Optimal incentive design for collaborative primary - IEEE Xplore

www.ietdl.org Published in IET Communications Received on 31st December 2012 Revised on 18th April 2013 Accepted on 26th May 2013 doi: 10.1049/iet-com.2012.0826

Special Issue: Cooperative Wireless and Mobile Communications ISSN 1751-8628

Optimal incentive design for collaborative primary– secondary transmission with primary data-rate constraints Insook Kim, Dongwoo Kim Department of Electrical Engineering and Computer Science, Hanyang University, Ansan, Kyeonggi 425-791, Korea E-mail: [email protected]

Abstract: In this study, the authors investigate a maximum incentive that a primary user (PU) can charge a secondary user (SU) on SU’s power consumption devoted to delivering its own signal. As a constraint, the PU also should achieve its data rate at a predefined level. Working as a half-duplex relay, SU hears PU’s signal in the first phase and sends its own signal superimposed on PU’s signal in the second phase. SU optimises power allocation between the two signals in the second phase, accounting for the expense-benefit trade-off in sending its own signal. Depending on SU’s receiver mode or the number of SU’s receiving nodes, SU’s decision on power allocation is differently optimised. The authors also provide optimal incentive for PU by optimal pricing for different occasions. The optimality is proved in this study. Since the optimality is purchased in practice when PU has perfect knowledge of the SU’s response on the PU-issuing price, the authors also propose an efficient interactive price-searching protocol between distributed PU and SU. Numerical investigations are given to verify and illustrate the optimality of the proposed price as well as the protocol.

1

Introduction

Among approaches of dynamic spectrum sharing, spectrum underlay allows primary and secondary users (PUs and SUs) to share the available spectrum simultaneously [1]. It requires that the spectrum access by SUs should not intervene in PU’s transmission. To protect PU, transmit power of SUs is usually restricted so that the interference caused by SUs to PU could be kept below a predetermined interference limit [2, 3]. Alternatively, the higher priority is given to PU by imposing a minimum rate constraint for PU to have a desired fraction of its achievable rate [4]. However, a certain level of the intervention is being allowed in the above approaches. On the other hand, the purpose of works in [5], [6] is not to degrade PU’s outage or data rate at all by letting SU act as a relay of PU. SU is allowed to transmit its own signal superimposed on PU’s signal only if it does not diminish PU’s achievable rate. In [7], the original data rate of PU is also guaranteed by incorporating multiple SUs as relays forming a virtual antenna array, where the signals from PU and SUs themselves occupy the respective time slots. In these schemes, due to the channel gain obtained by SUs working as relays [8], PU’s transmission is ‘genuinely’ protected and SU can find a room to send its data. One of the important issues open to evaluate these spectrum-sharing techniques is how to attract PU to participate in the collaboration. Especially for the collaboration using the superposition coding, we in this paper investigate an incentive model where PU charges SU IET Commun., 2013, Vol. 7, Iss. 17, pp. 1993–2003 doi: 10.1049/iet-com.2012.0826

on its power consumption devoted to delivering its signal. Pricing on secondary transmit power is also considered in [9] to allocate uplink transmit powers in a single-cell code division multiple access (CDMA) systems. Given the price issued by PU, SU has to decide whether to take it and then join the collaboration as a relay. If the SU takes the price, it also can deliver its own data. For SU, the payment is its cost of accessing the PU’s spectrum. However, PU regards the payment as the compensation of its effort to establish the collaboration. In this paper, we provide an optimal price that maximises PU’s income while keeping PU’s data rate at a given level and optimise the power allocated to SU’s signal, which also maximises SU’s benefit in the collaboration. We first deal with a simple network model that consists of a pair of transmit–receive PUs and a pair of transmit–receive SUs and then extend it to include multiple receiving SUs. Each of the receiving SUs is assumed to work in either of two different modes depending on its receiver complexity and radio environments: an interference cancellation (IC) mode and a non-IC (NIC) mode. The receiver status affects the incentive design. In an earlier conference version of this paper [10], an optimal pricing is provided for the simple network model only with an IC-mode SU. In the multiple-SU model, it is interesting to see that selecting a receiving SU with the greatest channel gain from the transmitting SU is not always but usually optimal in terms of PU’s incentive as well as SU’s revenue. We also provide a condition for which the greatest-channel selection is not optimal. The condition is expressed with 1993

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www.ietdl.org two channel gains: one is the greatest among IC-mode receivers and the other is the greatest among NIC-mode receivers. The condition is also used to determine an optimal price that maximises PU’s incentive. When increasing the number of receiving SUs, an opportunistic selection could increase a SU’s revenue but the increase cannot be taken by SU wholly. It is shared by PU and SU by the pricing mechanism, which is also an interesting point to be addressed in this paper. Numerical examples are given to verify and illustrate the optimality of the proposed incentive design. Since the PU determines the price and a SU then acts as a ‘price-follower’ with an optimal power allocation, the considered incentive model falls into a category of well-known ‘Stackelberg games’. PU optimises its pricing strategy based on the knowledge of the reaction of SU. However, since PU and SU are remote radio nodes, it is not clear how PU gathers the necessary knowledge in practice. In this paper, we also propose a price-searching protocol in which PU obtains the knowledge from SU’s response on the amount of power to be consumed. When the protocol terminates properly, PU finally finds an optimal price. Forced termination of the protocol also gives PU a sub-optimal price. The performance of the proposed protocol is investigated with numerical examples.

