A Spatially Adaptive Random Access - Semantic Scholar
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Exploiting Regional Differences: A Spatially Adaptive Random Access
arXiv:1403.3891v1 [cs.IT] 16 Mar 2014
Dong Min Kim and Seong-Lyun Kim
Abstract In this paper, we discuss the potential for improvement of the simple random access scheme by utilizing local information such as the received signal-to-interference-plus-noise-ratio (SINR). We propose the spatially adaptive random access (SARA) scheme in which the transmitters in the network utilize different transmit probabilities depending on the local situation. In our proposed scheme, the transmit probability is adaptively updated by the ratio of the received SINR and the target SINR. We investigate the performance of the spatially adaptive random access scheme. For the comparison, we derive an optimal transmit probability of the conventional random access scheme in which all transmitters use the same transmit probability. We illustrate the performance of the spatially adaptive random access scheme through simulations. We show that the performance of the adaptive scheme surpasses that of the optimal conventional random access scheme and the CSMA/CA scheme. The convergence property of the proposed scheme is analyzed using the submodular game.
Index Terms Random access, distributed scheduling, SINR-based interference model, stochastic geometry, submodular game.
I. Introduction A. Spatially Adaptive Random Access (SARA) In this paper, we propose a spatially adaptive random access (SARA) scheme. Each node i behaves as follows: The authors are with the Radio Resource Management and Optimization Laboratory, School of Electrical and Electronic Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, Korea (email: [email protected]; [email protected]).
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1) Initialize transmit probability with the largest value, φmax . 2) Compute the average signal-to-interference-plus-noise-ratio (SINR), Γk(i) (t), during period T as follows: 1 Γk(i) (t) ≈ T q=1 T X
Gi,k(i) Pi , P Gu,k(i) Pu j
t > T,
u∈Ti (t−q),
where the notation k(i) denotes the receiver associated with transmitter i. The notation Gi, j denotes the channel gain from node i to node j. The term Pi represents the transmit power of transmitter i. The term Ti j (q) denotes the j-th subset of the concurrent transmission nodes when node i transmits at time slot q. 3) Update transmit probability φi as follows: (
(
) ) Γk(i) (t) φi (t + 1) = min max φmin , , φmax , β
(1)
where the notations φmin and φmax represent minimum and maximum values of the transmit probability, respectively. The notation β denotes a target SINR threshold. SARA is a variant of ALOHA, where each transmitter updates the transmit probability depending on the local situation. We verify the convergence property using the submodular game ([1], [2]). SARA improves the average received SINR with very little message passing in the network. Our simulation results show that the performance of SARA is even better than a carrier sense multiple access with collision avoidance (CSMA/CA). B. Motivation and Related Works The ALOHA protocol [3] is the most well-known distributed random access scheme. The transmit probability controls the operation of the ALOHA protocol. In [4], the authors derive an optimal transmit probability under the protocol model, where the transmission fails if two or more nodes are transmitting simultaneously. To improve the performance of an ALOHA network, researchers conducted several studies using a simple protocol model to achieve proportional fair and max-min fair [5]–[7]. In [8]–[10], the authors investigated optimal random access approaches achieving network utility maximization using a family of α-fair utility functions [11] in the protocol model. Due to the characteristics of the wireless channel [12], however, the receiver March 18, 2014
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may successfully receive the signal if the concurrent transmitters are far away. The physical model [13] considers the effect of such accumulated multi-user interference. In practice, interfering nodes are randomly located. In this regard, stochastic geometry [14] is a useful mathematical tool to model such randomness. In [15], the authors provide a stochastic geometry-based analytical framework of ALOHA. In a recent study [16], the authors investigated an adaptive ALOHA using a SINR model from a stochastic geometry point of view. The authors of [16] focus on achieving proportional fairness while we concentrate on maximizing the area spectral efficiency. Refer to [17]–[24] and references therein for readers interested in such an approach. In [25], the authors investigate the SINR-based random access protocol. Later, in [26], the authors proposed an adaptive interference pricing scheme to find a local optimal solution of the network utility maximization problem. They adopted a game theoretic framework ([27], [28]) to analyze multiple access control (MAC). The proposed approaches in [25] and [26] require a large number of message exchanges among the transmitters to inform their transmit probabilities to the others while SARA requires very little message passing in the network. CSMA/CA is more advanced than ALOHA in that it has the ability to adapt the local situation through carrier sensing and exponential backoff. The conventional ALOHA-like random access cannot behave adaptively because the transmit probability is fixed by a single optimal value. The optimal values of transmit probabilities are different in dense and sparse environments, and all nodes in the network should not have the same transmit probability. Let us assume that the nodes are deployed as shown in Figure 1. About 200 communication pairs are randomly distributed in a rectangular area. In this case, the node density is 0.02, and all transmitters have a fixed transmission probability. The nodes in subarea A are located in a relatively sparse environment; their transmissions may not be interfered. On the other hand, the nodes in subarea B are located in a relatively dense environment, and the transmissions of the nodes in B would frequently fail due to heavy interference. The nodes in A may want to utilize a relatively large transmit probability, and in B, a small probability. To improve the performance of such an ALOHA-like random access scheme, we devise SARA, which adjusts its operation according to the local circumstance. The main contributions of this paper are summarized as follows: •
We proposed a distributed SARA scheme, where the average received SINR is improved without frequent message passing with other nodes in the network.
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λ=0.02, φ =0.121 50 40 30 20 10 0
A
-10 -20 -30
B
-40 -50 -50
0
50
Fig. 1. A snapshot of network topology. The circles represent transmitters, and the connected pentagrams represent associated receivers. The arrows represent active communication pairs. The node density, λ, is 0.02 and the transmit probablity, φ, is 0.121. The nodes in subarea A are located in a relatively sparse environment. On the other hand, the nodes in subarea B are located in a relatively dense environment.
•
We present the convergence property of the proposed scheme using the submodular game.
•
We show the performance of the proposed scheme is better than that of the ALOHA and CSMA/CA schemes.
The rest of the paper is organized as follows. In Section II, we describe the system model. In Section III, we investigate our SARA scheme. In Section IV, we analyze the the convergence property of the proposed scheme. We verify the performance through simulations in Section V. Section VI concludes the paper. II. System Model As shown in Figure 1, a random wireless network of a single radio channel is considered, where each transmitter is associated with a receiver over a shared wireless channel. The transmitters are distributed according to a homogeneous Poisson point process (PPP) with intensity λ. If the March 18, 2014
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TABLE I Key mathematical notations.
