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Delay Performance of a Broadcast Spectrum Sharing Network in Nakagami-m Fading Fahd Ahmed Khan, Student Member, IEEE, Kamel Tourki, Member, IEEE, Mohamed-Slim Alouini, Fellow, IEEE, and Khalid A. Qaraqe, Senior Member, IEEE
Abstract—In this paper, we analyze the delay performance of a point-to-multipoint secondary network (P2M-SN), which is concurrently sharing the spectrum with a point-to-multipoint primary network (P2M-PN). The channel is assumed to be independent but not identically distributed (i.n.i.d.) and has Nakagami-m fading. A constraint on the peak transmit power of the secondary-user transmitter (SU-Tx) is considered, in addition to the peak interference power constraint. The SU-Tx is assumed to be equipped with a buffer and is modeled using the M/G/1 queueing model. The performance of this system is analyzed for two scenarios: 1) P2M-SN does not experience interference from the primary network (denoted by P2M-SN-NI), and 2) P2M-SN does experience interference from the primary network (denoted by P2M-SN-WI). The performance of both P2M-SN-NI and P2M-SN-WI is analyzed in terms of the packet transmission time, and the closed-form cumulative density function (cdf) of the packet transmission time is derived for both scenarios. Furthermore, by utilizing the concept of timeout, an exact closed-form expression for the outage probability of the P2M-SN-NI is obtained. In addition, an accurate approximation for the outage probability of the P2M-SN-WI is also derived. Furthermore, for the P2M-SN-NI, the analytic expressions for the total average waiting time (TAW-time) of packets and the average number of packets waiting in the buffer of the SU-Tx are also derived. Numerical simulations are also performed to validate the derived analytical results. Index Terms—Cognitive radio, delay performance analysis, multi-user, outage performance, queuing theory, underlay spectrum sharing.
I. I NTRODUCTION
T
HE DEMAND for high-data-rate wireless communication services is growing. Achieving a high data rate requires more wireless spectra. However, the wireless spectrum has now become scarce as most of it has already been allocated for various services. Recent measurement studies have shown that
Manuscript received July 17, 2012; accepted July 3, 2013. Date of publication July 17, 2013; date of current version March 14, 2014. This work was supported in part by King Abdullah University of Science and Technology and in part by the Qatar National Research Fund through National Priorities Research Program under Grant NPRP 5-250-2-087. This paper was presented in part at the IEEE Symposium on New Frontiers in Dynamic Spectrum Access Networks (DySPAN 2012), Bellevue, WA, USA, October 16–19, 2012. The review of this paper was coordinated by Dr. D. Zhao. F. A. Khan and M.-S. Alouini are with the Computer, Electrical, and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia (e-mail:
[email protected];
[email protected]). K. Tourki and K. A. Qaraqe are with the Electrical and Computer Engineering Program, Texas A&M University at Qatar, 23874 Doha, Qatar (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2013.2273621
the wireless spectrum is greatly underutilized [2]. By utilizing the spectrum efficiently, it is possible to fulfill the demand for the increasing number of wireless services. As a consequence, cognitive radio has been proposed to improve the utilization of the spectrum [3], [4]. In cognitive radio, the spectrum utilization is improved through spectrum sharing in which the primary networks share its spectrum with a secondary network [4]. Various protocols have been proposed for spectrum sharing. One such protocol is where the secondary network uses the spectrum only when the primary network is not using the spectrum. This approach requires robust spectrum sensing algorithms that sense the spectrum perfectly (for details, see [5] and references therein). However, ideal spectrum sensing cannot be achieved in practice, and in the case of missed detection, the primary network experiences severe interference. Another approach of spectrum sharing is the underlay approach in which the secondary network is allowed to transmit concurrently with a primary network using the spectrum of the primary network if it does not cause harmful interference to the primary receiver. Thus, by limiting the interference from the secondary network, an acceptable level of performance of the primary network can be guaranteed, and the secondary network can also communicate and improve the utilization of the spectrum. In addition, it is essential that the secondary network satisfies a certain quality-of-service (QoS) requirement. The demand for delay-sensitive wireless services, such as voice over Internet Protocol services and video streaming services, is increasing [6]. These services are required to satisfy a delay QoS. However, due to the mobility of the user and the various impairments caused by the wireless channel, such as multipath fading and shadowing, this delay QoS is not always satisfied, and in the cognitive setting, satisfying the delay QoS becomes even more challenging due to the spectrum sharing constraints. Therefore, it is essential to characterize the delay QoS performance of a cognitive network and use this characterization to improve the network performance. Recently, much research is being done on cross-layer design and optimization of delay-sensitive cognitive networks. The concept of effective capacity that was proposed in 2003 has been utilized to characterize and optimize the performance of the cognitive network. Effective capacity can be interpreted as the maximum constant arrival rate that can be provided by the channel while the delay QoS requirement is satisfied [7]. An optimal power-and-rate-allocation scheme to maximize the effective capacity of the underlay cognitive network under average interference constraint was proposed in [8] and
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KHAN et al.: DELAY PERFORMANCE OF BROADCAST SPECTRUM SHARING NETWORK IN NAKAGAMI-m FADING
[9]. In [10], the authors considered the peak transmit power constraint in addition to the average interference constraint to maximize the effective capacity and derived the powerallocation policy. In [11], another power-allocation policy was derived with the objective of maximizing the effective capacity of an underlay spectrum sharing network, considering only a peak interference power constraint. In [12], the interference power from the primary user was also taken into account while deriving the power-allocation policy to maximize the effective capacity. With regard to the analysis of cognitive networks with a delay QoS requirement, a different approach has been adopted in [13]–[15], where the performance of a cognitive network under a delay QoS constraint is investigated based on the packet transmission time. In [13]–[15], the secondary-user transmitter (SU-Tx) is assumed to be equipped with a buffer and is modeled using an M/G/1 queue model. It was shown that the packet transmission time depends on the fading and the power-allocation policy employed by the SU-Tx. A powerallocation policy was derived considering only a peak interference power constraint. Based on this power-allocation policy, the probability density function (pdf) and the cumulative density function (cdf) of the packet transmission time were derived. Using the pdf and the cdf of the packet transmission time, performance measures (obtained from queueing theory), such as the total average waiting time (TAW-time) of packets in the system and the channel utilization, were analyzed [16]. In addition, the probability of outage performance was also analyzed, where an outage is declared if the packet is delayed, i.e., the packet transmission time exceeds a certain predefined threshold. In [13] and [14], the authors considered a point-to-point primary network sharing the spectrum with a point-to-point secondary network (P2P-SN), where the channel was assumed to have independent and identically distributed (i.i.d.) Rayleigh fading and Nakagami-m fading, respectively. In [15], the authors extended their previous work in [13], by considering a point-to-multipoint primary network (P2M-PN) sharing the spectrum with a point-to-multipoint secondary network (P2M-SN) for an i.i.d. Rayleigh fading channel. In all these works, only a peak interference power constraint was considered. However, in reality, there is a limit on the maximum power at which a node can transmit, and the power-allocation policy has to take it into account. Furthermore, in these previous works, the interference from the primary network to the secondary network was ignored; thus, the analysis does not depict a realistic scenario. In this paper, we analyze the performance (based on the packet transmission time) of the P2M-SN, where the channels have independent but not identically distributed (i.n.i.d.) Nakagami-m fading. This channel fading model is more generic and includes the channel fading models considered in [13]–[15] as its subcases. Furthermore, we also consider a peak transmit power constraint at the SU-Tx in addition to the peak interference power constraint and derive a power-allocation policy satisfying these constraints. Moreover, the performance of the secondary network degrades due to the interference from the primary network [17]–[19]. Therefore, in this paper, we also
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Fig. 1. System model diagram.