2 2.1

System model Collaboration model

We assume that there is a single frequency channel with a normalised bandwidth of 1 Hz. PU has a right to access the channel so that a primary transmitter (PT) is allocated radio resources such as a unit time-slot to send its own data to a primary receiver (PR). On the other hand, SUs are not allowed to use the time slot but are looking for an opportunity to access the channel under collaboration with PUs. Let Ps be transmit power of SU. If the collaboration is established, a time slot is divided into two equal-length subslots and a secondary transmitter (ST) works as a half-duplex relay for PU. In the first subslot, ST receives and decodes data from PU. In the second subslot, ST re-encodes and then sends the PU’s data superimposed on its own data. ST allocates transmit power αPs (0 ≤ α ≤ 1) and (1–α)Ps to its own and the PU’s data, respectively. For the successful collaboration, the following conditions should be satisfied: † SU decodes the PU’s data successfully. † The collaboration should not reduce the data rate of PU. † SU during the collaboration should pay λ to PU for a unit power it allocates to its own signal. † The benefit obtained by the collaboration should be positive for both PU and SU. λPsα is then an incentive for the collaboration of PU that allows SU to share the bandwidth. When λ is given to a SU, which decides whether it joins the collaboration and pays the price. Our main interest lies in investigating optimal λ that maximises PU’s incentive and its impact on SU’s revenue and power allocation. We consider two network scenarios, respectively, according to the formation of SUs (Fig. 1): (a) a single pair of an ST and a secondary receiver (SR) and (b) an ST with multiple SRs among which the ST opportunistically selects an SR. 1994 & The Institution of Engineering and Technology 2013

Fig. 1 System model a Single pair of an ST and an SR and b ST with multiple SRs among which the ST opportunistically selects an SR

2.2 Two-phase signal model during the collaboration Let x1 and x2 be complex symbol vectors for the messages transmitted by PT and ST, respectively. Let h0, h1, h2, h3 and h4 be the complex channel coefficients between PT– PR, PT–ST, PT–SR, ST–PR and ST–SR links, respectively. The channels are assumed to be static during the following two-phase transmission. In phase 1, PT broadcasts signal x1 with transmit power Pp, and ST and SR receive the signal. Also, PR overhears x1. Let ηST, ηSR and ηPR be the noises at ST, SR and PR, respectively. ST, SR and PR, respectively, receive y(1) ST = h1 Pp x1 + hST

(1)

y(1) ST = h2 Pp x1 + hSR

(2)

= h y(1) 0 Pp x1 + hPR ST

(3)

where superscript (1) denotes the phase 1. ST and SR decode x1. Moreover, ST relays x1 in phase 2 but SR keeps x1 in memory. PR does not process y(1) PR in phase 1. In phase 2, ST transmits superposed signal (1 − a)Ps x1 + aPs x2 . Let ηPR be the noise at destination PR. PR and SR receive the phase-2 signal y(2) PR = h3

(1 − a)Ps x1 + aPs x2 + hPR

(4)

y(2) SR = h4

(1 − a)Ps x1 + aPs x2 + hSR

(5)

respectively, where superscript (2) denotes the phase 2. We assume that the channel hi can be estimated at the respective receivers by preamble-aided channel estimation techniques. Thus PR knows the value of h3 as well as h0. Finally, PR uses maximal ratio combining to combine the signals from PT in phase 1 and from ST in phase 2, and decodes the message. SR subtracts x1 obtained in phase 1 IET Commun., 2013, Vol. 7, Iss. 17, pp. 1993–2003 doi: 10.1049/iet-com.2012.0826

www.ietdl.org from y(2) SR and finally obtains x2 from ySR = y(2) SR − h4 (1 − a)Ps x1 = h4 aPs x2 + hSR

Otherwise, SR should treat x1 contained in y(2) SR as a noise and the achievable data rate is then (6)

If SR fails to obtain x1 in phase 1, the above cancellation is not possible [If we assume optimal-combining between the two primary signals received in phase 1 and phase 2 at SR, respectively, there is more chance to cancel x1 in (6). This obviously will improve the performance of SU. However, to have an optimal power allocation considering the combining of primary signals at SR, ST needs further channel information between PT and SR as well as ST and SR. We leave this issue for future study] and it decodes x2 by treating the interfering x1 as a noise. We assume that ηST, ηPR and ηSR are the complex additive Gaussian noises with zero mean and variance σ 2. We also assume that the transmitted symbols have zero mean and unit variance. For notational convenience, let γ0, γ1, γ2, γ3 and γ4 denote |h0|2, |h1|2, |h2|2, |h3|2 and |h4|2, respectively. 2.3

Data rates and power constraint

When ST allocates power fraction α for its own signal, the primary data rate at PR is

g0 Pp (1 − a)g3 Ps 1 Rp (a) = log2 1 + 2 + s ag3 Ps + s2 2

(7)

Let R0 denote the achievable data rate between PT–PR without the collaboration, that is, R0 = log2(1 + γ0Pp/σ 2). The collaboration is possible only if Rp (a) ≥ R0

(8)

1 + g3 Ps /s2 a0 = 1 − g3 Ps /s2 2 g0 Pp /s2 + g0 Pp /s2 × 2 1 + g0 Pp /s2 + g0 Pp /s2

(11)

In the following, IC-SR and NIC-SR mean SR with IC and without IC, respectively. In addition, IC-SR or NIC-SR is sometimes used to denote a pair of ST and IC-SR or ST and NIC-SR, respectively, unless otherwise stated.

3 Optimal power allocation and pricing for a single pair of SUs Let SU have positive return ω per secondary data rate and δ denote an indicator of either IC or NIC, δ ∈ {IC, NIC}. The net revenue of the SU then can be modelled as Usd (a; l) = vRds (a) − laPs

(12)

In the following, ‘utility’ is interchangeably used to say SU’s revenue in (12) or PU’s incentive λαPs unless confusing. 3.1

IC-SR case

Given λ, ST with IC-SR finds optimal power allocation through solving max

0≤a≤a0

UsIC (a;

v ag4 Ps l) = log2 1 + − laPs 2 s2

(13)

We have the following optimal power allocation from [10].

R′p

IET Commun., 2013, Vol. 7, Iss. 17, pp. 1993–2003 doi: 10.1049/iet-com.2012.0826

0 ≤ l ≤ lL

lL ≤ l ≤ lU

(14)

l ≥ lU

where

≥ R0 . In this case,

1 aPs g4 = log2 1 + s2 2

⎧ a0 , ⎪ ⎪ ⎨ 1 v 1 s2 a∗ = − , ⎪ ⎪ Ps 2 ln 2 l g4 ⎩ 0,

(9)

is a maximally allocable fraction of power to SU’s own signal. If α0 ≤ 0, SU has no room in transmit power for sending its own signal without degrading the PU’s original data rate and then the collaboration cannot be established [It is also interesting to let the PU trade its data rate to increase its income in dynamic spectrum-sharing environments. In this paper, however, we concentrate on a cognitive radio network in which SU is allowed to access the PU’s spectrum only when guaranteeing the PU’s original data rate achievable without SUs.]. Otherwise, at least (1–α0)Ps is devoted to relaying primary signal x1 to meet the rate condition in (8). Let R′p denotes the data rate between PT–SR in phase 1, that ′ 2 is, Rp = (1/2) log2 1 + g2 Pp /s . To apply the IC, SR