λ
Node density
A
Area of interesting region
k(i)
Associated receiver of transmitter i
rt
Communication distance
P
Transmit power
φi
Transmit probability of transmitter i
φmin
Minimum transmit probability
φmax
Maximum transmit probability
Gi, j
Channel gain from node i to node j
α
Path loss exponent
γT j
Instantaneous SINR of receiver k(i)
Ti
Superset of concurrent interfering nodes when node i transmits
i
Ti
j
j-th subset of interfering nodes when node i transmits
β
Target SINR threshold
ri
Data rate of transmitter i
η
Area spectral efficiency
ps
Success probability
∗
φ
Optimal transmit probability
N
Set of transmitters
Γk
Average SINR
Φ
Vector of transmit probabilities of all transmitters
Φ−i
Vector of transmission probabilities of all transmitters except node i
region is finite (of size A), this means that nodes are independent and identically distributed with a uniform distribution in the region with a given average number of nodes (λA). Each transmitter, i, always has ample data to send. Each associated receiver, k(i), is located at a distance of rt from the transmitter i and the direction is random. The receivers also follow the homogeneous PPP by the the displacement theorem [17]. We assume that the time is slotted and synchronized so that transmissions begin with a time slot and continue during the slot length. The transmitter/receiver pair can be changed over the time. However, we focus on a snapshot of the overall communication process, where the network topology is fixed during each slot. The transmitter i attempts to send its data with transmit probability φi . The channel gain from node i to node j, Gi, j , depends on the distance between the transmitter and the receiver with path March 18, 2014
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loss exponent α and Rayleigh fading. The stochastic process of the wireless channel is ergodic. With a single shared channel, the concurrent transmissions cause cochannel interference. The instantaneous SINR of receiver k(i), γT j , is given by i
γT j = P
Gi,k(i) Pi , Gu,k(i) Pu + Wk(i)
i
u∈Ti
Ti j ∈ Ti ,
(2)
j
where we consider the interference limited network. Then, the noise power term Wk(i) is omitted from Eq. (2) and we deal with the signal-to-interference-ratio (SIR). The notation Ti denotes the superset of concurrent transmission nodes (interfering nodes) when node i transmits. When there are n transmitters in the networks, the cardinality of Ti is 2n−1 . The notation Ti j (q) denotes the j-th subset of the simultaneously transmitting nodes when node i transmits at time slot q. If a specific time slot does not matter, the notation Ti j (q) is simplified to Ti j . For example, assume that the network consists of three pairs {1, 2, 3}. If transmitter 1 is active, the superset of concurrent transmitting nodes, T1 , is {{}, {2}, {3}, {2, 3}}, and T11 = {}, T12 = {2}, T13 = {3}, T14 = {2, 3}. For a given target SINR threshold β, transmission is successful if γT j ≥ β is satisfied. We i
assume that the data rate of each transmitter, i, is a function of β, ri = log (1 + β), where we assume a unit bandwidth. III. Maximizing Area Spectral Efficiency A. Original Problem An area spectral efficiency (ASE) η is the sum of data rates per unit bandwidth in the unit area ([29], [30]). To focus on a network-wide performance, we use η as a performance metric. To maximize η, we formulate an optimization problem as follows: X h i max η = log2 (1 + β) E 1γk(i) ≥β
(3)
i
s.t.
φmin ≤ φi ≤ φmax ,
∀i,
where 1γk(i) ≥β denotes the indicator function defined as 1 if γk(i) ≥ β, otherwise 0. The term i P h E 1γk(i) ≥β represents the expected value of the number of successfully transmitting nodes in i
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the unit area. The notation γk(i) denotes the random variable of the instantaneous SINR of the receiver k(i). The objective function of Eq. (3) is a function of φi ’s as follows: 2n−1 Y Y X h i X X (1 − φm )1γ j ≥β , φl E 1γk(i) ≥β = Ti n j o j=1 j i i
(4)
m∈N\ Ti ,i
l∈Ti
where N is a set of all transmitters. The detailed derivation is in Appendix A. To maximize Eq. (4), we should find all γT j ’s. This means that we should compute all combinations of i
interferers. This is a combinatorial optimization problem which becomes harder to solve as the number of nodes in the network increases. B. Relaxed Problem In an effort to solve Eq. (3) dispersively, we decompose the problem as follows: 2n−1 Y Y X (1 − φm )1γ j ≥β max φl Ti j n j o j=1 l∈Ti
s.t.
(5)
m∈N\ Ti ,i
φmin ≤ φi ≤ φmax .
Each node i solves its own problem. However, we still should need all γT j ’s for each node i. As i
we mentioned in Section II, the number of combinations is 2
n−1
. To make the problem solvable
in a distributed way, we relax the problem as follows: max Ui (Φ)
(6)
s.t. φmin ≤ φi ≤ φmax , where Φ is a vector of the transmit probabilities of all nodes. We define Ui (Φ) as a utility function of node i as follows:1 X 2n−1 Y Y 1 2 1 j (1 ) Ui (Φ) = min − φ φ , φ γ φ m l max i − φi . T i 2 β j=1 l∈T j m∈N\nT j ,io i
(7)
i
Node i, who transmits with the probability φi , obtains the reward as a form of the former part of the right hand side of Eq. (7). The reward is a ratio of the average SINR to the target SINR. As we achieve a higher average SINR, the reward increases. The bad effect on the network 1
It is noted that the part of equation related with φmin is omitted for the simplicity.
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(increasing contention and interference) is assessed as the latter part of Eq. (7). Our utility function has a property to penalize the occurrence of the interference. It makes sure that the radio spectrum resources are efficiently shared. Our algorithm, shown in Section I-A, maximizes the utility function, Eq. (7). Using Eq. (6), our proposed update algorithm, Eq. (1), can be written as: φi (t + 1) X 2n−1 Y Y 1 j (1 (t)) (t) , φ . γ − φ φ = min max m l T i β n o j j j=1 l∈T m∈N\ T ,i i
(8)
i
To obtain the exact value of γT j , the nodes in the network need to frequently exchange i
the message with neighbor nodes to track the network topology information. To reduce this complexity, the time-averaged SINR can be used to update the transmit probability, as shown in Section I. A: Γk(i)
T Gi,k(i) Pi 1 X Gi,k(i) Pi P ≈ . = E P Gu,k(i) Pu T q=1 Gu,k(i) Pu j j u∈Ti ,
(9)
u∈Ti ,
If the average SINR is larger than the target β, the network situation is favorable for that communication pair. The pair may be isolated from the others. Therefore it is highly probable that the transmission of this transmitter will not interfere with the communications of the others. To promote more chances to transmit, the transmit probability is set to the maximum. On the other hand, as the ratio is less than φmax , the pair experience a highly contending situation. The transmit probability should be lowered to resolve the contention according to the ratio of the average SINR to the target SINR. The average SINR computation is done by the receiver. The receiver should notify its transmitter of the average SINR when the receiver transmits the acknowledgement signal or any kind of control signal (piggybacking). The convergence property of the proposed algorithm is given in the next section. IV. Convergence Property of SARA In this section, the convergence property of SARA is verified using the submodular game ([1], [2]). The non-cooperative game model suitable for our purpose is formulated as follows: N, {Φ} = {φi |φmin ≤ φi ≤ φmax , ∀i ∈ N} , {Ui (Φ)}i∈N , March 18, 2014
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where N is the set of transmitters. The set {Φ} is a strategy space. Ui (Φ) is a utility function for transmitter i. The medium access probability vector can be decomposed as Φ = {φi , Φ−i }, where φi is the medium access probability of node i and Φ−i is a medium access probability vector of all nodes except for node i. The medium access probability vector Φ∗ is said to be a Nash equilibrium if no unilateral deviation by any node is improvable for that node: Ui φ∗i , Φ∗−i ≥ Ui φi , Φ∗−i , 0 ≤ φi ≤ 1, φ∗i , φi , ∀i. In our game, there exists a Nash equilibrium characterized by the following equation: X 2n−1 Y Y 1 ∗ ∗ ∗ j . , φ γ φi = min 1 − φ φ max m l T i n j o β j=1 j l∈Ti
(11)
(12)
m∈N\ Ti ,i
The detailed derivation is in Appendix B.
