beyond that range no MAI at all is assumed to be induced. By fact, the main reason behind the. MAI model we adopt here is that a (very) large number of ...
1
Space-Division Multiple-Access for Wireless MIMO Networks: A Game Theoretic Approach
Enzo Baccarelli, Mauro Biagi, Cristian Pelizzoni, Nicola Cordeschi
{enzobac, biagi, pelcris, cordeschi }@infocom.uniroma1.it
Enzo Baccarelli, Mauro Biagi, Cristian Pelizzoni and Nicola Cordeschi are with INFO-COM Dept., University of Rome ”La Sapienza”, via Eudossiana 18, 00184 Rome, Italy. Ph. no. +39 06 44585466 FAX no. +39 06 4873330.This work has been partially supported by the Italian National project: ”Wireless 8O2.16 Multi-antenna mEsh Networks (WOMEN)” under grant number 2005093248. 29 agosto 2006
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Abstract Optimized Space-Division Multiple-Access (SDMA) strategies for Multiple-Input Multiple-Output (MIMO) Ad-hoc wireless networks are proposed in this paper, where noncooperating transmit/receive nodes are assumed to be affected by spatially colored Multiple Access Interference (MAI). The final task is to maximize the information throughput (in bits/slot) conveyed by each peer-to-peer link active in the network. For this purpose, the SDMA problem is modelled as an uncooperative strategic game, and the Game Theory approach is adopted for characterizing the nodes’ interactions and deriving out the conditions for the ”Nash Equilibrium” of the overall Ad-hoc networking game. The main contributions of this paper may be so summarized. Firstly, we develop a fully distributed scalable and asynchronous SDMA scheme combining both power-allocation and spatial data-shaping that maximizes (in a competitive sense) the information throughput sustained by each active peer-to-peer link under both Best-Effort and Contracted QoS access policies. Secondly, we characterize the convergence property of the proposed SDMA scheme under low and high traffic load offered to the network. Thirdly, based on the proposed SDMA scheme, we present two Connection Admission Procedures (CAPs) attaining an optimized (in a competitive sense) trade-off between number of allowed connections and throughput requested by each connection. Finally, for corroborating the carried out performance analysis, we present several numerical tests supporting superiority (in terms of conveyed peer-to-peer throughput) of the proposed SDMA scheme over more conventional collision-avoiding MAC schemes, such as TDMA and CSMA/CA.
Index Terms Multiple Antennas, Games Theory, MAI, SDMAC, Power-Allocation, Competitive Optimality, CAP, Self-reconfiguration, Fault-tolerant.
I. I NTRODUCTION Due to their capability to operate without any centralized infrastructure support, Ad-hoc wireless networks are considered to be main candidate to provide the distributed radio access facilities requested to support emerging high-throughput Personal Communication Services (PCSs) [9]. In order to meet the resulting QoS demands, lastly the utilization of the so-called ”smartantennas” in Ad-hoc networks gained consideration [9,23]. The term ”smart antennas” covers, indeed, a broad variety of Multi-Antenna Technological platforms that differ both in performance and transceiver complexity, such as the Switched-Beam and the Digital Adaptive Array (DAA) 29 agosto 2006
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antenna systems [23,26]. A Switched-Beam antenna system exhibits a pre-assigned antenna array pattern that can be pointed out to any of a (small) number of spatial directions [23,26]. The ability of such antennas to convey power in a specified direction provides a directive gain that can be exploited for extending range (e.g., system coverage), or reducing radiated power [23]. However, due to their limited signal processing capabilities, switched-beam antenna systems are not able to adaptively null out MAI typically affecting Ad-hoc networks [23]. An adaptive array receiver constructively combines multiple copies of the desired received signal, so to give arise to array gain, which is the increase factor in the average SNR measured at the receiver output [26]. Furthermore, when the receive antennas are sufficiently far apart, then the likelihood of simultaneous deep fades decreases, so that an adaptive array receiver is able to provide also diversity gain [23,26]. Finally, an adaptive array receiver may attenuate the signal from an interfering source (adaptive nulling). Transmit DAA systems can also provide array and diversity gains, so to increase those already provided by receiver DAA platforms [17]. In addition, a transmit DAA system may generate multiple co-channel data streams, so to give arise to spatial multiplexing gain [10]. However, for achieving multiplexing gain, DAA systems must be employed at both ends of a point-to-point communication link, giving arise to the Multiple-Input Multiple-Output (MIMO) technology [10]. The analysis and optimization (e.g., maximization) of the information throughput conveyed by each link active in an Ad-hoc MIMO network impaired by MAI, fading and MIMO channel-estimation errors are the focus of this work. The main output resulting from the carried out analysis is the optimized design of a novel fully distributed and asynchronous SDMA protocol (see Table III) allowing network nodes to maximize (in a competitive sense) the information throughput sustained by each peer-to-peer link active in the network. A. Flexibility Characteristics of the MIMO Physical Layer To attain the above mentioned goals, the SDMA protocol we develop exploits some flexibility characteristics that are unique to MIMO Physical Layer. These characteristics are Adaptive 29 agosto 2006
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Resource Usage, Range-vs.-Throughput Trade-off, Adaptive Interference Suppression, Robustness to Multipath fading [10,26]. Adaptive Resource Usage: A peer-to-peer MIMO link is able to exploit any radio resource not already spent to suppress MAI for increasing either array gains of the active streams, or the overall number of the spatially multiplexed streams (multiplexing gain) [21]. Range-vs.-Throughput Trade-off : Instead of splitting the overall data flow into several parallel independent streams to be simultaneously radiated by transmit antennas, correlated streams can be output by transmit antennas, so to achieve transmit diversity gain [12]. In turn, this gain may be exploited for achieving range extension (e.g., increased system coverage), or power minimization, or better reliability, as desired. Flexible MAI Suppression: Regardless of the location of interfering sources, the receiver of a MIMO link is able to suppress interfering streams, as long as it has sufficient number of degrees of freedom to do so [23]. In principle, even in the worst case of (r − 1) received interfering streams, a receiver equipped with r antennas is still able to properly detect the desired data conveyed by a single transmit stream [10,12]. Robustness to Multipath Fading: Peer-to-peer MIMO systems do not require line-of-sight (LOS) propagation, and are able to leverage multipath productively, so to give arise to multiplexing gain [12]. Hence, MIMO systems effectively work in rich scattering and multipath environments [10]. B. Motivations for novel distributed SDMA schemes Since Ad-hoc networks are characterized by topology-depending time-varying MAI and work without any central controller [9], an effective SDMA protocol should be able to adaptively exploit (in a combined way) the above mentioned flexibility characteristics of the MIMO Physical Layer by operating in a fully distributed, scalable and asynchronous way. Currently, Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) is ”de-facto” the MAC protocol considered for Ad-hoc networks [9]. Interestingly enough, a simple extension of CSMA/CA for 29 agosto 2006
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MIMO links can be designed that can provide an s min {t, r}-fold improvement in throughput performance compared to a Single-Input Single-Output (SISO) network (t is the number of antennas equipping each transmit node, while r indicates the number of antennas present at each receive node). We refer to this simple extension of CSMA/CA as CSMA/CA(s) [27]. Essentially, CSMA/CA(s) operates the same fashion as conventional CSMA/CA, except that all transmissions are performed using s independent parallel streams, so to attain spatial multiplexing gain. Such a protocol is still collision-free [27], and, when compared to default single-antenna (e.g., SISO) CSMA/CA, it is able to attain s times the throughput performance as the latter [27]. Thus, while this s-fold improvement guaranteed by the collision-free CSMA/CA(s) is, indeed, quite appealing, the key question we give insight in this contribution is: Is it possible for a more ”smart” SDMA scheme to attain better throughput performance by working in a fully distributed, scalable, and asynchronous fashion? The main conclusion arising from this contribution is that the answer to above question is, indeed, yes. Roughly speaking, the key-rationale behind this conclusion is that CSMA/CA(s) is still a collision-free MAC scheme, so that it is not able to fully exploit the advantages arising from the above mentioned flexible MAI-suppression capability of the MIMO Physical Layer. C. Proposed Contributions In this contribution we show that the potential gain arising from flexible MAI mitigation may be attained only in conjunction with adaptive radio resource usage (e.g., adaptive spatial multiplexing). In fact, the combined exploitation at the MAC layer of these two characteristics offered by MIMO Physical Layer requires that : i) the information throughput conveyed by each MAI-impaired peer-to-peer active link is evaluated; and ii) a (possibly) non-orthogonal (e.g., collision-affected) scalable SDMA scheme able to exploit the capture effect is developed and implemented in fully distributed and asynchronous fashion. Furthermore, in order to take full advantage from the potential offered by MIMO Physical Layer, in the proposed SDMA scheme each transmit node acquires and uses Channel State 29 agosto 2006
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Information (CSI) to perform both optimized power-control and (statistical) spatial-shaping of the transmitted multiplexed streams. Hence, in this contribution the topic concerning the optimized estimate of the path-gain coefficients of each MIMO link active in the network is also addressed, and the effects of errors possibly affecting the channel-estimations available at the transmit/receive nodes is explicitly taken into account in the developed SDMA protocol. Since no cooperation is assumed among nodes, we resort to the formal framework of the strategic non cooperative Game Theory [13] for modelling the mutual interactions between network nodes. Although the Game Theory approach has been already employed for solving power-control problems in wireless networks [1,4,5,20], nevertheless, at the best of the authors’ knowledge, till now the application of the Game Theory for the analysis and optimized design of distributed SDMA schemes results, indeed, to be new [27,28]. Specifically, the main appealing features of the SDMA scheme we propose may be so summarized. •
The proposed SDMA scheme combines (in an optimized way) power-control and spatial signal-shaping (in a statistic sense) of the multiplexed data streams to be transmitted. So doing, it allows the maximization (in a competitive sense) of the information throughput sustained by each peer-to-peer active link. This feature of the presented SDMA scheme is still retained even when the channel estimates available at the transmit/receive nodes are affected by errors.
