of objective functions is contained in the vector FR) = F 1,11 R),...,F M11,1 1 R),... ) F 1,YZ R),...,F MYZ,Y Z R). 7) Given the definitions above and being R TOT the overall available transmission rate, i.e., shared by all Z entities, the following constraint must hold: Y R y R TOT. 8) y=1 Transmission rate allocation is defined as an MOP roblem through the following, which must be solved under constraint 8) that defines the feasibility region: R ot =R 11,ot,R 21,ot,...,R Y1,ot,... R 1Z,ot,R 2Z,ot,...,R YZ,ot) =argmin FR) R R y 0 y [1,Y ] [1,Z]. 9) The set of solutions derived from 9) is called the POP set. In general, getting the overall POP set is not simle, but the structure of the objective functions hels take the decision in some cases. For examle, it is simle to rove that given roblem 9), subject to the constraint 8), if all objective functions are strongly decreasing [19], i.e., decreasing for all its variables and strictly decreasing for at least one function and one variable, then a solution R is a POP if and only if the solution is on the constraint boundary, i.e., Y R y = R TOT. 10) y=1 This is the case we have consered in [4] and [16]. It is also true that, given the inequality constraint 8), if all objective functions are decreasing, all the oints on the constraint boundary are POP solutions, but not all POP solutions necessarily belong to the constraint, as well as oints for which Y R y <R TOT 11) y=1 can be POP solutions. The strongly decreasing assumtion concerning the objective-function vector is quite tyical because common erformance functions alied in telecommunication networks such as PLP, acket delay, and acket jitter are quantities that decrease their values when the allocated caacity value increases. This is not true if other imortant metrics are also used: ower and rocessing and comutation effort. It is simle to rove that, given roblem 9) and constraint 8), if at least one function is strongly increasing, i.e., increasing for all its variables and strictly increasing for at least one variable, all the oints inse the feasibility region and on the constraint boundary may be POPs. F. L -Problem-Based Caacity Allocation Criterion Otimal allocations are chosen among POPs, and each POP is otimal from the Pareto viewoint. Nevertheless, for oerative reasons, it is necessary to choose one solution i.e., one transmission rate allocation). A ossibility, used also in this aer, is selecting a single POP minimiing the distance, in the sense of L -roblem [19], with a reference goal oint. The ea is to allocate transmission rate, within the POP set 9), so that the value of each objective function is as close as ossible to its eal value. The set of eal rates [i.e., the eal vector 12)] is defined as comosed of the eal decision variable vector elements y, for which F k,y attains the otimum value and may be known having information about the features of the objective functions, as discussed in Section III-E and as exlained in the following. This definition of the eal transmission-rate set is not the only ossible choice, e.g., if hard constraints on metrics were given, the eal vector may contain the minimum rate allocations to assure these constraints, i.e., = 1 1,,RF k,y 2 1,,...,RF k,y Y 1,,... ) 1 Z,,RF k,y 2 Z,,...,RF k,y Y Z, k [1,M y ] y [1,Y ] [1,Z]. 12) Each element y, can assume a value between 0 and R TOT, indeendently of any hysical constraint and of the values of the other comonents of vector 7). It is called eal utoian) for this. For examle, if a generic objective function is decreasing versus transmission rate, it is obvious that it is eal allocating all the ossible rate R TOT, whereas if it is increasing versus rate, it is eal to allocate no rate at all. The values of vector 12) are consered known in the remainder of this aer. The vector

5 3316 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 5, MAY 2016 in the following contains each objective function attaining its eal value: ) ) F = F 1,11,,...,F k,y,,... R F 1,1 1 F MYZ,Y Z, )) R F M YZ,Y Z. 13) The otimal transmission rates allocated based on the roosed L RA criterion are reorted in R all =R 11,all,R 21,all,...,R Y1,all,...,R 1Z,all,... R 2Z,all,...,R YZ,all) =arg min Ĵ R) 14) R R ot where Y M y Ĵ R) = Fk,y w k,y ) y=1 k=1 + F k,y, subject to the constraint reorted in 8) and where M y w k,y =1,w k,y >0 k [1,M y ] k=1 ) 1 15) y [1,Y ] [1,Z] 16) so to assure that the solution is chosen in the POP set defined by 9), i.e., to guarantee the Pareto otimality of the solution, as roven in the Aendix. In ractice, we select the oint inse the POP set that minimies the -norm i.e., the distance ) with resect to the utoia oint. The use of weights w k,y and of different norms allows allocating transmission rate to virtual entities by differentiating the imortance of the erformance metrics for different virtual entities u to neglecting one or more metrics if necessary. Section VII contains a comarative erformance analysis, which is carried out by varying weight combination. From the oerative viewoint, 15) can be simlified because the exonent 1/) can be droed without affecting the L -roblem solution. L -roblems with or without the mentioned exonent are equivalent for 1 [19,. 