RIC s (a)

1 ag4 Ps = log2 1 + (1 − a)g4 Ps + s2 2

Lemma 1: If λ is given, an optimal power allocation for the problem in (13) is

or equivalently, if α ≤ α0, where

requires good channel in γ2 such that the secondary data rate at SR is

RNIC s (a )

(10)

⎧ v 1 ⎪ ⎪ ⎨ lL = 2 ln 2 a P + s2 /g 0 s 4 v 1 ⎪ ⎪ ⎩ lU = 2 ln 2 s2 /g4

(15)

For PU, the payment by SU can be seen as the compensation of the collaboration. If SU optimises its utility as in Lemma 1, the incentive for PU by the collaboration is then given by ⎧ lPs a0 , ⎪ ⎪ ⎨ v s2 Up = − l, ⎪ ⎪ 2 ln 2 g4 ⎩ 0,

0 ≤ l ≤ lL

lL ≤ l ≤ lU

(16)

l ≥ lU

PU tries to optimise price λ in order to maximise its incentive, being aware that its decision will affect the power allocation made by SU. Let λ* denote an optimal price to maximise the primary utility Up in (16). Since Up is continuous, increasing in λ ≤ λL and decreasing in λL ≤ λ ≤ λU, λ* = λL 1995


www.ietdl.org maximises Up. Thus we have the following lemma without proof. Lemma 2: When SU allocates transmit power to maximise its own utility, an optimal pricing policy by PU is λ* = λL and the maximum incentive for PU is Up∗ = lL a0 Ps . 3.2

Given λ, the power optimisation problem for ST with NIC-SR becomes max

0≤a≤a0

v ag4 Ps − laPs l) = log2 1 + 2 (1 − a)g4 Ps + s2 (17)

UsNIC is convex, it has a global minimum at α such that

Since ∂UsNIC /(∂a) = 0. An optimal α* that maximises UsNIC (a)

then exists at the boundary of feasible region satisfying 0 ≤ α ≤ α0. Since U NIC(0) = 0, the maximum utility is achieved at α0 if UsNIC (a0 ) . 0. Otherwise, such SU cannot obtain positive utility and the collaboration fails. Thus, optimal power allocation for the problem in (17) is

a∗ =

a0 , 0,

0 ≤ l , lNIC l ≥ lNIC

(18)

where

1 + s2 /g4 Ps v

lNIC = log 2a0 Ps 2 (1 − a0 ) + s2 /g4 Ps

(19)

PU for NIC-SR then has the incentive UpNIC =

lPs a0 , 0,

0 ≤ l , lNIC l ≥ lNIC

(20)

It should be noted that maxl UpNIC does not have an optimal solution since UpNIC increases monotonically over an open interval of λ. Letting l˜ = λNIC–ε for certain small positive ε, PU has suboptimal price l˜ that can be applied practically. The effect of ε on PU’s incentive is discussed shortly in the following subsections. 3.3

l(k) l(j) r (j) a l − ar l(k) Ps (21) (k) (j) l −l −1 (k)

s2 l(k) l g4 = − 1 ar l(j) − (k) ar l(k) Ps l(k) − l(j) l − l(j)

v = (2 ln 2)

NIC-SR case

UsNIC (a;

† If αr(λ (i)) = αr(λ ( j )) and αr(λ (i)) ≠ 0, then αr(λ (i)) = αr(λ ( j )) = α0. † If PT has any two distinct responses such that 0 < αr(λ ( j )) ≠ αr(λ (k)) < α0, then these responses are from ST pairing with IC-SR and PT can obtain ω and γ4 and hence λL by putting αr(λ ( j )) and αr(λ (k)) into (14)

Price-searching protocol

Looking for the collaboration, PT initially broadcasts λ and γ0. Since ST can estimate γ3 and γ4 by exchanging signals with SR and overhearing certain signals from PR, respectively, ST then computes α0, λL, λU or λNIC with prior knowledge on Pp and σ 2. After that, ST determines the optimal power allocation α* as in either (14) or (18), and sends it to PT as an agreement for the collaboration if the power allocation is positive. Let us denote it αr(λ), which is sent to PT as a response. If PT is satisfied with αr(λ), the collaboration successfully starts. Otherwise, PT can try to send other prices to further increase its revenue. If PT knows Ps and σ 2 of SU a priori and the channels are static during the time span of interest, PT can learn about the optimal price from SU’s response as follows. Let λ (i)(i = 1, 2, …) be the ith price sent to ST and αr(λ (i)) be the corresponding response from ST. We assume that λ (i) ≠ λ ( j ) if i ≠ j. 1996 & The Institution of Engineering and Technology 2013

(22) If ST is paired by IC-SR, by choosing λ’s wisely so as to obtain the above conditions, PT can estimate the response αr from ST as well as optimal price λ* = λL by (21) and (22). PT then optimises its incentive. When ST is paired by NIC-SR, direct estimation on ω and γ4 as the above cannot be tried but using bisectional or incremental search on λNIC seems to work effectively. Once an optimal price is obtained, it is unique and does not change during the time span on which the channels are static. Table 1 provides a price-searching protocol in detail. It is based on bisectional search that guides interactive pricing trials of PU to find an optimal price of either λL or l˜ , which is referred to an interactive bisectional search protocol (IBSP) in this paper. In the protocol, λsmall and λbig denote a lower and an upper limit of the price, respectively, that are to be set by PU. If αr(λsmall ) = 0 at the initialisation step, IBSP stops and the collaboration fails. We assume that αr(λsmall ) = α0 and αr(λbig) = 0. Since IBSP starts with λleft ( = λsmall ) < λright( = λbig) and at every iteration i, λ (i) (λleft + λright)/2 successively replaces either λleft or λright, it converges geometrically and terminates within log2|(λbig– λsmall )/ε| iterations. The ‘count’ variable in IBSP logs the number of prices λ (i) that satisfy λL < λ (i) < λU for IC-SR. If count = 2, IBSP terminates with two prices between λL and λU and obviously obtain the optimal price λL from (21) and (22). When count is set to 1 by αr(λ (i)) > 0 and αr(λ (i)) ≠ α0 at Table 1 IBSP Initialise

