The Capacity of Energy-Constrained Mobile Networks with Wireless ...

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The Capacity of Energy-Constrained Mobile Networks with Wireless Power Transfer Seung-Woo Ko, Seung Min Yu and Seong-Lyun Kim

Abstract—We derive the throughput of an energy-constrained mobile network, where wireless charging vehicles recharge the battery of each mobile node. We find that the throughput is closely related to the ratio between the number of nodes n and the number of wireless charging vehicles m. Through analytic and numerical results, we show that the scaling law of the throughput ( ( ) min(1, m ) ) n is Θ min 1, m c , where c is constant (0 < c < 1). n Index Terms—Capacity, mobility, wireless power transfer, Markov chains. Fig. 1. Graphical representation of a wireless random network with wireless charging vehicles.

I. I NTRODUCTION ITH rapid increase of wireless devices, measuring the capacity of wireless random networks is important. In their seminal work [1], Gupta and Kumar claimed √ that the per node throughput for each S-D pair follows Θ(1/ n log n)1 , where n is the number of nodes in the network. Later, Glossglauser and Tse showed that by exploiting mobility of each node, the per node throughput becomes constant regardless of the node density [2]. Even though such constant throughput is achieved at the cost of long delay [3], various scheduling and routing strategies exploiting mobility have been proposed [4][5]. These works assume that the available transmission energy of each node is unlimited, which is not the case in practice. After the battery runs out, the throughput falls to zero. Wireless power transfer (WPT) is an emerging batterycharging technique that does not require plugs or wires to charge the batteries of mobile devices. There were attempts to develop practical WPT methods. In 2007, Kurs et al. suggested a novel method called magnetic resonant coupling [6]-[7]. They experimentally showed that, when two devices are tuned to the same resonant frequency, electric power is transferred from one to the other with high efficiency. The main purpose of this letter is to incorporate WPT into a wireless random network and to revisit the throughput scaling law of the energy-constrained network. In [8]-[10], the authors claimed that WPT can prolong the lifetime of wireless sensor networks. In their scenario, a wireless charging vehicle (WCV) periodically visits every sensor node to charge the battery of each node. When each node is mobile, it is difficult to plan a deterministic visiting path of WCV. Instead, we assume the scenario that WCVs randomly move and transfer energy to

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This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0088483). The authors are with the Radio Resource Management & Optimization Laboratory, Department of Electrical and Electronic Engineering, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 120-749, Korea. Email: {swko, smyu, slkim}@ramo.yonsei.ac.kr. 1 We recall that the following notation: (i) f (n) = O(g(n)) means that there exists a constant c and integer N such that f (n) ≤ cg(n) for n > N . (ii) f (n) = Θ(f (n)) means that f (n) = O(g(n)) and g(n) = O(f (n)).

mobile nodes when they are in close proximity. We prove that when m WCVs are employed, the scaling law of the per ( ( m ) min(1, m ) ) n node throughput follows Θ min 1, n c , where c is constant (0 < c < 1). In particular, in order to maintain the constant throughput of Θ(1) as in [2], at least Θ(n) WCVs should be used. II. N ETWORK D ESCRIPTION Consider a wireless random network where n nodes are randomly located in a unit area. Time is divided into equal length slots. In each time slot, nodes move around the network according to the i. i. d mobility model. In this model, nodes can move anywhere without being restricted by their previous location. All nodes have the same common transmission range r (see Figure 1). For the interference model, we adopt the protocol model from [1]. Transmitter i successfully delivers a packet to receiver j when the following conditions are satisfied: • The distance between them is no more than r. • The distances between node j and the other transmitting nodes are no less than r. If the transmission range r is too large, the transmission often fails as there are many interfering nodes. In order to avoid excessive interference, we set r to the average distance to the nearest node in the unit area: ∫ √1 π Γ(n) 1 ( ) √ r= (1 − πx2 )n−1 dx = (1) 1 ≈ 2 n, 2Γ n + 2 0 ∫∞ where Γ(z) = 0 tz−1 e−t dt is the gamma function. A pair of source and destination nodes is given randomly. Let us denote the source and the destination of node i by s(i) and d(i), respectively. Unless there is the destination node in its transmission range r, the packet should be delivered by a relay node. In this letter, the transmission policy follows the two-phase routing similar to that in [2]: • Mode switch. In each time slot, a node becomes a transmitter with probability q or a receiver with probability 1 − q.