analyze the impact on the performance of the P2M-SN due to interference from the P2M-PN.1 Specifically, the exact closed-form expression of the cdf of the packet transmission time is derived for both the P2MSN-WI and P2M-SN-NI. In addition, the outage probability expressions are also derived.2 For deriving the outage probability for the P2M-SN-NI, in [15], it is assumed that the transmission times are independent. This assumption is not valid as the power-allocation scheme, which was employed at the SU-Tx, results in correlated transmission times to each user. Thus, taking into account this correlation, the exact closed-form expression for the outage performance of the P2M-SN-NI is obtained. However, for the P2M-SN-WI, an accurate approximation of the outage probability is obtained. Furthermore, for the P2M-SN-NI, the closed-form pdf of the transmission time is derived. Using the pdf, the TAW-time of packets and the average waiting time of packets in the buffer are analyzed. Furthermore, numerical simulation are also carried out to validate the analytical result. The remainder of this paper is organized as follows. The system model is explained in Section II. Various queueing theory aspects and the related performance measures are discussed in Section III. In Section IV, the performance of the P2M-SNWI and P2M-SN-NI is analyzed. The numerical results are presented in Section V. Finally, the main results are summarized in Section VI. II. S YSTEM M ODEL Consider a spectrum sharing network, as shown in Fig. 1, in which the primary network allows the secondary network to use the spectrum if the interference level caused by the secondary networks is limited. The primary network consists of one transmitter, which is denoted by PU-Tx, and K receivers, which is denoted by PU-Rx. The SU network also consists of one transmitter, which is denoted by SU-Tx, and L receivers, which is denoted by SU-Rx. The channel power gain from the SU-Tx to the kth PU-Rx is denoted by gk , where k = 1, 2, . . . , K. The channel power gains from the SU-Tx to the lth SU-Rx is denoted by hl and 1 We denote the P2M-SN with interference from P2M-PN as P2M-SN-WI and the P2M-SN without interference from P2M-PN as P2M-SN-NI. 2 Outage is declared if the packet is delayed, i.e., the packet transmission time exceeds a certain predefined threshold.
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the channel power gain from the PU-Tx to the lth SU-Rx is denoted by ζl , where l = 1, 2, . . . L. The channel is assumed to have i.n.i.d. Nakagami-m fading and the channel power gains are assumed to be i.n.i.d. with gamma distribution. The pdf of a gamma distribution with parameters ma, b and Ωa, b is expressed as [20] ma, b ma, b 1 ma, b fa, b (x) = xma, b −1 exp −x Γ(ma, b ) Ωa, b Ωa, b (1) where a ∈ {g, h, ζ} and b ∈ {1, . . . K} if a = g, or b ∈ {1, . . . L} if a ∈ {h, ζ}. For example, mg, 3 and Ωg, 3 are the parameters of fading of the channel power gain from the SU-Tx to the third PU-Rx. Assuming that ma, b is an integer, the cdf can be expressed as ma, b i ma, b −1 m γ ma, b , x Ωa, ma, b xi −x Ω a, b b a, b = 1− e Fa, b (x) = Γ(ma, b ) Ωa, b i! i=0 (2) where γ(·, ·) is the lower incomplete gamma function [21, Eq. (8.350.1)], and Γ(·) is the gamma function [21, Eq. (8.310.1)]. III. S ECONDARY-U SER T RANSMITTER W ITH B UFFER AND P ERFORMANCE M EASURES In actual communication systems, the transceivers are allocated buffers to store packets before transmission. This helps to control the flow and processing of the packets in the network. Considering this practical scenario, it is assumed that the SU-Tx is equipped with a buffer that stores the packets before transmission. The packets arrive at the buffer by a Poisson process with arrival rate λ and are served in a first-in–first-out manner [13]–[15]. The time it takes for the packets to reach the lth SU-Rx after leaving the buffer is called packet transmission time and is denoted by Tl . The transmission time is inversely proportional to the transmission rate of the channel [13]–[15], [22], i.e., Tl =
¯ B U = B log2 (1 + γl ) loge (1 + γl )
(3)
where U is the length of the packet, B is the system bandwidth, ¯ = (U loge (2))/B, and γl is the signal-to-noise ratio at the B lth SU-Rx, which is given as γl =
hl PT N0
(4)
where PT denotes the transmit power of the SU-Tx, and N0 is the variance of the zero-mean additive white Gaussian noise at each of the terminal in the network. In case of interference from the PU-Tx, the transmission time depends on the signal-to-interference-plus-noise ratio (SINR) and can be obtained from (3) by replacing γl with the instantaneous SINR Υl to give l =
¯ B loge (1 + Υl )
(5)
where Υl is the SINR at the lth SU-Rx and is given as Υl =
PT hl P p ζ l + N0
(6)
where Pp denotes the transmit power of the PU-Tx. From (3)–(6), it can be observed that the transmission time depends on the channel power gain and the power-allocation scheme employed at the SU-Tx. A. Outage Analysis Outage probability can be defined to analyze the performance of the P2M-SN. We assume a similar setup considered in [13]– [15]. Then, an outage is declared, and the packet is dropped if the packet transmission time exceeds tout . Thus, in this case, the outage probability is similar to packet loss probability. For the lth SU-Rx, the probability of outage can be expressed as PS, l (tout ) = Pr{Tl > tout } = 1 − FTl (tout )
(7)
where FTl (·) denotes the cdf of Tl .3 In the case of a broadcast channel, the probability that υ SU receivers out of a total of L SU receivers are in outage is calculated. Let PM, υ, L (·) denote the probability that υ SU receivers out of L SU receivers are in outage and can be calculated as PM, υ, L (tout ) =
(Lυ) υ
1−FTzq, i (tout )
L−υ
1
q=1 i1 =1
FTyq, i (tout ) 2
i2 =1
(8) where Z = {1, . . . , L} is the set of indexes denoting the L SU receivers channels, zq denotes the qth set containing indexes of the channels that are in outage, and yq is also the qth set containing indexes outage. zq is the L of channels that are not inL qth set from the υ sets containing one of the υ combinations of elements of set Z, and yq = Z − zq is the set that contains the elements of Z that are not in zq . zq, i1 denotes the i1 th element of zq , and yq, i2 denotes the i2 th element of yq . The P2M-SN can be also considered in outage when any of the L SU receivers are in outage and can be expressed as PA, L (tout ) =
L
PM, υ, L (tout ).
(9)
υ=1
B. Queueing Analysis The system model considered here is analogous to an M/G/1 queueing system in which there is only one server, the packet interarrival times at the buffer are exponentially distributed, and the service time has a general distribution [13]–[16], [23]. Some common performance measures to analyze the performance of such queueing systems are the TAW-time, i.e., the average time taken by a packet from entering the buffer to reaching the SU-Rx; the average waiting time in the buffer (AWBtime); and the average number of packets in the buffer [16]. 3 Here, without loss of generality, we use T and T l q, l to define various performance measures. Note that Tl and Tq, l can be replaced with l and q, l to obtain corresponding performance expressions for the P2M-SN-WI.
KHAN et al.: DELAY PERFORMANCE OF BROADCAST SPECTRUM SHARING NETWORK IN NAKAGAMI-m FADING
For an M/G/1 queue, the TAW-time can be calculated using Pollaczek–Khinchin’s equation [16] E[Wl ] = E[Tl ] + E[Tq, l ]
(10)
where E[Wl ] is the TAW-time of packets at the SU-Tx for the lth SU-Rx, and E[Tq, l ] is the AWB-time, which is given as [16]
λE Tl2 (11) E[Tq, l ] = 2(1 − ρl ) where ρl = λE[Tl ] is known as channel utilization, and E[Tli ], i ∈ {1, 2} denotes the first and second moments of packet transmission time, respectively. The average number of packets Nl in the buffer at any given time can be calculated using Little’s law as Nl = λE[Tq, l ]. Another important notion in this regard is the notion of stability. The queueing system is considered to be stable if and only if the arrival rate is less than the average service rate, i.e., λ < (1/E[Tl ]) [16]. So far, here, we have only discussed the performance measures of a queueing system for only the lth SU-Rx, i.e., there is only one SU-Rx available. For a broadcast channel, in which there are L SU receivers, one can get the approximate performances by averaging all the mentioned performance measures over the L SU receivers. Thus, the mean TAW-time (M-TAWtime) for broadcast channel can be approximated as 1 E[Wl ]. L L
E[W ] =
(12)
l=1
Similarly, the mean average waiting time of the packets in the buffer can be expressed as 1 E[Tq, l ]. L L
E[Tq ] =
(13)
l=1
We call this the mean AWB-time (M-AWB-time). The average number of packets N in the buffer at any given time can be calculated as N = (λ/L) L l=1 E[Tq, l ]. The performance for the worst case can be also analyzed to determine the performance of a broadcast channel, i.e., the performance measures for the SU-Rx with the poorest channel power gain. For this case, the performances can be given by (10), (11), and Nl , where l denotes the SU-Rx with the worst channel condition. In the case of the broadcast channel, the system can be considered to be stable if and only if the arrival rate is less than the average service rate of the SU-Rx with the worst channel. IV. P ERFORMANCE A NALYSIS Power-Allocation Policy: A peak power constraint is imposed by the primary network on the secondary network, i.e., the peak interference power to any PU-Rx should be below a certain threshold Qpk . The peak interference power constraint at the kth PU-Rx can be expressed as PS (g1 , g2 , . . . , gK , h1 , h2 , . . . , hL )gk ≤ Qpk
(14)
where PS (g1 , g2 , . . . gK , h1 , h2 , . . . hL ) is the transmit power of the SU-Tx. It can be noted that the transmit power of the
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SU-Tx is adapted depending on the channel power gains of the network so that the interference power remains below threshold Qpk . In the worst-case scenario, the transmit power of the SU should be chosen such that the interference level at the PU-Rx with the best channel should be at most Qpk . This can be expressed as PS (g1 , g2 , . . . , gK , h1 , h2 , . . . , hL ) ≤
Qpk . maxk {gk }
(15)
Thus, the transmit power of the SU-Tx is chosen as PS (g1 , g2 , . . . gK , h1 , h2 , . . . hL ) =
Qpk . maxk {gk }
(16)
From (16), it can be noted that, if the interference link is very poor, then the power allocated by the SU-Tx is very high. Although, theoretically, this results in large capacity, but in reality, the SU-Tx can transmit only with limited power and has a peak power constraint, i.e., PS (g1 , g2 , . . . , gK , h1 , h2 , . . . , hL ) ≤ Pmax .