Step 1 Step 2

Step 3

Let δ = NIC λ (0) = λsmall Try λ(0) and obtain αr(λ(0)) from SU Let λleft = λ (0) and λright = λbig Let count = 0, ε be a small positive real number, and c(1) = c(2) = 0 lleft + lright Let i = 1 and l(i) = 2 Try λ (i) and obtain αr(λ (i)) from SU If αr(λ (i)) = 0, then let λright = λ (i) Otherwise, if αr(λ(0)) ≠ αr(λ (i)), then let δ = IC, c(count) = αr(λ (i)), count = count + 1 and λright = λ (i); if αr(λ (0)) = αr(λ (i)), then λleft = λ (i). If δ = IC and count ≥ 2, stop the procedure and obtain λL with c(1) and c(2) from (21) and (22) lleft + lright Otherwise, let l(i+1) = 2 If |λ (i + 1)–λ (i)| < ε, then stop the procedure Otherwise, i = i + 1 and repeat Step 1


www.ietdl.org certain iteration i, this power response is certainly from ST with IC-SR. In this case, choosing the next price in the neighbourhood of λ (i) gives higher probability of increasing count to 2 since λL < λ (i) < λU. For example, choosing λ (i + 1) = βλleft + (1–β)λright with 0 < β < 0.5 accelerates the termination with IC-SR. Without meeting count = 2 until termination, IBSP can terminate with a near-optimal solution λleft as the following. Proposition 1: Let ε be a constant parameter that controls the termination of IBSP in Step 3. When the procedure terminates with the condition |λ (i + 1)–λ (i)| < ε, λleft is a near-optimal price in a sense that λL–2ε < λleft ≤ λL if SR is an IC-SR or λNIC– 2ε < λleft < λNIC if SR is an NIC-SR. Proof: Please see Appendix 1

□

For NIC-SR, when IBSP terminates, PU takes price λleft and λleft > λNIC–2ε by Proposition 1. If PU wants to have l˜ = lNIC − 1 by setting ε, choosing ε ≤ ε/2 ensures lleft ≥ l˜ , where λleft is closer to λNIC than l˜ . The bandwidth used to exchange these price/response information should be counted as a cost of implementing the collaboration and then the incentive given to PU is possibly reduced by this overhead. We, however, leave the details of overhead analysis for future study. Regarding to this issue, we provide numerical investigation about how many exchanges are required for PU to achieve an optimal price in Section 5. Forced termination of IBSP within a given finite number of iterations also can be considered in order to limit such exchanging overheads. In this case, compared with Proposition 1, the quality of solution λleft is bounded by |λleft–λL| ≤ |λright–λleft| or |λleft–λNIC| ≤ |λright– λleft|. It is also noted that the PU’s incentive that results from the forced termination is probably less than the optimal one but SU normally has greater revenue.

allocation. Let S denote an index set of SRs. For given λ, S can be decomposed into three disjoint sets S1 = i|a∗i = a0 , S3 = i|a∗i = 0

S2 = i|0 , a∗i , a0

and

(25)

It is noted that Sj ( j = 1, 2, 3) depends on the given price λ but d S = γ4,k, SRj gives the maximum secondary utility. After some straightforward arithmetic operations, we can obtain the following inequality for γ4,j and γ4,k, which makes NIC IC Us,k . Us,j ,

g4,j

g4,k Ps + s2 s2 (2/v)l a∗j −a0 Ps , ∗ 2 −1 aj Ps (1 − a0 )g4,k Ps + s2 (29)

g4,j

1 − 1 − a0 g4,j Ps /s2

and

In summary of this Section 4.2, we now have the following result without proof. Lemma 4: Suppose that IC- and NIC-SRs coexist and let k = arg maxi[S {g4,i }. If k is IC-SR, then selecting SRk with a∗k optimises the secondary utility. Otherwise, k is NIC-SR. Let j = arg maxi[SIC γ4,k is sufficient but not necessary for SRj in IC mode to be an optimal selection. That is, although γ4,k > γ4,j, SRj sometimes can be optimal. For SRk in NIC mode to be an optimal selection, the following is required

g4,k .

(28)

1

NIC SRj and SRk give the maximum utility in SIC 1 and S1 ,

over all i [ we have

v a∗j g4,j Ps ∗ − la∗j Ps aj ; l = log2 1 + 2 s2 (a) v a g Ps ≥ log2 1 + 0 4,j − la0 Ps 2 s2 a0 g4,j Ps v . log2 1 + − la0 Ps 2 (1 − a0 )g4,j Ps + s2 (b) v a0 g4,k Ps . log2 1 + − la0 Ps 2 (1 − a0 )g4,k Ps + s2 NIC = Us,k a∗j ; l

s2 g4,j , 1 − a0 Ps (27)

The second inequality in (27) is equivalent to γ4,jα0Ps/σ2 < α0/ (1–α0), where the left-hand side is signal-to-noise-ratio (SNR) of SRj and the RHS is not greater than unity when α0 ≤ 1/2. Thus, the second inequality in (27) implies that SNR of SRj should be less than 0 dB.

4.2.2 Case SIC 1 = ∅: If S2 = ∅ and all the SRs in S1 are NIC-SRs, SRi with the greatest γ4,i is an optimal selection, which is analogous to the case that all are IC-SRs stated in Section 4.1. Now we assume that S2 = ∅. Let j = arg maxi[S2 {g4,i } and k = arg maxi[SNIC {g4,i }, SRj 1

and SRk then give the maximum secondary utility in S2 and SNIC 1 , respectively. 1998 & The Institution of Engineering and Technology 2013

satisfy (27) or (29) according to j [ SIC or j [ S2 , 1 respectively, SRk with aNIC maximises the secondary k utility. If the conditions in (27) and (29) are not satisfied, SRj with a∗j maximises the secondary utility. 4.3

Optimal pricing

Let λL,k, λU,k and λNIC,k be the thresholds used by the selected SRk to determine the optimal power allocation in (14) and (18). Let l˜ k = lNIC,k − 1 for sufficiently small positive ε. Let j = arg max g4,i |di = IC and i[S

k = arg max g4,i |di = NIC

(30)

i[S

Then PU’s incentive is maximised when it selects SU that gives the following price l∗ = max lL,j , l˜ k

