Cross-Layer Design of Wireless Multihop Networks over Stochastic Channels with Time-Varying Statistics
11 Cross-Layer Design of Wireless MultihopNetworks over Stochastic Channels withTime-Varying Statistics
Eleni Stai , Michail Loulakis , Symeon Papavassiliou School of Electrical & Computer Engineering School of Applied Mathematical & Physical SciencesNational Technical University of Athens, Athens, Zografou, 15780, Greece.Emails: [email protected], [email protected], [email protected]
Abstract
Network Utility Maximization is often applied for the cross-layer design of wireless networksconsidering known wireless channels. However, realistic wireless channel capacities are stochasticbearing time-varying statistics, necessitating the redesign and solution of NUM problems to capturesuch e ff ects. Based on NUM theory we develop a framework for scheduling, routing, congestion andpower control in wireless multihop networks that considers stochastic Long or Short Term Fadingwireless channels. Specifically, the wireless channel is modeled via stochastic di ff erential equationsalleviating several assumptions that exist in state-of-the-art channel modeling within the NUM frameworksuch as the finite number of states or the stationarity. Our consideration of wireless channel modelingleads to a NUM problem formulation that accommodates non-convex and time-varying utilities. Weconsider both cases of non orthogonal and orthogonal access of users to the medium. In the first case,scheduling is performed via power control, while the latter separates scheduling and power control andthe role of power control is to further increase users’ optimal utility by exploiting random reductionsof the stochastic channel power loss while also considering energy e ffi ciency. Finally, numerical resultsevaluate the performance and operation of the proposed approach and study the impact of severalinvolved parameters on convergence. Index Terms a r X i v : . [ c s . S Y ] A ug Wireless multihop networks; Network Utility Maximization; Stochastic wireless channels; Nonconvex utilities; Time-varying utilities; Non stationarity; Transient phenomena; Long Term Fading;Short Term Fading;
I. I ntroduction
Network Utility Maximization (NUM) is a very popular tool in the communications researchcommunity, for cross-layer design and optimization of wireless networks. Typically, a utilityfunction is assigned to each network flow (source-destination pair), and the sum of all utilitiesover the network is maximized, subject to network stability constraints. Most approaches inliterature applying NUM, consider ideal or stationary and ergodic wireless channels. However,under realistic conditions, fading occurring in wireless channels hampers the performance inwireless communications, leading to stochastic, i.e. time and space varying, random (thus un-known), wireless link capacities possibly bearing time-varying statistics. Therefore, it becomesnecessary to reformulate / redesign and solve the basic NUM problem for incorporating realisticstochastic wireless channel conditions by addressing non-stationarity issues, and also transientphenomena occurring when the network operates for a finite time interval.This paper aims at treating this problem by proposing a novel optimization framework forjoint congestion control, routing, scheduling and power control based on NUM, under stochasticpossibly non-stationary Long Term Fading (LTF) or Short Term Fading (STF) wireless channels.Congestion control determines the optimal sources’ data production rates, routing determinesthe optimal routes of the flows within the network, while the set of transmitting links and theircorresponding transmission powers are chosen based on scheduling and power control. Theproposed optimization framework refers to a finite network operation and it is developed fortwo cases, where the first one deals with non-orthogonal access to the wireless medium andthe second deals with orthogonal access to the wireless medium. In the first case, schedulingis performed via power control, while in the latter case, scheduling and power control areseparated. On the one hand, non-orthogonal access leads to a very hard to solve cross terminalpower control / scheduling problem in the physical layer. On the other hand, orthogonal accessto the medium allows for more e ffi cient and distributed power control in the physical layer,although introducing the need of the NP-hard computation of all the independent sets of thenetwork graph for scheduling. Importantly, in the case of orthogonal access, the role of power control is to further boost link capacity and consequently source rates by exploiting good channelstates and to save energy when the channel state is destructive for the transmitted signal.The structure of the rest of the paper is as follows. Section II presents the related literaturewhile summarizing the basic contributions of this paper. Then, Section III describes the consid-ered STF and LTF channel models derived with the use of stochastic di ff erential equations, whileSection IV presents the considered system model. Sections V and VI focus on the analysis andsolution of the proposed optimization framework in the cases of non-orthogonal and orthogonalaccess to the wireless medium respectively. Finally, in Section VII numerical results are presentedto evaluate the proposed approach and Section VIII concludes the paper.II. R elated W ork & C ontributions Several works exist in the literature targeting at incorporating the stochastic wireless channelsin the NUM problem’s formulation and solution. In [1], the channel quality is expressed viathe SIR (Signal-to-Interference-Ratio), since the latter is a ff ected by interference due to paralleltransmissions and LTF. It is assumed that the LTF parameters are deterministic and slowlyvarying, allowing for the algorithms which perform joint congestion and power control toconverge in the meantime of their change. In [2], [3], the case of composite fading (LTFand STF) is examined, considering channel conditions that vary faster than the algorithm’sconvergence, via the use of outage-probabilities in the NUM problem formulation. In [2], [3],STF follows Rayleigh and Nakagami distributions respectively, where however the statistics ofthe distributions remain invariant. A di ff erent approach is followed in [4], [5], where NUM isextended to Wireless NUM (WNUM), with random channel conditions. WNUM leads to policiesfor controlling the network by responding optimally to the change of the channel state, basedon random samples without a priori knowledge of the wireless channels’ statistics. Although thestatistics of the channel may be unknown, the latter is considered as stationary and ergodic.In the sequel, in [6], NUM is employed to perform, joint congestion control, power control,routing and scheduling assuming that the channel fading process is stationary and ergodic. Simi-larly in [7], [8], joint congestion control, routing and scheduling is performed in the framework ofNUM while assuming that there is a finite number of channel states. In these works, the networkfunctions in time slots, where during each time slot the channel state remains stable and changesrandomly and independently on the boundary of time slots. Finally, in [9] the convergence of primal-dual algorithms for solving NUM is studied under wireless fading channels with time-varying parameters (and thus statistics). Time-varying statistics of wireless channels lead totime-varying optimal solutions of the NUM problem necessitating the study of how well thesolution algorithms track the changes in the optimal values. However, it is assumed that thechannel fading parameters vary following a Finite State Markov Chain.In a nutshell, in the existing body of research work in literature, the wireless channel modelingin the framework of NUM is characterized by one or combinations of the following assumptions:(a) The channel process is stationary and ergodic. (b) The statistics of the wireless channel arefixed in time (and known), or vary at slower rate than the one of the network control algorithms’convergence. (c) The statistics of the channel change according to a Finite State Markov Chain.All previous approaches are not capable of capturing and tracking complex time and spacevariations in the propagation environment of realistic systems [10]. In [10], wireless channelmodels for both LTF and STF are introduced based on Stochastic Di ff erential Equations (SDEs)[11] in order to capture higher order dynamics of the wireless channel. In this case the wirelesschannel is modeled via stochastic processes which may have time-varying statistics. By meansof SDEs, it is possible to express an LTF or an STF channel capturing both space and timevariations [10], [12], as it will be described in detail in Section III.In our paper, the NUM problem is reformulated and solved using the SDE model to capturethe wireless channel state. Emphasis is placed on LTF especially for demonstration purposes.Due to the possible non-stationarity of the wireless channel we cannot formulate the NUMproblem based on the stationary mean values of the involved optimization variables (e.g. [6],[7], [13]). On the contrary we will adopt the stochastic optimal control problem’s formulation[14] based on expected values over time integrals, thus also allowing for the consideration of afinite time duration of the network’s operation. In [15], we proposed a preliminary version ofthis approach focused on congestion and power control in the case of orthogonal access to thewireless medium. The basic contributions of this paper can be summarized as: • We develop a framework for the cross-layer design and control of the operation of wirelessmultihop networks, i.e. for congestion and power control, routing and scheduling, overwireless channels (LTF or STF) that are stochastic but not necessarily stationary. • The proposed problem formulation adopts a (more realistic) finite duration of the network’soperation (e.g. corresponding to the case of finite battery levels of the wireless nodes or of finite flows) where the wireless channel may still operate in a transient state, even thoughthere may exist a limiting stationary distribution. The extension to an infinite duration ofthe network’s operation is discussed. • Utilities are not necessarily convex functions but adopt a more general form (more specif-ically a continuous di ff erentiable one), contrary to the related papers in literature (e.g.[6], [7], [13]) that assume convex utility forms. This fact is important since it allows foraddressing the case of real time tra ffi c modeled, for example, by sigmoidal utilities [16].Zero duality gap is analytically proven in this case of general utilities, following a techniquethat leverages from the wireless channels’ continuous stochastic modeling. • The proposed problem formulation (expected values over time integrals) allows for theadoption of time-varying utility functions. This serves the purpose of evolving users’ pref-erences / needs and is also aligned with the finite network duration, i.e. nodes may desire toproduce significantly less data close to the end of the network operation. To the best of ourknowledge, this fact has not been addressed in the literature. • Power control is shown to further boost link capacity and thus source rates by exploitinggood channel conditions while saving energy in case of destructive channel conditions. • Finally, we prove an interesting theorem in the case of LTF, elucidating a basic advantagewith respect to the optimal users’ utilities when exploiting the random channel fluctuations,compared to the conventional NUM problem formulation (e.g. [7], [13]). Specifically, it isproven that, contrary to what is possibly expected, a higher value of the di ff usion coe ffi cientof the wireless channels’ power loss leads to higher optimal sum of users’ utilities, factthat cannot be captured by the conventional NUM modeling approach. We also show vianumerical evaluations that a higher di ff usion coe ffi cient achieves simultaneously a reducedpower consumption leading to energy e ffi ciency. This result emphasizes the importance ofutilizing a more realistic power loss model such as the one of an SDE, as opposed to themean power loss model used in the conventional NUM problem formulation.III. B ackground on W ireless C hannel M odeling via SDE s The objective of this section is to briefly describe the SDE-based LTF and STF channel models,developed in the literature, and their assumptions, upon which, the optimization problems of thefollowing sections will be formulated and solved.