68]. As a consequence L RA is written and solved by using the following: R all =R 11,all,R 21,all,...,R Y1,all,... R 1Z,all,R 2Z,all,...,R YZ,all) =arg min J R) 17) R R ot Y M y J R) = Fk,y w k,y ) y=1 k=1 + F k,y, ). 18) From the imlementation viewoint, the roosed method can be alied to traditional dynamic time-division multile-access TDMA) method usually alied to satellite systems [20]. Indeed, TDMA is a method used to enable multile ESs to transmit intermittently on the same frequency, but with the timing of their transmissions so arranged that the bursts do not overlay when they arrive at the satellite but arrive in sequence and thus are all successfully received by the receivers. The oeration of TDMA requires an outlink control to all the ESs that contains some control information. This outlink carrier also had a frame structure that roves accurate timing information for all the ESs: the burst time lan BTP). This aroach include an MS, which is often called network control center, that tells each ES what articular time slot to use in the TDMA frame, and this time lan information is broadcast to all ESs eriodically. In general, the BTP may be fixed, to allocate each ES a articular roortion of the total TDMA frame time, or may be dynamic, whereby the time slot allocated is adjusted in resonse to the rate needs of each ES. The latter aroach is comatible with the allocation method roosed in this aer: Each ES communicates its traffic arameters values i.e., the arameters needed to define the objective functions), and through the same outlink control channel, the TDMA BTP is broadcast to inform all ESs with the timing lan obtained by running the roosed allocation aroach. This BTP might be alied unchanged if the allocation does not require different rate distribution, or it might be changed every few seconds according to the allocation result. This means that the roosed L RA criterion does not need articular imlementation solution and can be easily alied to the wely emloyed TDMA-based satellite networks. IV. TRANSMISSION RATE BOUND A. Condition for the Existence of the Transmission Rate Bound L RA exressed in 17) and 18) is a function of the rate vectorr. Nevertheless, if the erformancemetricsare decreasing functions of the transmission rate such as acket loss, as in this aer, and delay), the related comonent of the eal vector 12) is the overall available transmission rate R TOT.Asa consequence, R TOT is a arameter of the cost 18) that may be written as J R,R TOT ). It is imortant to remark that R TOT is not a variable of the minimiation rocess in 18), but it is a arameter influencing the value 18), as should be clearer from Section V-C, where the form of 18) will be secified. This section determines the conditions for the existence of a finite bound, R bound, indeendent of R TOT, where the sum of the transmission rates allocated by L RA converges when R TOT tends to. The conditions for the existence of a unique solution R all that minimies J R,R TOT ) are the following. Condition 1: The solution must reresent a coordinate of an equilibrium oint, i.e., at least a vector R all so that J R 11,R 21,...,R Y1,...,R 1Z,R 2Z,...,R YZ,R TOT ) R y = J R,R TOT ) R y = 0 y [1,Y ] [1,Z]. 19)

6 BISIO et al.: L RA PROBLEM-BASED TRANSMISSION RATE ALLOCATION 3317 Condition 2: The Hessian matrix of roblem 17) only with resect to the vector R all, HR all ) must be ositive semefinite in the same oint in which Condition 1 is verified, i.e., The constraint is defined by the overall amount R TOT available transmission rate in 8). of det [HR all )] 0 R y [0,R TOT ). 20) Moreover, if R all =R 11,all,R 21,all,...,R Y1,all,...,R 1Z,all, R 2Z,all,...,R YZ,all) must be indeendent of R TOT when R TOT, the following condition must hold. Condition 3: lim R y,allr TOT )=Ry bound R TOT y [1,Y ] [1,Z]. 21) Due to Condition 1, it is clear that Condition 3 is equivalent to J R,R TOT ) lim = J,y R) < R TOT R y y [1,Y ] [1,Z]. 22) In ractice, the limits of the artial derivatives of function J R,R TOT ) as R TOT aroaches to infinity must exist, must be finite, and must be function only of the rate vector R. If Ry bound exists y [1,Y ] [1,Z],thevalueR bound is defined as R bound = Y Ry bound,r bound <R TOT. 23) y=1 1) Imact of the Transmission Rate Bound on the Allocation Scheme: If R TOT <R bound, the global transmission rate allocated by L RA is R TOT in ractice, all the available transmission rate is used). If R TOT R bound, L RA globally allocates a global transmission rate below R bound but infinitesimally close to it. If R TOT = R bound ɛ, whereɛ is an arbitrarily small ositive number, L RA allocates a global transmission rate of R TOT, whereas if R TOT = R bound, L RA allocates globally a transmission rate R all : R bound ɛ<r all < R bound. Practically, if R TOT R bound, the L -roblem roves aroximately the same solution i.e., the caacity allocation among entities does not change meaningfully by increasing R TOT ), which means that, when R TOT R bound, the system erformance does not ractically change even if the overall available transmission rate indefinitely grows. In ractice, given a certain R TOT if the value of R bound is lower or equal to R TOT, it is ossible to avo allocating the amount of transmission rate [R TOT R bound ], without erformance detriment. V. O BJECTIVE FUNCTIONS AND COST FUNCTIONS In the remainder of this aer, each