•
The proposed SDMA scheme is fully scalable, distributed, and asynchronous. Furthermore, the proposed SDMA scheme is self-reconfiguring, since each transmit node dynamically updates the spatial shaping of the radiated streams, so to match to the changes of the network topology induced by nodes mobility. In addition, the proposed SDMA scheme is also fault-tolerant, thus meaning that it is self-reconfiguring when nodes leave or new nodes join the network.
•
The proposed SDMA scheme allows to implement both Best Effort and Contracted QoS access policies for sustaining multiple QoS traffic classes. In particular, when the QoS users requirements (measured in terms of requested throughput) cannot be sustained by the
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network, the proposed SDMA scheme automatically shifts the working point of the network (represented by the delivered throughput) to the next-close-sustainable one. •
Several numerical tests are provided for supporting the carried out analysis. They lead to the conclusion that the proposed SDMA scheme outperforms the conventional collision-free (e.g., orthogonal) MAC schemes (such as CSMA/CA(s), MIMO TDMA, MIMO FDMA, MIMO CDMA) in terms of peer-to-peer conveyed throughput, specially in MAI-limited application scenarios.
Finally, the fully distributed and scalable feature of the presented SDMA scheme is employed to develop two novel Connection Admission Procedures (CAPs) for Ad-hoc networks. The proposed CAPs are able to optimally balance (in a competitive sense) the number of allowed connections (fairness property) and the QoS requirements of each connection. D. Organization of the work The remainder of this paper is organized as follows. The system model is given in Sect.II, while the evaluation of conveyed information throughput of Ad-hoc networks affected by MAI is carried out in Sect.III. In Sect.IV the optimized transmit power-allocation is characterized when a single transmit/receive pair is impaired by ”static” (e.g., time-invariant) MAI. Thus, after shortly reviewing in Sect.V the MAI model of [16] for Ad-hoc networks, in Sect.VI we present an iterative Game Theory-based SDMA algorithm for the optimal (e.g., throughput maximizing) distributed and asynchronous power-allocation and signal-shaping of all peer-topeer active links. Analytical conditions for the convergence of the network state towards a stable point (e.g., the Nash Equilibrium of the underlying strategic Game) are also provided in Sect.VI and proved in the final Appendices. The performance of the proposed SDMA scheme in terms of conveyed network throughput is tested in Sect.VII, where its self-reconfiguring and faulttolerating capabilities are also pointed out. The above mentioned distributed CAPs are presented in the conclusive Sect.VIII, where some final hints for future investigations are also suggested. Before proceeding, few words about the adopted notation. In the sequel, capital letters denote 29 agosto 2006
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matrices, lower-case underlined symbols are for vectors, while characters overlined by arrow mean block-matrices and block-vectors. Apexes ∗ , T ,
→
†
are used for denoting conjugation,
transposition and conjugate-transposition respectively, while lower-case letters indicate scalar quantities. Furthermore, det [A] and T ra[A] mean determinant and trace of the matrix A [a1 ... am ], while vect(A) is the (block) vector obtained by the ordered stacking of the columns of matrix A. Finally, Im denotes the (mxm) identity matrix, ||A||E is the Euclidean norm of the matrix A [6], A ⊗ B represents the Kronecker product [6], 0m is the m-dimensional zero-vector, lg indicates natural logarithm and δ(m, n) is used for denoting the Kroenecker delta. II. T HE N ETWORK M ODEL The considered scenario is a wireless Ad-hoc network [9], where multiple autonomous transmitreceive nodes need simultaneously to be linked over a limited-size hot-spot cell, so that they are affected by MAI. Specifically, the (complex base-band equivalent) point-to-point radio channel from a transmit node Tx to a receive one Rx is sketched in Fig.1. It is composed by transmit and receive units equipped with t ≥ 1 and r ≥ 1 antennas respectively, and by a MIMO slow-variant Rayleigh flat faded radio channel1 impaired by (additive) MAI caused by the neighboring active transmit nodes. Each path gain hji ∈ C1 , 1 ≤ j ≤ r, 1 ≤ i ≤ t, from transmit antenna i to receive one j is modelled as a zero-mean unit-variance complex random variable (r.v.) [2,3,11] and these gains may be assumed mutually independent when the antennas are properly far apart (see [10]). Furthermore, for low-mobility applications, the path gains may be also considered time-invariant over T ≥ 1 signalling periods. The resulting ”block-fading” model describes the main features of several packet-based networks, serving quasi-static (or nomadic) users [11,16]. The MAI effect on the link of Fig.1 is dependent on the network topology [16], and in the considered scenario it is assumed to be constant over (at least) the transmission time of an overall packet [16]. Anyway, since the path gains {hji } and MAI may assume different values 1
By referring to Fig.1, the assumption of flat fading may be considered reasonable when the RF bandwidth Bw of the radiated
signals is less than the coherence bandwidth Bc of the MIMO forward channel of Fig.1 [10].
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over adjacent packets, we suppose that Tx and Rx in Fig.1 do not know them at the beginning of each packet transmission. Hence, we assume that all packets are composed by T ≥ 1 slots, where the first TL ≥ 0 are used by the receiver for learning the MAI statistics, the second Ttr ≥ 0 are employed for estimating the path gains {hji } of the MIMO channel (see Sect.II.A), and the last Tpay T − Ttr − TL are devoted to convey payload data (see Sect.II.B). During the Learning Phase, the transmitter Tx of Fig.1 is off, so that the sampled (vector) signal received by Rx is just given by the combined effect of MAI and (white) thermal noise. The resulting (r × r) covariance matrix Kd of the overall disturbance (e.g., MAI plus thermal noise) may be evaluated by Rx during the Learning Phase via a sample average of the received signals and, at the end of the Learning Phase, it is communicated back to Tx via the (ideal)2 feedback link of Fig.1. A comprehensive description of the Learning Phase may be found in [11,14,24] and, due to space limitation, will be not replicated here. A. The Training Phase On the basis of the MAI covariance matrix Kd received at the end of the Learning Phase, during the Training Phase the transmit node of Fig.1 optimally shapes the pilot streams { xi (n) ∈ C1 , TL + 1 ≤ n ≤ TL + Ttr }, 1 ≤ i ≤ t, to be used to estimate the (r × t) path gains {hji } of MIMO forward channel of Fig.1. The resulting (sampled) signals { yj (n) ∈ C1 , TL + 1 ≤ n ≤ TL + Ttr , 1 ≤ j ≤ r} received by the j-th antenna of Rx during this phase may be modelled as [10,11]
1 yj (n) = √ hji x i (n) + dj (n), TL + 1 ≤ n ≤ TL + Ttr , 1 ≤ j ≤ r , t i=1 t
(1)
j (n) ) is given where the corresponding overall disturbance (e.g., MAI vj (n) plus thermal noise w by dj (n) vj (n) + w j (n), TL + 1 ≤ n ≤ TL + Ttr , 1 ≤ j ≤ r, 2
(2)
About this assumption it is reasonable to assume a separate channel for feedback signalling. The non-ideality is taken in
account in [11] and it is not considered here.
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is independent from the path gains {hji } of the forward MIMO channel. The radiated pilot streams { xi (n)} in (1) are assumed power-constrained as in [10,11] t 1 || xi (n)||2 ≤ P, TL + 1 ≤ n ≤ TL + Ttr , t i=1
(3)
with P being the maximum power level radiable by each transmit antenna during the Training Phase. Hence, the resulting signal to interference-plus-noise ratio (SINR) γ j measured at the receive antenna j equates (see eqs.(2), (3)) γ j = P/kjj , 1 ≤ j ≤ r,
(4)
where kjj is the j-th diagonal entry of the MAI covariance matrix Kd . As shown in [10], the Ttr ×r (complex) samples in (1) received at the output of all receive antennas during the Training ≡ [y ...y ] defined as in [10,11] Phase may be collected into a (Ttr × r) observation matrix Y 1 r = √1 XH + D, Y t
(5)
[x1 ...xt ] is the (Ttr × t) matrix gathering the transmitted pilot symbols { xi (n)} in (1), where X H [h1 ...hr ] is the (t × r) matrix composed by the path gains {hji } in (1) of the considered [d1 ...dr ] is composed by the disturbance samples MIMO channel, and the (Ttr × r) matrix D {dj (n)} in (2) arising from MAI plus noise. Thus, directly from (3), it follows that the pilot in (5) must satisfy the power constrain matrix X †
X ] ≤ tTtr P. T ra[X
(6)
˜ in (5) are employed by the receive node Rx As detailed in [11], the training observations Y ˆ E{H|Y} of the for computing the Minimum Mean Square Error (MMSE) matrix estimate H MIMO channel matrix H in (5). At the end of the Training Phase (e.g., at n = TL + Ttr ), ˆ is communicated back by Rx to the transmitter Tx of Fig.1 the resulting matrix estimate H via the (ideal) feedback link of Fig.1. The resulting MMSE channel estimation errors {εji ˆ ji − hji , 1 ≤ j ≤ r, 1 ≤ i ≤ t} are zero-mean, complex, independent identically distributed h (i.i.d.) Gaussian random variables with variance ˆ ji −hji ||2 } = (1+a/t)−1 , for any i, j σε2 E{||εji ||2 } ≡ E{||h 29 agosto 2006
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where a
Ttr P˜ T ra[K−1 d ] r
(see [11] for more details on this point).