(31)

where λL,j = 0 or l˜ k = 0 if j or k is null, respectively. From the PU’s utility in (14) or (18), it is obvious that price λ* in (31) is optimal. However, this price cannot be always realised when the response of SU is made by choosing SR that maximises IET Commun., 2013, Vol. 7, Iss. 17, pp. 1993–2003 doi: 10.1049/iet-com.2012.0826

www.ietdl.org SU’s utility Us(λ). If lL,j , l˜ k , PU wants ST with SRk to establish the collaboration with l˜ k . However, when γ4,k < γ4, j, SRj always responds by Lemma 4 if lU ,j . l˜ k and thus PU should choose λL,j to maximise its incentive. When γ4,k > γ4, j but γ4,k fails to satisfy (29), SRj still becomes a candidate that gives incentive (v/(2 ln 2)) − l˜ k (s2 /g4,j ) to PU. In this case, PU reluctantly decreases the price until γ4,k obtains the condition in (29) as equality. Let Δ, if any, be such a decrement in price from l˜ k . At the equality condition, SRj gives (v/(2 ln 2)) − (l˜ k − D)(s2 /g4,j ) and SRk gives

˜lk − D Ps a0 . PU can choose the greater of two incentives. As a result, PU cannot always obtain the ‘greedy’ incentive expected by choosing the price in (31) unless it can fix the selection of SU. It should be noted again that such Δ does not always exist. If lL,j ≥ lNIC,k = l˜ k − 1, then PU selects λL,j and achieves the desired incentive since all NIC-SRs cannot join the collaboration. Thus, if λL,j ≥ λNIC, then λL,j is optimal. If l˜ ≤ lL,j , lNIC , λL,j is a suboptimal choice. In both of the above cases, PU achieves the desired incentive λL,jα0Ps. The following lemma formally states optimal price when λL,j < λNIC. Lemma 5: Let j and k be defined as in (30). If λL, j < λNIC, the following λ* is either optimal or suboptimal, which gives incentive λ*α0Ps to PU g4,k Ps + s2 s2 ln 2 if g4,j ≤ log2 −1 a0 Ps 1 − a0 g4,k Ps + s2 g4,k Ps + s2 s2 ln 2 if log2 −1 a0 Ps 1 − a0 g4,k Ps + s2 l∗ = ⎪ ⎪ g4,k ⎪ ⎪ ⎪ , g4,j , ⎪ 2 ⎪ (1 − a ) g ⎪ 0 4,k Ps /s + 1 ⎪ ⎪ ⎪ g ⎪ 4,k ⎪ , ⎩ lL,j , if g4,j ≥ (1 − a0 )g4,k Ps /s2 + 1 ⎧ ⎪ ⎪ ⎪ l˜ k , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨l 1 ,

Table 2 DIBSP Initialise

Step 1 Step 2

Step 3

Step 4

Step 5

Let δ = NIC λ (0)=λsmall Try λ (0) and obtain αr(λ (0)) from SU Let λleft = λ (0) and λright = λbig Let ε be a small positive real number lleft + lright Let i = 1 and l(i) = 2 Try λ (i) and obtain αr(λ (i)) from SU If αr(λ (i)) = 0, then let λright = λ (i) Otherwise, if αr(λ (0)) ≠ αr(λ (i)), then if δ = NIC, then let λnextleft = λ (i) and λnextright = λright, let δ = IC, and λright = λ (i); if αr(λ (0)) = αr(λ (i)), then λleft = λ (i) lleft + lright Let l(i+1) = 2 If |λ (i + 1) − λ (i)| < ε , then stop phase 1; Otherwise, let i = i + 1 and repeat Step 1 If phase 1 stops and δ = IC, then continue Step 4; Otherwise, stop the procedure lnextleft + lnextright Let, l(i+1) = 2 and let i = i + 1 If |λ (i + 1)–λ(i)| < ε, then stop phase 2; Otherwise, try λ (i) and obtain αr(λ (i)) from SU If αr(λ (i)) = 0, then let λnextright = λ (i); Otherwise, let λnextleft = λ (i) Repeat Step 4.

possibly greater than the price currently found. DIBSP introduces two new variables λnextleft and λnextright for the second-phase search of Step 4 and Step 5 in Table 2. When DIBSP terminates, PU can choose either of λleft and λnextleft, which provides the greater incentive among λleftαr(λleft)Ps and λnextleftαr(λnextleft)Ps.

(32) 1 = l − is a where l 1 is a suboptimal price and furthermore l root that makes (29) as an equality. Proof: Please see Appendix 6.

□

If lL,j , l˜ k , from Lemma 5, PU cannot always achieve the greedy incentive l˜ k a0 Ps that is possibly obtained when PU can select the SR. In this case, PU’s incentive is generally 1 , l˜ k . reduced since lL,j , l 4.4

5

Numerical results

For numerical investigation in this section, we assume that Pp/σ 2 = Ps/σ 2 = 10 dB. The secondary income per data rate ω is set to ω = 1.

Modified IBSP

When multiple SRs exist, PT does not know which SR is paired to the present ST’s response for given λ. Especially, although PT can detect αr(λ) from ST paired with IC-SR when 0 < αr(λ) < α0, PT cannot try to apply (21) and (22) directly since it is unclear that such responses are from the same SR. However, IBSP still works if some modifications are made. We propose a dual implementation of IBSP (DIBSP) as in Table 2. DIBSP consists of two phases: phase 1 to Step 3 and phase 2 to Step 5. In phase 1, DIBSP assumes that the responses are from IC-SRs. In phase 2, it tries to look for responses from NIC-SRs. Compared with IBSP, the count variable is removed and the estimation to obtain λL is not processed. When Step 3 of DIBSP terminates with δ IC, then the procedure restarts to seek, if any, l˜ k that is IET Commun., 2013, Vol. 7, Iss. 17, pp. 1993–2003 doi: 10.1049/iet-com.2012.0826

Fig. 2 Optimality of prices λL and λNIC for a single pair of ST and SR; γ0 = –19 dB, γ3 = 0 dB, γ4 = –10 dB 1999


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Fig. 3 CDF of the number of iterations needed to terminate IBSP