hysical entity reresents one ES that transmits through a satellite channel and is roved with a single buffer i.e., one virtual entity each) that receives a secific transmission rate allocation. As a consequence, hysical and virtual entities are not differentiated y ). Each consered entity is reresented by two objective functions: PLP, i.e., F 1,1 = P loss R ) and TP, i.e., F 2,1 = W tx R ). A. Packet Loss Probability Function The PLP model used in this aer consers a transmission control rotocol TCP)-based traffic. It has been defined, including all arameters value, in [21], and it is reorted in P loss R )= k N 2 CR R RTT l + Q ) 2 24) k is a constant deending on TCP arameters, and N is the number of active MNs. Each MN of station generates a single TCP connection for the th station. In ractice, there are N TCP connections for each th station. Q is the buffer sie for the MSG of the th station. RTT is the RTT, and l is the TCP acket sie exressed in bytes. The arameters values used in the erformance evaluation are secified in Section VII. CR and R are the code and transmission rate allocated to the th station, resectively. Channel conditions vary over time, and in this aer, the exerienced carrier-to-noise ratio C/N) for each station reresents the satellite channel status. C/N) includes both a free-sace loss FSL) comonent, also used in Section V-B to model TP, and a rain attenuation comonent, which is ignored concerning the TP. Each ES suorts different code rates CR deending on the channel status. Code rates are assigned in this aer to allow consering feasible the assumtion that acket losses are only due to congestion because channel errors may be consered negligible due to the alication of suitable code rates. This assumtion allows consering PLP and TP indeendent of each other. In the following, we rewrite 24), and it will be useful in Section VI to simlify mathematical tractability: P loss R )= A D R + Q ) 2 25) where A N )=k N 2,andD CR )=CR RTT)/l are ositive constants. B. Transmission Power Function By assigning a bandwth W, the model of the TP TP of the th station is reorted in ) TP h,r )= 2 R 1 W 1. 26) h The constant h > 0, which is defined the following, takes into account the arameters whose numerical values for erformance evaluation are contained in Section VII related to the link budget, i.e., th station transmission antenna gain G T, satellite receiver antenna gain G R common for each station), Boltmann s constant k, and noise temerature T consering additive white Gaussian noise), channel bandwth W = W TOT, and FSL): 1 h = k T W FSL G T G R. 27)

7 3318 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 5, MAY 2016 The TP function 26) is obtained by inverting the Hartley Shannon law as follows: where C R = W log N ) FSL ) 28) ) FSL C = G T G R TP N k T W FSL = TP 29) h is the carrier-to-noise ratio [22], due to FSL comonent. C. Analytical Definition of the Objective Function Vector FR) and of the Cost J ) Both objective functions 24) and 26) are continuous and differentiable in R so assuring the existence of a solution of the L roblem. The analytical definition of the objective function vector FR), which is introduced in 7), is reorted in the following, with y, R 11 = R 1, R 12 = R 2,andR 1 = R FR) = A 1 D 1 R 1 + Q 1, 2 R 1 B 1 ) 1 h 1,... A ) ) Z D Z R Z + Q Z ) 2, 2 R Z 1 B 1. 30) h According to 13), the utoia values for the emloyed objective functions are F 1,1, = A /D R TOT + Q ) 2 and F 2,1, = 0. Consequently, as assumed at the beginning of Section IV, J ) in 18) is a function of the vector R and has the overall available transmission rate R TOT as a arameter. J R,R TOT ) is exlicitly indicated in J R,R TOT )= + w 1,1 ) A D R TOT + Q ) 2 + w 2,1 A D R + Q ) 2 2 R B 1 ) 1 h ). 31) VI. VERIFICATION OF THE TRANSMISSION-RATE-BOUND EXISTENCE CONDITIONS Here, our aim is to check the transmission-rate-bound existence conditions resented in Section IV for the formulation of J R,R TOT ) exressed in 31). A. Condition 1 Concerning Condition 1, we show that the derivative J R,R TOT )/ R y assumes here a value equal to ero at least in one oint R all ) by alying the Bolano Theorem. Being J R,R TOT )/ R ) continuous, because it is a derivative of R 2 functions, defined for R [0,R TOT ] R and J R,R TOT ) R A = w 1,1 D R + Q ) 2 A D R TOT + Q ) 2 ) 1 2A D ) 1 2 R D R +Q ) 3 +w B ln2) 1 2,1 2 R B 1 h B 32) where J 0,R TOT ) A A = w 1,1 R Q 2 + D R TOT +Q ) 2 ) 1 2A D Q 3 < 0 33) J R TOT,R TOT ) ) = w 2,1 2 R TOT 1 B 1 R 2 RTOT B ln2) 1 h > 0. 34) B Equation 19) has, at least, one solution, and Condition 1 is satisfied. B. Condition 2 2 J R,R TOT ) R 2 = w 1,1 1) 1) + A D R + Q ) 2 + ) 2 A 2A D D R TOT +Q ) 2 D R +Q ) 3 A D R + Q ) 2 A D R TOT +Q ) 2 6A D 2 )) D R +Q ) 4 + w 2,1 R )2 ) 2 B ln2) 1 2 R B 1 2 h B ) 1 R )) ) + 2 R 1 2 B ln2)) 2 1 B 1 h B 2. 35) Equation 35) deends only on R. As a consequence, the Hessian is a diagonal matrix, as shown in 36) and 37) at the bottom of the next age. Equation 37) is a roduct of only ositive quantities R. Matrix 36) is ositive semi-definite R. This imlies that Condition 2 is satisfied. ) 2