B. The Payload Phase At the beginning of the Payload Phase (e.g., at n = TL + Ttr + 1), the Tx node of Fig.1 ˆ matrices, and it is going to transmit has already acquired the current values of both Kd and H the current packet M (see Fig.1). For this purpose, it has to ”shape” the corresponding payload streams {φi (n) ∈ C1 , TL + Ttr + 1 ≤ n ≤ T }, 1 ≤ i ≤ t, and then radiate them. The corresponding sampled signals {yj (n) ∈ C1 , TL + Ttr + 1 ≤ n ≤ T }, 1 ≤ j ≤ r, received at Rx may be modelled as [10,11] 1 yj (n) = √ hji φi (n) + dj (n), t i=1 t
TL + Ttr + 1 ≤ n ≤ T, 1 ≤ j ≤ r,
(7)
where the sequences dj (n) vj (n) + wj (n), 1 ≤ j ≤ r, denote the overall disturbances (e.g., MAI vj (n) plus thermal noise wj (n)) received during n-th slot of the Payload Phase. The statistical properties of {dj (n)} are the same ones we have described for the Learning Phase, and the disturbance samples in (7) are independent from the path gains {hji } and radiated payload streams {φi } [10]. Furthermore, after assuming the payload streams power-constrained as in [2] 1 E{||φi (n)||2 } ≤ P, TL + Ttr + 1 ≤ n ≤ T, t i=1 t
(8)
then the resulting SINR γj measured at the output of j-th receive antenna equates γj = P/kjj , 1 ≤ j ≤ r,
(9)
where kjj is the j-th diagonal entry of MAI-plus-noise covariance matrix Kd . Now, let y(n) [y1 (n)...yr (n)]T be the (r × 1) column vector collecting the r scalar quantities in (7), and φ(n) [φ1 (n)...φt (n)]T be the (t×1) column vector of the (sampled) payload streams radiated by the Tx node. Thus, from (7) they are related as in 1 y(n) = √ HT φ(n) + d(n), TL + Ttr + 1 ≤ n ≤ T, t
(10)
where {d(n) [d1 (n)...dr (n)]T , TL +Ttr +1 ≤ n ≤ T } is the temporally-white spatially-colored Gaussian disturbance sequence with spatial covariance matrix given by Kd . Furthermore, from 29 agosto 2006
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eq.(8) the (t × t) spatial covariance matrix Rφ E{φ(n)φ(n)† } of the t-dimensional signal vector radiated by Tx over a slot time is power-constrained as in T ra[Rφ ] E{φ(n)† φ(n)} ≤ tP, TL + Ttr + 1 ≤ n ≤ T.
(11)
Finally, by stacking the Tpay observed vectors in (10) into the corresponding (Tpay r × 1) block T → vector − y yT (TL + Ttr + 1) ...yT (T ) , we can compactly express the Tpay relationships in (10) as follows:
1 − − → → − → y = √ [IT pay ⊗ H]T φ + d , t
(12)
→− − → E{ d ( d )† } = ITpay ⊗ Kd ,
(13)
T → − where the (block) covariance matrix of the disturbance (block) vector d dT (TL + Ttr + 1) ...dT (T ) equates
T → − while the squared average Euclidean norm of the (block) vector φ φT (TL + Ttr + 1) . . . φT (T ) in (12) collecting the transmitted payload signals is bounded as in (see (11)) →† − − → E{ φ φ } ≤ Tpay tP.
(14)
III. P ER - NODE I NFORMATION T HROUGHPUT IN THE PRESENCE OF CHANNEL - ESTIMATION ERRORS AND SPATIALLY COLORED
MAI
The block-fading model of Sect.II used for modelling the MIMO channel of Fig.1 results to be information stable [8], so that the corresponding Shannon capacity C (nats/slots) dictates the ultimate information throughput conveyed by the MIMO link Tx → Rx of Fig.1 during the Payload Phase [7]. Therefore, by resorting to quite standard approaches [7], the average Shannon’s capacity C of the MIMO channel (12) can be expressed as ˆ ˆ H)d ˆ H, ˆ (nats/slot), C = E{C(H)} ≡ C(H)p( ˆ = where p(H)
1 π(1−σε2 )
rt exp
(15)
1 ˆ†ˆ − (1−σ 2 T ra[H H] ε)
is the Gaussian probability density function
ˆ [11], and the random variable (pdf) of the channel estimates H ˆ C(H)
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sup Tpay − − → →† − → φ :E{ φ φ }≤tTpay P
→ ˆ − − → I y ; φ |H , (nats/slot)
(16)
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is the Shannon capacity of the MIMO link Tx → Rx of Fig.1 conditioned on the current values ˆ of the available channel estimates. Finally, I(·; ·|·) in (16) denotes the mutual information H ˆ = H ), operator [7]. Unfortunately, when no exact channel estimates are available (e.g., H → − the pdf of the input signals φ attaining the sup in (16) is currently unknown, even for the simplest case of spatially white MAI [2,10]. Anyway, we know that Gaussian distributed input ˆ = signals achieve the sup in (16) also when (H H) as long as the length Tpay of the payload phase (largely) exceeds the number t of transmit antennas (see [10] about this asymptotic result). Therefore, in the sequel we proceed to evaluate the sup in (16) under the assumption of Gaussian distributed input signals. In this case, the Tpay components {φ(n) ∈ Ct , TL +Ttr +1 ≤ n ≤ T } in → − (10) of the overall signal vector φ in (12) are uncorrelated zero-mean proper complex Gaussian vectors, with correlation matrix Rφ meeting (11). The corresponding information throughput − → ˆ → ˆ 1 sup I − , (nats/slot), (17) TG (H) y ; φ |H Tpay T ra[R ]≤P t φ conveyed by the MIMO channel (12) for Gaussian input signals generally falls below the ˆ in (16), so that we have TG (H) ˆ ≤ C(H). ˆ However, the above inequality Shannon’ Capacity C(H) ˆ =H is satisfied as equality when at least one of the above cited two operating conditions (e.g., H ˆ in (17), we remark that, in general, and/or Tpay >> t) is met. Therefore, going to evaluate TG (H) − → ˆ → the conditional mutual information I − in (16) resists closed-form computation [10]. y ; φ |H However, in [11] the following result is proved. Proposition 1. Let the spatial correlation matrix Rφ in (11) be assigned. Thus, the resulting − → ˆ → conditional mutual information I − in (16) of the MIMO channel (12) when Gaussian y ; φ |H input signals are employed is given by the following closed-form relationship (see [11]): − → ˆ → I − y ; φ |H = 2 1 −1/2 ˆ T ˆ ∗ K−1/2 +σε2 P K−1 −lg det Irt + σε Tpay (K−1 )∗ ⊗R , (18) = Tpay lg det Ir + Kd H Rφ H d d d φ t t when at least one of the following conditions is met: a) 29 agosto 2006
both Tpay and t are large;
(19) DRAFT
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b)
ˆ approaches H; H
(20)
c)
all SINRs γj , 1 ≤ j ≤ r, in (14) vanish.
(21)
A proof of Proposition 1 is given in [14] and, for sake of brevity, will be not replicated here. IV. O PTIMIZED P OWER -A LLOCATION AND S IGNAL -S HAPING IN THE PRESENCE OF COLORED
MAI AND C HANNEL -E STIMATION ERRORS
Therefore, according to (17), let us carry out the power-constrained sup of the conditional throughput in (18). Towards this end, we begin to indicate as Kd = Ud Λd U†d ,
(22)
the Singular Value Decomposition (SVD) of the MAI spatial covariance matrix Kd , where Λd diag{µ1 , ..., µr },
(23)
is the (r × r) diagonal matrix composed by the magnitude-ordered singular values of Kd . Thus, after introducing the (t × r) matrix
∗
−1/2
ˆ K AH d
Ud ,
(24)
ˆ and spatial MAI Kd , let accounting for the effects of both imperfect channel estimate H A = UA DA V†A ,
(25)
be the corresponding SVD, where UA and VA are unitary matrices, and DA diag{k1 , ..., ks , 0t−s },
(26)
is the (t × r) diagonal matrix built up by the s min{r, t} magnitude-ordered singular-values k1 ≥ k2 ≥ ... ≥ ks > 0 of A. Finally, let us also introduce the following dummy positions: αm
2 µm km , 1 ≤ m ≤ s; t(µm + P σε2 )
βl
σε2 Tpay , 1 ≤ l ≤ r. tµl
(27)
Thus, it can be proved [11,14] that an application of the Kuhn-Tucker conditions [7] allows to compute the powers {P (m), 1 ≤ m ≤ t} to be radiated by t antennas of Tx in Fig.1 for 29 agosto 2006
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achieving the sup in (17). These optimized powers are detailed by the following Proposition 2. Proposition 2. Let us assume that at least one of conditions (19), (20), (21) is met. Thus, for m = s + 1, ..., t, the radiated powers {P (m)} achieving the sup in (17) vanish, while for m = 1, ..., s they are given by the following expressions: σ 2 P t 2 ≤ 1+ ε ] ; P (m) = 0, when km + σε2 T ra[K−1 d µm ρ
1 1 rρβmin βmin L − 1 + {βmin L}2 + 4βmin ρ − , P (m) = − 2βmin αm αm Tpay
where βmin
(28)
σ 2 P t 2 when km > 1+ ε ] , (29) + σε2 T ra[K−1 d µm ρ r ρ − α1m . Furthermore, the nonnegative min{βl , l = 1, .., r} and L 1 − Tpay
scalar parameter ρ in (28), (29) is set so to satisfy the following power constrain (see eq.(8)):
P (m) ≤ P t,
(30)
m∈I(ρ)
where I(ρ) {m = 1, ..., s :
2 km
σε2 P t −1 2 + σε T ra[Kd ] }, > 1+ µm ρ
(31)
is the set composed by m indexes meeting the inequality in (29). Finally, the resulting optimal spatial correlation matrix Rφ (opt) shaping the signal radiated by Tx in Fig.1 is aligned along the right-eigenvectors of matrix A according to Rφ (opt) = UA diag{P (1), ...P (s), 0t−s }U†A ,
(32)
so that the resulting maximum information throughput in (17) can be evaluated via the following closed-form relationship: ˆ = TG (H)
s r σε2 P 1 1+ + lg 1 + βl P (m) (nats/slot). lg(1 + αm P (m)) − µm Tpay l=1 m=1 (33)
r m=1
A proof of the results reported by Proposition 2 may be found in [11,14].