Our game model can be categorized by a submodular game model because the utility function satisfies submodularity as follows: ∂2 Ui (Φ) ≤ 0, ∀v ∈ N\i, i ∈ N. ∂φi ∂φv The detailed derivation is in Appendix C. In a submodular game, if the transmit probability of each transmitter is updated by the best response strategy, the iterative update can converge to a Nash equilibrium. The definition of the best response is as follows: Bi (Φ−i ) = arg max Ui (φi , Φ−i ) ,
(13)
φi
φi (t + 1) = Bi (Φ−i (t)) .
(14)
The utility function (7) is concave and differentiable with φi . Therefore it is maximized when ∂U i (Φ) ∂φi
= 0. The corresponding φi is X 2n−1 Y 1 φi = min φl β j=1 j l∈Ti
Y
(1 − φm )γT j , φmax . o i
(15)
n j m∈N\ Ti ,i
Eq. (8) is the same form of Eq. (15). Thus, the proposed algorithm is a best response iterative update algorithm. In submodular game, if the transmit probability of each transmitter is updated by the best response strategy starting with initial value, the iterative update converges to Nash equilibrium. or at least oscillates between two values [2]. In the next section, the performance of the proposed random access scheme is evaluated. March 18, 2014
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10
15
1 10
0.7147
5
0.214
0
0.001 0.0105 -5
0.0476 -10
-15 -15
Fig. 2.
-10
-5
0
5
10
15
There are 11 communication pairs in the 30-by-30 rectangular area. Two communication pairs are relatively isolated
from the others.
V. Performance Evaluation A. Average SINR Validation and Convergence Simulation To evaluate the accuracy of Eq. (9), we conducted a simple simulation. As shown in Figure 2, a total of 11 transmitter/receiver pairs are distributed on the 30 m by 30 m area. The communication distance between a transmitter/receiver pair is 5 m. The transmit power is 30 dBm. The target threshold is 3 dB. Figure 3 shows the exact SINR (ensemble average) in Eq. (23) and the time-averaged SINR in Eq. (9). Figure 3 shows that the time-averaged SINR can approximate the exact ensemble average. The small differences are caused by the fading characteristics of wireless channel. Figure 4 shows the updated transmit probabilities. The updated probabilities using the timeaveraged SINR are almost the same as the updated probabilities using the ensemble-averaged SINR (with frequent message passing), as shown in Figure 4. In our system, the success of a transmission is determined by an instantaneous SINR and the March 18, 2014
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15 10
Average SINR [dB]
5 0 -5 -10 -15 -20 -25
Fig. 3.
ensemble average time average 1
2
3
4
5 6 7 Node Index
8
9
10
11
The average SINR of randomly distributed nodes. The exact SINR is estimated by time averaged SINR.
target SINR β. The instantaneous SINR changes with a small-time-scale (milliseconds) due to the Rayleigh fading, which is independent with a spatial random distribution of nodes in the network. To get rid of the effect of fading and to reflect the distribution of nodes, we utilize the average SINR, which varies with a large-time-scale (seconds). The average is taken by each of the nodes during the buffered period T . The ensemble average is more accurate than the time average. When the wireless channel is ergodic, the time average with a sufficient period can approximate the ensemble average. In this regard, the time-averaged SINR value is an indicator of the network condition. If the average SINR is lower than the target SINR, there could be many transmitters contending for the opportunity to transmit. Figure 5 shows the time scale dynamics of the transmit probabilities while applying SARA, where the algorithm converges within a few iteration.
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1 w/ message passing wo/ message passing
0.9
Transmission Probability
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Fig. 4.
1
2
3
4
5 6 7 Node Index
8
9
10
11
The updated transmission probabilities.
B. Optimal Transmit Probability of Conventional ALOHA We analyze the performance of the conventional random access (CRA) scheme (i.e., ALOHAlike random access). In this case, all transmitters utilize the same transmit probability φ. In a stochastic geometry point of view, the ASE can be expressed as the product of the successfully transmitting node density and data rate as follows [24]: η = λφ log (1 + β) p s ,
(16)
where the success probability, p s , of ALOHA is derived as follows ([31, Proposition 2.1]): p s = exp −λφrt2 β2/α ρ (α) , (17) 2 . With Eq. (17), we can rewrite the ASE η as a function of φ as follows: where ρ (α) = 2πα csc 2π α (18) η (φ) = λφ log (1 + β) exp −λφrt2 β2/α ρ (α) . As shown in Figure 6, there is an optimal φ that maximizes the ASE of ALOHA Eq. (18): φ∗ = arg max λφ log (1 + β) p s .
(19)
φ
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β=10dB 1 0.9
Transmission Probability
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1
Fig. 5.
2
3
4
5
6 iteration
7
8
9
10
11
The transmit probabilities converge.
The solution of Eq. (19), φ∗ , is obtained as follows: φ∗ = where ρ (α) =
2π2 α
csc
2π α
1 λrt2 β2/α ρ (α)
,
(20)
. The detailed derivation is in Appendix D.
By substituting Eq. (20) into Eq. (18), we have the maximum ASE η∗ of ALOHA as follows: η∗ = 0.3679
log (1 + β) . rt2 β2/α ρ (α)
(21)
What is interesting in Eq. (21) is that the maximum ASE η∗ of ALOHA is independent of node density λ. This is because the optimal transmit probability achieving the maximum ASE decreases at the rate of 1/λ. This scaling characteristic is consistent with that of the protocol model, in which the optimal transmit probability scales with 1/N when there are a total of N transmitters. In the physical model, the effect of target SINR β and path-loss exponent α are counted. As we mentioned earlier, the original problem (3) is hard to solve. The optimal transmit probability (20) is just a one-dimensional local optimal solution of the original problem. FigMarch 18, 2014
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λ=0.02, β=3dB
−3
4
x 10
r =5 t
rt=7.5 r =10 t
2
Area Spectral Efficiency [bits/Hz/m ]
3.5 3 2.5 2 1.5 1 0.5 0
Fig. 6.
0
0.2
0.4 0.6 Transmit Probability
0.8
1
The area spectral efficiency as a function of transmit probability. The node density, λ, is 0.02 and the target SINR, β,
is 3 dB.
ure 7 shows a more general case, where there are two transmit probabilities (φ1 , φ2 ) in the network. We obtain Eq. (20) when φ1 = φ2 . However, the global optimal exists elsewhere. The previous framework cannot improve the performance more than Eq. (20) while there is room for improving. Our approach can improve the performance. C. Large-scale Network Simulation To quantify the performance of SARA, we conducted a large-scale network simulation. In a 100 m×100 m area, various numbers of nodes are distributed according to the node density. The node density varies from 0.005 (sparse case) to 0.06 (dense case). The communication distance is 5 m. The transmission power is 30 dBm and the noise power is -70dBm.2 2
In the simulation, the received SINR is used instead of the received SIR. This is because the received SINR value may
diverge when the interference does not exists at a certain time slot. This frequently happens when the node density is sparse.
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Fig. 7.
The area spectral efficiency as a function of transmit probabilities. The node density, λ, is 0.02 and the target SINR,
β, is 3 dB.