5.1

Single pair of SUs

Fig. 2 illustrates the optimality of the proposed prices in Lemma 2 and l˜ shown below (20) for IC- and NIC-SR, respectively. In this figure, the utility a function of PU and SUs are plotted as a function of price λ, respectively. We assume that γ0 = –19 dB, γ1 = –7 dB, γ3 = 0 dB, γ4 = – 10 dB. Up is optimised at λL = 0.0387 as shown in the figure. Up(λL) = 0.3348 and UsIC (lL ) = 0.1152. For NIC-SR, we assume the same channel to IC-SR but IC is disabled. The same convention is used for NIC-SR throughout this section. UsNIC (l) increases until λ < λNIC = 0.0473. Thus l˜ = lNIC − 1 is suboptimal for small positive ε. Compared with IC-SR, UsIC (l) . UsNIC (l) for 0 ≤ λ < λU = 0.0721. UsIC (l) − UsNIC (l) can be regarded as a positive gain that is obtained when NIC-SR employs the IC function on its receiver, which reduces PU’s incentive. Thus IC-SR looks to have a better position in the negotiation than NIC-SR as the price-follower along with the price-leading PU. Fig. 3 reports how fast IBSP terminates. In the figure, cumulative distributions of the number of iterations when

Fig. 4 Performance of forced termination of IBSP 2000 & The Institution of Engineering and Technology 2013

IBSP terminates are provided. We assume that γ0 = –19 dB, γ1 = –7 dB and γ3 = –10 dB. For simulation, 100 000 γ4’s are randomly generated between − 20 and 0 dB. We have set λsmall = 10–1λL or 10–1λNIC and λsmall = 10λL or 10λNIC, respectively. The termination condition is set to ε 10–6 and 10–3, respectively. For the number of iterations less than three, the probability is 0. For NIC-SR and IC-SR terminated with count ≤ 1, the number of iterations is about 20 and 10, respectively. The wise search of optimal price based on (21) and (22) in IBSP reduces the number of iterations to five for more than 71.42 and 71.31% of the simulated cases for ε = 10–6 and 10–3, respectively. In Fig. 3, β is a parameter used to search the next price, which is introduced in Section 3.3 to accelerate the termination of IBSP with IC-SR. When IC-SR is found, letting β = 0.1 further reduces the number of iterations. Fig. 4 reports how forced termination affects the optimality of IBSP. When the number of iterations reaches ‘iter’, IBSP is forced to terminate. We have set iter = 5 and 8, respectively. After termination, Up and Us are computed with λleft and plotted in the figure. Up∗ with an optimal price is also given as a reference. The utilities are simulated as a function of γ4. For the simulation, 100 000 α0’s are randomly generated between 0.2 and 0.8, and average utilities are reported. For IC-SR, iter = 5 returns the utility close to Up∗ when γ4 > –10 dB. iter = 8 gives PU’s incentive very close to Up∗ over the whole range of tested γ4. For NIC-SR, iter = 5 cannot find a good price compared with the case of IC-SR. More iterations than 5 as iter = 8 are needed to find a near-optimal price. By letting iter = 5, IBSP terminates earlier than iter = 8, which gives better Us and worse Up. Compared with IC-SR, NIC-SR seems to have more benefit from the forced termination. 5.2

SUs from multiple SRs

Fig. 5 illustrates the optimality of the proposed price in IC Lemma 5 when multiple SRs exist. The utility of PU, Us,j NIC and Us,k are plotted as a function of λ, where indices j and k are selected as in (30). We assume γ4,j = –10 dB with which the RHS of inequality in (29) is also given for comparison. The other channel powers are common to SRj and SRk, and assumed to be γ0 = –19 dB, γ1 = –7 dB and γ3 = –10 dB. Since lL,j = 0.0411 , l˜ k = 0.0543, PU wants to fix k as the selection and tries to set l˜ k as the

Fig. 5 Optimality of the proposed price when multiple SRs exist; γ0 = –19 dB, γ1 = –7 dB, γ3 = –10 dB, γ4j = –10 dB IET Commun., 2013, Vol. 7, Iss. 17, pp. 1993–2003 doi: 10.1049/iet-com.2012.0826

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Fig. 6 PU’s greedy incentive against price-determining incentive when multiple SRs exist;

Fig. 7 Performance of forced DIBSP as the degradation of PU’s incentive from its optimal value; iter = 5 and 8 are tested, respectively

a γ3 = –15 dB, b γ3 = 0 dB

IC ˜ NIC ˜ price. However, at l˜ k , since Us,j lk . Us,k lk ST selects IC-SR

j to maximise its own revenue and responds with aj l˜ k = 0.3274. Since IC-SRj is selected, λL,j is better than l˜ k for both PU and SU. If PU decreases the price from IC NIC l˜ k , all of Up, Us,j and Us,k increase as in the figure. Furthermore, RHS of (29) also increases. During the = 0.0460 where γ4,j is equal to RHS of increment, at l (29), ST could

change

its selection

from IC-SRj to NIC-SRk IC NIC is either 0.2617 or 0.3476 since Us,j l = Us,k l . Up l if either of SRj and SRk is selected, respectively. Since PU has the greater incentive when NIC-SRk is selected with , l 1 = l − l 1 for small positive 1 is a suboptimal price for PU. Fig. 6 provides optimal primary and secondary utilities (Up∗ and Us∗ , respectively) as a function of the number of SRs. Upmax denotes PU’s greedy incentive if it can select and fix an SR and then sets the price as in (31). For the simulation, among N SRs, a half of the SRs are assumed IC-SRs and the other half are NIC-SRs. SRs are randomly generated to have γ4,i between − 40 and 0 dB. The utilities are averaged over 10 000 experiments. Two cases γ3 = –15 and 0 dB are tested and the results are shown in Figs. 6a and b, respectively. In both the cases, when the number of SRs increases the utilities log-likely. Us∗ increases more than Up∗ since increasing number of SRs has a similar effect as increasing γ4. Compared with case γ3 = –15 dB, case γ3 = 0 dB shows the greater gap between Up∗ and Upmax , which implies that PU’s desired selection based on (31) does not coincide with SU’s response more frequently. For case γ3 = –15, Up∗ and Upmax are well matched. This is because more probably λNIC,k > λL,j where j and k are defined in (30) if γ3 is big and hence so is α0. If λNIC,k > λL.j, Lemma 5 is applied and PU has less chance to achieve the greedy incentive. Fig. 7 provides the performance of forced DIBSP. DIBSP is forced to terminate when the number of iterations reaches iter. We have tested iter = 5 and 8, respectively. After termination, a price that gives the greater among αr(λleft)λleft and αr(λnextleft)λnextleft is selected, and the corresponding Up and Us are plotted in the figure. Letting ε = 10–6, DIBSA is IET Commun., 2013, Vol. 7, Iss. 17, pp. 1993–2003 doi: 10.1049/iet-com.2012.0826