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V. A T OPOLOGY-BASED MAI MODEL FOR M ULTI -A NTENNA AD - HOC N ETWORKS To test the actual effectiveness of the power-allocation and signal-shaping results previously reported in Proposition 2, we consider the application scenario of Fig.2 that captures the keyfeatures of the spatial MAI affecting MIMO Ad-hoc networks [9,16]. Specifically, according to [16], the Ad-hoc network of Fig.2 is composed by n∗ noncooperative, mutually interfering, pointto-point3 links Txf → Rxf , 1 ≤ f ≤ n∗ , so that the signal vector received by node Rxg is the combined effect of the desired MIMO signal radiated by Txg and the (n∗ − 1) interfering MIMO signals generated by all other transmit nodes. In detail, each transmit node Txf is assumed to be equipped with tf antennas, while rf is the number of the antennas equipping the corresponding Rxf node. Therefore, after denoting as l(f, g) the length of the link Txf → Rxg , thus the resulting rg -dimensional disturbance d(g) (n) in (10) received by Rxg may be modelled as in (see [16]) N l(g, g) 4 1 d(g) (n) = (34) √ χ(f, g)HT (f, g)φ(f ) (n) + w(g) (n), tg l(f, g) f =1,f =g where w(g) (n) denotes the thermal noise, φ(f ) (n) is the tf -dimensional (Gaussian distributed) signal radiated by Txf , and χ(f, g) captures the shadowing effects4 impairing the link Txf → Rxg . The (tf × rg ) matrix H(f, g) in (34) denotes the Rice-distributed fast-fading phenomena of the link Txf → Rxg . More in detail, according to the fast-fading spatial interference model developed in [16], this channel matrix H(f, g) in (34) may be modelled as in (see [16]) k(f, g) 1 H(sp) (f, g) + H(sc) (f, g), 1 ≤ f, g ≤ n∗ , H(f, g) ≡ 1 + k(f, g) 1 + k(f, g)
(35)
where k(f, g) ∈ [0, +∞) is the Rice-factor of the link Txf → Rxg , while the elements of the (tf × rg ) matrix H(sc) (f, g) are mutually independent zero-mean unit-variance Gaussian r.v.s, accounting for the scattering phenomena present on the link Txf → Rxg . Furthermore, according 3
Without loss of generality we assume transmitters and receivers separately. In a realistic scenario each node can act as
transmit/receive one. So in this analysis the same results can be obtained by exchanging transmitters and receivers. 4
Without loss of generality, we may assume the coefficient χ(f, g) falls into the interval [0, 1]. Specifically, χ(f, g) = 1
represents the worst case where MAI induced by Txf on Rxg is maximal.
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to [16], the (tf × rg ) matrix H(sp) (f, g) in (35) accounts for the (MIMO) specular component of the signal received by Rxg , and it can be modelled as in (see [16]) H(sp) (f, g) ≡ a(f, g)bT (f, g), 1 ≤ f, g ≤ n∗ ,
(36)
where a(f, g) and b(f, g) are the (tf × 1) and (rg × 1) column vectors describing the specular array responses of Rxg and Txf , respectively [16]. In particular, when regularly spaced linear arrays with isotropic elements are used, the above vectors may be directly evaluated as [16] a(f, g) = [1, exp(j2πν cos(θa (f, g))), ... exp(j2πν(rf − 1) cos(θa (f, g)))]T ,
(37)
b(f, g) = [1, exp(j2πν cos(θd (f, g))), ... exp(j2πν(rg − 1) cos(θd (f, g)))]T ,
(38)
where θa (f, g), θd (f, g) are the arrival and departure angles of the link Txf → Rxg (see Fig.2), while ν is the antennas spacing in multiple of the RF wavelength5 . Thus, basing on the model (g)
(35), (36), the resulting MAI covariance matrix Kd measured at Rxg is given by the following relationship:
† (g) ∆ Kd = E d(g) (n) d(g) (n) ≡
+
⎧ ⎪ ⎨ n∗ ⎪ ⎩ f =1 f =g
⎧ ⎪ ⎨ ⎪ ⎩
N0 +
l(g,g) 4 n∗
l(f,g)
⎫ ⎪ ⎬
χ2 (f,g) P (f ) Irg + 1+k(f,g) ⎪
⎭ ⎫ ⎪ ⎬ 4 (f ) ∗ l(g,g) k(f,g) χ2 (f,g) T T a(f, g)b (f, g)R b (f, g)a (f, g) , l(f,g) 1+k(f,g) tf φ ⎪ ⎭ f =1 f =g
(39)
where P (f ) is the power radiated by Txf , N 0 (watt/Hz) is the thermal noise level at Rxg , and † (f ) (f ) (f ) R E φ (n) φ (n) is the spatial covariance matrix of the MIMO signals radiated φ by Txf . The resulting spatial MAI model in (39) implies that each transmit node induces MAI on all receive nodes different from the intended one. We consider this model, indeed, more realistic than those generally adopted by the IEEE802.11-oriented literature (see, for example, [29] and references therein), where the transmission range of any transmit node is supposed circular and 5
Several measures support the conclusion that ν values of the order of 1/2 generally suffice to meet the above mentioned
uncorrelation assumption among rays impinging the receive antennas, at least in application scenarios as those here considered where the terminals are (approximately) co-located at the same level over the ground [10,16,17]. 29 agosto 2006
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beyond that range no MAI at all is assumed to be induced. By fact, the main reason behind the MAI model we adopt here is that a (very) large number of interferes might cause negligible MAI individually, but their aggregate effect could straightly affect the quality of the on-going desired transmissions. The numerical results we present in the sequel support, indeed, this conclusion. VI. T HE S PATIAL P OWER -A LLOCATION M ULTI -A NTENNA (SPAM) G AME FOR AD - HOC NETWORKS
In an Ad-hoc network with multiple no cooperating mutually interfering peer-to-peer links, the MAI correlation matrix seen by each receive node varies with the signal correlation matrices generated by all (e.g., desired plus interfering) transmit nodes. Being power-allocation and signalshaping performed by each transmit node depending on the MAI covariance matrix measured by the corresponding receiver, thus a change in the transmit correlation matrix of one link induces changes in the signal correlation matrices of all other links. Hence, to properly model this nodes interaction, we resort to the analytical framework of the Strategic Game Theory [5]. We recall that a noncooperative and strategic game G N, A, {ug } has three components [5,13]: a finite set N {1, 2, . . . , n∗ } of players, a set Ag , g ∈ N of possible actions for each player and a set of utility functions. Specifically, after denoting as A A1 × A2 × . . . × An∗ the space of action profiles [13], let us indicate as ug : A → R the g-th player’s utility function. Thus, after indicating by a ∈ A an action profile, by ag ∈ Ag the players g’s action in a and by a−g the actions in a of the other (n∗ − 1) players, we can say that ug (a) ≡ ug (ag , a−g ) maps6 each action profile a into a real number [13]. In particular, in a strategic noncooperative game each player chooses a suitable action a•g from his action set Ag so to maximize its utility function, according to the following game rule [13]:
a•g ≡ maxag ∈Ag ug (ag , a−g ).
(40)
Therefore, since there is no cooperation among the players, it is important to ensure the dynamic stability of the overall game. A concept related to this issue is the so-called Nash 6
The notation ug (ag , a−g ) emphasizes that the g-th player controls only own action ag , but his achieved utility depends also
on the actions a−g taken by all other players [5,13]. 29 agosto 2006
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Equilibrium (NE). Simply stated, a Nash Equilibrium is an action profile a at which no player may gain by unilaterally deviating [5,13]. So, an NE is a stable operating point of the Game, because no player has any profit to change his strategy [4,5]. More formally, an NE is an action profile a such that for all ag ∈ Ag the following inequality is satisfied [5, 13]: ug (a g , a−g ) ≥ ug (ag , a−g ), ∀g ∈ N, ∀ag ∈ Ag .