A. Long Term Fading (LTF)
LTF consists of path loss and shadowing [17]. Path loss is due to the dissipation of thetransmitted power and the e ff ects of the propagation channel, while shadowing is caused byobstacles between the transmitter and the receiver. LTF depends on the geographical area andoccurs in sparsely populated or suburban areas. Before describing the dynamic in time LTFmodel for the wireless channels [10], we recall the conventional LTF model, where the powerloss PL along a given link ( i , j ) between nodes i and j in Euclidean distance d i j , is given [17]: PL ( d i j )[ dB ] = PL ( d )[ dB ] + γ log (cid:0) d i j d (cid:1) + ˜ Z , d i j ≥ d , (1)where γ is the power loss exponent and depends on the wireless propagation medium, d is thereference distance, PL ( d ) is the expected power loss on the reference distance, and ˜ Z ∼ N (0; σ ),is a gaussian random variable with zero mean and variance σ , used to model any uncertainty inthe propagation environment. Note that the statistics (mean (denoted as PL ( d i j )) and variance)of the conventional LTF model are invariant in time.In the following, we describe the extension of the LTF model to dynamically changingconditions in time, as it is developed in [10]. Specifically, the random variable, PL ( d i j )[ dB ],of Eq. (1), becomes a stochastic process denoted as { X i j ( t ) } t ≥ t ([dB]), where t represents time.Time dependence is used to capture time variations of the propagation environment due to e.g.movement of objects and scatterers in the area surrounding the network. In a similar spirit withEq. (1), { X i j ( t ) } t ≥ t , represents the power lost by the signal during a transmission from i to j ata particular distance d i j . Although, { X i j ( t ) } t ≥ t depends on the distance d i j , we do not explicitlymodel this dependence as the network considered is static.In [10], { X i j ( t ) } t ≥ t , ∀ ( i , j ), are modeled as solutions of mean reverting linear SDEs, given as: dX i j ( t ) = β i j ( t ) (cid:0) γ i j ( t ) − X i j ( t ) (cid:1) dt + δ i j ( t ) dW i j ( t ) , X i j ( t ) ∼ N ( PL ( d i j )[ dB ]; σ ) , (2)where { W i j ( t ) } t ≥ t , ∀ ( i , j ), are independent standard Brownian motions defined over a filteredprobability space ( Ω , F , {F t } t ≥ t , P ) and each one being independent of the corresponding X i j ( t ). {F t } t ≥ t is the filtration produced by X i j ( t ), ∀ ( i , j ), and the Brownian motions themselves. Foreach ( i , j ), γ i j ( t ) is the power loss level X i j ( t ) is attracted to, β i j ( t ) is the positive speed of thisadjustment and finally, δ i j ( t ) is the di ff usion coe ffi cient of the SDE, determining the “noise” ofthe channel. The parameters β i j ( t ) , γ i j ( t ) , δ i j ( t ), ∀ ( i , j ), are assumed to be deterministic and can be estimated directly from signal measurements following the approaches in [26], [18], [19],which can be implemented online, i.e. while receiving the signal measurements. The existenceof a strong solution to the SDE (2) is satisfied if the relation T (cid:82) s (cid:110) β i j ( t ) (cid:12)(cid:12)(cid:12) γ i j ( t ) (cid:12)(cid:12)(cid:12) + δ i j ( t ) (cid:111) dt < ∞ , ∀ ( i , j )holds [11]. The time dependent attenuation coe ffi cient (in squared magnitude) equals to: a i j ( t ) = e − ln1010 X i j ( t ) = e KX i j ( t ) , ∀ ( i , j ) , K = − ln1010 . In [10], it is shown that when all the parameters of theSDE (2) are time independent, its solution tends to the conventional LTF model (Eq. (1)) as t → ∞ (which is stationary). In general when the parameters of the SDE (2) change with time, X i j ( t ) is gaussian with time-varying statistics and a stationary distribution may not exist. B. Short Term Fading (STF)
In a similar spirit as LTF, in [12], [18], [10] a stochastic model for STF wireless channels hasbeen developed, alleviating the assumption of stationarity. This kind of signal fading is due tothe constructive and destructive addition of multipath components [17] created from reflections,di ff ractions and scattering and usually occurring in densely built-up areas. The statistics of theSTF models usually applied in the literature (e.g. Rayleigh, Nakagami, Ricean, etc. [2], [17]),are assumed constant over local areas (i.e. at a microscopic level) [12]. However, STF wirelesschannels are of stochastic nature with time varying statistics mainly due to the continuous andarbitrary change of the propagation environment if the transmitter, the receiver or objects betweenthem move. The latter is the main reason why in this paper, we adopt a stochastic process withtime varying statistics for modeling STF channels.For the models developed in [12], [18], the inphase and quadrature components of the wirelessfading channels are assumed conditionally uncorrelated gaussian random variables (thus condi-tionally independent). In the case of flat fading, the multipath components are not resolvableand can be considered as a single path. Then, the inphase, I , and quadrature, Q , componentsover one link (e.g. ( i , j )) can be realized as: dX I ( t ) = A I ( t ) X I ( t ) dt + B I ( t ) dW I ( t ) , X I ( t ) , I ( t ) = C I X I ( t ) , dX Q ( t ) = A Q ( t ) X Q ( t ) dt + B Q ( t ) dW Q ( t ) , X Q ( t ) , Q ( t ) = C Q X Q ( t ) , where X I ( t ) , X Q ( t ) are the state vectors of the inphase and quadrature components and { W I ( t ) } t ≥ t , { W Q ( t ) } t ≥ t are independent standard Brownian motions corresponding to the inphase and thequadrature components respectively, defined over a filtered probability space ( Ω , F , {F t } t ≥ t , P ). The same model describes every link with di ff erent parameter values A I ( t ) , A Q ( t ), B I ( t ), B Q ( t ), C I , C Q and (independent) Brownian motions W I ( t ) , W Q ( t ), but this fact is not explicitly modeledfor ease of presentation. The attenuation coe ffi cient (in squared magnitude) is [19] a i j ( t ) = I ( t ) + Q ( t ) . As in the case of LTF, the coe ffi cients A I ( t ) , A Q ( t ) , B I ( t ) , B Q ( t ) , C I , C Q can be obtaineddirectly via signal measurements following the methodology proposed in [19], [18] using the EMalgorithm together with Kalman filtering. This model leads to time-varying mean and variance forthe inphase and the quadrature components and thus for the STF wireless channel and includesthe Ricean, Rayleigh and Nakagami distributions as special cases [12].In the rest of the paper, the vectors X ( t ) , W ( t ) denote collectively (for all links) the channelstates and the Brownian motions respectively at time t . Note that we assume that the wirelesschannels are uncorrelated. This assumption is also made in [20], where it is argued that inter-linkcorrelations do not impact the network capacity region and therefore the maximum utility.IV. S ystem M odel & A ssumptions We consider a static wireless multihop network with N nodes and E directed links formingthe set E . The network serves F overlaying flows (source-destination pairs) over a finite duration(lifetime) [ s , T ]. At time t ∈ [ s , T ], λ di ( t ) data (e.g. packets) are produced from the source node i forits destination node d . Let S r ( d ) be the set of sources for node d . Then, λ di ( t ) = , ∀ i (cid:60) S r ( d ) , ∀ t ∈ [ s , T ]. We denote with r di j ( t ) the communication tra ffi c on the link ( i , j ) for destination d at time t ∈ [ s , T ]. Then, R ( t ) , Λ ( t ) , denote collectively the variables { r di j ( t ) } ∀ d , ( i j ) , { λ di ( t ) } ∀ i , d , respectively,at time t . The set R ( i , d ) consists of the one-hop (out-)neighbors of node i which are allowedto serve as next-hop nodes towards d according to the routing protocol under consideration. Ifthere are no routing constraints, we consider R ( i , d ) = N out ( i ), where N out ( i ) = { j | ( i , j ) ∈ E} . Also, r di j ( t ) =