8 BISIO et al.: L RA PROBLEM-BASED TRANSMISSION RATE ALLOCATION 3319 TABLE I OBJECTIVE FUNCTION PARAMETERS VALUES TABLE II APPLIED CODE RATES C. Condition 3 From 22), we have J R,R TOT ) lim R TOT R ) A 1 2A D = w 1,1 D R + Q ) 2 D R + Q ) 3 + w 2,1 2 R B ) 1 2 R B ln2) 1 1 h <. B 38) The limits of the artial derivatives in 38) exist, are finite, and do not deend on R TOT. Condition 3 is satisfied. VII. PERFORMANCE ANALYSIS The scenario consered in this erformance evaluation has been imlemented through the ns-2 simulator. It is comosed of Z ESs, including MNs that transmit TCP traffic over a common GEO satellite channel through the MSG. TCP traffic features are secified at the beginning of Section V-A. The overall duration of the simulation is 300 s. The transmission rate allocation is erformed each 5 s the allocation eriod). Used values for objective functions arameters see Sections V-A and B) are secified in Table I. The channel status C/N), which assumes the values, ket constant in each allocation eriod, as reorted in Table II together with the code rates CR. Function 31) has been used with = 2, and the emloyed rocedure to find the minimum is based on a classical dynamic rogramming algorithm [23]. To ractically imlement the allocation rocedure, the transmission rate to be allocated is discrete, i.e., dived in units called minimum allocation units MAUs) set to 128 kb/s for the results reorted in the following. It means that the erformed allocations are not the exact solutions of the minimiation roblem but a very good aroximation. A consequence of using MAUs is that, when the existence of the transmission rate bound has been checked through simulations in Section VII-B, the allocated transmission rate behavior versus R TOT is not asymtotic. The sum of the allocations of the Z stations coinces with R bound.theemloyment of very small MAUs, which is not shown in this aer for the sake of brevity, highlights the asymtotic behavior. Due to the need to fully understand L RA behavior, erformance analysis from Section VII-A to D) is erformed by using Z = 2 ES. The number of stations is varied in Section VII-E). A. L RA Reaction to Channel Status Changes Here, the aim is to check the reactive behavior of L RA for different weight configurations in the case of channel status variations. The erformed tests conser the two ESs with the following carrier-to-noise ratio: Station 1 S1) has 5 db for the overall duration of the simulation 300 s), and Station 2 S2) has 5 db in the first 150 s and 0 db for the rest of the simulation. Related code rates CR are consequently chosen according to Table II. The channel status is consered known when the allocation algorithm acts. The overall available transmission rate R TOT is set equal to 4 Mb/s. TP, in watts, is comuted through 26). The acket loss referenced as acket loss rate PLR) has been comuted at each allocation eriod as the ratio between the number of lost and sent ackets. Three weights configurations are alied to both stations = 1and = 2) to differentiate the imortance of the objective functions: w 1,1 = 0.1 and w 2,1 = 0.9; w 1,1 = 0.5 and w 2,1 = 0.5; and w 1,1 = 0.9 and w 2,1 = 0.1. Two additional weight configurations have been imlemented but will be used in Section VII-B. H J R,R TOT )) = det H J R,R TOT ))) = Z 2 J R 1,R TOT ) R J R,R TOT )... R J R,R TOT ) R 2 2 J Z R Z,R TOT ) R 2 Z 36) > 0. 37)

9 3320 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 5, MAY 2016 Fig. 3. Allocated transmission rate to stations S1 and S2 for different weights; S2 channel status changes at t = 150 s 30th allocation eriod). Fig. 5. TP by S1 and S2 for different weights and by using the allocated transmission rates in Fig. 3. Fig. 4. PLR of stations S1andS2 for different weights by using the allocated transmission rates in Fig. 3. Fig. 6. Overall transmission rate allocated to S1 ands2 versusr TOT for different weights configurations. Fig. 3 reorts L RA transmission rate allocations over time, for the two stations S1 ands2. If t<150 s, both stations receive the same amount of transmission rate that deends on the alied weight configuration: When w 1,1 increases and stresses the acket loss imortance, each station receives a larger amount of transmission rate; when w 2,1 makes the TP more rewarded, less of a rate is allocated to each station, thus saving ower. This general comment is still true for t 150, but in this case, the rate allocated to S2 grows to comensate the enaliing channel status and the related emloyment of a secific code rate. S1 receives the same amount of rate of the t 150 case, excet for w 2,1 = 0.1, where the rate allocated to S1 must decrease because the overall allocated rate is equal to R TOT and increasing R 12 imlies, obviously, decreasing R 11. Fig. 4 shows the PLR of both stations obtained by allocating the transmission rates in Fig. 3. PLR is aroximately the same for both stations because the rate allocated to S2, enalied by the channel status, is larger than the rate assigned to S1 to comensate the reduction of the rate emloyed for information bits due to the use of a more rotective code rate. Fig. 4 allows checking the effect of the weight configurations on the PLR. Similar comments may be reorted for Fig. 5, which shows the TP of the two stations by using again the transmission rate allocations in Fig. 3. In ractical use, it is hard to strike a balance between these two metrics. The use of different weights assures more flexibility in the allocation roblem, to