(17)
Note that this constraint was not considered in [13]–[15]. Thus, one can combine the constraints of (16) and (17) to obtain the power-allocation policy as
Qpk PCSI = min Pmax , (18) maxk {gk } where PCSI is a short-form notation of PS (g1 , g2 , . . . gK , h1 , h2 , . . . , hL ). It can be noted from (18) that PCSI needs instantaneous channel state information (CSI) of the interference links gk , where k = 1, . . . K. A. Secondary Network Without Primary Interference 1) Finding the Distribution of Service Time: Using (3), the cdf of Tl can be expressed as
¯ B < t = Pr {γl > Ξ(t)} (19) FTl (t) = Pr loge (1 + γl ) ¯ where Ξ(t) = exp(B/t) − 1. By substituting PT = PCSI in (4), we obtain Q hl min Pmax , gpk 0 (20) γl = N0 where g0 = maxk {gk }. By substituting (20) in (19), the cdf of Tl conditioned on g0 can be expressed as ⎧ ⎫ ⎨ ⎬ Ξ(t)N0 FTl |g0 (t) = Pr hl > Q ⎩ ⎭ min Pmax , gpk 0 ⎛ ⎞ Ξ(t)N 0 ⎠ . (21) = 1 − Fh, l ⎝ Q min Pmax , gpk 0 The unconditional cdf of Tl can be obtained by averaging over the pdf of g0 , i.e., fg0 (·), as ⎛ ⎛ ⎞⎞ ∞ Ξ(t)N 0 ⎠⎠ fg0 (g0 )dg0 . (22) FTl (t) = ⎝1−Fh, l ⎝ Qpk min Pmax , g0 0
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j j h, l −1 m 1 mh, l N0 κn, i Ξ(t) + j! Ωh, l Qpk j=0 G0 ∞ m N −g0 Bn +Ξ(t) Q 0 Ω h, l pk h, l × e
Using [24, Eq. (12)], fg0 (·) can be expressed as fg0 (x) =
κn, i e−xBn xAn, i −1 (An, i − xBn )
(23)
G0
n −1 nj ij nj where κn, i = K (mg,jj /(ij !)nj ), j=1 (−1) (mg, j /Ωg, j ) K K Bn = j=1 nj (mg, j /Ωg, j ), An, i = j=1 ij nj , θK is the set of all possible K bit binary numbers, nj is the jth bit of the binary number n ∈ θK , and G0 is a shorthand notation for m K −1 mg, 1 −1 mg, 2 −1 . . . iKg,=0 . i1 =0 i2 =0 n∈θK As (22) contains min{Pmax ,(Qpk/g0 )}, the integration needs to be split over separate limits, and it can be expressed as Qpk Pmax
N0 FTl (t) = 1 − Fh, l Ξ(t) fg0 (g0 )dg0 Pmax 0 ∞ N0 + g0 1 − Fh, l Ξ(t) fg0 (g0 )dg0 Qpk
Qpk Pmax
∞ +
N0 fg0 (g0 )dg0 Pmax 0 N0 g0 1−Fh, l Ξ(t) fg0 (g0 )dg0 . (24) Qpk
Qpk Pmax
By substituting the value of the pdf of g0 and the cdf of hl in (24), we obtain j N0 mh, l 1 mh, l −Ξ(t) Pmax Ωh, l e j! Ω h, l j=0 j Qpk N0 × Ξ(t) Fg 0 Pmax Pmax
mh, l −1
FTl (t) =
mh, l −1
fTl (t) =
j=0
×
G0
δlj
A
× g0 n, i
+j−1
(An, i − g0 Bn ) dg0
(25)
where Fg0(x) denotes the cdf of g0, which is given by [24, Eq. (11)] κn, i e−xBn xAn, i . (26) Fg0 (x) = G0
By substituting values from (26) in (25) and solving the resulting integration, we obtain the cdf of the transmission time as follows: mh, l −1
FTl (t) =
j=0
+ ×
κn, i
G0 mh, l −1
G0
Qpk
Pmax
1 − Fh, l Ξ(t)
=
Qpk Pmax
j=0
An, i
(Ξ(t)ηP )j −Ξ(t)δl ηP −P Bn An, i e P j!(δl )−j
1 j δ κn, i (Ξ(t)ηQ )j j! l
Γ (An, i + j, (Bn + Ξ(t)δl ηQ ) P )
−Bn
(Bn + Ξ(t)δl ηQ )(An, i +j) Γ (An, i + j + 1, (Bn + Ξ(t)δl ηQ ) P )
(Bn + Ξ(t)δl ηQ )(An, i +j+1) (27)
where Γ(·, ·) denotes the upper incomplete gamma function [21, Eq. (8.350.2)], δl = (mh,l )/(Ωh,l ), P = Qpk /Pmax , ηP = N0/ Pmax, and ηQ = N0/Qpk. The pdf is obtained by taking derivative of the cdf in (27) to yield (28), shown at the bottom of the page. 2) Outage Analysis: By substituting (27) in (7), the probability of outage PS,l (tout ) of the lth SU-Rx in the P2M-SN-NI is obtained. For the broadcast channel, the probability that υ SU receivers out of a total of L SU receivers are in outage is given by (8).
¯ P κn, i Bη j!
1 −P Bn An, i j−1 e P ηP (Ξ(t) + 1)e−Ξ(t)δl ηP Ξ(t)j−1 (Ξ(t)δl ηP − j) t2 j−1 Ξ(t)j−1 ηQ 1 + An, i (Ξ(t) + 1) (Ξ(t)δl ηQ An, i − jBn ) Γ (An, i + j, (Ξ(t)ηQ δl + Bn ) P ) P t2 (Ξ(t)δl ηQ + Bn )An, i +j+1 (Ξ(t)ηQ )j −1+j+An, i −Ξ(t)δl ηP −P Bn (Ξ(t) + 1) e (Ξ(t)δl ηP + P Bn ) +δl t2 (Ξ(t)δl ηQ + Bn )An, i +j (Ξ(t)ηQ )j−1 1 (Ξ(t) + 1) (Ξ(t)δl ηQ (1 + An, i ) − jBn ) − Bn P t2 (Ξ(t)δl ηQ + Bn )An, i +j+2 × Γ (An, i + j + 1, (Ξ(t)ηQ δl + Bn ) P ) (Ξ(t)ηQ )j j+An, i −Ξ(t)δl ηP −P Bn (28) (Ξ(t) + 1) e (Ξ(t)δl ηP + P Bn ) +δl t2 (Ξ(t)ηQ δl + Bn )An, i +j+1
KHAN et al.: DELAY PERFORMANCE OF BROADCAST SPECTRUM SHARING NETWORK IN NAKAGAMI-m FADING
In (8), it is assumed that the transmission times of all the SU receivers are independent. However, in the case of the CSIbased power allocation in (18), the transmission times to all the SU receivers are not independent due to the dependence of the power-allocation scheme on g0 . Conditioned on g0 , the transmission times are independent, and the outage can be expressed similar to (8) as PM, υ, L|g0 (tout ) =
(Lυ) υ
1 − FTzq, i
q=1 i1 =1
×
L−υ
mh,yq,1 −1 mh,yq,2 −1 mh,yq ,L−υ −1 . . . vL−υ =0 , v1 =0 v2 =0 Hξ2 denotes L−υ L−υ vj ξv,q = j=1 (1/vj !)(mh,yq,vj/Ωh,yq,vj ) , Wv = j=1 vj , and Xq = L−υ i2 =1 (mh,yq, i2 /Ωh,yq, i2 ). By substituting the values of
where
ξ1 (q), ξ2 (q), and Δ(tout , g0 ) in (30), PM,υ,L|g0 is given by PM,υ,L|g0
1
=
|g0 (tout )
FTyq, i
2
|g0 (tout ).