0, if j (cid:60) R ( i , d ) and r dii ( t ) = , r ddi ( t ) = , ∀ i , d , t ∈ [ s , T ]. Furthermore, we assume that thetransmitter of the link ( i , j ), i.e. the node i , transmits with power P i j ( t ) at time t ∈ [ s , T ]. P ( t )expresses collectively the transmission powers of all links at time t ∈ [ s , T ].Each source node associates its satisfaction for its produced data for destination d , λ di , at time t ∈ [ s , T ], with a time-varying continuous di ff erentiable utility function U di ( λ di , t ). Several utilityfunctions used in literature belong in this category, such as the strictly convex and increasing a -fair utility, including the logarithmic one [21]. It is further assumed that U di ( λ di , t ) is increasingwith λ di and uniformly bounded as t → ∞ . Also, a (continuous di ff erentiable) cost function, J i j ( P i j ), is assigned to each directed link ( i , j ) with respect to its transmission power P i j ( t ), t ∈ [ s , T ]. In literature, J i j ( P i j ) is often assumed to be a strictly convex function [6].The proposed cross-layer framework includes routing, scheduling, power and congestion con-trol. Routing (network layer) determines the amount of tra ffi c for each destination that will beserved by every link, by optimizing R ( t ), ∀ t ∈ [ s , T ]. Scheduling and power control (MAC andphysical layers) determine which links are going to transmit and their transmission power byoptimizing P ( t ), ∀ t ∈ [ s , T ]. Finally, congestion control (transport layer) optimizes Λ ( t ), ∀ t ∈ [ s , T ].Therefore, the proposed cross-layer scheme aims at determining the optimal values of the controlvariables R ( t ) , Λ ( t ) , P ( t ), ∀ t ∈ [ s , T ], according to an optimality criterion designed with the aidof the utility and cost functions defined above, considering the channel models of Section III.Due to the considered underlying channel processes, the control variables R , Λ , P should be ingeneral defined as stochastic processes. Let us define the value range for each λ di , U λ = [0 , λ max ],and the corresponding feasible set U λ = (cid:8) λ : [ s , T ] × Ω → U λ : λ is {F t } t ≥ s adapted (cid:9) . Then, Λ ∈U F λ . Similarly, we define the value range for each r di j , U r = [0 , R max ], and the correspond-ing feasible set U r = (cid:8) r : [ s , T ] × Ω → U r : r is {F t } t ≥ s adapted (cid:9) . Then, R ∈ U E × ( N − r . Finally,we define the value range for each P i j , U P = [0 , P max ], and the corresponding feasible set U P = (cid:8) P : [ s , T ] × Ω → U P : P is {F t } t ≥ s adapted (cid:9) . Then, P ∈ U EP . We will use E s , x to denoteexpectations given the initial condition X ( s ) = x .At this point, we distinguish two cases with respect to the access to the wireless medium.Based on the two types of access, two cross-layer problems are developed. The first case concernsnon-orthogonal access to the wireless medium, in which the transmitters are allowed to accessthe wireless medium simultaneously (one frequency carrier is assumed), while the interferingtransmissions are considered as noise. For this case we define the Signal-to-Interference-plus-Noise-Ratio ( S INR ) for the link ( i , j ) as follows: S INR i j ( t ) = a i j ( t ) P i j ( t ) N + (cid:80) ( k , l ) ∈I i j a k j P kl ( t ) , where I i j denotesthe subset of E containing the links that interfere with the link ( i , j ). N (Watts) stands for theaverage background noise at the receiver’s ( j ) side and a i j ( t ) is defined in Section III dependingon the fading type. The capacity of the link ( i , j ) is given by the Shannon’s formula in bits / sec as C i j ( P ( t )) = B i j log (1 + S INR i j ( t )), where B i j ( Hertz ) is the wireless channel’s bandwidth at link( i , j ). The second case refers to the orthogonal access to the wireless medium, where only non-interfering links can access simultaneously the wireless medium. Orthogonal access decreasesthe complexity of the proposed framework’s operation in the physical layer (power control), as it will be shown in later sections, while it is nearly optimal when interference is strong [6]. Inthis case, the connectivity graph of the wireless multihop network is important in identifyingthe feasible schedules. Based on the latter, the finite set of all possible independent sets of links(i.e. links that do not interfere with each other) is constructed. Only links belonging to the sameindependent set can access the wireless medium simultaneously. In this case, the capacity of thelink ( i , j ) is a concave function of P i j , given in bits / sec as C i j ( P i j ( t )) = B i j log (cid:18) + a i j ( t ) P i j ( t ) N (cid:19) .V. N on - orthogonal A ccess to the M edium A. Problem Formulation & Analysis
According to the discussion in Section IV, the optimization framework, denoted as P , isformulated as follows. P : = max Λ ∈U F λ , R ∈U E × ( N − r , P ∈U EP E s , x (cid:90) Ts (cid:88) i , d : i ∈ S r ( d ) U di ( λ di ( t ) , t ) − (cid:88) ( i , j ) J i j ( P i j ( t )) dt s . t . E s , x (cid:90) Ts λ di ( t ) dt + (cid:90) Ts (cid:88) j : i ∈R ( j , d ) r dji ( t ) dt ≤ E s , x (cid:90) Ts (cid:88) j ∈R ( i , d ) r di j ( t ) dt , ∀ i , d (3) E s , x (cid:90) Ts (cid:88) d r di j ( t ) dt ≤ E s , x (cid:34)(cid:90) Ts C i j ( P ( t )) dt (cid:35) , ∀ ( i , j ) ∈ E (4) E s , x (cid:90) Ts (cid:88) j ∈N outi P i j ( t ) dt ≤ P i , max , ∀ i (5)The objective function expresses the trade-o ff between the accumulated for all sources util-ities / satisfaction of producing data and the accumulated cost due to power consumption forlink transmissions. Thus, its maximization targets at improving energy e ffi ciency by maximiz-ing source rates while penalizing the cost of power consumption for achieving them when J i j (cid:44) , ∀ ( i , j ), i.e. when power control is applied. The first constraint relates to the flowconservation at each node and for each destination, used to ensure high throughput for theexamined time interval [ s , T ]. The second constraint relates to the capacity restriction (right side)due to power, channel and interference limitations for each link. Finally, the third constraint,relates to a limitation on the total power consumption of each node for the examined timeinterval, [ s , T ], (left side) according to its energy storage, denoted as P i , max > It is important to note that we consider continuous time network operation (thus, using timeintegrals), for ease of presentation, due to the continuous time evolution of the channel state(Section III). The case of discrete time network operation can be obtained trivially by replacingthe integrals (cid:82) Ts by sums (cid:80) t = N L t = where N L is the number of time slots in [ s , T ] considered forthe network operation and each denoted by t . In this case the channel state will be sampled ateach time slot as described in the subsequent sections. Furthermore, for considering an infinite T , we should also divide every integral by T , thus considering time averages.This problem is non-convex due to the forms of the capacity, utility and cost functions. Evenif the utility and cost functions were concave and convex respectively as commonly assumed inliterature, the problem would still be non-convex due to the capacity function forms. However,we will prove that its duality gap is zero, which is an important fact as it renders the Lagrange(dual)-based optimization method optimal. The latter allows for devising e ffi cient algorithmicsolutions as it leads to a separable optimization problem with respect to the variables of eachlayer while using the Lagrange multipliers for the communication between adjacent layers forachieving a cross-layer optimal solution. Let µ di ≥ , ∀ i , d , l i j ≥ , ∀ ( i , j ) ∈ E , ν i ≥ , ∀ i be theLagrange multipliers associated with the constraints (3), (4), (5) correspondingly. Denote with L the whole set of the Lagrange multipliers. Then, the dual function is formulated as follows: L A ( L ) = max Λ ∈U F λ , R ∈U E × ( N − r , P ∈U EP E s , x (cid:90) Ts (cid:88) i , d : i ∈ S r ( d ) U di ( λ di ( t ) , t ) − (cid:88) ( i , j ) J i j ( P i j ( t )) dt − (cid:88) i , d µ di E s , x (cid:90) Ts λ di ( t ) dt + (cid:90) Ts (cid:88) j : i ∈R ( j , d ) r dji ( t ) dt − (cid:90) Ts (cid:88) j ∈R ( i , d ) r di j ( t ) dt − (cid:88) ( i , j ) ∈E l i j E s , x (cid:90) Ts (cid:88) d r di j ( t ) dt − (cid:90) Ts C i j ( P ( t )) dt − (cid:88) i ν i E s , x (cid:90) Ts (cid:88) j ∈N outi P i j ( t ) dt − P i , max . (6)Consequently, the dual problem of P is defined as D : = inf L ( L A ( L )). Theorem 1:
The problem P has zero dual gap, i.e. if P ∗ its optimal value and D ∗ the optimalvalue of the dual problem, then P ∗ = D ∗ .To prove this theorem we proceed in analogy with Theorem 1 in [22]. The proof relies on thefact that the channel’s cumulative distribution function (cdf) is continuous and thus no channelrealization has strictly positive probability. It uses the definition of nonatomic measures and theLyapunov’s convexity theorem [22]. Proof:
To prove the zero duality gap, we consider a perturbed version of the problem P ,obtained by perturbing the constraints used to define the Lagrangian. Let P ( ∆ ) be the functionthat assigns to each perturbation set ∆ = ( { ∆ i , d } ∀ i , d , { ∆ i , j } ∀ ( i , j ) , { ∆ i } ∀ i ), the solution of the followingperturbed optimization problem. P ( ∆ ) = max Λ ∈U F λ , R ∈U E × ( N − r , P ∈U EP E s , x (cid:90) Ts (cid:88) i , d : i ∈ S r ( d ) U di ( λ di ( t ) , t ) − (cid:88) ( i , j ) J i j ( P i j ( t )) dt s . t . E s , x (cid:90) Ts λ di ( t ) dt + (cid:90) Ts (cid:88) j : i ∈R ( j , d ) r dji ( t ) dt − (cid:90) Ts (cid:88) j ∈R ( i , d ) r di j ( t ) dt ≤ ∆ i , d , ∀ i , d (7) E s , x (cid:90) Ts (cid:88) d r di j ( t ) − C i j ( P ( t )) dt ≤ ∆ i , j , ∀ ( i , j ) ∈ E , E s , x (cid:90) Ts (cid:88) j ∈N outi P i j ( t ) dt − P i , max ≤ ∆ i , ∀ i (8)i.e. the constraints can be violated by ∆ amounts. In order to prove zero duality gap, we shouldshow that the function P ( ∆ ) is a concave function of ∆ [22].Let ∆ = ( { ∆ i , d } ∀ i , d , { ∆ i , j } ∀ ( i , j ) , { ∆ i } ∀ i ), ∆ = ( { ∆ i , d } ∀ i , d , { ∆ i , j } ∀ ( i , j ) , { ∆ i } ∀ i ) be two arbitrary sets ofperturbations with respective optimal values P = P ( ∆ ), P = P ( ∆ ) and respective solutions( Λ , R , P ) and ( Λ , R , P ). Then, for an arbitrary a ∈ [0 , ∆ = a ∆ + (1 − a ) ∆ and for feasible solutions ( ˆ Λ , ˆ R , ˆ P ), i.e. satisfying the constraints (7), (8), we need to show P ( ˆ ∆ ) = P ( a ∆ + (1 − a ) ∆ ) ≥ aP ( ∆ ) + (1 − a ) P ( ∆ ) . (9)Consider the set of all possible state ( X ) realizations H and the Borel field, B , on H . For A ∈ B , let E As , x be the expected value restricted on channel realizations included in A . We definethe following measures. θ id ( A ) = (cid:34) E As , x (cid:34)(cid:90) Ts U di ( λ di ( t ) , t ) dt (cid:35) , E As , x (cid:34)(cid:90) Ts U di ( λ di ( t ) , t ) dt (cid:35)(cid:35) , ∀ i , d : i ∈ S r ( d ) , (10) φ id ( A ) = (cid:34) E As , x (cid:34)(cid:90) Ts λ di ( t ) dt (cid:35) , E As , x (cid:34)(cid:90) Ts λ di ( t ) dt (cid:35)(cid:35) , ∀ i , d , (11) w i j ( A ) = (cid:34) E As , x (cid:34)(cid:90) Ts C i j ( P ( t )) dt (cid:35) , E As , x (cid:34)(cid:90) Ts C i j ( P ( t )) dt (cid:35)(cid:35) , ∀ ( i , j ) ∈ E , (12) v i j ( A ) = (cid:34) E As , x (cid:34)(cid:90) Ts J i j ( P i j ( t )) dt (cid:35) , E As , x (cid:34)(cid:90) Ts J i j ( P i j ( t )) dt (cid:35)(cid:35) , ∀ ( i , j ) ∈ E , (13) ξ i j ( A ) = (cid:34) E As , x (cid:34)(cid:90) Ts P i j ( t ) dt (cid:35) , E As , x (cid:34)(cid:90) Ts P i j ( t ) dt (cid:35)(cid:35) , ∀ ( i , j ) ∈ E , (14) while we also define θ id ( ∅ ) = φ id ( ∅ ) = w i j ( ∅ ) = v i j ( ∅ ) = ξ i j ( ∅ ) =
0. These measures are nonatomic[22] since the channel cdf is continuous and all the control variables are bounded. Thus, there areno channel realizations with positive measure, i.e., θ id ( A ) = φ id ( A ) = w i j ( A ) = v i j ( A ) = ξ i j ( A ) = A ∈ H . Let W ( A ) be the vector measure expressing collectively all measures θ id , φ id , w i j , v i j , ξ i j , which is also obviously nonatomic. Then from Lyapunov’s convexity theorem[22], the range of W is convex. Therefore, the value w = aW ( H ) + (1 − a ) W ( ∅ ) = aW ( H ) belongsto the range of values of W . As a result there exists A ∈ B such that W ( A ) = aW ( H ), i.e., θ id ( A ) = a θ id ( H ) , φ id ( A ) = a φ id ( H ), etc. Moreover, due to the additivity of measures, for thecomplement of A , A c , we have W ( A c ) = W ( H ) − W ( A ) = (1 − a ) W ( H ). Then, we define thefollowing controls for the new perturbation ˆ ∆ .ˆ r di j ( t ) = ar di j ( t ) + (1 − a ) r di j ( t ) , ∀ d , ( i , j ) ∈ E , t ∈ [ s , T ] , P − a . s . (15)ˆ λ di ( t ) = λ di ( t ) within A λ di ( t ) within A c ˆ P i j ( t ) = P i j ( t ) within A P i j ( t ) within A c ∀ i , d : i ∈ S r ( d ) , ∀ ( i , j ) ∈ E , t ∈ [ s , T ] , P − a . s . Obviouslysince Λ , Λ ∈ U F λ , R , R ∈ U E × ( N − r , P , P ∈ U EP , it also holds that ˆ Λ ∈ U F λ , ˆ R ∈ U E × ( N − r , ˆ P ∈U EP . Now, based on the above, we check if the controls defined above for the perturbation ˆ ∆ satisfy Ineqs. (7), (8) and if Ineq. (9) holds. For the constraint (7), we have: E s , x (cid:90) Ts ˆ λ di ( t ) + (cid:88) j : i ∈R ( j , d ) ˆ r dji ( t ) − (cid:88) j ∈R ( i , d ) ˆ r di j ( t ) dt = E A s , x (cid:34)(cid:90) Ts λ di ( t ) dt (cid:35) + E A c s , x (cid:34)(cid:90) Ts λ di ( t ) dt (cid:35) ++ E s , x (cid:90) Ts (cid:88) j : i ∈R ( j , d ) (cid:16) ar dji ( t ) + (1 − a ) r dji ( t ) (cid:17) dt − (cid:90) Ts (cid:88) j ∈R ( i , d ) (cid:16) ar di j ( t ) + (1 − a ) r di j ( t ) (cid:17) dt = a E s , x (cid:90) Ts λ di ( t ) dt + (cid:90) Ts (cid:88) j : i ∈R ( j , d ) r dji ( t ) dt − (cid:90) Ts (cid:88) j ∈R ( i , d ) r di j ( t ) dt + (1 − a ) E s , x (cid:90) Ts λ di ( t ) dt + (cid:90) Ts (cid:88) j : i ∈R ( j , d ) r dji ( t ) dt − (cid:90) Ts (cid:88) j ∈R ( i , d ) r di j ( t ) dt ≤ a ∆ i , d + (1 − a ) ∆ i , d = ˆ ∆ i , d . (16)With respect to the first constraint in (8), we have: E s , x (cid:90) Ts (cid:88) d ˆ r di j ( t ) dt − E s , x (cid:34)(cid:90) Ts C i j ( ˆ P ( t )) dt (cid:35) = a E s , x (cid:90) Ts (cid:88) d r di j ( t ) dt − (cid:90) Ts C i j ( P ( t )) dt + (1 − a ) E s , x (cid:90) Ts (cid:88) d r di j ( t ) dt − (cid:90) Ts C i j ( P ( t )) dt ≤ a ∆ i , j + (1 − a ) ∆ i , j = ˆ ∆ i , j . (17) Similarly, with respect to the second constraint in (8), we have: E s , x (cid:90) Ts (cid:88) j ∈N outi ˆ P i j ( t ) dt − P i , max = E A s , x (cid:90) Ts (cid:88) j ∈N outi P i j ( t ) dt + E A c s , x (cid:90) Ts (cid:88) j ∈N outi P i j ( t ) dt − P i , max = a E s , x (cid:90) Ts (cid:88) j ∈N outi P i j ( t ) dt + (1 − a ) E s , x (cid:90) Ts (cid:88) j ∈N outi P i j ( t ) dt − P i , max ≤ a ∆ i + (1 − a ) ∆ i = ˆ ∆ i . (18)Finally, P ( ˆ ∆ ) ≥ E s , x (cid:20)(cid:82) Ts (cid:16)(cid:80) i , d : i ∈ S r ( d ) U di ( ˆ λ di ( t ) , t ) − (cid:80) ( i , j ) J i j ( ˆ P i j ( t )) (cid:17) dt (cid:21) , i.e., P ( ˆ ∆ ) ≥ = E A s , x (cid:90) Ts (cid:88) i , d : i ∈ S r ( d ) U di ( λ di ( t ) , t ) − (cid:88) ( i , j ) J i j ( P i j ( t )) dt + E A c s , x (cid:90) Ts (cid:88) i , d : i ∈ S r ( d ) U di ( λ di ( t ) , t ) − (cid:88) ( i , j ) J i j ( P i j ( t )) dt = aP ( ∆ ) + (1 − a ) P ( ∆ ) , (19)which concludes the proof. B. Problem Solution
Since the dual gap corresponding to the problem P is null, we can obtain its optimal valueby solving its dual problem via a subgradient methodology [23]. The subgradient algorithm,repeatedly renews the Lagrange multipliers until converging to their optimal solutions. We use thesymbol η = { , , .. } for the repetitions of the subgradient algorithm. Then, the renewal equationsof the Lagrange multipliers take the form: µ di ( η + = µ di ( η ) + κ ( η ) · E s , x (cid:90) Ts λ d ∗ i ( t ) dt + (cid:90) Ts (cid:88) j : i ∈R ( j , d ) r d ∗ ji ( t ) dt − (cid:90) Ts (cid:88) j ∈R ( i , d ) r d ∗ i j ( t ) dt + , ∀ i , d , (20) l i j ( η + = l i j ( η ) + κ ( η ) · E s , x (cid:90) Ts (cid:88) d r d ∗ i j ( t ) dt − (cid:90) Ts C i j ( P ∗ ( t )) dt + , ∀ ( i , j ) ∈ E , (21) ν i ( η + = ν i ( η ) + κ ( η ) · E s , x (cid:90) Ts (cid:88) j ∈N outi P ∗ i j ( t ) dt − P i , max + , ∀ i . (22)where {} + denotes projection to [0 , ∞ ) and the values µ di (0) ≥ , ∀ i , d , l i j (0) ≥ , ∀ ( i , j ) ∈ E , ν i (0) ≥ , ∀ i , are considered given. The subgradient method is known to converge to a close neighbor-hood of the optimal values for the Lagrange multipliers if constant step-size, κ ( η ), is used, whilediminishing, non summable but square summable step size allows for convergence to the optimal values [23], [24]. The values of the stochastic processes λ d ∗ i , ∀ i , d , r d ∗ i j , ∀ ( i , j ) , d , P ∗ i j , ∀ ( i , j ) , arecomputed while obtaining the dual function (Eq. (6)) with the current solution for the Lagrangemultipliers, i.e. for the iteration η . For performing this maximization, we can observe that onecannot achieve a better objective value than the one achieved by choosing at each time t ∈ [ s , T ],each control variable optimally as a function of X ( t ) and the current values of the Lagrangemultipliers (iteration η ). Therefore, at the iteration η , given µ di ( η ) , ∀ i , d , l i j ( η ) ∀ ( i , j ) ∈ E , ν i ( η ) , ∀ i ,the controls are computed as follows: • The optimal λ d ∗ i , ∀ i , d : i ∈ S r ( d ), at the transport layer, are computed source-wise by ϑ U di ( λ di , t ) ϑλ di − µ di ( η ) = , ∀ t ∈ [ s , T ] , (23)while taking into account that λ d ∗ i ∈ U λ . From Eq. (23), it is observed that λ d ∗ i is time-varying but not random as it depends only on the deterministic Lagrange multiplier µ di ( η )(which is constant in time t ∈ [ s , T ]) and the deterministic time-varying utility function U di but not on the channel state X ( t ). In the case that U di ( λ di , t ) , ∀ i , d : i ∈ S r ( d ) are invariantwith t then λ d ∗ i , ∀ i , d : i ∈ S r ( d ) are constants over [ s , T ], too. • The optimal routing variables r d ∗ i j , ∀ ( i , j ) , d (network layer) are computed bymax R (cid:88) d (cid:88) ( i , j ) | j ∈ R ( i , d ) r di j ( µ di ( η ) − µ dj ( η ) − l i j ( η )) , ∀ t ∈ [ s , T ] , (24)i.e. r d ∗ i j ( t ) = R max , ∀ t ∈ [ s , T ], if ( µ di ( η ) − µ dj ( η ) − l i j ( η )) >