Figure 8 shows the snapshot of the network topology in the case of λ = 0.02. Even though the same node density is applied, we can observe the regional variance of the population. Figure 8(a) and Figure 8(b) illustrate the dense part of the network. In Figure 8(a), the conventional ALOHA scheme is applied, and the transmitters highly overlap each other. On the other hand, in Figure 8(b) the transmitters are separated by utilizing the SARA scheme. In Figure 8(c) and 8(d), the sparse part of the network is depicted. Sine the transmit probability of the SARA scheme is adjusted by the number of strong interferers, the transmitters in the sparse situation try to transmit frequently while the transmitters using ALOHA do not. Figure 9 shows the average SINR performance of 600 nodes in the case of spatial averaging (Figure 9(a)) and SARA (Figure 9(b)). The variance of the average SINR with the conventional ALOHA scheme is 35.4476, and the variance of the average SINR with SARA scheme is 13.0263. This means that the conventional ALOHA scheme shows more severe regional performance differences than SARA scheme. Figure 10 shows the topology of the active transmitters of the network. In the case of the conventional ALOHA scheme, the active transmitters are overlapped (Figure 10(a)). On the other hand, with SARA, the active transmitters span the entire network (Figure 10(b)). It resembles March 18, 2014
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0
0
−5
−5
−10
−10
−15
−15
−20
−20
−25
−25 20
25
30
35
40
20
(a) Dense environment using ALOHA.
30
35
40
(b) Dense environment using SARA.
−10
−10
−15
−15
−20
−20
−25
−25
−30
−30
−35
−35 −45
25
−40
−35
−30
−25
−20
(c) Sparse environment using ALOHA.
−40
−35
−30
−25
−20
(d) Sparse environment using SARA.
Fig. 8. Snapshot of the dense and sparse environment using ALOHA and SARA (λ = 0.02, β = 3 dB, rt = 5 m, P = 30 dBm).
the topology of the CSMA/CA network. Figure 11 shows the ASE performance of the random access schemes. The spatially adaptive scheme surpasses the conventional ALOHA scheme. In most cases, the SARA scheme shows superior performance. The performance difference is severe for the highly dense networks. We also conducted the comparison with the CSMA/CA scheme. The carrier sensing range is set by doubling the transmission distance as a conventional setting. Surprisingly, the performance of SARA is better than that of the CSMA/CA scheme. This is because the adaptive behavior of SARA is superior than that of the CSMA/CA scheme. If the CSMA/CA protocol has functionality that adjusts its carrier sensing range based on the network situation, the performance of CSMA/CA would increase.
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VI. Concluding Remarks In this paper, we have shown the potential for improvement of the simple random access scheme by utilizing the received SINR. We investigated the performance of the spatial adaptive random access scheme. For the comparison, we analytically derived the optimal transmit probability of the ALOHA scheme in which all transmitters use the same transmit probability. We proposed the adaptive random access scheme in which the transmitters in the network utilize the different transmit probabilities depending on the situation. The transmit probability is adaptively updated by the ratio of the SINR and the target SINR. We illustrated the performance of the SARA scheme through simulation. We showed that the performance of the spatially adaptive scheme surpasses that of the ALOHA scheme and CSMA/CA scheme. The desirable research direction is to design a new backoff scheme using the proposed SARA scheme. Additionally, a possible research direction is to find the throughput maximization scheduling under the SINR rate-based interference model, where the instantaneous throughput of transmitter i, ri , is the function of the instantaneous SINR at the receiver k (i). That is, the data rate is ri = log 1 + γT j . With the adaptive modulation scheme, the data rate is selected according to i
the channel condition. In this case, the rule for adjusting the transmit probability may differ from that of the SINR-based interference model which is proposed in this paper. Appendix A. Expected value of the number of successfully transmitting nodes
Transmitters have transmit probabilities, therefore the concurrent transmission nodes are determined stochastically. When transmitter i is transmitting at a given time slot, the probability that the subset Ti j is selected as the concurrent transmission nodes is given by h i Pr transmitters in Ti j are active | node i is active Y Y (1 − φm ). φl = n j o j l∈Ti
The term Q n
Q
o j m∈N\ Ti ,i
March 18, 2014
l∈Ti j
(22)
m∈N\ Ti ,i
φl is the probability that all transmitters in set Ti j are transmitting, and the term
(1 − φm ) is the probability that all the transmitters, excluding those in Ti j and node i,
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18
are not transmitting. Using Eq. (22), the average SINR at the receiver of node i can be written as
X 2n−1 Y Y (1 − φm )γT j , φl Γk(i) (Φ) = φi i j=1 j n j o
(23)
m∈N\ Ti ,i
l∈Ti
where Φ denotes the vector of the transmission probabilities of all transmitters. If the topological information is given, then the γT j ’s are constant. The conditioned average SINR can be written i
as
2n−1 Y X φl Γk(i) (Φ−i | φi ) = j=1
l∈Ti
j
Y
(1 − φm )γT j , i o
(24)
n m∈N\ Ti j ,i
where Φ−i denotes the vector of the transmission probabilities of all transmitters except node i. i h Thus, the term E 1γk(i) ≥β is 2n−1 Y Y i X h (1 − φm )1γ j ≥β . (25) φl E 1γk(i) ≥β = Ti j n j o j=1 l∈Ti
m∈N\ Ti ,i
B. Existence of Nash equilibrium (12) The set of actions of transmitter i, {φi |φmin ≤ φi ≤ φmax }, is a nonempty compact convex subset of a Euclidian space and utility function Ui (Φ) is continuous and quasi-concave on the set of actions of transmitter i. According to Proposition 20.3 in [32], the existence of Nash equilibrium is guaranteed. By applying the first derivation to utility function (7), we obtain X 2n−1 Y Y ∂Ui (Φ) 1 j (1 ) − φi . , φ γ − φ φ = min max m T l i ∂φi β n o j j j=1 l∈T m∈N\ T ,i
(26)
i
i
We can easily verify that n−1 2 1 P Q Q j , φmax (1 ) > 0, φ < min φ − φ γ i l m ∂Ui (Φ) Ti β n o j j=1 l∈Ti j . m∈N\ Ti ,i ∂φi < 0, otherwise
(27)
Therefore, we can characterize Nash equilibrium for medium access probabilities of transmitters as
n−1 2 X Y Y 1 ∗ ∗ ∗ j , φmax , ∀i. γ 1 − φ φ φi = min m l Ti β j=1 l∈T j m∈N\nT j ,io i
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19
C. Submodularity of utility function (6) Let
X 2n−1 Y Y 1 (1 − φm )γT j . S = φl i β j=1 j n j o
(28)
m∈N\ Ti ,i
l∈Ti
The first derivative of utility function (6) is
∂Ui (Φ) = min {S , φmax} − φi . ∂φi Using Eq. (29), we can show that the second derivative of the utility function is 0, S ≥ φmax 2 ∂ Ui (Φ) n−2 2P = Q Q φ 1 ∂φi φv (1 − φm ) γ(+v) − γ(−v) , otherwise, l β j=1, l∈T j [−v] m∈N\nT j [−v],i,vo i
(29)
(30)
i
where Ti j [−v] denotes the j-th subset of superset Ti [−v], which is defined as all the combinations
of nodes concurrently transmitting with node i excluding node v. The notation for γ(+v) is Gi,k(i) Pi P Gu,k(i) Pu
γ(+v) =
(31)
u∈Ti j (−v)∪{v}
and the notation for γ(−v) is γ(−v) =
Gi,k(i) Pi . P Gu,k(i) Pu
(32)
j u∈Ti (−v)
As we can see, γ(+v) contains one more interferer than γ(−v) . Thus, we have γ(+v) < γ(−v) .