⎞ ⎞ )N Ξ(t out 0 ⎠ (Do, q + Xq )⎠ × exp ⎝− ⎝ Q min Pmax , gpk 0
(29)
ξ1 (q) × ξ2 (q)
(30)
q=1
where ξ1 (q) = υi1 =1 Fh,zq, i1 (Δ(tout , g0 )), ξ2 (q) = L−υ i2 =1 (1 − Fh,yq, i2 (Δ(tout , g0 ))), and Δ(tout , g0 ) = ((Ξ(tout )N0 )/ (min{Pmax , Qpk /g0 })). After substituting the cdf, ξ1 (q) can be expressed as mh,z υ q, i1 −Δ(tout ,g0 ) Ω h,zq, i 1 ξ1 (q) = 1−e i1 =1
mh,zq, i −1
1
×
i=0
mh,zq, i1 Ωh,zq, i1
⎛
i
×⎝
Ξ(tout )N0 ⎠ Q min Pmax , gpk 0
.
(35)
Λ1 (Qpk , Pmax , L, υ, tout ) Qpk Pmax
PM,υ,L|g0 fg0 (g0 )dg0
= 0
⎞ Δ(tout , g0 )i ⎠ . i!
⎞Co, p +Wv
To find the outage probability expression, PM,υ,L|g0 needs to be averaged over the pdf of g0 by performing integration similar to (24). For the interval (0, (Qpk /Pmax )), we obtain
=
(Lυ)
χo, p, q ξv,q e
−
Ξ(tout )N 0
Pmax
(Do, q +Xq )
q=1 Hξ1 Hξ2
(31)
C +W Ξ(tout )N0 o, p v × Pmax An, i Qpk Qpk − Pmax Bn × κn, i e Pmax
Using [24, Eq. (9)] and doing some algebraic manipulations, ξ1 (q) can be expressed as ξ1 (q) = χo, p, q e−Δ(tout ,g0 )Do, q Δ(tout , g0 )Co, p (32)
(36)
G0
and for the interval ((Qpk /Pmax ), ∞), we obtain
Hξ1
where θυ is the set of all possible υ bit binary numbers, oj is the jth bit of the binary number o ∈ θυ , χo, p, q = υ oj −1 (−1)oj (mh,zq, j /Ωh,zq, j )pj oj (mh,z /(pj !)oj ), Do, q = q, j j=1 υ υ Hξ1 is a j=1 oj (mh,zq, j /Ωh,zq, j ), Co, p = j=1 pj oj , and mh, 1 −1 mh, 2 −1 mh,υ −1 shorthand notation for o∈θυ p1 =0 p2 =0 . . . pυ =0 . Similarly, after substituting the cdf, ξ2 (·) can be expressed as ⎛ mh,y L−υ q, i2 −Δ(tout ,g0 ) Ω h,yq, i ⎝e 2 ξ2 (q) = i2 =1 mh,yq, i −1
×
χo, p, q ξv,q
q=1 Hξ1 Hξ2
To avoid confusion, the indexing in (29) is the same as that in (8). By substituting the values from (21) to that in (29), we obtain PM,υ,L|g0 (tout ) =
(Lυ)
⎛ ⎛
i2 =1
(Lυ)
1355
2 i=0
mh,yq, i2 Ωh,yq, i2
i
⎞ Δ(tout , g0 )i ⎠ . i!
Qpk Pmax
=
(Lυ)
(33)
κn, i χo, p, q ξv,q
q=1 Hξ1 Hξ2 G0
∞ ×
After doing some algebraic manipulations, ξ2 (·) is given by ξ2 (q) = ξv,q e−Δ(tout ,g0 )Xq Δ(tout , g0 )Wv (34) Hξ2
Λ2 (Qpk , Pmax , L, υ, tout ) ∞ PM,υ,L|g0 fg0 (g0 )dg0 =
e
−
Ξ(tout )N0 Qpk
Ξ(tout )N0 Qpk
Co, p +Wv
(Do, q +Xq )g0 −Bn g0
Qpk Pmax
C
× g0 o, p
+Wv +An, i −1
(An, i − g0 Bn )dg0 .
(37)
If υ = L, then for certain cases, the argument of the exponential function becomes 0, and the integral in (37) is not available in closed form. However, for υ < L, the integral is available in closed form and can be expressed in closed form using
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[21, Eq. (3.3512)]. Thus, for υ < L, we obtain (38), shown at the bottom of the page. Finally, for the P2M-SN-NI, the probability of outage of υ SU receivers when υ < L can be expressed as PM, υ, L (tout ) = Λ1 (Qpk , Pmax , L, υ, tout ) +Λ2 (Qpk , Pmax , L, υ, tout )
(39)
and the probability of outage when υ = L can be obtained as PM,L,L (tout ) = 1 −
L−1
PM, υ, L (tout ).
(40)
υ=0
Similar to (9), the probability that any SU-Tx in the P2M-SNNI is in outage is given by PA, L (tout ) =
L
Similarly, E[Tl2 |Tl < tout ] can be obtained using E
Tl2 |Tl
< tout
1 = Pr{Tl < tout }
tout t2 fTl (t)dt.
(45)
0
By substituting the pdf from (28) into that of (44), applying the transformation of variable of integration x = Ξ[t], and solving the resulting integral, we obtain E[Tl |Tl < tout ] mh, l −1 j κn, i ¯ P Bη = δl Pr{Tl < tout } j=0 j! G0
∞
×
PM, υ, L (tout ) = 1 − PM,0,L (tout ). (41)
1 log(1 + x) Ξ(tout ) ×
υ=1
Finding the Moments of Service Time: As was discussed earlier, the maximum time for any packet can be tout . If the packet is transmitted in a time less than tout , it is considered a successful transmission. However, as soon the packet transmission time reaches tout , the packet is dropped, and a new transmission is started. Therefore, the moments of transmission time can be found using total law of expectation [15], i.e.,
E Tli = E Tli |Tl < tout Pr{Tl < tout }
+ E Tli |Tl ≥ tout Pr{Tl ≥ tout }
= E Tli |Tl < tout FTl (tout ) + tiout (1 − FTl (tout ))
P n, i ηPj−1 xj−1 (xδl ηP − j)e−P Bn −xδl ηP An, i A +j j + ηQ δl P n, i e−P Bn − Bn P j−1 j−1 j −xδl ηP ηQ x x e + × (xδl ηQ +Bn ) P (xδl ηQ + Bn )An, i +j+1 × An, i (xδl ηQ An, i − jBn ) A
× Γ (An, i + j, (xδl ηQ + Bn )P ) (xδl ηQ (1 + An, i ) − jBn ) − Bn (xδl ηQ + Bn )
×Γ(An, i +j +1, (xδl ηQ +Bn )P )
dx.
(42)
(46)
where i = 1 for the first moment, and i = 2 in the case of the second moment. Let T¯l denote the transmission time of the packet, given that Tl < tout ; then, the pdf of T¯l can be found as
1 f (t), if 0 ≤ t < tout (43) fT¯l (t) = Pr{Tl Ξ(t) ⎩ ⎭ P p ζ l + N0
Qpk = Pr min Pmax , hl > Ξ(t)ℵl (48) g0
j−1 j−1 ηQ x xj e−xδl ηP + (xδ η +Bn ) P (xδl ηQ + Bn )An, i +j+1 l Q × An, i (xδl ηQ An, i − jBn )
×
× Γ (An, i +j, (xδl ηQ +Bn )P ) (xδl ηQ (1 + An, i ) − jBn ) − Bn (xδl ηQ + Bn )
where ℵl = Pp ζl + N0 . Following similar steps to those in the case without interference, after averaging over g0 , we obtain (49), shown at the bottom of the page, where ΞQ (t) = (Ξ(t)δl )/Qpk , and ΞP (t) = (Ξ(t)δl )/Pmax , Substituting ℵl = Pp ζl + N0 and doing some algebraic manipulations yield (49) as well as (50), also shown at the bottom of the page. Fl (·) can be obtained by averaging over the pdf of the interference channel as ∞ (51) Fl (t) = Fl |ζl (t, x)fζl (x)dx.