(41)
A. The considered networking Game Let us focus now on the Ad-hoc network of Fig.2 composed by n∗ mutually interfering transmit/receive Multi-Antenna units. The ultimate task of the g-th transmit/receive pair is to maximize the information throughput TG (g), g = 1, .., n∗ , sustained by the corresponding link Txg → Rxg via suitable power-allocation and shaping of the signals radiated by Txg . Since the signals radiated by the g-th transmitter induces MAI over all other receivers {Rxi , i = g} and the Ad-hoc nature of the network does not allow transmitters to exchange information (e.g., the transmitters do not cooperate), we may model the interaction between transmit/receive pairs active over the network as a noncooperative strategic game [1,4,5]. Specifically, in the considered Ad-hoc networking scenario of Fig.2, the players’ set N is composed by the n∗ transmit/receive pairs, while the set Ag of actions available to the the g-th player is the set of all the covariance (g)
matrices {R } meeting the power constraint (11), so we can pose φ (g)
(g)
Ag ≡ {R : 0 ≤ T ra[R ] ≤ tg Pg }, g = 1, ..., n∗ . φ φ
(42)
This means that the generic action ag of Txg consists in the transmission of a Gaussian distributed (g)
payload sequence with covariance matrix R . Furthermore, the utility function ug (.) for the gφ th transmit/receive pair is the conditional throughput conveyed by the g-th link, so that we can write (see eq.(18)) 1 − →(g) ˆ − (1) (g) (n∗ ) ug (a) ug (R , ..., R , ..., R ) ≡ I → y (g) ; φ |H g φ φ φ Tpay 1 (g)−1/2 ˆ T (g) ˆ ∗ (g)−1/2 (g)−1 )Hg R Hg (Kd ) + σε2 (g)P (g) (Kd ) ≡ lg det Irg + (Kd φ tf 29 agosto 2006
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−
1 Tpay
lg det Irg tg
σε2 (g)Tpay (g) −1 ∗ (g) + ((Kd ) ) ⊗ R , φ tg (g)
(43) (i)
where the g-th MAI covariance matrix Kd depends on the spatial covariance matrices {R , i = φ g} of the signals radiated by the interfering transmitters as detailed by (39). About the rule of (g)•
the game, each player (e.g., transmitter Txg ) chooses the action R maximizing the throughput φ (43) conveyed by own link, so we can write (see (45))
1 →(g) ˆ (g) − → − R I y ; φ |Hg , g = 1, ...n∗ . ≡ arg max φ (g) Rφ ∈Ag Tpay (g)•
(44)
Before proceeding, few remarks about the considered network model of Fig.2. Firstly, routing is no considered in our analysis. In fact, we focus on single-hop (e.g., shortcut) transmissions, since the performance of the SDMA scheme we go to present depends only on the singlehop links quality. In fact, the main target of the proposed SDMA scheme is to maximize (in a competitive sense) the information throughput of each single-hop peer-to-peer link in the presence of MAI, fading, path loss, and channel estimation errors. Secondly, we assume that at any time, the packet generated by each transmit node is intended for a single receive node only, thus meaning that broadcasting and multicasting are out of the scope of our analysis. B. A Competitive Optimal distributed SDMA Algorithm under the Best Effort and ContractedQoS access Policies In this sub-Section we present the algorithm for an optimized SDMA for the networking scenario of Fig.2. Before proceeding, some remarks about the considered QoS policies are in order. We consider the QoS from an information throughput point of view. Specifically, in Adhoc networks with no centralized controllers it may be not possible to guarantee to any user the requested QoS. Thus, in place of guaranteed users’ QoS, it is more reasonable, indeed, to resort to the concept of contracted QoS, defined according to predefined multiple QoS classes. Specifically, the SDMA algorithm we present attempts to achieve the target throughput classes dictated by the MAC layer and, if these classes are not achievable due to the MAI, the algorithm attempts to achieve the next lower QoS classes by decreasing the throughput requested by the 29 agosto 2006
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users. From this point of view, the Best Effort strategy is a particular case of the contracted QoS one, where the number of QoS classes approaches infinity. The SDMA algorithm for achieving the maximal throughput over the g-th link under the above mentioned contracted QoS policy is reported in Table I. It must be run by each transmit/receive pair active over the network of Fig.2. In particular, in Table I the Steps from 0 to 11 are set-up (z)
procedures and eigen/singular values computations, while T RT H (nats/slot) at the Step 0 is the target throughput defining the z-th QoS class. Step 12 verifies that the Game is playable (e.g., the Nash Equilibrium exists; see Sect.VI.B), while Steps 13 and 14 set up the ρ parameter, I(ρ) and the step size ∆ requested to carry out the power-allocation procedure. The condition at Step 15 assures that the power meets the constraint (11), and the Steps from 16 to 18 perform the competitive optimal power-allocation and spatial signal-shaping for the link Txg → Rxg . In the Steps from 18 to 22 the convergence of the SDMA algorithm towards the NE is checked, and at Step 23 the maximized information throughput sustained by g-th link is evaluated. Finally, Step 24 checks if the achieved throughput is compliant with the QoS requirement. If it is compliant, then the game stops. Otherwise, Txg reduces the overall radiated power of an assigned step-size (z)
∆l and restarts the game. If the obtained throughput is below the requested one T RT H , the (z−1)
transmitter Txg restarts the game with a target throughput T RT H lower than the original one (z)
T RT H . C. Distributed and Asynchronous implementation of the SPAM Game Let us assume that the SDMA algorithm reported in Table I is iteratively run (possibly, in an asynchronous way) by all transmit/receive pairs active over the network of Fig.2. Specifically, (1)
after measuring the impairing MAI covariance matrix Kd , the first Tx1 → Rx1 pair begins to update its power-allocation and signal-shaping by running the algorithm of Table I. Thus, this algorithm is successively run by the second pair Tx2 → Rx2 , the third one, etc. Hence, the
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algorithm is applied again by the first pair, the second one and so on7 . Formally, in the fully asynchronous and distributed implementation of the SPAM Game, the g-th transmit/receive pair of Fig.2 executes the power-control and signal-shaping algorithm of Table I at time instances given by the set Υg {tg1 , tg2 , tg3 ...} with tgi < tg( i+1) . Thus, after indicating by Υ {τ1 , τ2 , τ3 , ...} the overall set of updating instants Υ1 Υ2 ... Υn∗ sorted in increasing order, the asynchronous and distributed implementation of the considered SPAM Game generates the sequence of power-allocations and signal-shapings following the iterative procedure detailed in Table II. Thus, about the asynchronous and distributed implementation of the SPAM Game reported in Table II, the key questions are: •
Does Nash Equilibrium exist for the SPAM Game?
•
Is the Nash Equilibrium unique?
•
Does the above iterative algorithm converge towards the Nash Equilibrium?
The following Proposition 3 gives sufficient conditions for the existence, uniqueness and achievement of the Nash Equilibrium. Proposition 3 - By referring to the asynchronous and distributed implementation of the SPAM Game reported by Table II, let following three conditions be met: σ 2 (g)P (g) t (g) −1 (g)2 2 km > 1 + ε (g) + σ (g)T ra[(K ) ] , 1 ≤ m ≤ min{rg , tg }, 1 ≤ g ≤ n∗ ; (45) ε d (g) ρ µm rg ≥ tg , Tpay >> tg > 1
and/or
σε2 (g) → 0,
1 ≤ g ≤ n∗ ;
(46)
1 ≤ g ≤ n∗ .
(47)
Thus, the Nash Equilibrium of the SPAM Game of Table II exists and is unique. Furthermore, the distributed and asynchronous implementation of the SPAM Game of Table II converges to 7
In an asynchronous implementation of the game, the updating ordering may also change from time to time, possibly in a
random way [13].