meet the reference of the service rover concerning erformance metrics. For examle, by observing Figs. 3 5, setting w 1,11 = 0.1 allows allocating rates to get relevant ower saving but a PLR close to 0.1, which is too high for most alications. On the other hand, setting w 1,11 = 0.09 assures PLR of about 0.02 but larger TP. Tuning weights allows driving L RA allocations suitably. B. Transmission Rate Bound and L RA Comarison With a Method Aimed Only at PLP Minimiation A key concet of this aer is the transmission rate bound R bound to which the overall allocated transmission rate the sum of the allocated transmission rates) converges if R TOT. All the tests in Figs. 6 8 assume a random channel status uniformly distributed among all ossible levels in Table II. Again, the channel status is consered known when L RA acts. Each value in Figs. 6 8 reresents the average of the values obtained by a number of simulation runs sufficient to guarantee a confence interval of 10% with a confence level of 95%. Fig. 6 shows the overall transmission rate allocated to the two stations versus R TOT whose value varies in the interval [1 10] Mb/s. Different weight configurations are consered. The allocation of the overall available transmission rate R TOT has been reorted as a reference to allow an immediate comarison with the allocation method that, keeing the same structure of the L RA, minimies only PLP in 24) and ignores TP in 26). Let this allocation method be referenced as MIN-PLP in the remainder of this aer. As exected, the allocated rate stays on the constraint R TOT if R TOT R bound and converges to R bound when R TOT if R TOT >R bound.infig.6, the allocated caacity if R TOT >R bound is exactly R bound.

10 BISIO et al.: L RA PROBLEM-BASED TRANSMISSION RATE ALLOCATION 3321 Fig. 7. PLR versus R TOT variation by using the allocated transmission rates in Fig. 6. other hand, the TP see Fig. 8) required by MIN-PLP is higher than the one required by L RA. This is quite obvious, but the real challenge comes from the numerical values. Is the gain concerning PLR a by a relevant ower increase) assured by using full available rate really erceived by users? For examle, looking at Figs. 7 and 8, when R TOT = 7Mb/s,L RA [w 1,1 = 0.9,w 2,1 = 0.1] guarantees PLR = 0.02 and TP = 0.16 W; full R TOT use assures a minimum PLR = 0.01 but imlies using TP = 0.86 W. Is this relevant additional ower alied at benefit of the users or is it only a waste? The answer is obviously in service level secifications, where erformance requirements agreed uon between user and service rover are stated, as well as in the ower availability of the rover and in its cost. Fig. 8. TP versus R TOT variation by using the allocated transmission rates in Fig. 6. This is due to the used MAU value, as reviously discussed. If R TOT >R bound, increasing R TOT does not modify the L RA solution. Fig. 6 allows checking numerically the effect of the weight configurations on the transmission rate allocation andonthevalueofr bound, which ranges from 4.1 Mb/s, for w 1,11 = 0.1, to 1 Mb/s, when w 1,11 = 0.9. Fig. 7 shows the average PLR of the overall system comosed of two stations versus R TOT by using the allocated transmission rates reorted in Fig. 6. It is clear that, if R TOT > R bound, the PLR does not change when R TOT increases. A similar comment may be reorted for Fig. 8, which shows the TP used by the overall system versus R TOT :IfR TOT > R bound, the TP does not change if R TOT increases. As commented for Figs. 4 and 5, a roer tuning of weights can adat L RA allocations to match alication requirements. The TP is constantly lower than 0.1 W when R bound < R TOT, and the overall allocated caacity is R bound.ifthe allocations should follow R TOT availability, the TP would grow exonentially. Another imortant observation, which is val also if concerning the Section VII-A, is that rivileging acket loss with resect to TP through suitable weight configurations does not imly a huge increase of the TP. Figs. 7 and 8 allow a comarison between L RA and MIN-PLP that otimies PLR ignoring TP and obviously uses all available transmission rate R TOT. As event in Fig. 7, and as exected, PLR obtained through MIN-PLP decreases when R TOT increases, and its values, assumed by a full use of the available transmission rate, are lower than the PLR values guaranteed by L RA. Onthe C. Performance Enhancement Analysis Let the quantity R ref be the transmission rate allocated to the th station by the MIN-PLP method. Obviously, Z Rref = R TOT. Let the quantities PLR R ref) and TP R ref ) be, resectively, the PLR and the TP of the th station obtained by allocating R ref. With the transmission rate allocated by L RA being R all in 17) because hysical and virtual entities are undifferentiated y = ), as sa at the beginning of Section V, we can write R all =R 1,all,...,R,all,...,R Z,all ),wherer,all is the transmission rate allocated by L RA to the th ES. It is true that R,all R bound 39) but if R TOT R bound R,all R bound. 40) In the resented results, we have equality for the exlained reasons linked to MAUs. We define the quantity transmission rate gain TRG) 41) as the ercentage of saved transmission rate obtained by using L RA with resect to MIN-PLP, i.e., R TOT Z R,all 100 TRG =. 41) R TOT A ractical consequence of the TRG obtained through L RA is the ossibility to serve additional stations over the same channel, without increasing R TOT and without a degradation or with a forecast and exected degradation) of the erformance erceived by the users. Even if the real number deends on the channel conditions, an estimation of the average number of additional stations allowed by L RA is contained in the following: N R TOT, ) Z R,all,Z = R TOT R,all ) Z R,all 42)