×Γ(An, i +j +1, (xδl ηQ +Bn )P )
dx. (47)
The integrals in (46) and (47) can be numerically evaluated using mathematical software, such as Matlab, as follows: By substituting the values from (27) and (46) into those of (42) and taking i = 1, we obtain the first moment of the transmission time of packets. Similarly, by substituting the values from (27) and (47) into those of (42) and taking i = 2, we obtain the second moment of the transmission time of packets. The TAW-time for a single SU-Rx case can be found by substituting the obtained moments into the equation obtained after substituting (11) into (10). By substituting the moments in (11), we obtain the M-AWB-time for the single SU-Rx case. Similarly, for the case of multiple SU receivers, the M-TAW-time and M-AWB-time can be obtained using (12) and (13).
mh, l −1
Fl |ℵl (t) =
j=0
G0
(Ξ(t))j κn, i j!(δl )−j
0
Substituting the pdf and the cdf into those of (51) and doing algebraic manipulations, Fl (t) can be expressed as (52), shown at the bottom of the next page, where R(p, q, r, l) = (1/Γ(mζ,l ))(mζ,l/Ωζ,l )mζ,l = An, i +j, α∞ mζ,l −1 −((mζ,l/Ωζ,l )+p)x x e (x+q)r dx, and S(p, q, r, a, b, c) = 0∞ mζ,l mζ,l −1 −(mζ,l /Ωζ,l )x x e /Γ(mζ,l )(x + q)−r ) 0 ((mζ,l /Ωζ,l ) p (Γ(p, (ax + b)c)/(ax + b) )dx. R(·, ·, ·, ·) and S(·, ·, ·, ·, ·, ·) are given in closed form in Appendixes A and B, respectively. Note that S(p, ·, ·, ·, ·, ·) is valid for p > 0. Note that, from (52), p = α; thus, p can be zero, and the term S(α, ·, ·, ·, ·, ·)
A P n, i e−P Bn
×
An, i
Fl |ζl (t, ζl ) =
j=0
G0
(Ξ(t))j κn, i j!(δl )−j
A
P n, i j Pmax
ℵl Pmax
j e
−ΞP (t)ℵ
+
ℵl Qpk
j
(Bn + ΞQ (t)ℵl )An, i +j Γ (An, i + j + 1, (Bn + ΞQ (t)ℵl ) P )
(49)
(Bn + ΞQ (t)ℵl )An, i +j+1
j e−P Bn (Pp ζl + N0 )j e−ΞP (t)(Pp ζl +N0 ) + Q−j pk (Pp ζl + N0 )
×
Γ (An, i + j, (Bn + ΞQ (t)ℵl ) P )
−Bn
mh, l −1
1357
An, i
Γ (An, i + j, (Bn + ΞQ (t)(Pp ζl + N0 )) P )
−Bn
(Bn + ΞQ (t)(Pp ζl + N0 ))An, i +j Γ (An, i + j + 1, (Bn + ΞQ (t)(Pp ζl + N0 )) P ) (Bn + ΞQ (t)(Pp ζl + N0 ))An, i +j+1
(50)
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is indeterminate. However, it can be observed that term S(α, ·, ·, ·, ·, ·) is being multiplied with An, i , which is also zero when α = 0. Therefore, (52) can be accurately evaluated by replacing S(α, ·, ·, ·, ·, ·) with any constant c when α = 0. The pdf can be found by taking the derivative of the cdf in (52). 2) Outage Analysis: By substituting (52) in (7), the probaint (tout ) of the lth SU-Rx in the P2M-SN-WI bility of outage PS,l is obtained. From (5) and (6), it can be noticed that the transmission time now depends on an additional random variable due to the interference link. In this case, conditioned on g0 and ℵl , the transmission times are independent, and the probability that υ SU receivers out of a total of L SU receivers in a P2M-SN-WI are in outage can be similarly expressed as in (8), i.e.,
=
1 − Fzq, i
q=1 i1 =1
1
|g0 ,ℵl (tout )
L−υ i2 =1
Fyq,i
2
To avoid confusion, the indexing in (53) is the same as the indexing in (8). By substituting values from (21) into those of (53), we obtain int PM,υ,L|g (tout ) 0 ,ℵl
=
mh,zq,1 −1
ξ¯1 (q, tout )ξ¯2 (q, tout )
(54)
¯ z (tout, g0)), ξ¯2(q, tout) = where ξ¯1(q,tout)= υi1 =1 Fh,zq,i1 |g0(Δ q,i1 L−υ ¯ ¯ (1 − F ( Δ (t , g yq, i1 out 0 ))), and Δl (tout , g0 ) = h,yq, i1 |g0 i2 =1 (Ξ(tout )ℵl )/(min{Pmax , Qpk /g0 }). After substituting the cdf, ξ¯1 (·, ·) can be expressed as υ ¯ −Δ (t ,g )δ ¯ ξ1 (q, tout ) = 1 − e zq, i1 out 0 zq, i1
mh,yq, i −1
×
i=0
δzi q, i
1
i!
¯ z (tout , g0 ) Δ q, i1
i
.
(55)
By using [24, Eq. (9)] and after doing some algebraic manipulations, ξ¯1 (·, ·) can be expressed as ¯ ¯ out , g0 )Co,p (56) χℵ,o,p,q e−Δ(tout ,g0 )Dℵ,o,q Δ(t ξ¯1 (q, tout ) =
mh, l −1
j=0
¯ out, g0) = Δ(t is a shorthand nota-
δyi q, i
2
i!
i=0
i ¯ Δyq, i2 (tout , g0 ) .
(57)
Hξ¯2
mh,y −1 mh,y −υ−1 where Hξ¯ is shorthand notation of v1=0q,1 . . . vL−υq,L , =0 2 L−υ vj L−υ vj ¯ ξℵ,v,q = j=1 (δyq,j /vj !)ℵyq, j , Xℵ,q = j=1 ℵyq, j δyq, j , and ¯ Wv = L−υ j=1 vj . Substituting values of ξi (·, ·) from (56) and (58) into (54) yields int PM,υ,L|g (tout ) 0 ,ℵl
=
(Lυ)
χℵ,o,p,q ξ¯ℵ,v,q
q=1 Hξ¯ Hξ¯ 1 2
×e
¯ out ,g0 )(Xℵ,q +Dℵ,o,q ) −Δ(t
¯ out , g0 )Co, p +Wv . Δ(t
(59)
The unconditioned outage probability can be obtained from int (·) as PM,υ,L|g 0 ,ℵl ∞ int PM, υ, L (tout )
=
∞ int PM,υ,L|g (tout ) 0 ,ℵl
... 0
0
×fg0 (g)fζ (ζ1 , . . . , ζL )dgdζ1 . . . dζL
(60)
where fζ (ζ1 , . . . , ζL ) is the joint pdf of ζl , ∀ l. By substituting ℵl = (Pp ζl + N0 ) and after doing some algebraic manipulaint (·) can be expressed as tions, PM,υ,L|g 0 ,ζl int (tout ) PM,υ,L|g 0 ,ζl
=
(Lυ)
¯ out , g)Co, p +Wv χo, p, q ξ¯v,q Δ(t
q=1 Hξ¯ Hξ¯ 1 2
Hξ¯1
Fl (t) =
2
×
i1 =1
1
oj
mh,zq,υ −1
i2 =1
q=1
mh,zq, i −1
Hξ¯1
j=1 (−1)
. . . pυ =0 . Similarly, after tion for p1 =0 o∈θυ substituting the cdf, ξ¯2 (·, ·) can be expressed as L−υ ¯ −Δ (t ,g )δ ¯ ξ2 (q, tout ) = e yq, i2 out 0 yq, i2
|g0 ,ℵl (tout ).