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Phase 1. In odd time slots, if node i becomes a transmitter and there is at least one receiver within r, node i can transmit its packet to one of them. This receiver node can be d(i). Phase 2. In even time slots, if node i becomes a receiver and there is at least one transmitter having a packet for node i within r, node i can receive the packet from one of them. This transmitter node can be s(i).

III. A N E NERGY-C ONSTRAINED M OBILE N ETWORK WITH W IRELESS P OWER T RANSFER A. Throughput of the Energy-Constrained Network In [2], the throughput of the two-phase routing is defined as follows: Definition 1. (Throughput) Let Mi (t) be the number of node i packets that node d(i) receives during t time slots. We say that a long-term per node throughput of Λ(n) is feasible for every S-D pair if: 1 lim inf Mi (t) ≥ Λ(n), (2) t→∞ t Hereafter, the long-term per node throughput is abbreviated as the throughput. When a transmitter tries to forward a packet, a constant amount of energy is required. We denote this as one unit of energy. A node is active when it has more than one unit of energy. Otherwise, the node is inactive. Definition 2. (Active probability) The active probability pon denotes the probability that a node has at least one unit of energy. Proposition 1. Under two-phase routing, the throughput Λ(n) of the energy-constrained network is: ( ) π π 1 Λ(n) = · q · pon · e− 4 qpon · 1 − e 4 (−1+q) (3) 2 Proof. Appendix 2 The throughput Λ(n) is closely related to the active probability pon . When the energy of each node is unlimited, the active probability pon becomes 1. According to Proposition 1, the throughput Λ(n) of (3) is: ( ) π π 1 Λ(n) = · q · e− 4 q · 1 − e 4 (−1+q) = Θ(1), (4) 2 which is reduced to the result of [2]. If the energy of each node is limited and there is no recharging, then the throughput Λ(n) becomes zero because there is no active node (pon = 0). The active probability pon is determined by the recharging mechanism. In the following subsections, we will explain how wireless charging vehicles can charge node batteries. We then analyze the active probability pon and the corresponding throughput Λ(n). B. Recharging Mechanism by Wireless Charging Vehicles The maximum battery capacity of each node is set to L units of energy. A node cannot transmit a packet when its battery is exhausted. In order to recharge the battery, m WCVs are employed in the network. WCVs exploit magnetic resonance coupling to recharge nodes. WCVs freely move around the

Fig. 2. State diagram of the number of units of energy (L = 5, E = 3). Probabilities pc , pt and β(i) are given in Equations (6), (7) and (9), respectively.

network according to the i. i. d mobility model. The maximum energy transfer rate from WCV to a node is E units of energy per time slot. When the distance between a node and a WCV is d, recharged units of energy v(d) is:   0 if R1 < d v(d) = k if Rk+1 < d ≤ Rk , k = 1, 2, · · · , E − 1. (5)  E if 0 < d ≤ RE , where Rx = {d: E · τ (d) = x} and τ (d) denotes the energy transfer efficiency that is a non-increasing function of d (0 ≤ τ (d) ≤ 1). Let us define charging range as the maximum distance that a node receives at least one unit of energy from a WCV. Under our recharging mechanism of (5), the charging range is R1 . With magnetic resonance coupling, each WCV can charge at most u nodes at a time. • When there are less than u nodes within R1 , WCV selects all of them. • When there are more than u nodes within R1 , WCV randomly selects u nodes among them. Each WCV returns to the service station to recharge its own energy before its battery runs out. This means that all WCVs always have sufficient energy. C. The Process of Charging and Consuming Energy Let us denote by pc and pt the conditional probabilities that a WCV charges a node given the battery of the node is not full, and a node transmits a packet given the node has at least one unit of energy in its battery, respectively. The probabilities pc and pt are expressed as a function of the number of nodes n, the number of WCVs m, the maximum number of nodes charged by one WCV at a time u, the transmission range r and the charging range R1 (See Equations (6) and (7)). The detailed derivations are given in Appendix. Note that under the i. i. d mobility model, the distance between two nodes is independent of their previous locations. The probabilities pc and pt reflect this independence, and they are constant regardless of time. Let us denote by X the number of times that a node is charged during t time slots. The number X follows a binomial distribution, which approaches a Poisson distribution if t is sufficiently large and pc is sufficiently small. On average, a node is charged λ = pc t times. Let us denote by Y the interval between adjacent transmissions. The interval Y follows a geometric distribution, which is the discrete analogue of an exponential distribution. The mean duration, µ1 is p1t t . From these approximations, the number of units of energy in

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{ ( )n )}m u( pc = 1 − 1 − πR1 2 F (u − 2; n − 1, πR1 2 ) − 1 − 1 − F (u − 1; n, πR1 2 ) n [ ] { } 2 n−1 pt = q · 1 − 1 − (1 − q)πr

(6) (7)

∑k ( ) where F (k; n, p) = i=0 ni pi (1 − p)n−i is the cumulative distribution function (CDF) of the binomial distribution with parameters k, n and p.