terminated when the number of iterations is about 40 without the forced termination, the result of which is denoted by Up(ε = 10–6) in the figure. Up∗ and Us∗ with an optimal price are also given as references. The utilities are simulated as a function of the number of users. For the simulation, α0’s are randomly generated between 0.2 and 0.8 and γ4,i’s are between − 20 and 0 dB. The points shown in the figure are average results over 100 000 samples. When iter 5, Up is much lower than Up∗ but SU enjoys greater revenue Us than Us∗ . When iter = 8, Up relatively increases to about 90% of Up∗ from the case of iter = 5. As a result of increasing incentive for PU, SU’s revenue diminishes. The resulting utilities are mostly increasing when the number of SRs increases except that Up with iter 5 decreases. This means that iter = 5 is not sufficient to exploit the diversity effect because of multiple SRs. The incentive gap between Up(ε = 10–6) and the result from iter = 8 is a bound that DIBSP can achieve with further iterations after 8. When the number of SRs increases, this gap also increases slightly.

6

Conclusions

We have provided an optimal incentive for PU by optimal pricing. This incentive attracts that PU allows SU to access its spectrum. Since SU is allowed as a relay, PU’s data rate achievable without the SU is not degraded. Although SU has two types of burden: paying money for using its transmit power to send its own data and relaying PU’s signal safely, SU joins the collaboration if it has positive revenue, which is also a chance to enlarge the sum of the utilities of PU and SU. In the protocol proposed for a practical purpose in this paper, PU searches the price by an interactive protocol but forced termination due to the bandwidth limitation results in a sub-optimal price. This limitation is shown more beneficial to SU, especially more to NIC-SR. The large number of SRs also increases the negotiation power of SU. We have provided analytical and numerical results for these new factors that should be investigated in designing incentives in spectrum-sharing environments. 2001


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Acknowledgments

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 (MEST-2010-0013544).

8

References

1 Hossain, E., Bhargava, V.K.: ‘Cognitive wireless communication networks’ (Springer, 2007, edited volume) 2 Li, T., Jayaweera, S.K.: ‘Dynamic spectrum leasing in cognitive radio networks via primary–secondary user power control games’, IEEE Trans. Wireless Commun., 2009, 8, (6), pp. 3300–3310 3 Song, Y.-K., Kim, D.: ‘Interactive admission and power control protocol for cooperative spectrum underlay in distributed cognitive radio networks’, IEICE Trans. Commun., 2011, E94-B, (10), pp. 2785–2795 4 Beyaztas, Z., Pandharipande, A., Gesbert, D.: ‘Optimum power allocation in a hierarchical spectrum sharing scheme’. Proc. IEEE ICC, 2008, pp. 97–101 5 Han, Y., Pandharipande, A., Ping, S.H.: ‘Cooperative decode-and-forwards relaying for secondary spectrum access’, IEEE Trans. Wirel. Commun., 2009, 8, (10), pp. 4945–4950 6 Shin, E.-H., Kim, D.: ‘Time and power allocation for collaborative primary–secondary transmission using superposition coding’, IEEE Commun. Lett., 2011, 15, (2), pp. 196–198 7 Simeone, O., Stanojev, I., Savazzi, S., Bar-Ness, Y., Spagnolini, U., Pickholtz, R.: ‘Spectrum leasing to cooperating secondary ad hoc networks’, IEEE J. Sel. Areas Commun., 2008, 26, (1), pp. 203–213 8 Lee, I.-H., Kim, D.: ‘Probability of SNR gain by dual-hop relaying over single-hop transmission in SISO Rayleigh fading channels’, IEEE Commun. Lett., 2008, 12, (10), pp. 734–736 9 Yu, H., Gao, L., Li, Z., Wang, X., Hossain, E.: ‘Pricing for uplink power control in cognitive radio networks’, IEEE Trans. Veh. Technol., 2010, 59, (4), pp. 1769–1778 10 Kim, I., Kim, D.: ‘Pricing and optimal power allocation in collaborative primary-secondary transmission using superposition coding’. Proc. IEEE TENCON, November 2010 11 Luo, Z.-Q., Yu, W.: ‘An introduction to convex optimization for communications and signal processing’, IEEE J. Sel. Areas Commun., 2006, 24, (8), pp. 1426–1438

9

Appendix

9.1

which also gives λL–2ε < λleft ≤ λL. When SR is NIC-SR, the proof is analogous to the above case of IC-SR, being noted that the update of λleft keeps λleft < λNIC for NIC-SR. 9.2

For given λ, by the definition of S1 and from S2 and the assumption that each SRi is IC-SR, we have the following three inequalities with respect to λ

v 1 ,l 2 ln 2 a0 Ps + s2 /g4,i2

lU ,i2 =

v 1 .l 2 ln 2 s2 /g4,i2

and

s2 , g4,i2 (v/2 ln 2)(1/l) s2 , ≤ g4,i1 ((v)/2 ln 2)(1/l) − a0 Ps

(35)

which completes the proof. 9.3

Proof of Proposition 3

First we consider the case of S1 = ∅ and S2 = ∅. For i [ S1 , from the definition of S1 , a∗i = a0 . Since the utility of each SRi, i [ S1 , Us,i is strictly increasing over γ4,i, SRk with k = arg maxi[S1 {g4,i } provides the maximum utility among SRi, i [ S1 . Now we consider the other case of S1 = ∅ and S2 = ∅. For i [ S2 , an optimal power allocation is 1 v 1 s2 = − Ps 2 ln 2 l g4,i

The utility function is then

v v 1 1 v + log2 g4,i Us,i = log2 2 2 2 ln 2 s l 2

v s2 +l − 2 ln 2 g4,i

(36)

Since

g4,i . gU =

which leads to λL–2ε < λleft ≤ λL since λleft ≤ λL. On the other hand, if λ (i) λright λ (i) = λright

s2 v/2 ln 2 (1/l)

for i [ S2 by Proposition 1, ∂Us,i v 1 s2 = − l 2 ∂g4,i 2 ln 2 g4,i g4,i

l

2002 & The Institution of Engineering and Technology 2013

lL,i2 =

(33)