(33)
Therefore, ∂2 Ui (Φ) ≤ 0. ∂φi ∂φv D. Derivation of Eq. (20) Point φ∗ is a strict local maximizer if it satisfies the following conditions (second order sufficient condition (SOSC)) [33]. =0 1) ∂η(φ) ∂φ φ=φ∗ 2 2) ∂ ∂η(φ)
Exploiting Regional Differences: A Spatially Adaptive Random Access
arXiv:1403.3891v1 [cs.IT] 16 Mar 2014
Dong Min Kim and Seong-Lyun Kim
Abstract In this paper, we discuss the potential for improvement of the simple random access scheme by utilizing local information such as the received signal-to-interference-plus-noise-ratio (SINR). We propose the spatially adaptive random access (SARA) scheme in which the transmitters in the network utilize different transmit probabilities depending on the local situation. In our proposed scheme, the transmit probability is adaptively updated by the ratio of the received SINR and the target SINR. We investigate the performance of the spatially adaptive random access scheme. For the comparison, we derive an optimal transmit probability of the conventional random access scheme in which all transmitters use the same transmit probability. We illustrate the performance of the spatially adaptive random access scheme through simulations. We show that the performance of the adaptive scheme surpasses that of the optimal conventional random access scheme and the CSMA/CA scheme. The convergence property of the proposed scheme is analyzed using the submodular game.
Index Terms Random access, distributed scheduling, SINR-based interference model, stochastic geometry, submodular game.
I. Introduction A. Spatially Adaptive Random Access (SARA) In this paper, we propose a spatially adaptive random access (SARA) scheme. Each node i behaves as follows: The authors are with the Radio Resource Management and Optimization Laboratory, School of Electrical and Electronic Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, Korea (email: [email protected]; [email protected]).
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1) Initialize transmit probability with the largest value, φmax . 2) Compute the average signal-to-interference-plus-noise-ratio (SINR), Γk(i) (t), during period T as follows: 1 Γk(i) (t) ≈ T q=1 T X
Gi,k(i) Pi , P Gu,k(i) Pu j
t > T,
u∈Ti (t−q),
where the notation k(i) denotes the receiver associated with transmitter i. The notation Gi, j denotes the channel gain from node i to node j. The term Pi represents the transmit power of transmitter i. The term Ti j (q) denotes the j-th subset of the concurrent transmission nodes when node i transmits at time slot q. 3) Update transmit probability φi as follows: (
(
) ) Γk(i) (t) φi (t + 1) = min max φmin , , φmax , β
(1)
where the notations φmin and φmax represent minimum and maximum values of the transmit probability, respectively. The notation β denotes a target SINR threshold. SARA is a variant of ALOHA, where each transmitter updates the transmit probability depending on the local situation. We verify the convergence property using the submodular game ([1], [2]). SARA improves the average received SINR with very little message passing in the network. Our simulation results show that the performance of SARA is even better than a carrier sense multiple access with collision avoidance (CSMA/CA). B. Motivation and Related Works The ALOHA protocol [3] is the most well-known distributed random access scheme. The transmit probability controls the operation of the ALOHA protocol. In [4], the authors derive an optimal transmit probability under the protocol model, where the transmission fails if two or more nodes are transmitting simultaneously. To improve the performance of an ALOHA network, researchers conducted several studies using a simple protocol model to achieve proportional fair and max-min fair [5]–[7]. In [8]–[10], the authors investigated optimal random access approaches achieving network utility maximization using a family of α-fair utility functions [11] in the protocol model. Due to the characteristics of the wireless channel [12], however, the receiver March 18, 2014
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may successfully receive the signal if the concurrent transmitters are far away. The physical model [13] considers the effect of such accumulated multi-user interference. In practice, interfering nodes are randomly located. In this regard, stochastic geometry [14] is a useful mathematical tool to model such randomness. In [15], the authors provide a stochastic geometry-based analytical framework of ALOHA. In a recent study [16], the authors investigated an adaptive ALOHA using a SINR model from a stochastic geometry point of view. The authors of [16] focus on achieving proportional fairness while we concentrate on maximizing the area spectral efficiency. Refer to [17]–[24] and references therein for readers interested in such an approach. In [25], the authors investigate the SINR-based random access protocol. Later, in [26], the authors proposed an adaptive interference pricing scheme to find a local optimal solution of the network utility maximization problem. They adopted a game theoretic framework ([27], [28]) to analyze multiple access control (MAC). The proposed approaches in [25] and [26] require a large number of message exchanges among the transmitters to inform their transmit probabilities to the others while SARA requires very little message passing in the network. CSMA/CA is more advanced than ALOHA in that it has the ability to adapt the local situation through carrier sensing and exponential backoff. The conventional ALOHA-like random access cannot behave adaptively because the transmit probability is fixed by a single optimal value. The optimal values of transmit probabilities are different in dense and sparse environments, and all nodes in the network should not have the same transmit probability. Let us assume that the nodes are deployed as shown in Figure 1. About 200 communication pairs are randomly distributed in a rectangular area. In this case, the node density is 0.02, and all transmitters have a fixed transmission probability. The nodes in subarea A are located in a relatively sparse environment; their transmissions may not be interfered. On the other hand, the nodes in subarea B are located in a relatively dense environment, and the transmissions of the nodes in B would frequently fail due to heavy interference. The nodes in A may want to utilize a relatively large transmit probability, and in B, a small probability. To improve the performance of such an ALOHA-like random access scheme, we devise SARA, which adjusts its operation according to the local circumstance. The main contributions of this paper are summarized as follows: •
We proposed a distributed SARA scheme, where the average received SINR is improved without frequent message passing with other nodes in the network.
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λ=0.02, φ =0.121 50 40 30 20 10 0
A
-10 -20 -30
B
-40 -50 -50
0
50
Fig. 1. A snapshot of network topology. The circles represent transmitters, and the connected pentagrams represent associated receivers. The arrows represent active communication pairs. The node density, λ, is 0.02 and the transmit probablity, φ, is 0.121. The nodes in subarea A are located in a relatively sparse environment. On the other hand, the nodes in subarea B are located in a relatively dense environment.
•
We present the convergence property of the proposed scheme using the submodular game.
•
We show the performance of the proposed scheme is better than that of the ALOHA and CSMA/CA schemes.
The rest of the paper is organized as follows. In Section II, we describe the system model. In Section III, we investigate our SARA scheme. In Section IV, we analyze the the convergence property of the proposed scheme. We verify the performance through simulations in Section V. Section VI concludes the paper. II. System Model As shown in Figure 1, a random wireless network of a single radio channel is considered, where each transmitter is associated with a receiver over a shared wireless channel. The transmitters are distributed according to a homogeneous Poisson point process (PPP) with intensity λ. If the March 18, 2014
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TABLE I Key mathematical notations.