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the NE from any starting point. A Proof of this proposition is reported in the final Appendix II. VII. N UMERICAL TESTS ON THE CONVEYED NETWORK THROUGHPUT AND CONVERGENCE Numerical tests have been carried out in order to evaluate the the performance of the SPAM Game of Table II both in terms of achieved network throughput and self-reconfiguring/faulttolerating capability. The model of Sect.V has been adopted to (numerically) generate the MAI. The obtained results are detailed in the following sub-Sections. A. Conveyed Average Throughput and self-reconfiguring/fault-tolerating capability Fig.3 depicts the basic squared network considered for the tests. It is composed by two transmit/receive pairs equipped with t=4 and r=8 transmit/receive antennas and operating at SNR=10dB with Tpay = 120. The numerical tests have been carried out under the Best Effort policy. At the beginning (e.g., at iteration 0), only the first transmit/receive pair is assumed to be on (see Fig.3). Thus, by running the SPAM Game we obtain an average information throughput around 18 bits/slots for the first link (see Fig.4), while the throughput sustained by the second link Tx2 → Rx2 is (obviously) zero. Next, the link Tx2 → Rx2 turns on, so the throughput over the Tx1 → Rx1 link decreases (till to 13 bits/slots; see Fig.4), while the (average) throughput of the Tx2 → Rx2 link increases till the same value of 13 bits/slot. This point represents the (first) Nash Equilibrium for the considered squared topology and it has reached after 23 iterations (see Fig.4). Next, the network topology changes and a barrier is introduced between the the second transmitter and first receiver, so that χ2 (1, 2) = 0.6 while χ2 (2, 1) = 1. As it can be seen by Fig.4, the network self-reconfigures and new Nash Equilibrium (achieved at the 60th iteration) is characterized by different values of the achieved throughput over the active links. After, we considered an operating scenario with χ2 (1, 2) = 0.8 and χ2 (2, 1) = 1. In this case, the SPAM Game gives arise to an information throughput over the link Tx1 → Rx1 limited up to 14.3 bits/slot (see Fig.4). Next, we introduced an additional change in the network topology,
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so that both receivers do not suffer from MAI (e.g., χ2 (1, 2) = χ2 (2, 1) = 0 in Fig.3). In this operating condition, the sustained links throughput increase so to approach a new NE, where the throughput conveyed by both links equates 19.2 bits/slot (see Fig.4). Finally, we assumed that a third pair of trasmit/receive units switch on, so that the network assumes an hexagonal topology (see Fig.3). The new NE achieved by running the SPAM Game approaches 8.3 bits/slot for all active links (see Fig.4). About the convergence property, an interesting still open question concerns the convergence rate of the SPAM Game towards the NE for increasing values of the number k of performed iterations (see Table III). By fact, this question is still open and till now the convergence rate seems to resist, indeed, closed-form analytical evaluation. However, some considerations about this point may be found in [25]. B. The Achievable Throughput Region The set of simultaneous average throughput achieved by the n∗ peer-to-peer links Txg → Rxg , g = 1, .., n∗ active in the ad-hoc network of Fig.2 may be described by resorting to the concept of achievable throughput region [2,3]. Roughly speaking, for a given statistical description of the network links and a set of constraints on the network input statistics (power, pdf, etc.), the corresponding achievable throughput region of the overall network is the closure of all average information throughput n∗ -ples (TG (1), ..., TG (n∗ )) that can be simultaneously sustained by the peer-to-peer links Txg → Rxg , g = 1, ..., n∗ , active over the network [9,13]. Barring some partial contributions, till now no closed-form analytical formulas are available for the computation of the achievable throughput region of an interference network as that sketched in Fig.2 [12,18,19]. Thus, in this sub-Section we comment some results we have numerically obtained for a squared network composed by two (e.g., n∗ = 2) multi-antenna (e.g., t1 = t2 = r1 = r2 = 4) transmit/receive units. Specifically, Fig.5 reports the achievable throughput regions of the considered squared network for different values of the shadowing factors χ2 (1, 2) = χ2 (2, 1). These regions represent the 2-ples of average information throughput (TG (1), TG (2)) that the links active over the considered network may guarantee when the 29 agosto 2006
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proposed SPAM Game is run. After comparing the throughput regions achieved by the proposed SPAM Game with those of the CSMA/CA(s) orthogonal access method (see the inner square in Fig.5), we may conclude that at χ2 (1, 2) = χ2 (2, 1) < 0.7 (e.g.,in the presence of strong MAI) the proposed SPAM Game outperforms the CSMA/CA(s) one in terms of conveyed average throughput. C. SPAM Game-vs.-CSMA/CA: a throughput comparison The above conclusion is also supported by the dotted line of Fig.4 that reports the corresponding CSMA/CA(s) average throughput for the same previously considered networking scenarios 8 . An examination of Fig.4 shows that, although CSMA/CA(s) is, by fact, collision-free, nevertheless the average throughput are less than those guaranteed by SPAM Game, specially in MAI limited application scenarios. Overall, the SPAM Game-vs.-TDMA comparison of Fig.4 supports for the superiority of competitively optimal access strategies over collision-free ones, at least in networking scenarios where the spatial-dimension of the system may be efficiently exploited to perform MAI suppression. D. Convergence Property of the SPAM Game toward the nearest allowable operating point In actual application scenarios, the transmit/receive nodes are not aware in advance about the throughput region of Fig.5 sustainable by the network, neither this region may be analytically evaluated in closed-form. Thus, a key question concerns the convergence of the operating point (0)
(0)
of SPAM Game when the requested initial throughput (TG (1), TG (2)) fall out of the achievable throughput region of Fig.5. It can be proved (see Appendix III of [25]) that, under Best Effort (0)
(0)
policy, the operating point of the SPAM Game moves from (TG (1), TG (2)) and converges to 8
To really guarantee both collision-free (e.g., perfectly MAI-free) and fair access under the MAI model of Sect.V, in the
carried out numerical tests the CSMA/CA(s) scheme we implemented schedules a single peer-to-peer link at a time and activates the scheduled link at maximum allowed power for an n -th of the time. Furthermore, the throughput loss due to the exchange of RTS/CTS packets has been no accounted for in the reported numerical plots. Thus, being the implemented CSMA/CA(s) scheme collision-free (e.g., fully MAI-free), the corresponding throughput region of Fig.5 is the largest one attainable by the CSMA/CA(s) policy. 29 agosto 2006
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the point on the boundary of the throughput region at the minimum Euclidean distance from the (0)
(0)
initial (TG (1), TG (2)) point (see the dotted arrow of Fig.5). Likewise, under the Contracted (0)
(0)
QoS policy, the operating point of the SPAM Game moves from (TG (1), TG (2)) and converges (0)
(0)
to the point on the QoS grid at minimum distance from (TG (1), TG (2)) (see the dashed grid of Fig.5). VIII. D ISTRIBUTED C ONNECTION A DMISSION P ROCEDURES (CAP S ) AND C ONCLUSIONS Since the Ad-hoc networks do not adopt any centralized controller and the number of active nodes is random, it may be of interest to develop distributed and scalable CAPs, balancing QoS users’ requirements and aggregate networking throughput. In this Section, we propose two distributed CAPs for Ad-hoc networks based on the SPAM Game of Table II. The first one (referred as Hard Connection Admission Procedure (HCAP)) is devoted to benefit users asking for higher QoS classes, while the second one (referred as Soft Connection Admission Procedure (SCAP)) is designed to maximize the number of allowed connections. The HCAP’s flow chart is given in Table III. About this last, let us assume the network has just approached the NE, and a new connection service request is incoming with the QoS class equal to z. Thus, the algorithm given in Table I is implemented by the incoming (new) transmit node, which also evaluates its current interference covariance matrix Kd . Thus, the other nodes already joined the network compute their own interference covariance matrix too. If convergence is reached (e.g., no changes of MAI covariance matrices are recognized), then a new NE is approached. Otherwise, each user waits for a time zT which is function9 of his current QoS class. If the resulting MAI matrices change, then user with QoS class z needs to wait for a zT period before passing to next lower (z − 1) QoS class. Afterwards, the power-allocation algorithm starts again (see Table III). In Table IV the flow-chart of SCAP is shown. The SCAP approach is quite similar to HCAP. 9
The value assumed by the waiting time may be set (possibly in an adaptive way) by the MAC layer on the basis of the
maximum delay (e.g., latency) allowed for the (successful) transmission of each MAC PDU.
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The only difference is given by the waiting time (that is inversely proportional to the required QoS class) and by the reduced number of classes. Being the ultimate task of SCAP to maximize the overall number of allowed connections, in this approach the user with highest class is the first to reduce the request of QoS class. The numerical plots of Figs.7, 8 confirm the above mentioned properties of the proposed CAPs. Specifically, in Fig.7 the users’ number is evaluated and expressed as a function of QoS users’ percentage. The SCAP connection number is higher than HCAP one, while collision-free access methods (such as, TDMA) admit all requiring connections, regardless of QoS issues. The number of users when both SCAP and HCAP are employed is decreasing as the QoS user percentage is increasing. In Fig.8 the resulting aggregated network throughput is reported. Although the number of connections attained by HCAP is the lowest one, nevertheless the resulting network throughput is the highest one. The number of connections allowed by SCAP is greater than that attained via HCAP, but the network throughput is lower. Fig.8 shows that, in terms of aggregate network throughput, the TDMA gives the worst performance. Overall, the final conclusion that arises from the performance tests described in Sects. VII, VIII is that the proposed SPAM Game represents a distributed Multi-Antenna access strategy able to outperform (in terms of peer-to-peer throughput) the conventional collision-free ones. From this point of view, it is likelihood to retain that the results presented in this paper only grasp the tip of the iceberg and much remains to be done. Specifically, the effect of multi-hop routing and relays [24] on the performance of the proposed SPAM Game is a topic currently investigated by the authors. A PPENDIX I - E XISTENCE OF A NASH E QUILIBRIUM FOR THE SPAM G AME In order to prove the existence of a Nash Equilibrium for the distributed and asynchronous implementation of the SPAM Game reported in Table II, we report the following result from [1,13]. Proposition 4 - Existence of a Nash Equilibrium 29 agosto 2006
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Given an uncooperative strategic Game G N, A, {ug } , an NE exists if, for all g = 1, ..., n∗ : a) the set Ag is not empty, compact and convex;
(48)
b) the utility f uncion ug (a) is continuous over a ∈ A;
(49)
c) the utility f uncion ug (ag , a−g ) is quasi − concave10 in ag ∈ Ag f or any assigned a−g . (50) Thus, our task is to prove that the SPAM Game of Table II meets all above a), b), c) conditions. Condition (48) - For all g values, the set of actions Ag in (42) is limited between the null and (g)
(g)
R (max) matrix, where T ra[R (max)] ≡ tg Pg . This set is closed, being its boundary (e.g., φ φ (g) the null and the maximum R (max) matrix) included into the Ag set. As a consequence, the φ closure property makes the set Ag compact. Furthermore, Ag is also convex. In fact, after taking (g)
(g)
two elements R (1) and R (2) of Ag , then the resulting combined matrix φ φ (g)
(g)
(g)
Rφ λR (1) + (1 − λ)R (2), 0 ≤ λ ≤ 1, φ φ
(51)
also falls into Ag , since its trace (g)
(g)
(g)
T ra[Rφ ] = λT ra[R (1)] + (1 − λ)T ra[R (2)], φ φ
(52)
falls into the interval [0, tg Pg ]. Thus, the set Ag is convex.
Condition (49) - Since the function lg det [M] is continuous in the elements of the matrix M, in order to prove the continuity of the utility function ug (.) in (43) with respect to the (1)
(n∗ )
(matrix) arguments {R ...R }, it suffices to test the continuity of the two terms enclosed by φ φ (g) the squared brackets in (43). Since both these terms are continuous in the g-th argument R φ (g) −1 (i) and, in addition, Kd is also continuous in {R , i = g}, we conclude that ug (.) in (43) meets φ the continuity property (49).