(53)
(Lυ)
(Ξ(tout ))/(min{Pmax , Qpk /g0 }), and
υ
After doing some algebraic manipulations, ξ¯2 (·, ·) can be expressed as ¯ ¯ out , g0 )Wv (58) ξ¯v,q e−Δ(tout ,g0 )Xq Δ(t ξ¯2 (q, tout ) =
int (tout ) PM,υ,L|g 0 ,ℵl
(Lυ) υ
υ
j=1 oj δzq,j ℵzq,j , χℵ,o,p,q = oj−1 op oj ojpj υ (mh,zq,j/(pj ) )δzq,j j=1ℵzjq,jj , Co,p = υj=1 oj pj ,
where Dℵ,o,q =
G0
(Ξ(t))j κn, i j!(δl )−j
N0 −ΞP (t)N0 −P Bn j j e P R Ξ (t)P , , j, l + Q−j P p p pk Pp j Pp Pmax A
P n, i
N0 N0 × An, i S α, , j, ΞQ (t)Pp , (Bn + ΞQ (t)N0 ) , P − Bn S α + 1, , j, ΞQ (t)Pp , (Bn + ΞQ (t)N0 ) , P Pp Pp
(52)
KHAN et al.: DELAY PERFORMANCE OF BROADCAST SPECTRUM SHARING NETWORK IN NAKAGAMI-m FADING
×
υ
Again, due to the min{·, ·} function, the integration needs to be split as
¯
Pp j j e−Δ(tout ,g)oj δzq, j N0 o p
1359
j=1
×
L−υ
int PM, υ, L (tout ) = I1 (tout ) + I2 (tout )
¯ v Pp j e−Δ(tout ,g)N0 δyq, j
j=1
×
υ
ζzq, j +
j=1
×
L−υ
N0 Pp
N0 ζyq, j + Pp
j=1
o j p j
vj
¯
e−Δ(tout ,g)oj δzq, j Pp ζzq, j
0
∞
¯ e−Δ(tout ,g)Pp ζyq, j δyq, j .
I1 (tout ) can be obtained in closed form as
int (tout ) PM,υ,L|g 0
I1 (tout ) =
(Lυ) C +W Ξ(tout ) o, p v
¯ out , g) Δ(t
Co, p +Wv
×
χo, p, q ξ¯v,q
×
o p Pp j j e
×
¯ out ,g)oj δz −Δ(t N0 q, j
×
j=1 υ
¯ out , g)oj δz Pp , N0 , oj pj , zq, j × R Δ(t q, j Pp j=1 ×
×
¯
Pp j e−Δ(tout ,g)N0 δyq, j v
¯ out , g)Pp δy , N0 , vj , yq, j . R Δ(t q, j Pp j=1
L−υ
×
×
υ
o p
Pp j j e
Ξ(tout ) min
Qpk Pmax , g0
×
j=1 υ
Ξ(tout )
− v Pp j e
min
L−υ
Pp j e− v
Ξ(tout ) Pmax N0 δyq, j
Qpk Pmax , g0
⎞
Ξ(t ) N oj δzq, j Pp , 0 , oj pj , zq, j⎠ out × R⎝ Qpk Pp min Pmax , g0 j=1 ⎞ ⎛ L−υ ) Ξ(t N Pp δyq, j , 0 , vj , yq, j ⎠ . out × R⎝ Qpk Pp min P , j=1 max
R
L−υ
Ξ(tout ) N0 oj δzq, j Pp , , oj pj , zq, j Pmax Pp
R
Ξ(tout ) N0 Pp δyq, j , , vj , yq, j Fg0 (). Pmax Pp
To the best of the author’s knowledge, I2 (·) cannot be obtained in closed form; thus, one can use well-known numerical integration techniques to evaluate the integral. By applying the change of variable g0 = tan(x) and using the Gauss–Chebyshev rule, I2 (·) can be accurately evaluated as T π u L − bL I2 (tout ) = 1 − x2t1 T 2 t =1 1
2 int ×PM, υ, L (tout|tan(Xt1)) fg0 (tan(Xt1 )) (sec(Xt1 ))
(66) where bL=tan−1(), uL= π/2, Xt1 = ((uL −bL )/2)xt1 +((uL+ bL )/ 2), and xt1 = cos((2t1 − 1)π/ 2T ) [25, Eq. (25.4.38)]. Thus, for the P2M-SN-WI, the probability of outage of υ SU receivers, when υ < L, is given by (64). The probability of outage when υ = L can be obtained using
N0 δyq, j
⎛
υ
oj δzq, j N0
j=1 L−υ
Ξ(tout ) Pmax oj δzq, j N0
(65) (62)
⎛
−
o p
j=1
⎞Co, p +Wv ) Ξ(t ⎝ ⎠ out = χo, p, q ξ¯v,q Qpk min Pmax , g0 q=1 Hξ¯ Hξ¯ 1 2 (Lυ)
Pp j j e−
j=1
¯ out , g) = (Ξ(tout ))/ To average over g0 , substitute Δ(t (min{Pmax , Qpk /g0 }), and after doing some algebraic manipulations, this yields int (tout ) PM,υ,L|g 0
υ
χo, p, q ξ¯v,q
j=1
j=1 L−υ
Pmax
j=1
q=1 Hξ¯ Hξ¯ 1 2 υ
(64)
(61)
q=1 Hξ¯ Hξ¯ 1 2
(Lυ)
int PM,υ,L|g (tout )fg0 (g)dg. 0
+
By using the independence among ζl and integrating with int (·) is expressed as respect to ζl , PM,υ,L|g 0
=
int PM,υ,L|g (tout )fg0 (g)dg 0
=
g0
(63)
int (tout ) = 1 − PM,L,L
L−1
int PM, υ, L (tout ).
(67)
υ=0
Similarly, for the P2M-SN-WI, the probability that any SU-Rx is in outage is given as int PA, L (tout ) =
L υ=1
int int PM, υ, L (tout ) = 1 − PM,0,L (tout ). (68)
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Fig. 2. Probability of outage for the SU-Rx in a P2P-SN-NI. mg, K = 2 ∀ k, and mh,1 = 3.
Fig. 3.
TAW-time for the SU-Rx in a P2P-SN-NI. mg, K = 2, ∀ k mh,1 = 3.
Fig. 4.
AWB-time for an SU-Rx in a P2P-SN-NI. mg, K = 2, ∀ k mh,1 = 3.
Note that, for L = 1, the exact outage probability can be simply obtained using (52) as follows: int (tout ) = 1 − Fl (tout ). PM,1,1
(69)
Finding the Moments of Service Time: Following a similar procedure as in Section IV-A3, the moments of service time can be obtained for the P2M-SN-WI. Details are omitted here due to space limitations. V. N UMERICAL R ESULTS AND D ISCUSSION Here, numerical results based on Monte Carlo simulations are presented to verify the derived results. In obtaining these numerical results, B = 1 MHz, U = 4096 bits (512 B), tout = 10 ms, N0 = 1, and Ωa, b = 0 dB, ∀{a, b}. These parameters are fixed in the simulation, unless stated. A. Secondary Network Without Primary Interference 1) Single SU-Rx: The outage probability for a P2P-SN-NI is shown in Fig. 2. It can be observed that, as the allowed peak interference threshold level Qpk is relaxed, the outage probability decreases, and the SU performance improves. However, as the number of PU-Rx K is increased, the SU performance degrades, and the outage probability increases. Furthermore, if the maximum transmit power of the SU-Tx, i.e., Pmax , is limited, then at high Qpk , the outage probability becomes constant due to constant transmit power of the SU-Tx. Figs. 3 and 4 show the effect of increasing Qpk on the TAW-time and AWBtime for a P2P-SN-NI, respectively. It can be observed that, as Qpk increases, the TAW-time and the AWB-time decreases, and the packets are transmitted quickly. As the number of PU receivers K increases, both the TAW-time and the AWB-time increase. This happens because the SU-Tx has to reduce the transmit power to satisfy more PU receivers. Furthermore, increasing packet arrival rate λ increases the TAW-time and the AWB-time. Note that this increase in TAW-time is due to the increase in the
AWB-time. The transmission time does not depend on λ, as observed in (3). It can be also observed that, if the maximum transmit power of the SU-Tx, i.e., Pmax , is limited, then at high Qpk , the TAW-time and the AWBtime become constant due to constant transmit power of the SU-Tx. It can be noted that, at low Qpk , the TAW-time exceeds tout . This happens due to the AWB-time. One can obtain the transmission time by subtracting AWBtime from TAW-time. 2) Multiple SU-Rx: The probability of outage of υ SU receivers out of L = 4 SU receivers in a P2M-SN-NI is shown in Fig. 5. After a certain value of Qpk , it can be observed that the probability of outage decreases as Qpk increases. However, for lower Qpk , the probability of outage shows an inverse behavior for lower values of υ. This can be explained as follows. At low Qpk , it is very likely that majority of the SU-Rx are in outage, and it is very less likely that few SU-Rx are in outage. Therefore, one can observe in Fig. 5 that the probability that only one SU-Rx is in outage is lower compared with the probability that all four SU-Rxs are in outage.