λ
Node density
A
Area of interesting region
k(i)
Associated receiver of transmitter i
rt
Communication distance
P
Transmit power
φi
Transmit probability of transmitter i
φmin
Minimum transmit probability
φmax
Maximum transmit probability
Gi, j
Channel gain from node i to node j
α
Path loss exponent
γT j
Instantaneous SINR of receiver k(i)
Ti
Superset of concurrent interfering nodes when node i transmits
i
Ti
j
j-th subset of interfering nodes when node i transmits
β
Target SINR threshold
ri
Data rate of transmitter i
η
Area spectral efficiency
ps
Success probability
∗
φ
Optimal transmit probability
N
Set of transmitters
Γk
Average SINR
Φ
Vector of transmit probabilities of all transmitters
Φ−i
Vector of transmission probabilities of all transmitters except node i
region is finite (of size A), this means that nodes are independent and identically distributed with a uniform distribution in the region with a given average number of nodes (λA). Each transmitter, i, always has ample data to send. Each associated receiver, k(i), is located at a distance of rt from the transmitter i and the direction is random. The receivers also follow the homogeneous PPP by the the displacement theorem [17]. We assume that the time is slotted and synchronized so that transmissions begin with a time slot and continue during the slot length. The transmitter/receiver pair can be changed over the time. However, we focus on a snapshot of the overall communication process, where the network topology is fixed during each slot. The transmitter i attempts to send its data with transmit probability φi . The channel gain from node i to node j, Gi, j , depends on the distance between the transmitter and the receiver with path March 18, 2014
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6
loss exponent α and Rayleigh fading. The stochastic process of the wireless channel is ergodic. With a single shared channel, the concurrent transmissions cause cochannel interference. The instantaneous SINR of receiver k(i), γT j , is given by i
γT j = P
Gi,k(i) Pi , Gu,k(i) Pu + Wk(i)
i
u∈Ti
Ti j ∈ Ti ,
(2)
j
where we consider the interference limited network. Then, the noise power term Wk(i) is omitted from Eq. (2) and we deal with the signal-to-interference-ratio (SIR). The notation Ti denotes the superset of concurrent transmission nodes (interfering nodes) when node i transmits. When there are n transmitters in the networks, the cardinality of Ti is 2n−1 . The notation Ti j (q) denotes the j-th subset of the simultaneously transmitting nodes when node i transmits at time slot q. If a specific time slot does not matter, the notation Ti j (q) is simplified to Ti j . For example, assume that the network consists of three pairs {1, 2, 3}. If transmitter 1 is active, the superset of concurrent transmitting nodes, T1 , is {{}, {2}, {3}, {2, 3}}, and T11 = {}, T12 = {2}, T13 = {3}, T14 = {2, 3}. For a given target SINR threshold β, transmission is successful if γT j ≥ β is satisfied. We i
assume that the data rate of each transmitter, i, is a function of β, ri = log (1 + β), where we assume a unit bandwidth. III. Maximizing Area Spectral Efficiency A. Original Problem An area spectral efficiency (ASE) η is the sum of data rates per unit bandwidth in the unit area ([29], [30]). To focus on a network-wide performance, we use η as a performance metric. To maximize η, we formulate an optimization problem as follows: X h i max η = log2 (1 + β) E 1γk(i) ≥β
(3)
i
s.t.
φmin ≤ φi ≤ φmax ,
∀i,
where 1γk(i) ≥β denotes the indicator function defined as 1 if γk(i) ≥ β, otherwise 0. The term i P h E 1γk(i) ≥β represents the expected value of the number of successfully transmitting nodes in i
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7
the unit area. The notation γk(i) denotes the random variable of the instantaneous SINR of the receiver k(i). The objective function of Eq. (3) is a function of φi ’s as follows: 2n−1 Y Y X h i X X (1 − φm )1γ j ≥β , φl E 1γk(i) ≥β = Ti n j o j=1 j i i
(4)
m∈N\ Ti ,i
l∈Ti
where N is a set of all transmitters. The detailed derivation is in Appendix A. To maximize Eq. (4), we should find all γT j ’s. This means that we should compute all combinations of i
interferers. This is a combinatorial optimization problem which becomes harder to solve as the number of nodes in the network increases. B. Relaxed Problem In an effort to solve Eq. (3) dispersively, we decompose the problem as follows: 2n−1 Y Y X (1 − φm )1γ j ≥β max φl Ti j n j o j=1 l∈Ti
s.t.
(5)
m∈N\ Ti ,i
φmin ≤ φi ≤ φmax .
Each node i solves its own problem. However, we still should need all γT j ’s for each node i. As i
we mentioned in Section II, the number of combinations is 2
n−1
. To make the problem solvable
in a distributed way, we relax the problem as follows: max Ui (Φ)
(6)
s.t. φmin ≤ φi ≤ φmax , where Φ is a vector of the transmit probabilities of all nodes. We define Ui (Φ) as a utility function of node i as follows:1 X 2n−1 Y Y 1 2 1 j (1 ) Ui (Φ) = min − φ φ , φ γ φ m l max i − φi . T i 2 β j=1 l∈T j m∈N\nT j ,io i
(7)
i
Node i, who transmits with the probability φi , obtains the reward as a form of the former part of the right hand side of Eq. (7). The reward is a ratio of the average SINR to the target SINR. As we achieve a higher average SINR, the reward increases. The bad effect on the network 1
It is noted that the part of equation related with φmin is omitted for the simplicity.
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(increasing contention and interference) is assessed as the latter part of Eq. (7). Our utility function has a property to penalize the occurrence of the interference. It makes sure that the radio spectrum resources are efficiently shared. Our algorithm, shown in Section I-A, maximizes the utility function, Eq. (7). Using Eq. (6), our proposed update algorithm, Eq. (1), can be written as: φi (t + 1) X 2n−1 Y Y 1 j (1 (t)) (t) , φ . γ − φ φ = min max m l T i β n o j j j=1 l∈T m∈N\ T ,i i
(8)
i
To obtain the exact value of γT j , the nodes in the network need to frequently exchange i
the message with neighbor nodes to track the network topology information. To reduce this complexity, the time-averaged SINR can be used to update the transmit probability, as shown in Section I. A: Γk(i)
T Gi,k(i) Pi 1 X Gi,k(i) Pi P ≈ . = E P Gu,k(i) Pu T q=1 Gu,k(i) Pu j j u∈Ti ,
(9)
u∈Ti ,
If the average SINR is larger than the target β, the network situation is favorable for that communication pair. The pair may be isolated from the others. Therefore it is highly probable that the transmission of this transmitter will not interfere with the communications of the others. To promote more chances to transmit, the transmit probability is set to the maximum. On the other hand, as the ratio is less than φmax , the pair experience a highly contending situation. The transmit probability should be lowered to resolve the contention according to the ratio of the average SINR to the target SINR. The average SINR computation is done by the receiver. The receiver should notify its transmitter of the average SINR when the receiver transmits the acknowledgement signal or any kind of control signal (piggybacking). The convergence property of the proposed algorithm is given in the next section. IV. Convergence Property of SARA In this section, the convergence property of SARA is verified using the submodular game ([1], [2]). The non-cooperative game model suitable for our purpose is formulated as follows: N, {Φ} = {φi |φmin ≤ φi ≤ φmax , ∀i ∈ N} , {Ui (Φ)}i∈N , March 18, 2014
(10) DRAFT
9
where N is the set of transmitters. The set {Φ} is a strategy space. Ui (Φ) is a utility function for transmitter i. The medium access probability vector can be decomposed as Φ = {φi , Φ−i }, where φi is the medium access probability of node i and Φ−i is a medium access probability vector of all nodes except for node i. The medium access probability vector Φ∗ is said to be a Nash equilibrium if no unilateral deviation by any node is improvable for that node: Ui φ∗i , Φ∗−i ≥ Ui φi , Φ∗−i , 0 ≤ φi ≤ 1, φ∗i , φi , ∀i. In our game, there exists a Nash equilibrium characterized by the following equation: X 2n−1 Y Y 1 ∗ ∗ ∗ j . , φ γ φi = min 1 − φ φ max m l T i n j o β j=1 j l∈Ti
(11)
(12)
m∈N\ Ti ,i
The detailed derivation is in Appendix B.