10
See [15] for the definition and main properties of the quasi-concave functions. We only stress that a concave function is
also quasi-concave [15]. 29 agosto 2006
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Condition (50) - After recalling that a concave function is also quasi-concave [15], it suffices (g)
(g)
to prove that the utility function ug (.) in (48) is concave in R for any assigned Kd . To φ accomplish this task, we simply observe that the lg det[M] function is concave in M, so that (g)
ug (.) in (43) is concave in R when the term following the minus sign in (43) becomes φ negligible. An examination of (43) leads to the conclusion that this term becomes negligible ˆ are very reliable (e.g., σε2 (g) in (43) vanishes), or when when the available channel estimates H Tpay is large, or when all the inequalities in (45) are met. These last considerations complete the proof about the existence of a NE for the SPAM Game of Table II. A PPENDIX II - U NIQUENESS AND ACHIEVABILITY OF THE NE FOR THE SPAM G AME To prove the uniqueness of NE for the SPAM Game of Table II, we resort to some basic results reported in [1,4] about the so called standard functions. Formally, according to a current (i)
taxonomy [1,4], for any assigned (n∗ − 1)-ple of spatial covariance matrices {R , i = g}, the φ (g)• resulting maximizing R in (44) constitutes the so-called ”g-th terminal best response” to the φ (i) set {R , i = g} of interfering covariance signal matrices. Therefore, for any assigned n∗ -ple φ (i) (g)• {R , i = 1, ..., n∗ } we may collect the resulting terminals best responses {R , g = 1, ..., n∗ } φ φ • in (44) into the so called Matrix of Best Responses (MBR) [B ], formally defined as (see [1,4] for more details) (1)
(n∗ )
[B• ](R , ..., R ) [Rφ (1)• , ..., Rφ (n∗ )• ]T . φ φ
(53)
According to [1,4], from an analytical point of view the matrix [B• ](.) in (53) constitutes a standard function when it meets the following three properties: Positiveness: (1)
(n∗ )
(1)
(n∗ )
a) ∀a (R , ..., R ) ∈ A, then [B• ](R , ..., R ) 0; φ φ φ φ
(54)
Monotonicity: (g)
(g)
(1)
(n∗ )
(1)
(n∗ )
b) if R R , then [B• ](R , ..., R ) [B• ](R , ..., R ); φ φ φ φ φ φ
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30
Scalability: (n∗ )
(1)
(n∗ )
(1)
c) ∀ c ≥ 1, then c[B• ](R , ..., R ) [B• ](cR , ..., cR ), φ φ φ φ
(56)
where the expression QU (QU) means that Q−U is a definite (semidefinite) positive matrix. Therefore, since Theorem 1 of [4] assures the uniqueness of the NE when the corresponding MBR is a standard function (see also [1] for additional details on this topic), our next task is to prove that the MBR [B• ](.) in (53) of the considered SPAM Game meets all the properties (54), (55), (56). a) Positiveness - In order to prove the positiveness of [B• ](.) for the SPAM Game of Table II, (g)•
it suffices to test that all the best response matrices {R , g = 1, ..., n∗ } in (53) are definite φ (g) positive for any assigned MAI matrices {Kd , g = 1, ..., n∗ }. By examining the expression of (g)•
(g)•
reported in (32), we conclude that R is positive when tg ≤ sg min{rg , tg } and all the R φ φ powers {Pg∗ (m)} in (28)-(29) are strictly positive. In turn, both these conditions are met when the inequalities (45), (46) are fulfilled. b) Monotonicity - In order to prove the monotonicity property, we have to test the validity of the following inequalities: (g)
(g)
(g)
(g)
(g)
(g)
(g)
(g)
(g)
(g)
(g)
(g)
ug (R ; Kd ) ≤ ug (R ; Kd ), ∀Kd Kd , φ φ
(57)
ug (R ; Kd ) ≤ ug (Rφ ; Kd ), ∀Rφ R . φ φ
(58)
For testing (57), we simply note that, since the utility function in (43) is composed by the function log det (I + A), we have that this last can be re-written as: log((1 + λa )), where {λa } are the (g)
(g)
eigenvalues of A. Thus, log det (I + A) increases with {λa }. Now, when Kd Kd , we have (g)2
(g)2
that k m ≥ km , ∀m. This implies that the eigenvalues of the matrix in (43) act as a reference (g)−1
to the matrix Kd
(g)
(g)
. So, we have that, when Kd Kd , thus ug (.) decreases. By applying (g)
the same proof-arguments to the matrix R , we directly arrive at test the monotonic increasing φ (g) behavior of the function ug (.) in (43) with respect to R , and this proves the validity of the φ inequality (58).
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31
(i)
(g)
c) Scalability - For any assigned (n∗ − 1)-ple {R , i = 1, .., n∗ ; i = g}, let us indicate by Kd φ the resulting MAI covariance matrix computed according to the model (39) and let us denote (g)
(i)
as Kd the corresponding MAI matrix generated by the (n∗ − 1)-ple {cR , i = 1, .., n∗ ; i = g}. φ Therefore, by definition, the proof of the scalability property in (56) is equivalent to test the validity of the following inequality: (g)
(g)
(g)
(g)
c arg max {ug (R ; Kd )} arg max {ug (R ; Kd )}, g = 1, .., n∗ . φ φ R(g) R(g) φ ∈Ag φ ∈Ag
(59)
(g)
(g)
Now, we observe that, when Kd and Kd are computed according to the MAI model in (39), then they satisfy the following chain of inequalities: (g)
(g)
(g)
cKd Kd Kd .
(60) (g)
Therefore, since we have already proved that ug (.) in (43) increases for increasing Kd , the validity of (59) directly arises from the inequality chain (60). After proving the uniqueness of the NE for the SPAM Game in Table II, its reachability is directly guaranteed by the fact that the NE represents the unique stable operating point of the Game [13]. R EFERENCES [1] C.Saraydar, N.B. Mandayanan, D.J.Goodman, ”Efficient Power control via pricing in wireless data networks”, IEEE Trans. on Comm., vol.50, no.2, pp.291-303, Feb.2002. [2] A.B.Carleial, ”Interference Channels”, IEEE Trans. on Inf. Theory, vol.24, no.1, pp.60-70, Jan.1978. [3] T.M. Cover, J.A.Thomas, Elements of Information Theory, Wiley, New York, 1991. [4] R.Yates, ” A framework for Uplink Power control in Cellular Radio Systems”, IEEE Journ. of Sel. Areas on Comm., vol.13, pp.1341-1347, Sept.1995. [5] A.B.MacKenzie, S.B.Wicker, ”Game Theory in Communications: Motivation, Explanation and Application to Power Control”, Globecom 2001, pp.821-825. [6] P.Lancaster, M.Tismetesky, The Theory of Matrices, 2nd Ed., Academic Press, 1985. [7] R.G.Gallagher, Information Theory and Reliable Communication, Wiley, 1968. [8] S.Verdu’, T.S.Han,”A general Formula for channel Capacity”, IEEE Trans. on Inform. Theory, vol.40, no.6, pp.1147-1157, July 1994. [9] C.E.Perkins, Ad Hoc Networking, Addison Wesley, 2000. [10] A.J.Paulraj, D.A.Gore, R.U.Nabar, H.Bolcskei, ”An Overview of MIMO Communications: A Key to Gigabit Wireless”, Proc. of IEEE, pp.198-218, Feb. 2004.
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[11] E.Baccarelli, M.Biagi, ”Optimized Power Allocation and Signal Shaping for Interference-Limited Multi-Antenna ”ad-hoc” Networks”, PWC2003 proc., Sept.2003, pp.138-152. [12] L.Zheng, D.Tse, ”Diversity and Multiplexing: A fundamental Tradeoff in Multiple-Antenna channels”, IEEE Tr. on Inform.Theory, vol.49, no.5, pp. 1073-1096, May 2003. [13] M.J.Osborne, A.Rubinstein, A course in Game Theory, MIT Press 1994. [14] E.Baccarelli, M.Biagi, C.Pelizzoni ”Games Theory for Power allocation and Spatial Shaping in MIMO ad-hoc Networks”, INFO-COM Tec. report, available at website http : //inf ocom.uniroma1.it/ biagi/gamet.pdf [15] J.Ponstein, ”Seven kinds of convexity”, SIAM Rev., vol.9, no.1, pp.115-119, Jan.1967. [16] F.R.Farrokhi, G.J.Foschini, A.Lozano, R.A.Valenzuela,”Link-Optimal Space-Time Processing with Multiple Transmit and Receive Antennas”, IEEE Comm. Letters, vol.5, no.3, pp.85-87, March 2001. [17] A.Paulraj, R.Nabar, D.Gore, Introduction to Space-Time Wireless Communications, Cambridge Press, 2003. [18] M.Grossglauser, D.Tse, ”Mobility Increases the Capacity of ad-hoc Wireless Networks”, IEEE Infocom Proc, pp.1360-1369, 2001. [19] M.Gatspar, M.Vetterli, ”On the Capacity of wireless Networks: the relay case”, IEEE Infocm Proc, pp.1577-1586, 2002. [20] V.Srinivasan, P.Nuggehalli, C.F.Chiasserini, R.R.Rao, ”Cooperation in Wireless Ad Hoc Networks”, IEEE Infocom Proc., pp.808-817, 2003. [21] I.Koutsopoulos, T.Ren, L.Tassinulas, ”The Impact of Space Division Multiplexing on Resource Allocation: A Unified Approach”, IEEE Infocom Proc., pp.533-543, 2003. [22] F.Shad, T.D.Todd, V.Kezys, J.Litva, ”Dynamic Slot Allocation (DSA) in indoor SDMA/TDMA using a smart antenna base station”, IEEE/ACM Tr. on Networking, vol.9, no.1, pp.69-81 , Feb.2001 [23] G.Okamoto, Smart Antenna Systems and Wireless LANs, Kluver, 1999. [24] V.Kawadia, P.R.Kumar, ”Power Control and Clustering in Ad Hoc Netwroks”, IEEE Infocom Proc., pp.459-469, 2003. [25] E.Baccarelli, M.Biagi, C.Pelizzoni, ”Ad Hoc Game” INFO-COM Tec. report, available at website http
:
//inf ocom.uniroma1.it/ biagi/gameext.pdf [26] J.B. Andersen, ”Antenna Arrays in Mobile Communications: Gain, Diversity and Channel Capacity”, IEEE Ant. and Prop. Mag., vol.42, no.2, pp. 12.16, April 2000. [27] K. Sundaresan, R. Sivakumar, M.A. Ingram, T.-Y. Chang, ”Medium Access Control in Ad-Hoc Networks with MIMO Links: Optimmization Considerations and Algorithms”, IEEE Tr. on Mob. Comp., vol3, no.4, pp. 350-365, Oct.-Dec.2004. [28] M.Hu, J.Zhang, ”MIMO Ad Hoc Networks: Medium Access Control, Saturation throughput, and Optimal Hop Distance”, Journ. of Comm. and Netw. (JCN), vol.6, no.4, pp.317-330, Dec.2004. [29] R. Ramanathan, R.Rosales-Hain, ”Topology Control of Multi-Hop wireless Networks using transmit power adjustment”, Proc. IEEE INFOCOM’00, 2000.