11 3322 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 5, MAY 2016 Fig. 9. TRG in 41). Fig. 10. Average number of ossible additional ESs in 42) by using the allocated transmission rates in Fig. 6. where Z is the overall number of ESs. Fig. 9 shows TRG versus R TOT for different weights by using the transmission rates in Fig. 6. Fig. 10 shows the average number of ossible additional ESs in 42) versus R TOT, again for different weights and by using the allocated transmission rates in Fig. 6. D. Comarison With Other Aroaches We reort a erformance comarison among sum caacity maximiation SCM) in 2), PM in 3), and L RA. R TOT, which does not influence SCM and PM methods because they use bandwth and ower, has been set to R TOT = 3Mb/s. Moreover, in this case, the erformed tests conser two ESs characteried by random fading levels uniformly distributed among all ossible levels shown in Table II. The code rate is chosen consequently. Again, the fading level is consered known when the allocation is erformed. The obtained values are the average of the values obtained by a number of simulations that guarantee a confence interval of 10% with a confence level of 95%. Weights α in 2), β in 3), and w in 31) are fixed to 0.5 for each station i.e., α 1 = α 2 = β 1 = β 2 = w 1,11 = w 2,11 = 0.5). The total amount of bandwth W TOT emloyed by SCM and PM is ket constant and equal to 1 MH. Two values of ower availability P TOT 1 and 2 W) have been consered as SCM constraint. In the PM case, we emloyed two ossible transmission rate thresholds R th :1and 1.5 Mb/s. Table III reorts the globally allocated transmission rate, the metrics PLR and TP in watts), both referred to the overall system of two stations, as well as the execution time in seconds), for L RA, SCM,andPM. The minimum PLR, which is aroximately 0.03, is obtained by PM with R th = 1.5 Mb/s). L RA allows a satisfying PLR = On the contrary, SCM, concerning PLR, roves incomatible results with many alications. In ractice, SCM gives the overall bandwth W TOT to the less faded station, and as a consequence, the other station exeriences relevant losses. Concerning TP, the minimum value is obtained by L RA. SCM uses all available ower i.e., TP is equal to the ower constraint). PM, in the case of R th = 1.5 Mb/s, uses much more caacity and obviously more ower, whereas in the case of R th = 1 Mb/s, it uses less transmission rate, thus requiring less ower. This last case is interesting because, although it kees owerat0.08w,itassuresaplr= 4%. Consering these metrics, it is ossible to conclude that L RA, which guarantees PLR = 3.9% and TP = 0.1 W, and PM R th = 1 Mb/s rove a good erformance comromise. In summary, referring to the metrics PLR and TP, SCM seems not suitable for the oerative environment of this aer; maximiing the weighted sum of downlink transmission rates imlies getting allocations that rivilege less faded stations and that lead to huge PLR values. Working on weights may mitigate this drawback. PM can be very efficient in this environment. The constraint R th allows controlling the PLR. Minimiing the ower is the aim of the allocation scheme. The drawback is that R th udate and related PLR control are not automatically erformed within PM. R th is a arameter that PM uses as a threshold. Its comutation may be either heuristic or analytical, but it is not art of PM allocation. In L RA, even if it cannot assure to give a threshold on PLR and/or on TP because it does not use fixed constraints obviously excet for the maximum available transmission rate), minimiing, jointly, PLR and TP, allows getting allocations tunable through weights) that are a satisfactory erformance comromise. Another key oint to be consered for comarison is reresented by the comutational comlexity. The last row of Table III reorts the time sent during the overall simulation time of 300 s) for the consered resource allocations. With the allocation eriod being 5 s, each reorted value reresents the time needed to execute 60 allocations. Shown values conser also secific oerations of the simulator e.g., READ/WRITE log files) not included in real systems, but this is true for all consered resource allocation aroaches. PM assures the best erformance: 0.07/0.08 s. L RA requires a slightly higher execution time, i.e., 0.2 s. In SCM, the execution time is larger because, with this algorithm, both ower and bandwth are allocated, and there are two control variables involved in the otimiation roblem. E. L RA Performance Analysis by Using Multile Stations L RA erformance is analyed by using more than two ESs. The tests are erformed as done in Section VII-B assuming a random channel status, uniformly distributed among all levels shown in Table II, and assigning a consequent code rate for all involved stations whose number varies in the range Z =[2, 4, 6, 8, 10]. Values for Z = 2, which have already been shown in