Our game model can be categorized by a submodular game model because the utility function satisfies submodularity as follows: ∂2 Ui (Φ) ≤ 0, ∀v ∈ N\i, i ∈ N. ∂φi ∂φv The detailed derivation is in Appendix C. In a submodular game, if the transmit probability of each transmitter is updated by the best response strategy, the iterative update can converge to a Nash equilibrium. The definition of the best response is as follows: Bi (Φ−i ) = arg max Ui (φi , Φ−i ) ,
(13)
φi
φi (t + 1) = Bi (Φ−i (t)) .
(14)
The utility function (7) is concave and differentiable with φi . Therefore it is maximized when ∂U i (Φ) ∂φi
= 0. The corresponding φi is X 2n−1 Y 1 φi = min φl β j=1 j l∈Ti
Y
(1 − φm )γT j , φmax . o i
(15)
n j m∈N\ Ti ,i
Eq. (8) is the same form of Eq. (15). Thus, the proposed algorithm is a best response iterative update algorithm. In submodular game, if the transmit probability of each transmitter is updated by the best response strategy starting with initial value, the iterative update converges to Nash equilibrium. or at least oscillates between two values [2]. In the next section, the performance of the proposed random access scheme is evaluated. March 18, 2014
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10
15
1 10
0.7147
5
0.214
0
0.001 0.0105 -5
0.0476 -10
-15 -15
Fig. 2.
-10
-5
0
5
10
15
There are 11 communication pairs in the 30-by-30 rectangular area. Two communication pairs are relatively isolated
from the others.
V. Performance Evaluation A. Average SINR Validation and Convergence Simulation To evaluate the accuracy of Eq. (9), we conducted a simple simulation. As shown in Figure 2, a total of 11 transmitter/receiver pairs are distributed on the 30 m by 30 m area. The communication distance between a transmitter/receiver pair is 5 m. The transmit power is 30 dBm. The target threshold is 3 dB. Figure 3 shows the exact SINR (ensemble average) in Eq. (23) and the time-averaged SINR in Eq. (9). Figure 3 shows that the time-averaged SINR can approximate the exact ensemble average. The small differences are caused by the fading characteristics of wireless channel. Figure 4 shows the updated transmit probabilities. The updated probabilities using the timeaveraged SINR are almost the same as the updated probabilities using the ensemble-averaged SINR (with frequent message passing), as shown in Figure 4. In our system, the success of a transmission is determined by an instantaneous SINR and the March 18, 2014
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11
15 10
Average SINR [dB]
5 0 -5 -10 -15 -20 -25
Fig. 3.
ensemble average time average 1
2
3
4
5 6 7 Node Index
8
9
10
11
The average SINR of randomly distributed nodes. The exact SINR is estimated by time averaged SINR.
target SINR β. The instantaneous SINR changes with a small-time-scale (milliseconds) due to the Rayleigh fading, which is independent with a spatial random distribution of nodes in the network. To get rid of the effect of fading and to reflect the distribution of nodes, we utilize the average SINR, which varies with a large-time-scale (seconds). The average is taken by each of the nodes during the buffered period T . The ensemble average is more accurate than the time average. When the wireless channel is ergodic, the time average with a sufficient period can approximate the ensemble average. In this regard, the time-averaged SINR value is an indicator of the network condition. If the average SINR is lower than the target SINR, there could be many transmitters contending for the opportunity to transmit. Figure 5 shows the time scale dynamics of the transmit probabilities while applying SARA, where the algorithm converges within a few iteration.
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1 w/ message passing wo/ message passing
0.9
Transmission Probability
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Fig. 4.
1
2
3
4
5 6 7 Node Index
8
9
10
11
The updated transmission probabilities.
B. Optimal Transmit Probability of Conventional ALOHA We analyze the performance of the conventional random access (CRA) scheme (i.e., ALOHAlike random access). In this case, all transmitters utilize the same transmit probability φ. In a stochastic geometry point of view, the ASE can be expressed as the product of the successfully transmitting node density and data rate as follows [24]: η = λφ log (1 + β) p s ,
(16)
where the success probability, p s , of ALOHA is derived as follows ([31, Proposition 2.1]): p s = exp −λφrt2 β2/α ρ (α) , (17) 2 . With Eq. (17), we can rewrite the ASE η as a function of φ as follows: where ρ (α) = 2πα csc 2π α (18) η (φ) = λφ log (1 + β) exp −λφrt2 β2/α ρ (α) . As shown in Figure 6, there is an optimal φ that maximizes the ASE of ALOHA Eq. (18): φ∗ = arg max λφ log (1 + β) p s .
(19)
φ
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β=10dB 1 0.9
Transmission Probability
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1
Fig. 5.
2
3
4
5
6 iteration
7
8
9
10
11
The transmit probabilities converge.
The solution of Eq. (19), φ∗ , is obtained as follows: φ∗ = where ρ (α) =
2π2 α
csc
2π α
1 λrt2 β2/α ρ (α)
,
(20)
. The detailed derivation is in Appendix D.
By substituting Eq. (20) into Eq. (18), we have the maximum ASE η∗ of ALOHA as follows: η∗ = 0.3679
log (1 + β) . rt2 β2/α ρ (α)
(21)
What is interesting in Eq. (21) is that the maximum ASE η∗ of ALOHA is independent of node density λ. This is because the optimal transmit probability achieving the maximum ASE decreases at the rate of 1/λ. This scaling characteristic is consistent with that of the protocol model, in which the optimal transmit probability scales with 1/N when there are a total of N transmitters. In the physical model, the effect of target SINR β and path-loss exponent α are counted. As we mentioned earlier, the original problem (3) is hard to solve. The optimal transmit probability (20) is just a one-dimensional local optimal solution of the original problem. FigMarch 18, 2014
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λ=0.02, β=3dB
−3
4
x 10
r =5 t
rt=7.5 r =10 t
2
Area Spectral Efficiency [bits/Hz/m ]
3.5 3 2.5 2 1.5 1 0.5 0
Fig. 6.
0
0.2
0.4 0.6 Transmit Probability
0.8
1
The area spectral efficiency as a function of transmit probability. The node density, λ, is 0.02 and the target SINR, β,
is 3 dB.
ure 7 shows a more general case, where there are two transmit probabilities (φ1 , φ2 ) in the network. We obtain Eq. (20) when φ1 = φ2 . However, the global optimal exists elsewhere. The previous framework cannot improve the performance more than Eq. (20) while there is room for improving. Our approach can improve the performance. C. Large-scale Network Simulation To quantify the performance of SARA, we conducted a large-scale network simulation. In a 100 m×100 m area, various numbers of nodes are distributed according to the node density. The node density varies from 0.005 (sparse case) to 0.06 (dense case). The communication distance is 5 m. The transmission power is 30 dBm and the noise power is -70dBm.2 2
In the simulation, the received SINR is used instead of the received SIR. This is because the received SINR value may
diverge when the interference does not exists at a certain time slot. This frequently happens when the node density is sparse.
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Fig. 7.
The area spectral efficiency as a function of transmit probabilities. The node density, λ, is 0.02 and the target SINR,
β, is 3 dB.