From (8), we derive the active probability pon as follows: 1 (10) pon = 1 − π(0) = 1 − ( )i ∑L pc i=0 α(i, L) pt As L increases to infinity, pon (10) converges to: { pc γ pt if pc < pγt lim pon = (11) 1 if pc ≥ pγt L→∞ )i ( ∑E ∑j−1 where limL→∞ α(i, L) = E − j=1 k=1 β(k) = γ i . In other words, the value γ ppct is an upper bound of pon . D. Throughput with Wireless Power Transfer The throughput Λ(n) of (3) is determined by the active probability pon , which is a function of m, the number of WCVs, according to Equations (6) and (10). As m increases, each node has more chances to recharge its battery. Consequently, the active probability pon becomes higher and the corresponding Λ(n) increases. We summarize this relationship from the perspective of the scaling law using the following proposition: Proposition 2. Under two-phase routing, the throughput Λ(n) of the energy-constrained network scales with: ) ( ( m) m cmin(1, n ) , (12) Λ(n) = Θ min 1, n α(1,1)

where 0 < e− 4·a 1+α(1,1) < c < e− 4·a γ < 1 and a = 1 − π e− 4 (1−q) . Proof. The probability pon is a finite series with the component value ppct . As n increases, the ratio ppct becomes: ( ( )m )m 1 − 1 − nu 1 − 1 − nu pc )= → ( (13) π pt q·a q 1 − e− 4 (1−q) π·u

π·u

2 Due to the space limitation, we omit the derivation of (8). For interested readers, check http://ramo.yonsei.ac.kr/WPT.pdf.

0.05 0.045 Throughput Λ(n) (packet/time slot)

the battery can be designed as a Markov chain (Figure 2). The steady state probability that there are 0 ≤ x ≤ L units of energy in the battery, π(j), is described by2 : ( )x ( )x α(x, L) ppct α(x, L) µλ π(x) = ( )i = ∑ ( )i , (8) ∑L L pc λ α(i, L) α(i, L) i=0 i=0 µ pt )aj ( i ) ∏E ( ∑ ∑j−1 where α(i, L) = A(i) a1 ,··· ,aE 1 − β(k) , j=1 k=1 { } ∑E ∑E A(i) = (a1 , · · · , aE ) | j=1 aj = i and k=1 kak ≤ L and { 2 Ri −Ri+1 2 if i = 1, · · · , E − 1 2 (9) β(i) = R 2 R1 i if i = E R1 2

0.04 0.035

Simulation (m=Θ(n) and u=2) Analysis (m=Θ(n) and u=2) Simulation (m=Θ(1) and u=2) Analysis (m=Θ(1) and u=2) Simulation (m=Θ(n) and u=1) Analysis (m=Θ(n) and u=1) Simulation (m=Θ(1) and u=1) Analysis (m=Θ(1) and u=1)

0.03 0.025 0.02 0.015 0.01 0.005 0 100

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500 600 700 The number of nodes n

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Fig. 3. The throughput Λ(n) with various numbers of WCVs (m) and the maximum number of nodes simultaneously charged by one WCV (u).

ln(1− q·a ) − ln(1− q·a γ ) The ratio ppct is less than γ1 if m < ln 1− uγ ≈ n . u ( n) From Equations (10) and (11), we make lower-bound (L = 1) and upper-bound (L = ∞) of pon as follows:

α(1, 1) · ppct α(1, 1) pc pc · < ≤ pon < γ 1 + α(1, 1) pt 1 + α(1, 1) · ppct pt

(14)

Therefore, the scaling law of pon is equivalent to that of ppct ≈ (m) − ln(1− q·a γ ) m u , n · q·a = Θ n . On the other hand, if m ≥ n · u we obtain the following inequality: α(1, 1) ·

1 γ

1 + α(1, 1) ·

1 γ