12 BISIO et al.: L RA PROBLEM-BASED TRANSMISSION RATE ALLOCATION 3323 TABLE III COMPARISON AMONG RESOURCE ALLOCATION APPROACHES Fig. 11. Overall transmission rate allocated to different number of stations versus R TOT,usingL RA with w 1,11 = w 2,11 = 0.5 and MIN-PLP. Fig. 13. TP versus R TOT variation by using the allocated transmission rates in Fig. 11. Fig. 12. PLR versus R TOT variation by using the allocated transmission rates in Fig. 11. Section VII-B, are reorted again for the sake of comarison. L RA weights are set as follows: w 1,1 = w 1,1 = 0.5. Fig. 11 shows the overall transmission rate allocated to the Z stations for L RA and MIN-PLP. The trend is exactly the same for each Z value, and the comments reorted for Fig. 6 are still val. Additionally, we can say that, given that MIN- PLP globally allocates R TOT indeendently of the number of stations, each station receives less transmission rate when Z increases. The effect on the erformance metrics is clear in Figs. 12 and 13, which show, resectively, PLP and TP values referred to the overall system comosed of Z stations and corresonding to the allocations in Fig. 11: PLP increases, and TP decreases with the number of stations. L RA behavior is the same if R TOT <R bound,butwhenr TOT R bound, L RA allocations converge to R bound, whose value, as obvious from 23), increases with Z, the number of stations. Consequently, L RA PLP Fig. 12), when R TOT R bound, is also indeendent of the number Z of stations. L RA TP values are shown in Fig. 13 for different Z values. VIII. CONCLUSION This aer has introduced the L RA, which is aimed at allocating transmission rates among the ESs of a satellite network suited to be emloyed in emergency situations. The network is comosed of different MNs accessing the satellite channel through an MSG. The obtained allocation is reresentative of a comromise between acket loss and TP, which are simultaneously consered as erformance metrics. This aer has highlighted the existence of a rate bound indeendent of the overall available rate R TOT,towhichtheL RA allocations converge when R TOT. The roosed erformance analysis is obtained through simulations for scenarios comosed of ESs characteried by different fading conditions. It analyses the L RA reaction to changes in the channel status, comares L RA with a method aimed only at PLP minimiation, evaluates the ractical effect of the rate bound on the erformance metrics, comares L RA with two allocation schemes in the literature, and shows L RA behavior in the case of multile ESs. The obtained results allow the conclusion that L RA roves satisfactory erformance both in terms of comutational comlexity and of PLP and TP. The obtained values are comatible with the requirements of most ractical alications. APPENDIX For the sake of comleteness, the roof of the Pareto otimality of the L roblem solution, if M y k=1 w k,y = 1, w k,y > 0 k [1,M y ], y [1,Y ], [1,Z], is now reorted. It is worth noticing that the roosed roof is well known [19]. It has been formulated here by using the model roosed in Section III-D. The solution of the weighted L -roblem when 1 << ) is Pareto otimal if either the solution is unique or all the weight coefficients are ositive. We start from the

13 3324 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 5, MAY 2016 second condition and remember that the L RA roblem, which is defined in 15), can be written as follows: Y M y arg min Fk,y w k,y ) subject to y=1 k=1 + F k,y, ) Y M y RTOT. 43) y=1 k=1 Let ot be a solution of the roblem roosed in 43), with w k,y > 0 k [1,...,M y ] y [1,...,Y ] and [1,...,Z]. Suosing that ot is not a POP, should exist such that ) F k,y ) Fk,y, ) ) F k,y F k,y, ot ot k [1,...M y ], y [1,...Y ] [1,...Z] 44) ) F k,y ) Fk,y, ) ) < F k,y F k,y, for at least one k [1,...,M y ],y [1,...Y ] [1,...,Z]. 45) Since w k,y > 0, k [1,...,M y ] y [1,...,Y ] [1,...,Z], wehave Y M y Fk,y w k,y ) Fk,y, y=1 k=1 < Y M y y=1 k=1 1 ) Fk,y w k,y ot F k,y, ) ). 46) This contradicts the assumtion that ot is a solution of the weighted roblem; thus, ot must be Pareto otimal. q.e.d.) REFERENCES [1] K.-D. Lee, An efficient real-time method for imroving intrinsic delay of caacity allocation in interactive geo satellite networks, IEEE Trans. Veh. Technol., vol. 53, no. 2, , Mar [2] K. Kumaran and H. Viswanathan, Joint ower and bandwth allocation in downlink transmission, IEEE Trans. Wireless Commun., vol. 4, no. 3, , May [3] X. Wang and G. Giannakis, Power-efficient resource allocation for timedivision multile access over fading channels, IEEE Trans. Inf. Theory, vol. 54, no. 3, , Mar [4] I. Bisio and M. Marchese, Minimum distance bandwth allocation over sace communications, IEEE Commun. Lett., vol. 11, no. 1, , Jan [5] I. Bisio and M. Marchese, Power saving bandwth allocation over geo satellite networks, IEEE Commun. Lett., vol. 16, no. 5, , May [6] Y. Kawamoto, H. Nishiyama, N. Kato, N. Yoshimura, and N. Kadowaki, Packet transfer delay minimiation by network-we equaliation of unbalanced trafficload in multi-layered satellite networks, in Proc. IEEE 77th VTC Sring, Jun. 2013, [7] L. T. 