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33
Multiple Access Interference
1
Tx
Source Message
M
Space Time Encoder and Modulator with t antennas
1
h11 h21
2
2
Demodulator, Detected channel Message estimator and decoder with r ˆ antennas
M
hrt
t
Rx
r
MIMO FORWARD-CHANNEL H
Hˆ Kd
Fig. 1.
Hˆ
K
FEEDBACK LINK
d
ˆ and impaired by MAI with spatial Multi-Antenna system equipped with imperfect (forward) channel estimates H
covariance matrix Kd .
Tx1
Rx1
θd (1, g ) l(1,g)
Tx2
Rx2
θd (2, g ) l(2,g)
l(g,g)
Fig. 2.
θa (2, g )
θ a (n , g )
Rxg
*
...
... Txn*
l(n*,g)
...
...
θ a (1, g )
Txg
θ d (n* , g )
Rxn*
The model [16] considered for generating MAI.
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34
Rx2
Tx2
Tx2 Tx1
Tx1
Rx1
Tx1
Rx2
Rx2
Tx2
Tx1
Tx2
Tx2 Tx1
Rx1
Rx1
Tx3 Rx3 Rx1
Rx2
Rx1
Tx1
Rx2
Rx1
Fig. 3. The network topology sequence considered in Sect.VII for the numerical tests. Txg → Rxg , g = 1, 2, 3 are the desired links, while Txg → Rxf , g = f are the interfering ones.
CSMA/CA(4)
Fig. 4.
Average Information Throughput achieved by the SPAM Game and CSMA/CA(4) under the Best Effort policy for the
network topology sequence of Fig.3.
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35
(z)
0. Set the target throughput T RT H of the z-th QoS Classes ; (g)
(g)
1. Initialize R (new) := R (old) = [0tg ×tg ]; φ φ 2. fl(g)=1; 3. TG (g) = 0; (g) 4. α(g) (P˜ Ttr /rg )T ra[(K )−1 ]; d
5. σε2 (g) (1 + α(g) /tg )−1 ; (g)
6. Compute and sort the r eigenvalues of Kd ; ˆ Tg ; ˆ ∗g K(g)−1 H 7. Compute the SVD of H d (g)2
8. Sort the s min(r, t) eigenvalues {k1 (g)
(g) (g)2
(g)2
, ..., ks
(g)
} of Kd ;
(g)
9. αm µm km /tg (µm + P (g) σε2 (g)); (g)
10. βl
(g)
σε2 (g)Tpay /µl tg ;
(g)
(g)
(g)
11. µmin min1≤l≤r {µm }, βmax (g)2
12. if km
(g)
> (µm + P (g) σε2 (g))
2 σε (g)
2 σε (g)Tpay (g)
õmin tg
(g)
rTpay (g)
µmin µm
;
for all m and fl(g)=1
{ 13. Set ρ(g) := 0 and I(ρ(g) ) := ∅; 14. Set the step size ∆; (g) 15. While P (m) < P t do (g) g m∈I(ρ ) { 16. Update ρ(g) = ρ(g) + ∆; 17. Update the set I(ρ(g) ) via eq. (31); 18. Compute the powers and the covariance matrix via eq.(28), (29), (32); (g)
(g)
19. Set Ψ(g) := R (new) − R (old); φ φ (g) 20. If (||Ψ||2E ≤ 0.05||R (g) (old)||2E ) φ 21. then fl(g)=0, else fl(g)=1; (g)
(g)
22. R (old):=R (new) φ φ } 23. Evaluate TG (g) via (33) for the g-th link; (z)
24. if TG (g) = T RT H stop; else (z)
25. if TG (g) > T RT H reduce the radiated power P (g) and go to Step 1, else (z)
26. if TG (g) < T RT H lower the target class to z-1 and go to Step 1; } TABLE I A
PSEUDO - CODE FOR THE IMPLEMENTATION OF THE POWER - ALLOCATION AND SIGNAL - SHAPING ALGORITHM FOR THE
g- TH TRANSMITTER / RECEIVER PAIR UNDER THE ” CONTRACTED Q O S”
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POLICY.
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36
For all k such that τk ∈ Υ { For all terminals g ∈ N such that τk ∈ Υg (g)
{ Evaluate the MAI matrix Kd
;
(g)• → − (g)• ; and R Run the algorithm of Table I so to compute φ φ (g)• → − ; Radiate the signal vector φ
} } TABLE II A
PSEUDO - CODE FOR THE
A SYNCHRONOUS AND D ISTRIBUTED IMPLEMENTATION OF THE SPAM G AME OF TABLE I.
χ2 = 0
χ 2 = 0.3 χ 2 = 0.4 χ 2 = 0.7 χ2 = 1 CSMA/CA(4)
Fig. 5.
Regions of the average information throughput achieved by the proposed SPAM Game and CSMA/CA(4) squared
network and different values of the shadowing factors χ2 (1, 2) ≡ χ2 (2, 1) ≡ χ2 .
1. Equilibrium for the network composed by (n∗ − 1) pairs of transmit/receive nodes; ∗
2. New Request from user n∗ with QoS Class equal to z (n ) ; 3. Power allocation for g-th user , g = 1, ..., n∗ ; 4. The user g waits for z (g) T for the network equilibrium, g = 1, ..., n∗ ; 5. If the network is in equilibrium, go to Step 1; 6. else z (g) := z (g) − 1, g = 1, ..., n∗ ; 7. Go to Step 3. TABLE III H ARD C ONNECTION A DMISSION P ROCEDURE (HCAP).
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37
CSMA/CA(4)
Fig. 6.
Regions for the information throughput achieved by the proposed SPAM Game and CSMA/CA(4) for a squared
network with χ2 (1, 2) ≡ χ2 (2, 1) ≡ 0.4, Rice factors equal to 10, P (1) = P (2) = 10−3 W , N
(1) 0
=N
(2) 0
= 10−4 W . All the
transmit nodes are equipped with t=r=4 transmit/receive antennas. The dotted-dashed grid represents the throughput allowed by the considered QoS classes.
1. Equilibrium for the network composed by (n∗ − 1) pairs of transmit/receive nodes; ∗
2. New Request from user n∗ with QoS Class equal to z (n ) ; 3. Power allocation for the g-th user , g = 1, ..., n∗ ; 4. The g-th user waits for (zmax − z (g) )T for the network equilibrium, g = 1, ..., n∗ ; 5. If the network is in equilibrium, go to Step 1; 6. else z (g) := z (g) − 1, g = 1, ..., n∗ ; 7. Go to Step 3. TABLE IV S OFT C ONNECTION A DMISSION P ROCEDURE (SCAP).
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38
12
10
Number of Active Users
8
6
4
TDMA SCAP HCAP
2
0
0
0.1
0.2
0.3
0.4 0.5 0.6 QoS users’ Percentage
0.7
0.8
0.9
1
Fig. 7. Number of connected users for a random network topology. Performance comparison between TDMA, SCAP and HCAP. The QoS-classes are the same considered in Fig.5. Furthermore, χ2 (f, g) ≡ 0.4, Rice factors equal to 10, and P (g) = 10−3 W , N
(g) 0
= 10−4 W have been considered for all users. All the transmit/receive nodes are equipped with r=t=4 antennas.
Aggregate Network Throughput (bits/slot)
120
100
80
60
40
TDMA SCAP HCAP
20
0
Fig. 8.
0
0.1
0.2
0.3
0.4 0.5 0.6 QoS users’ Percentage
0.7
0.8
0.9
1
Aggregate Network Throughput for the same random network topology of Fig.6. Performance comparison between
TDMA, SCAP and HCAP. The QoS-classes are the same considered in Fig.5 while the system parameters are the same specified in Fig.6.
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