KHAN et al.: DELAY PERFORMANCE OF BROADCAST SPECTRUM SHARING NETWORK IN NAKAGAMI-m FADING
Fig. 5. Probability that υ SU-Rxs are in outage in a P2M-SN-NI. K = 3, mg, K = k ∀ k, mh,1 = 1, mh, 2 = 3, mh,3 = 2, and mh,4 = 1.
Fig. 6. Probability that any SU-Rx is in outage in a P2M-SN-NI. K = 3, mg, K = k ∀ k, and mh, l = 2 ∀ l.
In addition, in this case, at high Qpk , the probability of outage becomes constant depending on the value of Pmax . The probability that any SU-Rx in a P2M-SN-NI is in outage is shown in Fig. 6. It can be observed that the outage probability decreases as Qpk increases. As the number of SU-Rx (L) increases, it becomes more probable that some SU-Rx are in outage. Therefore, the probability that any SU-Rx is in outage increases when L increases. In addition, in this case, at high Qpk , the probability of outage becomes constant, depending on the value of Pmax . The TAW-time for all the SU receivers and the M-TAW time for a P2M-SN-NI are plotted in Fig. 7 for three SU receivers. Similarly, trends can be observed, as were observed for the single SU-Rx case. The TAW time and the M-TAWtime decrease with increasing Qpk . Each SU-Rx has its own TAW-time depending on the channel condition. If the channel is good, then the TAW-time is less, and vice versa. At high Qpk , the TAW-time and, thus, the M-TAW-time, becomes constant depending on the value of Pmax . Similarly, trends can be observed for AWB-time and M-AWB-time. However, those results are omitted due to space limitations.
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Fig. 7. TAW-time and M-TAW-time for multiple SU-Rxs in a P2M-SN-NI. K = 3, mg, K = 2 ∀ k, mh,1 = 3, mh, 2 = 2, mh,3 = 1, and λ = 50.
Fig. 8. Probability of outage for a SU-Rx in a P2P-SN-WI, where mg, K = 2 ∀ k, mζ,k = 2 ∀ k, and mh,1 = 3.
B. Secondary Network With Primary Interference 1) Single SU-Rx: Fig. 8 shows the outage performance of the P2P-SN-WI. The curves in Fig. 8 are plotted using (64). In the simulations, we have considered the case where the PU-Tx transmits with the maximum available power i.e., Pmax i.e., Pp = Pmax . It can be observed that interference from the primary network results in severe degradation in performance of the secondary network. The outage probability of the secondary network increases as the interference power from the PU-Tx increases. Furthermore, it can be observed that increasing Pmax does not always improve the secondary network performance. This happens because the PU-Rx interference constraint limits the transmit power of the SU-Tx, whereas the interference power from the PU-Tx increases with increasing Pmax . However, when the PU-Rx interference constraint is relaxed (Qpk is large), increasing Pmax improves the outage performance. 2) Multiple SU-Rx: The probability of outage of υ SU receivers out of L = 3 SU receivers in a P2M-SN-WI is shown in Fig. 9. The curves in Fig. 9 are plotted using (64)
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was analyzed based on the packet transmission time, and two scenarios were considered: 1) The P2M-SN does not experience interference from the P2M-PN, and 2) the P2M-SN does experience interference from the P2M-PN. For both scenarios, based on the timeout concept, the outage performance of the secondary networks was analyzed, and mathematical expressions for outage probability of the P2M-SN were obtained. Furthermore, the TAW-time of packets and the average number of packets waiting in the buffer of the SU-Tx were also analyzed. A PPENDIX A F UNCTION R(p, q, r, l)
Fig. 9. Probability that υ SU-Rxs are in outage in a P2M-SN-WI, where Pp = Pmax = 20 dB, K = 3, L = 3, mg, K = 2 ∀ k, mζ,k = 2 ∀ k, mh,1 = 1, mh, 2 = 3, and mh,3 = 2.
R(·, ·, ·, ·) can be expressed as follows: mζ,l mζ,l ∞ m − Ω ζ,l +p x Ωζ,l ζ,l mζ,l −1 R(p, q, r, l) = x e (x+q)r dx. Γ(mζ,l ) 0
(70) By using binomial expansion, we have mζ,l 1 mζ,l R(p, q, r, l) = Γ(mζ,l ) Ωζ,l ∞ m r − Ω ζ,l +p x r r−r1 ζ,l mζ,l +r1 −1 × x e dx. q r1 r =0 1
(71)
0
Solving integration using [21, Eq. (3.351.3)] yields mζ,l 1 mζ,l R(p, q, r, l) = Γ(mζ,l ) Ωζ,l −(mζ,l+r1 ) r mζ,l r r−r1 × Γ(mζ,l +r1 ) +p . q r1 Ωζ,l r =0
(72)
1
Fig. 10. Probability that any SU-Rx is in outage in a P2M-SN-WI, where Pmax = Pp dB, K = 3, L = 3, mg, K = 2 ∀ k, mζ,k = 2 ∀ k, mh,1 = 1, mh, 2 = 3, and mh,3 = 2.
and (67). Similar to Fig. 5, for lower Qpk , the probability of outage shows an inverse behavior for lower values of υ. The probability that any SU-Rx is in outage is shown in Fig. 10. The curves in Fig. 10 are plotted using (68). It can be observed that the outage probability decreases as Qpk increases. Similar to Fig. 6, the probability that any SU-Rx is in outage increases when L increases, and at high Qpk , the probability of outage becomes constant depending on the value of Pmax . Similar to Fig. 8, it can be observed that increasing Pmax increases the outage probability when the Qpk is low. It can be noted that, in all the figures, the analytic results match well with the simulation results. VI. C ONCLUSION The delay performance of a P2M-SN, which shares the spectrum with a P2M-PN, has been analyzed for an i.n.i.d. Nakagami-m-faded channel. The performance of the network
A PPENDIX B Function S(p, q, r, a, b, c) S(·, ·, ·, ·, ·, ·) can be expressed as follows: S(p, q, r, a, b, c) mζ,l m ∞ mζ,l ζ,l xmζ,l −1 e− Ωζ,l x Ωζ,l Γ (p, (ax + b)c) = dx. −r Γ(mζ,l )(x + q) (ax + b)p
(73)
0
By using binomial expansion, we have mζ,l mζ,l r Ωζ,l r r−r1 S(p, q, r, a, b, c) = q Γ(mζ,l ) r =0 r1 1 ∞ m − ζ,l x Γ (p, (ax + b)c) × xmζ,l +r1 −1 e Ωζ,l dx. (ax + b)p
(74)
0
Using [21, Eq. (8.352.2)], expressing Γ(n + 1, x) = n!e−x n jc 4 and doing some algebraic manipulations jc =0 x /jc !, 4 Note
that using this expansion requires that p is an integer and that p > 0.