, 21

, 21

v 1 ≥ l, 2 ln 2 a0 Ps + s2 /g4,i1

a∗i

We assume that SR is IC-SR. Since λleft is updated by λ (i) only when αr(λ (0)) = αr(λ (i)), which keeps αr(λleft) = αr(λ (0)) α0, λleft ≤ λL. On the other hand, since λright is updated by λ (i) only when αr(λ (i)) = 0 or αr(λ (i)) ≠ αr(λ (0)), which keeps αr(λright) < α0, λright > λL. Consequently, λleft ≤ λL < λright. Suppose that IBSA terminates with |λ (i + 1)–λ(i)| < ε. If λ (i) λleft,

left − lL ≤ lright − lleft = 2lright − 2l(i+1) = 2l(i+1) − l(i)

lL,i1 =

From the inequalities, we have


|lleft − lL | ≤ lright − lleft = 2l(i+1) − 2lleft = 2l(i+1) − l(i)


(34)

(v/(2 ln 2))g4,i − ls2 = .0

2 g4,i

(37)


www.ietdl.org Thus, Us,i is also a strictly increasing function over γ4,i for i [ S2 . Therefore, SRk with k = arg maxi[S2 {g4,i } maximises the secondary utility if S1 = ∅ and S2 = ∅ and, which completes the proof. 9.4

lU ,j =

Us,i1 l; g4,i1 ≤ Us,i2 l; g4,i2

a∗i g4,i Ps v log2 1 + 2 2 2 − la∗i2 Ps 2 s a∗i2 g4,i1 Ps v , log2 1 + − la∗i2 Ps 2 s2 =

Proof of Lemma 3

For given λ, let Let k = arg maxi[S {g4,i }. i1 = arg maxi[S1 {g4,i } and i2 = arg maxi[S2 {g4,i }. If S1 is not empty, k [ S1 since lL,k ≥ lL,i1 , which means that k = i1 and k is an optimal selection by Propositions 2 and 3. If S1 is empty but S2 is not empty, then k [ S2 since lU ,k ≥ lU ,i2 , which means that k = i2 and k is an optimal selection by Proposition 2. If both S1 and S2 are empty, then any selection returns zero utility and k is also an optimal selection, which completes the proof. Proof of Lemma 5

When lL,j , l˜ k , we have the following three disjoint cases: 1. lU ,j ≤ l˜ k , NIC IC 2. lU ,j . l˜ k and Us,k (lL,j ) . Us,j (lL,j ),

3. lU ,j . l˜ k and

NIC Us,k (lL,j )

≤

IC Us,j (lL,j ).

IC NIC (l) and Us,k (l) are linearly decreasing Recall that Us,j

over lL,j ≤ l ≤ min{lU,j , l˜ k }. Then there exists at most one IC NIC λ such that Us,j (l) = Us,k (l) on the above interval if NIC IC Us,k (lL,j ) = Us,j (lL,j ) or lU,j = l˜ + 1.


v g4,j 2 ln 2 s2

and

g Ps + s 2 v 4,k l˜ k = log −1 2a0 Ps 2 1 − a0 g4,k Ps + s2 case (1) lU ,j ≤ l˜ k is equivalent to g4,k Ps + s2 s2 ln 2 g4,j ≤ log2 −1 a0 Ps 1 − a0 g4,k Ps + s2

(39)

(38)

Since a∗i1 ≥ a∗i2 , a∗i2 is also a feasible power allocation for i1 and gives greater utility than a∗i1 to i1 , which makes a

contradiction. Thus Us,i1 l; g4,i1 . Us,i2 l; g4,i2 .

9.6

For case (1), l∗ = l˜ k

When l˜ k is given by PU, if lU ,j ≤ l˜ k SRk always responds

NIC ˜ IC ˜ (lk ) . Us,j lk = 0. And hence l∗ = l˜ k . Since since Us,k


For given λ, let i1 [ S1 and i2 [ S2 and let us denote a∗i1 and a∗i2 as the corresponding optimal power allocation. Then g4,i1 . g4,i2 by Proposition 2. Suppose that

Us,i1 l; g4,i1 ≤ Us,i2 l; g4,i2 . Then

9.5

9.7

9.8

1 For case (2), l∗ = l

NIC IC If lU ,j . l˜ k and Us,k (lL,j ) . Us,j (lL,j ), there exists l

NIC IC = Us,j l . If between λL,j and l˜ k such that Us,k l NIC IC ≤ l , l˜ k , then Us,k (l) ≤ Us,j (l) (the equality holds l ) and SRj responds, in which PU’s optimal strategy only at l IC NIC , then Us,j . If lL,j , l ≤ l (l) ≤ Us,k (l) (the is taking l ) and SRk always responds, in equality holds only at l . Combining which PU’s optimal strategy also takes l NIC IC conditions lU ,j . l˜ k and Us,k (lL,j ) . Us,j (lL,j ), we have g4,k Ps + s2 s2 ln 2 log2 a0 Ps 1 − a0 g4,k Ps + s2 g 4,k − 1 , g4,j , (40) 1 − a0 g4,k Ps /s2 + 1

0 Ps or , PU obtains the incentive either la l according to which of SRk or SRj (v/(2 ln 2)) − (s2 /g4,j )l 0 Ps ≥ (v/(2 ln 2)) − (s2 /g4,j )l if responds. Since la 1 = l − 1 by taking 0 , 1 , l − lL,j is a ≥ lL,j , l l At

suboptimal price to which SRk responds. 9.9

For case (3), λ* = λL,

j

NIC IC IC NIC If lU ,j . l˜ k and Us,k (lL,j ) ≤ Us,j (lL,j ), Us,j (l) ≥ Us,k (l), ˜ always holds over lL,j ≤ l ≤ lk (the equality holds only at NIC IC (lL,j ) = Us,j (lL,j )). In this case, SRj always λL,j if Us,k

responds and PU gets incentive la∗j (l)Ps = v/(2 ln 2)− (s2 /g4,j )l , la0 Ps for lL,j ≤ l ≤ l˜ k . Thus λL,j maximises PU’s incentive for case (3), which is equivalent to

g4,j ≥ (g4,k / 1 − a0 g4,k Ps /s2 + 1).

2003