0, while it is observed (from Eq.(24)) that each r d ∗ i j is constant in time. • The optimal transmission power values at the physical layer, P ∗ , are computed via solving:max P (cid:88) ( i , j ) (cid:16) − J i j ( P i j ) + l i j ( η ) C i j ( P ) − ν i ( η ) P i j (cid:17) , ∀ t ∈ [ s , T ] , (25)while considering the U P constraints (Section IV). From Eq. (25), it is observed that P isa stochastic process since the link capacity C i j , ∀ ( i , j ) depends on the stochastic process X ( t ) representing the wireless channels (Section IV).The computation of the expected values involved in the Lagrange multipliers’ renewal equationsrequires Monte Carlo simulations. However, since the controls Λ ∗ , R ∗ , have been shown to bedeterministic, the expected value involved in the renewal equation of the Lagrange multipliers µ di , ∀ i , d (Eq. (20)) is superfluous. For the rest of the Lagrange multipliers, the expected valuesare obtained via the following algebraic computations. Firstly, the processes { X i j ( t ) : t ≥ } are discretized [25]. We compute δ t = T − sn , where n is adesign parameter representing the number of samples of the channel over the time interval [ s , T ]of the network’s operation, and we sample on the time instants { τ b = s + b · δ t } b = n . In view of(2) it is not hard to see that for every i , j and b we have X i j ( τ b ) = ρ i j ( b ) X i j ( τ b − ) + ζ i j ( b ) + Z i j ( b )where ρ i j ( b ) = exp( − (cid:90) τ b τ b − β i j ( s ) ds ) ζ i j ( b ) = (cid:90) τ b τ b − β i j ( s ) γ i j ( s ) exp (cid:0) − (cid:90) τ b s β i j ( q ) dq (cid:1) ds and Z i j ( b ) = (cid:90) τ b τ b − δ i j ( s ) exp (cid:0) − (cid:90) τ b s β i j ( q ) dq (cid:1) dW i j ( s ) . Note that { Z i j ( b ) } i , j , b are independent random variables with distributions N (0 , σ i j ( b )), where σ i j ( b ) = (cid:90) τ b τ b − δ i j ( s ) exp( − (cid:90) τ b s β i j ( q ) dq ) ds . The discretized scheme then becomes for every ( i , j ): X i j ( τ b ) = ρ i j ( b ) X i j ( τ b − ) + ζ i j ( b ) + σ i j ( b ) ξ i j ( b ) , b = , , . . . , n , X i j ( s ) = x i j (26)where { ξ i j ( b ) } i , j , b are independent samples from a standard normal random variable. After nu-merically computing the solution of the SDE (2), we compute P ∗ ( τ b ) for each b from Eq. (25)and we use a Riemann sum approximation for a sample of the integral (cid:82) Ts C i j ( P ∗ ( t )) dt , thatis (cid:82) Ts C i j ( P ∗ ( t )) dt (cid:39) (cid:80) n − b = C i j ( P ∗ ( τ b )) δ t , where C i j ( P ∗ ( τ b )) = B i j log + e KXi j ( τ b ) P ∗ i j ( τ b ) N + (cid:80) ( k , l ) ∈I i j e KXk j ( τ b ) P ∗ kl ( τ b ) (Sections III, IV). Finally, we repeat the above procedure M times to obtain M independentsamples of the preceding integral, and we average these samples to estimate the expectedcapacity of link ( i , j ) over the time interval [ s , T ], i.e. E s , x (cid:20)(cid:82) Ts C i j ( P ∗ ( t )) dt (cid:21) , appearing in theEq. (21). Similarly we obtain E s , x (cid:20)(cid:82) Ts P ∗ i j ( t ) dt (cid:21) . Since the computation of P ∗ from Eq. (25)involves the Lagrange multipliers’ values, the Monte Carlo computations of the expected valuesshould be performed at each iteration of the subgradient algorithm. Notably, ρ i j ( b ) , ζ i j ( b ) , σ i j ( b )are deterministic so we only need to compute them once. The optimal power allocation problem at the physical layer, i.e., the solution of Eq. (25)determines the complexity of the whole problem since everything else is simple algebraiccomputations. Indeed the computations of Eqs. (23), (24) can be distributed to the sources andlinks correspondingly. This cross-terminal optimization problem at the physical layer constitutesan important challenge in wireless networking [22] which is treated in other works in literature[22], [24] and is out of the scope of this paper. In the next section, we formulate and solvethe same problem in the case of orthogonal access to the wireless medium where the capacityfunctions take much simpler concave forms leading to tractable analytic solutions.We summarize below the steps of the algorithm proposed in this section for obtaining D ∗ .1) Initialize the Lagrange multipliers, η = , µ di (0) , ∀ i , d , l i j (0) , ∀ ( i , j ) ∈ E , ν i (0) , ∀ i .2) Compute λ d ∗ i , ∀ i , d , r d ∗ i j , ∀ ( i , j ) , d using Eqs. (23), (24) respectively.3) Compute the expected values E s , x (cid:20)(cid:82) Ts C i j ( P ∗ ( t )) dt (cid:21) , E s , x (cid:20)(cid:82) Ts P ∗ i j ( t ) dt (cid:21) , ∀ ( i , j ), as described.4) Compute µ di ( η + , ∀ i , d , l i j ( η + , ∀ ( i , j ) ∈ E , ν i ( η + , ∀ i from Eqs. (20), (21), (22) andset η ← η + , , C. Discussion
It is important to note that the time scale of the renewal of the Lagrange multipliers shouldbe distinguished from the time interval [ s , T ] of the network’s operation. In principle, the abovealgorithm should run o ff -line, i.e. prior to the network operation to determine the optimal sourcerates, routing variables and Lagrange multipliers and afterwards, the online network operationwill be designed based on these optimal values and the solution of the cross-terminal powerallocation problem of Eq. (25). Note that convergence of the dual variables close to their optimalvalues does not imply convergence of the primal variables except if the primal variables changecontinuously with respect to the optimal Lagrange multipliers (e.g. source rates). Following theapproach of [24], we can compute optimal routing variables while performing the subgradientiterations. Specifically let as assume that N o is the total number of subgradient iterations,while the index η ∈ , ..., N o − r di j ( t , N o ) = (cid:80) No − η = r d ∗ i j ( t ,η ) N o , ∀ ( i , j ) , d as optimal routing variables for each t ∈ [ s , T ], we can achieve aclose to the optimal value of P , using diminishing step size. This can be proven as in [24], if wefirst reformulate P in an equivalent form replacing the objective function by the optimization variable P (cid:48) and adding the constraint E s , x (cid:20)(cid:82) Ts (cid:16)(cid:80) i , d : i ∈ S r ( d ) U di ( λ di ( t ) , t ) − (cid:80) ( i , j ) J i j ( P i j ( t )) (cid:17) dt (cid:21) ≥ P (cid:48) .Then, obviously, Theorem 1 holds. More discussion on a possible (suboptimal) online imple-mentation of the proposed algorithm is made in Section VII.We also note that instantaneous values of the controls (online approach - as functions of X ( t ))during the network operation for such a problem may be obtained via a dynamic programmingsolution methodology (Hamilton-Jacobi-Bellman partial di ff erential equation (HJB pde)) [14]which adds dramatically to complexity for a wireless multihop network (specifically the solutionof the HJB pde is completely ine ffi cient [27], [26]).VI. O rthogonal A ccess to the M edium In this section, we redesign the problem P allowing only orthogonal access to the wirelessmedium, and thus leading to convex link capacity forms (Section IV) since the noise frominterference will become negligible. In order to achieve this, we introduce new optimizationvariables for each independent set ι , denoted as π ι , expressing the activation percentage of thecorresponding independent set at time t ∈ [ s , T ], and further satisfying the relations: (cid:80) ι π ι ( t ) ≤ , ≤ π ι ( t ) ≤ , ∀ ι, t ∈ [ s , T ] . In the following, Π stands for the collection of all π ι , ∀ ι and I n isthe number of the independent sets of the network’s connectivity graph. Since the channel stateis a stochastic process, similarly to the definition of the rest of the control variables (SectionIV), we define the value range for each π ι , U π = [0 , U π = (cid:8) π : [ s , T ] × Ω → U π : π is {F t } t ≥ s adapted (cid:9) . Then, Π ∈ U Π = {U I n π : (cid:80) I n ι = π ι ( t ) ≤ , ∀ t ∈ [ s , T ] } .The new optimization variables impose time-sharing among the independent sets, thus theyrender the interference levels negligible and the capacity of each link ( i , j ) is given by the concavefunction in Section IV. The time share corresponding to the link ( i , j ) at t ∈ [ s , T ], is given by (cid:80) ι :( i , j ) ∈ ι π ι ( t ). The new optimization problem P is formulated as: P : = max Λ ∈U F λ , R ∈U E × ( N − r , P ∈U EP , Π ∈U Π E s , x (cid:90) Ts (cid:88) i , d : i ∈ S r ( d ) U di ( λ di ( t ) , t ) − (cid:88) ( i , j ) (cid:88) ι :( i , j ) ∈ ι π ι ( t ) J i j ( P i j ( t )) dt s . t . E s , x (cid:90) Ts λ di ( t ) dt + (cid:90) Ts (cid:88) j : i ∈R ( j , d ) r dji ( t ) dt ≤ E s , x (cid:90) Ts (cid:88) j ∈R ( i , d ) r di j ( t ) dt , ∀ i , d (27) E s , x (cid:90) Ts (cid:88) d r di j ( t ) dt ≤ E s , x (cid:90) Ts (cid:88) ι :( i , j ) ∈ ι π ι ( t ) C i j ( P i j ( t )) dt , ∀ ( i , j ) ∈ E (28) E s , x (cid:90) Ts (cid:88) j ∈N outi (cid:88) ι :( i , j ) ∈ ι π ι ( t ) P i j ( t ) dt ≤ P i , max , ∀ i (29)The formulation of P is similar to the one of P (Section V), with the di ff erence that in P , wehave introduced the new optimization variables Π ∈ U Π , the link capacities are concave and thelink transmission powers, P i j ( t ), the link costs with respect to the transmission powers, J i j ( P i j ( t )),and the link capacities, C i j ( P i j ( t )), are replaced by their e ff ective values (cid:80) ι :( i , j ) ∈ ι π ι ( t ) P i j ( t ), (cid:80) ι :( i , j ) ∈ ι π ι ( t ) J i j ( P i j ( t )), (cid:80) ι :( i , j ) ∈ ι π ι ( t ) C i j ( P i j ( t )), correspondingly [6]. P is non-convex due to theappearance of the control variables in multiplicative form in the objective function and theconstraints (Eqs. (28), (29)) in addition to the general forms of the utility and cost functions.However, in a similar way as for P , it can be shown that P has a zero duality gap. Let µ di ≥ , ∀ i , d , l i j ≥ , ∀ ( i , j ) ∈ E , ν i ≥ , ∀ i , be the Lagrange multipliers associated with theconstraints (27), (28), (29), respectively. Denote with L the whole set of the Lagrange multipliers.Then, the dual function is formulated as follows: L A ( L ) = max Λ ∈U F λ , R ∈U E × ( N − r , P ∈U EP , Π ∈U Π E s , x (cid:90) Ts (cid:88) i , d : i ∈ S r ( d ) U di ( λ di ( t ) , t ) − (cid:88) ( i , j ) (cid:88) ι :( i , j ) ∈ ι π ι ( t ) J i j ( P i j ( t )) dt − (cid:88) i , d µ di E s , x (cid:90) Ts λ di ( t ) dt + (cid:90) Ts (cid:88) j : i ∈R ( j , d ) r dji ( t ) dt − (cid:90) Ts (cid:88) j ∈R ( i , d ) r di j ( t ) dt − (cid:88) ( i , j ) ∈E l i j E s , x (cid:90) Ts (cid:88) d r di j ( t ) − (cid:88) ι :( i , j ) ∈ ι π ι ( t ) C i j ( P i j ( t )) dt − (cid:88) i ν i E s , x (cid:90) Ts (cid:88) j ∈N outi (cid:88) ι :( i , j ) ∈ ι π ι ( t ) P i j ( t ) dt − P i , max . (30)Then, the dual problem is defined as D : = inf L ( L A ( L )). Theorem 2:
The problem P has zero dual gap.The proof is briefly described in Appendix B, provided as a supplementary file, as it is verysimilar to the proof of Theorem 1. Now, we obtain the optimal value of the problem P viasolving its dual. The renewal equations of the Lagrange multipliers are given by ( η = { , , .. } ): µ di ( η + = µ di ( η ) + κ ( η ) · E s , x (cid:90) Ts λ d ∗ i ( t ) dt + (cid:90) Ts (cid:88) j : i ∈R ( j , d ) r d ∗ ji ( t ) dt − (cid:90) Ts (cid:88) j ∈R ( i , d ) r d ∗ i j ( t ) dt + , ∀ i , d , (31) l i j ( η + = l i j ( η ) + κ ( η ) · E s , x (cid:90) Ts (cid:88) d r d ∗ i j ( t ) dt − (cid:90) Ts (cid:88) ι :( i , j ) ∈ ι π ∗ ι ( t ) C i j ( P ∗ i j ( t )) dt + , ∀ ( i , j ) ∈ E , (32) ν i ( η + = ν i ( η ) + κ ( η ) · E s , x (cid:90) Ts (cid:88) j ∈N outi (cid:88) ι :( i , j ) ∈ ι π ∗ ι ( t ) P ∗ i j ( t ) dt − P i , max + , ∀ i . (33)with given values µ di (0) ≥ , ∀ i , d , l i j (0) ≥ , ∀ ( i , j ) ∈ E , ν i (0) ≥ , ∀ i .The optimal values of the stochastic processes λ d ∗ i , ∀ i , d , r d ∗ i j , ∀ ( i , j ) , d , for each η are computedby Eqs. (23), (24) correspondingly and the same observations hold. Regarding the optimal values P ∗ i j , ∀ ( i , j ) ∈ E , π ∗ ι , ∀ ι , for each η , they are obtained by solving ∀ t ∈ [ s , T ]:max P , Π (cid:88) ( i , j ) (cid:88) ι :( i , j ) ∈ ι π ι (cid:16) l i j ( η ) C i j ( P i j ) − J i j ( P i j ) − ν i ( η ) P i j (cid:17) = max P , Π (cid:88) ι π ι (cid:88) ( i , j ) ∈ ι (cid:16) l i j ( η ) C i j ( P i j ) − J i j ( P i j ) − ν i ( η ) P i j (cid:17) , (34)which constitutes a maximum weight matching problem over the independent sets. Specifically,for its solution, each link ( i , j ) computes the stochastic process P ∗ i j ( t ), t ∈ [ s , T ] (depending onthe state’s, X , path) by maximizing − J i j ( P i j ) + l i j ( η ) C i j ( P i j ) − ν i ( η ) P i j , i.e. solving − ϑ J i j ( P i j ) ϑ P i j + l i j ( η ) ϑ C i j ( P i j ) ϑ P i j − ν i ( η ) = , ∀ t ∈ [ s , T ] , (35)while taking into account the U P constraints (Section IV). Then, each link ( i , j ) ∈ E is assigned aweight equal to W e ( i , j , t ) = (cid:16) − J i j ( P ∗ i j ( t )) + l i j ( η ) C i j ( P ∗ i j ( t )) − ν i ( η ) P ∗ i j ( t ) (cid:17) and finally the independentset ι ∗ maximizing the sum (cid:80) ( i , j ) ∈ ι W e ( i , j , t ) receives π ι ∗ ( t ) =
1, while π ι ( t ) = ι (cid:44) ι ∗ at t ∈ [ s , T ].In this paper, we assume that ties break arbitrarily, however, a study on how to break ties canbe found in [6].The computation of the expected values involved in the Lagrange multipliers follows thelines of the Monte Carlo simulations described in Section V and the algorithm for solving D is similar to the one in Section V. Note that Eq. (35) can be solved link-wise in a verye ffi cient manner, contrary to Eq. (25) which involves the complex cross-terminal problem. Theobservations regarding the convergence to the optimal Lagrange multipliers and primal valuesare of similar nature to the ones of Section V. At this point we study the solution of Eq. (35)in more detail in order to gain more insight regarding the optimal power control. Let us assumeLTF and convex link costs of the form J i j ( P i j ) = V P i j , where V > i , j ) ∈ E .Then, for a given η , the solution of Eq. (35) is explicitly given by: P ∗ i j ( X i j ( t )) = max (cid:8) , min { ˜ P ∗ i j ( X i j ( t )) , P max } (cid:9) , where ˜ P ∗ i j ( X i j ( t )) = − (cid:32) N e − KX i j ( t ) + ν i ( η )2 V (cid:33) + (cid:115)(cid:32) N e − KX i j ( t ) + ν i ( η )2 V (cid:33) − (cid:32) ν i ( η ) N e − KX i j ( t ) V − l i j ( η ) B i j V log(2) (cid:33) (36)Note that the optimally controlled power at η never exceeds the value P ∗ max i j = (cid:115)(cid:32) ν i ( η )2 V (cid:33) + l i j ( η ) B i j V log(2) − ν i ( η )2 V , (37) irrespectively of the P max value, while this value is achieved asymptotically when X i j ( t ) → −∞ .Note also that P ∗ i j ( X i j ( t )) → X i j ( t ) → ∞ . Therefore, the optimal power control exploitslow power loss values created by random fluctuations to increase the link capacity as much aspossible, thus positively a ff ecting the flows’ throughput (rates). On the other hand, transmissionpower is not wasted when the power loss is high. Note that in case of orthogonal access to themedium the aim of power control is not scheduling (which is defined via the Π variables) butto take advantage of the channel when it is favorable and to avoid depleting energy when thechannel is destructive.In the following, we assume no power control and scheduling and we prove an interestingcounter-intuitive theorem regarding the relation between the achieved utility by the network andthe channel’s di ff usion coe ffi cient in the case of LTF. Absence of power control means that in P , J i j ( t ) = , ∀ ( i , j ) , t ∈ [ s , T ], the constraint of Eq. (29) is dropped and finally, P i j ( t ) , ∀ ( i , j ) , t ∈ [ s , T ]are constant and predefined. Absence of scheduling means that the time share of each link isconstant and predefined, e.g. (cid:80) ι :( i , j ) ∈ ι π ι ( t ) = ζ i j ≥ , ∀ ( i , j ) , t ∈ [ s , T ]. A. Utilities as Increasing Functions of the Channel’s Di ff usion Coe ffi cient in the case of LTF We examine the e ff ect of the channel’s di ff usion coe ffi cient, on the users’ optimal utility. Con-trary to what is perhaps expected, we prove that an increase of a channel’s di ff usion coe ffi cient(SDE (2)) leads to increased achieved users’ utility. This result emphasizes the importance ofa more realistic model for the power loss such as the one of SDE (2), as opposed to the meanpower loss model used in the conventional NUM problem formulation. Theorem 3:
Let us consider two networks 1, 2, one noisier than the other, but otherwiseidentical. Precisely, for k ∈ { , } , ∀ ( i , j ) , ∀ t ∈ [ t , T ] , dX ki j ( t ) = β i j ( t ) (cid:0) γ i j ( t ) − X ki j ( t ) (cid:1) dt + δ ki j ( t ) dW ki j ( t ) , X ki j ( t ) = x i j , , and δ i j ( t ) ≤ δ i j ( t ) . (38)If J ∗ k , k ∈ { , } , is the optimal objective value achieved for each network, then J ∗ ≤ J ∗ . Proof:
The solutions of the SDEs in Eq. (38) can be written, ∀ ( i , j ), as X ki j ( t ) = X deti j ( t ) + (cid:90) tt δ ki j ( s ) e − (cid:82) ts β i j ( r ) dr dW ki j ( s ) , k ∈ { , } , (39)where X deti j ( t ) = e − (cid:82) tt β i j ( s ) ds x i j + (cid:82) tt β i j ( s ) γ i j ( s ) e − (cid:82) ts β i j ( r ) dr ds is the solution for a noiseless channel.Note that E t , x (cid:2) X ki j ( t ) (cid:3) = X deti j ( t ) , k ∈ { , } . By Proposition 3 . X i j ( t ) ( c ) ≤ X i j ( t ) , ∀ t ∈ [ t , T ], where ( c ) ≤ stands for partialordering in the convex order. That is, for every convex function ψ we have E t , x (cid:104) ψ (cid:0) X i j ( t ) (cid:1)(cid:105) ≤ E t , x (cid:104) ψ (cid:0) X i j ( t ) (cid:1)(cid:105) , ∀ t ∈ [ t , T ]. One such convex function is Shannon’s formula for the channel’scapacity C i j , i.e. x → B i j log (cid:18) + e Kx P i j N (cid:19) . Hence, E t , x (cid:104) C i j ( t ) (cid:105) ≤ E t , x (cid:104) C i j ( t ) (cid:105) , ∀ t ∈ [ t , T ] . (40)In particular, let us denote by L k , k ∈ { , } , the set of the deterministic Λ ∈ U F λ , R ∈ U E × ( N − r that satisfy the constraints of P imposed on each network: E t , x (cid:90) Tt λ di ( t ) dt + (cid:90) Tt (cid:88) j : i ∈R ( j , d ) r dji ( t ) dt ≤ E t , x (cid:90) Tt (cid:88) j ∈R ( i , d ) r di j ( t ) dt , ∀ i , d , (41) E t , x (cid:90) Tt (cid:88) d r di j ( t ) dt ≤ E t , x (cid:34)(cid:90) Tt ζ i j C ki j ( t ) dt (cid:35) , ∀ ( i , j ) ∈ E , k = { , } . (42)In view of Rel. (40) we have that L ⊆ L and therefore, as asserted, J ∗ = max Λ , R ∈ L E t , x (cid:90) Tt (cid:88) i , d U di ( λ di ( t ) , t ) dt ≤ max Λ , R ∈ L E t , x (cid:90) Tt (cid:88) i , d U di ( λ di ( t ) , t ) dt = J ∗ . (43) Remark 1:
Note that C i j (cid:0) δ i j ( . ) (cid:1) ≥ B i j E A (cid:104) log (cid:0) + P i j e KXi j ( t ) N (cid:1)(cid:105) ≥ P ( A ) B i j log P i j N + B i j K log2 E A (cid:2) X i j ( t ) (cid:3) ,where A = (cid:8) X i j ( t ) ≥ E [ X i j ( t )] (cid:9) . Since X i j ( t ) are gaussian, P ( A ) = / E A [ X i j ( t )] = E [ X i j ( t )] + (cid:113) V ( X i j ( t ))2 π where V ( X i j ( t )) is the variance of X i j ( t ) given by (cid:82) ts δ i j ( r ) exp( − (cid:82) tr β i j ( q ) dq ) dr .Therefore, the link capacity may take arbitrarily large values if (cid:82) Ts δ i j ( t ) dt is su ffi ciently large.VII. N umerical R esults In this section, we present and discuss indicative numerical results evaluating the proposedschemes focusing on the case of orthogonal access to the medium and LTF. After describing theevaluation setting and some general observations, we illustrate numerically Theorem 3 and thebehavior of the proposed framework in case of congestion control and routing (i.e. optimization in transport and network layers). The latter is then compared with the behavior of the joint routing,scheduling, congestion and power control scheme (i.e. cross-layer optimization). In addition,we examine the behavior of the proposed framework in the case of time varying utilities andthe possibility of applying the solution procedure of the dual problem D , during the networkoperation (online deployment).We consider a wireless multihop network of N =
16 nodes, forming a 4 × γ i j ( t ) , β i j ( t ) , δ i j ( t ), as wellas the initial states X i j ( s ) = x are identical for every link ( i , j ). Here, we consider M =
200 paths, N = . W , s =
0. Furthermore, ∀ ( i , j ) , τ b , B i j = Hz , γ i j ( τ b ) = γ (cid:18) + . e ( − τ bn ) sin(10 π τ b n ) (cid:19) [10], X i j (0) = γ , n = β i j ( τ b ) = γ = dB unless di ff erently mentioned. Moreover, δ i j ( τ b ) = δ, ∀ ( i , j ) , τ b , where δ will be tuned in each numerical experiment. Note that the value n =
500 determines the sampling rate for computing the Riemann sums that approximate the integralse.g. in Eqs. (32), (33) and it is chosen so that the Riemann sum is close to the correspondingintegral value, while simultaneously being small enough for trading-o ff the cost of sampling.Low sampling rate impacts performance since the Riemann sum does not converge to the actualvalue of the corresponding integral. On the contrary high sampling rate may induce extra costwithout o ff ering significant improvement in the Riemann sum’s accuracy in approximating thecorresponding integral. The periodic behavior of the LTF wireless channel parameters may bedue to an absorbing obstacle intervening periodically between the transmitter and the receiver (asin an example of [17]). In the absence of scheduling optimization, for each link ( i , j ) the value ζ i j is computed based on scheduling all the maximal independent sets of the network topology forequal percentage of time. Regarding the tra ffi c model, each node chooses a random destinationamong its non-physically connected nodes, and becomes the source for this destination. Initially,we consider logarithmic utilities, i.e. U di ( λ di ) = log( λ di ), a common choice in the literature tomodel elastic tra ffi c [29].Each numerical experiment for the determination of the optimal control variables and theoptimal Lagrange multipliers runs until convergence is ensured and specifically until the sumof the changes between consecutive values of the Lagrange multipliers over the preceding eightiterations is less than 0 . κ ( η ) = A (cid:48) η , ∀ η [23], where A (cid:48) = . A (cid:48) by one or more orders of magnitude our criterion of P o w e r l o ss va l u es Path 1 Path 2 Path 3 Path 4 γ ij (t) (a) δ = P o w e r l o ss va l u es Path 1 Path 2 Path 3 Path 4 γ ij (t) (b) δ = P o w e r l o ss va l u es γ ij (t)Path 1Path 2Path 3 (c) δ = β =
10 for τ b ≤ β = < τ b <
333 else β = convergence is satisfied too soon, impeding the subgradient algorithm approach to the globaloptimal values. On the other hand, increasing A (cid:48) or using constant values of κ ( η ) , ∀ η do notlead to convergence within a reasonable time interval.In order to obtain an intuition regarding the wireless channel’s stochastic behavior, Figs. 1(a),1(b) show typical sample paths of the solution to the SDE (2) for δ =
25 and δ =
50, respectively.As expected, we observe larger deviations of the power loss from γ i j ( · ) for the higher choice of δ . Consequently, the capacity may achieve higher values due to random fluctuations in this case.Note that although the curve of γ i j ( · ) tends to weaken to a line as time increases, it fluctuatesconsiderably for the duration of the network’s operation (transient state). Fig. 1(c) shows typicalsample paths of the solution to the SDE (2) for time varying speed of adjustment β i j ( τ b ). When β i j ( τ b ) attains low values (i.e. τ b ≤ γ i j ( τ b ), while the attraction to γ i j ( τ b ) becomes faster when β i j ( τ b ) =
500 (i.e. τ b > A. Congestion Control & Routing
In the sequel, we examine the case of applying only congestion control and routing, assumingthe transmitter of each link ( i , j ) has a constant power of P i j ( t ) = W , ∀ t ∈ [0 , T ]. Fig. 2(a) showsthe optimal source rates (i.e. after convergence since they change continuously with respect tothe optimal Lagrange multipliers) for di ff erent choices of the di ff usion coe ffi cient δ . It can beobserved that as δ increases, random fluctuations to higher capacity values are exploited to o ff erincreased optimal source rates, hence verifying numerically the statement of Theorem 3. It isimportant to note here that the noise level δ does not a ff ect the mean power loss, as indicated inthe proof of Theorem 3. In other words, if the mean power loss is used to determine capacity, O p t i m a l s ou r ce r a t es δ =50 δ : TV δ =20 δ =0 δ =5 (a) Optimal source rates (bits / sec) vs. δ . η ) µ L a g r a ng e m u l t i p li e r s (b) Convergence of the Lagrange multi-pliers { µ di } ∀ i , d . η ) l L a g r a ng e m u l t i p li e r s (c) Convergence of the Lagrange multi-pliers { l i j } ∀ ( i , j ) .Fig. 2. Congestion control & routing: Optimal source rates and convergence of the Lagrange multipliers. higher capacity values due to random fluctuations cannot be tracked and exploited for increasingthe source rates. Time-varying δ (TV) is also applied, and specifically δ = sin (10 π d / n ) + δ =
20 and δ =
50, thus leading to optimal source rates in betweenthe ones corresponding to these two values of δ . For benchmarking purposes, in Fig. 2(a) wehave added the case of δ =
0, which corresponds to time-varying but deterministic channels.We observe that the optimal source rates achieved under deterministic wireless channels are thelowest (approximately the same as in the case of a low value of noise, i.e. the case that δ = T on the time to convergence andthe achieved source rates. The value of n is adapted for each T as it is shown in Fig. 3(a). Weobserve that as the duration of the network’s operation, T , increases, the time to convergencealso increases (Fig. 3(a)), while the achieved arrival source rates decrease for all flows (Fig.3(b)). Finally, the behavior of the proposed algorithm in case of time varying β i j ( t ) with respectto the optimal source rates is shown in Fig. 3(c). We observe that when β i j ( t ) is high at thebeginning (close to time s =