Figure 8 shows the snapshot of the network topology in the case of λ = 0.02. Even though the same node density is applied, we can observe the regional variance of the population. Figure 8(a) and Figure 8(b) illustrate the dense part of the network. In Figure 8(a), the conventional ALOHA scheme is applied, and the transmitters highly overlap each other. On the other hand, in Figure 8(b) the transmitters are separated by utilizing the SARA scheme. In Figure 8(c) and 8(d), the sparse part of the network is depicted. Sine the transmit probability of the SARA scheme is adjusted by the number of strong interferers, the transmitters in the sparse situation try to transmit frequently while the transmitters using ALOHA do not. Figure 9 shows the average SINR performance of 600 nodes in the case of spatial averaging (Figure 9(a)) and SARA (Figure 9(b)). The variance of the average SINR with the conventional ALOHA scheme is 35.4476, and the variance of the average SINR with SARA scheme is 13.0263. This means that the conventional ALOHA scheme shows more severe regional performance differences than SARA scheme. Figure 10 shows the topology of the active transmitters of the network. In the case of the conventional ALOHA scheme, the active transmitters are overlapped (Figure 10(a)). On the other hand, with SARA, the active transmitters span the entire network (Figure 10(b)). It resembles March 18, 2014
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16
0
0
−5
−5
−10
−10
−15
−15
−20
−20
−25
−25 20
25
30
35
40
20
(a) Dense environment using ALOHA.
30
35
40
(b) Dense environment using SARA.
−10
−10
−15
−15
−20
−20
−25
−25
−30
−30
−35
−35 −45
25
−40
−35
−30
−25
−20
(c) Sparse environment using ALOHA.
−40
−35
−30
−25
−20
(d) Sparse environment using SARA.
Fig. 8. Snapshot of the dense and sparse environment using ALOHA and SARA (λ = 0.02, β = 3 dB, rt = 5 m, P = 30 dBm).
the topology of the CSMA/CA network. Figure 11 shows the ASE performance of the random access schemes. The spatially adaptive scheme surpasses the conventional ALOHA scheme. In most cases, the SARA scheme shows superior performance. The performance difference is severe for the highly dense networks. We also conducted the comparison with the CSMA/CA scheme. The carrier sensing range is set by doubling the transmission distance as a conventional setting. Surprisingly, the performance of SARA is better than that of the CSMA/CA scheme. This is because the adaptive behavior of SARA is superior than that of the CSMA/CA scheme. If the CSMA/CA protocol has functionality that adjusts its carrier sensing range based on the network situation, the performance of CSMA/CA would increase.
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VI. Concluding Remarks In this paper, we have shown the potential for improvement of the simple random access scheme by utilizing the received SINR. We investigated the performance of the spatial adaptive random access scheme. For the comparison, we analytically derived the optimal transmit probability of the ALOHA scheme in which all transmitters use the same transmit probability. We proposed the adaptive random access scheme in which the transmitters in the network utilize the different transmit probabilities depending on the situation. The transmit probability is adaptively updated by the ratio of the SINR and the target SINR. We illustrated the performance of the SARA scheme through simulation. We showed that the performance of the spatially adaptive scheme surpasses that of the ALOHA scheme and CSMA/CA scheme. The desirable research direction is to design a new backoff scheme using the proposed SARA scheme. Additionally, a possible research direction is to find the throughput maximization scheduling under the SINR rate-based interference model, where the instantaneous throughput of transmitter i, ri , is the function of the instantaneous SINR at the receiver k (i). That is, the data rate is ri = log 1 + γT j . With the adaptive modulation scheme, the data rate is selected according to i
the channel condition. In this case, the rule for adjusting the transmit probability may differ from that of the SINR-based interference model which is proposed in this paper. Appendix A. Expected value of the number of successfully transmitting nodes
Transmitters have transmit probabilities, therefore the concurrent transmission nodes are determined stochastically. When transmitter i is transmitting at a given time slot, the probability that the subset Ti j is selected as the concurrent transmission nodes is given by h i Pr transmitters in Ti j are active | node i is active Y Y (1 − φm ). φl = n j o j l∈Ti
The term Q n
Q
o j m∈N\ Ti ,i
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l∈Ti j
(22)
m∈N\ Ti ,i
φl is the probability that all transmitters in set Ti j are transmitting, and the term
(1 − φm ) is the probability that all the transmitters, excluding those in Ti j and node i,
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are not transmitting. Using Eq. (22), the average SINR at the receiver of node i can be written as
X 2n−1 Y Y (1 − φm )γT j , φl Γk(i) (Φ) = φi i j=1 j n j o
(23)
m∈N\ Ti ,i
l∈Ti
where Φ denotes the vector of the transmission probabilities of all transmitters. If the topological information is given, then the γT j ’s are constant. The conditioned average SINR can be written i
as
2n−1 Y X φl Γk(i) (Φ−i | φi ) = j=1
l∈Ti
j
Y
(1 − φm )γT j , i o
(24)
n m∈N\ Ti j ,i
where Φ−i denotes the vector of the transmission probabilities of all transmitters except node i. i h Thus, the term E 1γk(i) ≥β is 2n−1 Y Y i X h (1 − φm )1γ j ≥β . (25) φl E 1γk(i) ≥β = Ti j n j o j=1 l∈Ti
m∈N\ Ti ,i
B. Existence of Nash equilibrium (12) The set of actions of transmitter i, {φi |φmin ≤ φi ≤ φmax }, is a nonempty compact convex subset of a Euclidian space and utility function Ui (Φ) is continuous and quasi-concave on the set of actions of transmitter i. According to Proposition 20.3 in [32], the existence of Nash equilibrium is guaranteed. By applying the first derivation to utility function (7), we obtain X 2n−1 Y Y ∂Ui (Φ) 1 j (1 ) − φi . , φ γ − φ φ = min max m T l i ∂φi β n o j j j=1 l∈T m∈N\ T ,i
(26)
i
i
We can easily verify that n−1 2 1 P Q Q j , φmax (1 ) > 0, φ < min φ − φ γ i l m ∂Ui (Φ) Ti β n o j j=1 l∈Ti j . m∈N\ Ti ,i ∂φi < 0, otherwise
(27)
Therefore, we can characterize Nash equilibrium for medium access probabilities of transmitters as
n−1 2 X Y Y 1 ∗ ∗ ∗ j , φmax , ∀i. γ 1 − φ φ φi = min m l Ti β j=1 l∈T j m∈N\nT j ,io i
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i
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C. Submodularity of utility function (6) Let
X 2n−1 Y Y 1 (1 − φm )γT j . S = φl i β j=1 j n j o
(28)
m∈N\ Ti ,i
l∈Ti
The first derivative of utility function (6) is
∂Ui (Φ) = min {S , φmax} − φi . ∂φi Using Eq. (29), we can show that the second derivative of the utility function is 0, S ≥ φmax 2 ∂ Ui (Φ) n−2 2P = Q Q φ 1 ∂φi φv (1 − φm ) γ(+v) − γ(−v) , otherwise, l β j=1, l∈T j [−v] m∈N\nT j [−v],i,vo i
(29)
(30)
i
where Ti j [−v] denotes the j-th subset of superset Ti [−v], which is defined as all the combinations
of nodes concurrently transmitting with node i excluding node v. The notation for γ(+v) is Gi,k(i) Pi P Gu,k(i) Pu
γ(+v) =
(31)
u∈Ti j (−v)∪{v}
and the notation for γ(−v) is γ(−v) =
Gi,k(i) Pi . P Gu,k(i) Pu
(32)
j u∈Ti (−v)
As we can see, γ(+v) contains one more interferer than γ(−v) . Thus, we have γ(+v) < γ(−v) .
(33)
Therefore, ∂2 Ui (Φ) ≤ 0. ∂φi ∂φv D. Derivation of Eq. (20) Point φ∗ is a strict local maximizer if it satisfies the following conditions (second order sufficient condition (SOSC)) [33]. =0 1) ∂η(φ) ∂φ φ=φ∗ 2 2) ∂ ∂η(φ)