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Zhang, Jointly otimum ower and bandwth allocation for multi-rate FDM and FDMA over overloaded channels, in Proc. IEEE GLOBECOM Conf., Nov. 29 Dec. 3, 2004, vol. 1, [14] N. Krishnan, R. Yates, N. Mandayam, and J. Panchal, Bandwth sharing for relaying in cellular systems, IEEE Trans. Wireless Commun., vol. 11, no. 1, , Jan [15] X. Gong, S. Vorobyov, and C. Tellambura, Joint bandwth and ower allocation with admission control in wireless multi-user networks with and without relaying, IEEE Trans. Signal Process., vol. 59, no. 4, , Ar [16] I. Bisio and M. Marchese, Packet loss and delay combined otimiation for satellite channel bandwth allocation controls, in Proc. IEEE ICC, May 2008, [17] I. Bisio, S. Delucchi, F. Lavagetto, and M. Marchese, Caacity bound of mo-based allocation with acket loss and ower metrics in satellite communications systems, in Proc. IEEE GC Conf., Dec. 2012, [18] I. Bisio, S. Delucchi, F. Lavagetto, and M. Marchese, Comarison among resource allocation methods with acket loss and ower metrics in geostationary satellite scenarios, in Proc. IEEE ICC, 2013, [19] K. M. Miettinen, Nonlinear Multiobjective Otimiation. Boston, MA, USA: Kluwer, [20] H. Koraitim and S. Tohme, Performance analysis of multile access rotocols for multimedia satellite networks, IEEE J. Sel. Areas Commun., vol. 18, no. 9, , Se [21] I. Bisio and M. Marchese, Analytical exression and erformance evaluation of TCP acket loss robability over geostationary satellite, IEEE Commun. Lett., vol. 8, no. 4, , Ar [22] L. J. Iolito, Satellite Communications System Engineering. Chichester, U.K.: Wiley, [23] D. P. Bertsekas, Dynamic Programming: Deterministic and Stochastic Models. Englewood Cliffs, NJ, USA: Prentice-Hall, Igor Bisio S 04 M 07 SM 15) was born in Novi Ligure, Italy, in He received the Laurea degree in telecommunications engineering and the Ph.D. degree from the University of Genoa, Genoa, Italy, in 2002 and 2006, resectively. He is currently an Assistant Professor and a member of the Digital Signal Processing and Satellite Communications and Networking Laboratories, Deartment of Naval, Electrical, Electronic and Telecommunications Engineering, University of Genoa. He is the author of about 100 scientific aers, including international journals, international conferences, and book chaters. His main research interests include signal rocessing over ortable devices such as smarthones, context and location awareness, adative coding mechanisms, indoor localiation, security and e-health alications, and resource allocation and management for satellite and sace communication systems.

14 BISIO et al.: L RA PROBLEM-BASED TRANSMISSION RATE ALLOCATION 3325 Stefano Delucchi S 15) was born in Genoa, Italy, in He received the Bachelor s and Master s degrees in telecommunication engineering and the Ph.D. degree from the University of Genoa in 2007, 2010, and 2014, resectively. His master s thesis on resource allocation in satellite networks was develoed at the Digital Signal Processing Laboratory, University of Genoa, in collaboration with the Satellite Communications and Networking Laboratory SCNL). He has been a Postdoctoral Research Fellow with the SCNL and Telecommunications Networks and Telematics Laboratory, Deartment of Naval, Electrical, Electronic, and Telecommunications Engineering, University of Genoa. He is also currently with Aitek S..A, Genoa. His main research interests include logistic and transort communications, resource allocation and management for satellite communication systems, and quality of service over heterogeneous networks. Mario Marchese S 94 M 97 SM 04) was born in Genoa, Italy, in He received the Laurea degree cum laude) in electronic engineering and the Ph.D. degree in telecommunications from the University of Genoa in 1992 and 1996, resectively. He is currently an Associate Professor with the Deartment of Naval, Electrical, Electronic, and Telecommunications Engineering DITEN), University of Genoa. He is the Founder and is resonsible for the Satellite Communications and Networking Laboratory, DITEN. He is the author of about 250 scientific aers, including international journals, international conferences, book chaters, and the book QoS Over Heterogeneous Networks Wiley, 2007). His main research interests include satellite and radio networks, transort layer over satellite and wireless networks, quality of service and data transort over heterogeneous networks, and alications for smarthones. Fabio Lavagetto was born in Genoa, Italy, in He was the Vice Chancellor with resonsibility for research and technology transfer with the University of Genoa. Since 1995, he has also been the Head of Research with the Digital Signal Processing Laboratory and, since 2005, the Vice Chair of the Institute for Advanced Studies in Information Technology and Communication, University of Genoa. He is currently a Full Professor of telecommunications with the Deartment of Naval, Electrical, Electronic, and Telecommunications Engineering, University of Genoa. He is the author of over 100 scientific ublications in international journals and conferences. His main research interests include signal rocessing over ortable devices such as smarthones, context and location awareness, adative coding mechanisms, indoor localiation, security, and e-health alications. Mr. Lavagetto has served as the General Chair for several international scientific conferences.