0) the optimal rates are higher than when β i j ( t ) is initially low andincreases later in time. B. Joint Scheduling, Routing, Congestion & Power Control Scheme
In order to evaluate the joint scheduling, routing, congestion and power control scheme, weassume that each link can vary its transmission power between 1 W and 3 W = P i , max , ∀ i . Thenetwork topology and tra ffi c along with the rest of the parameters remain the same as in the (1,500) (2,1000) (4,2000) (8,4000) (10,5000)02000400060008000 Value of (T,n) T i m e t o c on ve r g e n ce (a) Impact of T on the time to conver-gence (iterations). O p t i m a l s ou r ce r a t es (1,500) (2,1000)(4,2000)(8,4000)(10,5000) (b) Impact of T on the optimal flows’rates (bits / sec). Flows’ IDs O p t i m a l s ou r ce r a t es β : 500, 100, 10 β : 100 β : 10, 100, 500 (c) Impact of time varying β i j ( t ) of the SDE(2) on the optimal flows’ rates (bits / sec).Fig. 3. Congestion control & routing: Study of the impact of T and time-varying β on convergence. In subfigure (c), blackcorresponds to β =
10 for τ b ≤ β =
100 for 166 < τ b <
333 else β = β =
500 for τ b ≤ β = < τ b <
333 else β =
10, and cyan corresponds to β = previous experiments. Fig. 4(a) depicts the optimal transmission power at each repetition ofchannel state’s sampling derived from Eq. (36) and the corresponding path of the link capacity. Itis observed that the optimal power increases when power loss decreases (thus capacity increases)and attains low values for high values of power loss, as expected from the analysis of Section VI.Therefore, the transmission power increases only when there is an opportunity for an importantcapacity improvement due to random dips of power loss. On the contrary, transmission power isnot wasted when the stochastic power loss does not support capacity increase. Fig. 4(b) showsthe optimal source rates (i.e. after convergence) for di ff erent choices of the di ff usion coe ffi cient δ . As in the previous case (Fig. 2(a)), it can be observed that as δ increases the optimal sourcerates increase for all flows. This is beyond the scope of Theorem 3, that does not account forpower control and scheduling in the optimization problem. Fig. 4(b) also includes the optimalsource rates when δ is time-varying (TV) and when channels are deterministic ( δ = τ b ) Optimal Power pathCapacity path (a) Capacity (bits / sec) & optimal trans-mission power paths (W) ( δ = Flows’ IDs O p t i m a l s ou r ce r a t es δ =50 δ =TV δ =20 δ =5 δ =0 (b) Optimal source rates (bits / sec) vs. δ . Flows’ IDs O p t i m a l s ou r ce r a t es Routing, Scheduling,Power & Congestion ControlRouting & Congestion Control (c) Comparison of optimal source rates(bits / sec) with and without schedulingand power control ( δ = η ) µ L a g r a ng e m u l t i p li e r s (d) Convergence of the Lagrange multi-pliers { µ di } ∀ i , d . η ) l L a g r a ng e m u l t i p li e r s (e) Convergence of the Lagrange multi-pliers { l i j } ∀ ( i , j ) .Fig. 4. Routing, Scheduling, Congestion & Power control. at the receiver’s side only. Furthermore, we study the expected per link transmission power overthe entire time interval [0 , T ] for the numerical experiments of Fig. 4. It can be easily computedthat it is equal to 1 . W for δ =
50, 1 . W for δ =
20, 1 . W for δ =
5, 1 . W for δ = W used to evaluatethe routing and congestion control scheme (Fig. 2(a)), thus achieving a more “green” networkoperation in addition to throughput improvement (Figs. 4(b), 4(c)) leading to energy e ffi ciency.On the other hand, we observe that as δ increases, the expected per link transmission powerdecreases, indicating the importance of taking randomness into account in operating wirelesschannels with energy e ffi ciency. C. Time Varying Utilities & Online Deployment
Finally, we study the case of time-varying utilities when applying routing and congestioncontrol. The utilities take the form U di ( λ di ) = log( λ di ) t , t ∈ [ s , T ] , s > ∀ i , d : i ∈ S r ( d ), i.e., theydecrease with time modeling the decreasing willingness of users to produce high data amounts τ b ) O p t i m a l s ou r ce r a t es (a) Optimal functions (bits / sec) to whichthe source rates converge ( δ =
20) fortime varying utilities. T i m e t o c on ve r g e n ce (b) Impact of n on the time to conver-gence (iterations) for time varying utili-ties. Value of n T i m e t o c on ve r g e n ce (c) Impact of n on the time to con-vergence (iterations) for time invariantutilities.Fig. 5. Routing & Congestion control: Time varying utilities & Impact of n on convergence. when approaching the end of the network’s operation. The optimal function to which the sourcerates converge over [ s , T ] is depicted in Fig. 5(a). We observe that as time increases, it dominatesover the Lagrange multiplier for the determination of the source rates (Eq. (23)).At this point we will make another interesting observation regarding the online application ofthe proposed approach (Section VI) during the network’s operation which is initially discussed inSection V. In order to obtain an online algorithm for the network control, i.e. during the networkoperation, we may consider that the network decisions are taken at τ b times when the channel issampled, while also considering that τ b ≡ η . Then, we should consider the optimal control valuesin the interval ( τ b , T ] as “predicted” and the ones in the interval [ s , τ b ) as “corrections”. However,we should study under what conditions convergence is achieved (in practice) early enough (forsmall τ b ) so that optimality with respect to the achieved value of P is not a ff ected. In Figs. 5(b),5(c), it is shown that when increasing n , the time for convergence (according to the imposedcriterion) of the proposed scheme is not significantly a ff ected for time invariant utilities whileit is a ff ected in a concave manner for time varying utilities. As a result, if n is large enough,the decisions taken at times τ b will converge fast enough compared to the whole duration T .Therefore, the online application of the proposed approach during the network operation will besuboptimal only at the beginning barely a ff ecting the optimal objective value of P .VIII. C onclusions In this paper we presented, analyzed and evaluated a framework of NUM for performingrouting, scheduling, congestion and power control under stochastic possibly non-stationary LTF or STF wireless channels modeled by SDEs. The continuous stochastic non-stationary wirelesschannels along with the consideration of transient phenomena lead to a problem formulationthat can also tackle non-convex and time-varying objective functions in an optimal way. Powercontrol aims at increasing users’ experience allowing for higher source rates while also improvingthe energy e ffi ciency. In the case of LTF, we prove that higher values of the di ff usion coe ffi cientof the power loss lead to higher optimal users’ utilities, a fact that cannot be captured by theconventional NUM problem’s formulation. Numerical results evaluate the latter along with theconvergence properties of our proposed algorithms and the e ff ect of diverse parameters on it.The e ffi ciency of power control and the conditions under which an online implementation of theproposed approach is possible are also investigated. Finally, our proposed NUM-based frameworkmay constitute a core for devising e ffi cient cross-layer algorithms for the network operation thatincorporate transient or non-stationary phenomena.IX. A cknowledgment This research is co-financed by the European Union (European Social Fund) and Hellenic national funds throughthe Operational Program ’Education and Lifelong Learning’ (NSRF 2007-2013). (under “ARISTEIA” 1260). M.L.acknowledges support from the NSRF Research Funding Program Thales: Optimal Management of DynamicalSystems of the Economy and the Environment MIS375586. R eferences [1] M. Chiang, “Balancing Transport and Physical Layers in Wireless Multihop Networks: Jointly Optimal Congestion Controland Power Control”, IEEE Journal on Selected Areas in Communications , Vol. 23, No. 1, pp. 104-116, Jan. 2005.[2] J. Papandriopoulos, S. Dey, J. S. Evans, “Optimal and Distributed Protocols for Cross-Layer Design of Physical andTransport Layers in MANETs”,
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