KHAN et al.: DELAY PERFORMANCE OF BROADCAST SPECTRUM SHARING NETWORK IN NAKAGAMI-m FADING
yield S(p, q,r, a, b, c)
mζ,l
p−1 r r r−r1 Γ(p) (ac)p1 = e−bc p q Γ(mζ,l ) r =0 p =0 r1 Γ(p1 + 1) a 1 1 −p+p1 ∞ mζ,l − ac+ Ω x b ζ,l × xmζ,l +r1 −1 x + e dx. a mζ,l Ωζ,l
0
(75) Solving the integration using [26, Eq. (2.1.3.1)] yields S(p, q, r, a, b, c) mζ,l p−1 r r mζ,l = r1 Ωζ,l r =0 p =0 1
1
Γ(p)Γ(mζ,l + r1 ) cp1 q r−r1 × Γ(mζ,l)Γ(p1 + 1) ap−p1
mζ,l +r1 −p+p1 b a
× e−bc Ψ mζ,l + r1 , mζ,l + r1 − p + p1 + 1; mζ,l b . ac + Ωζ,l a
(76)
Note that this is valid for integers p and p > 0. R EFERENCES [1] F. A. Khan, K. Tourki, M. -S. Alouini, and K. A. Qaraqe, “Delay analysis of a point-to-multipoint spectrum sharing network with csi based power allocation,” in Proc. IEEE Int. Symp. DYSPAN, Bellevue, WA, USA, Oct. 16–19, 2012, pp. 235–241. [2] ET Docket 03-108, Facilitating opportunities for flexible, efficient, and reliable spectrum use employing cognitive radio technologies 2003. [3] J. Mitola, III and G. Maguire, Jr, “Cognitive radio: Making software radios more personal,” IEEE Pers. Commun., vol. 6, no. 4, pp. 13–18, Aug. 1999. [4] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [5] Y. Zou, Y.-D. Yao, and B. Zheng, “Cooperative relay techniques for cognitive radio systems: Spectrum sensing and secondary user transmissions,” IEEE Commun. Mag., vol. 50, no. 4, pp. 98–103, Apr. 2012. [6] I. Akyildiz, W. Lee, M. Vuran, and S. Mohanty, “NeXt generation/ dynamic spectrum access/cognitive radio wireless networks: A survey,” Comput. Netw., vol. 50, no. 13, pp. 2127–2159, Sep. 2006. [7] D. Wu and R. Negi, “Effective capacity: A wireless link model for support of quality of service,” IEEE Trans. Wireless Commun., vol. 2, no. 4, pp. 630–643, Jul. 2003. [8] L. Musavian and S. Aissa, “Quality-of-service based power allocation in spectrum-sharing channels,” in Proc. IEEE GLOBECOM, New Orleans, LA, USA, Nov. 30–Dec. 4, 2008, pp. 1–5. [9] L. Musavian and S. Aissa, “Effective capacity of delay-constrained cognitive radio in Nakagami fading channels,” IEEE Trans. Wireless Commun., vol. 9, no. 3, pp. 1054–1062, Mar. 2010. [10] Y. Ma, H. Zhang, D. Yuan, and H. Chen, “Adaptive power allocation with quality-of-service guarantee in cognitive radio networks,” Comput. Commun., vol. 32, no. 18, pp. 1975–1982, Dec. 2009. [11] D. Li, “Effective capacity limits of cognitive radio networks under peak interference constraints,” in Proc. IEEE ICCT, Nanjing, China, Nov. 11–14, 2010, pp. 218–222. [12] S. Vassaki, M. Poulakis, A. Panagopoulos, and P. Constantinou, “Optimal power allocation under QoS constraints in cognitive radio systems,” in Proc. IEEE ISWCS, Aachen, Germany, Nov. 6–9, 2011, pp. 552–556. [13] H. Tran, T. Q. Duong, and H.-J. Zepernick, “Queuing analysis for cognitive radio networks under peak interference power constraint,” in Proc. IEEE ISWPC, Hong Kong, Feb. 23–25, 2011, pp. 1–5. [14] H. Tran, T. Q. Duong, and H.-J. Zepernick, “Performance of a spectrum sharing system over Nakagami-m fading channels,” in Proc. IEEE ICSPCS, Gold Coast, Qld., Australia, Dec. 13–15, 2010, pp. 1–6.
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[15] H. Tran, T. Q. Duong, and H.-J. Zepernick, “Delay performance of cognitive radio networks for point-to-point and point-to-multipoint communications,” EURASIP J. Wireless Commun. Netw., vol. 2012, p. 9, Jan. 2012. [16] S. Ross, Introduction to Probability Models. New York, NY, USA: Academic, 2007. [17] Y. Zou, J. Zhu, B. Zheng, and Y. Yao, “An adaptive cooperation diversity scheme with best-relay selection in cognitive radio networks,” IEEE Trans. Signal Process., vol. 58, no. 10, pp. 5438–5445, Oct. 2010. [18] P. Yang, L. Luo, and J. Qin, “Outage performance of cognitive relay networks with interference from primary user,” IEEE Commun. Lett., vol. 16, no. 10, pp. 1695–1698, Oct. 2012. [19] H. Kim, H. Wang, S. Lim, and D. Hong, “On the impact of outdated channel information on the capacity of secondary user in spectrum sharing environments,” IEEE Trans. Wireless Commun., vol. 11, no. 1, pp. 284– 295, Jan. 2012. [20] M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channels, 2nd ed. Hoboken, NJ, USA: Wiley, 2005. [21] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. New York, NY, USA: Academic, 1994. [22] N. Mehta, V. Sharma, and G. Bansal, “Performance analysis of a cooperative system with rateless codes and buffered relays,” IEEE Trans. Wireless Commun., vol. 10, no. 4, pp. 1069–1081, Apr. 2011. [23] J. Burdin and R. Landry, “Delay analysis of wireless Nakagami fading channels,” in Proc. IEEE GLOBECOM, New Orleans, LA, USA, Nov. 30–Dec. 4, 2008, pp. 1–5. [24] F. Yilmaz, A. Yilmaz, M.-S. Alouini, and O. Kucur, “Transmit antenna selection based on shadowing side information,” in Proc. IEEE VTC Spring, Budapest, Hungary, May 15–18, 2011, pp. 1–5. [25] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. New York, NY, USA: Dover, 1972. [26] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series: Volume 4, Direct Laplace Transforms. Boca Raton, FL, USA: CRC, 1990.
Fahd Ahmed Khan (S’10) received the B.Sc. degree in electrical engineering from National University of Science and Technology, Islamabad, Pakistan, in 2007 and the Master’s degree in communications engineering from Chalmers University of Technology, Gothenburg, Sweden, in 2009. He is currently working toward the Ph.D. degree with the Computer, Electrical, and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. His research interests include performance analysis of cooperative relaying networks and cognitive radio networks. Mr. Khan received the Academic Excellence Award from KAUST in 2010 and 2011 and the Best Poster Award at the IEEE Dynamic Spectrum Access Network Conference in 2012.
Kamel Tourki (S’05–M’08) was born in Antibes, France. He received the Engineering degree in telecommunications from the National School of Engineers of Tunis, Tunis, Tunisia, in 2003 and the Master’s and Ph.D. degrees from the University of Nice Sophia-Antipolis, Nice, France, in 2004 and 2008, respectively. Since August 2008, he joined Texas A&M University at Qatar (TAMUQ), Doha, Qatar, where he is currently an Assistant Research Scientist. His current research interests include wireless communication systems, cooperative communications, and cognitive radio networks. Dr. Tourki has been a member of technical program committees and a Session Chair of different IEEE conferences, including the IEEE Global Communications Conference; the IEEE International Conference on Communications; the IEEE Vehicular Technology Conference; the IEEE International Symposium on Personal, Indoor, and Mobile Radio Communications; and the IEEE Wireless Communications and Networking Conference. He is currently a member of the Editorial Board of the Recent Patents on Telecommunications and a Reviewer for several IEEE Journals. He received the Research Fellow Excellence Award from TAMUQ in April 2011, the Best Poster Award at the IEEE Dynamic Spectrum Access Network Conference, and the Outstanding Young Researcher Award from the IEEE Communications Society for the Europe–Middle East–Africa region in June 2013.
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Mohamed-Slim Alouini (S’94–M’98–SM’03– F’09) was born in Tunis, Tunisia. He received the Ph.D. degree in electrical engineering from the California Institute of Technology, Pasadena, CA, USA, in 1998. He served as a faculty member with the University of Minnesota, Minneapolis, MN, USA, and then with Texas A&M University at Qatar, Doha, Qatar. Since 2009, he has been a Professor of electrical engineering with the King Abdullah University of Science and Technology, Thuwal, Saudi Arabia. His current research interests include the modeling, design, and performance analysis of wireless communication systems.
Khalid A. Qaraqe (M’97–SM’00) was born in Bethlehem. He received the B.S. degree (with honors) from the University of Technology, Baghdad, Iraq, in 1986; the M.S. degree from the University of Jordan, Amman, Jordan, in 1989; and the Ph.D. degree from Texas A&M University, College Station, TX, USA, in 1997, all in electrical engineering. From 1989 to 2004, he has held a variety positions with many companies, and he has over 12 years of experience in the telecommunication industry. He has worked for Qualcomm, Enad Design Systems, Cadence Design Systems/Tality Corporation, STC, SBC, and Ericsson. He has worked on numerous Global System for Mobile Communications, codedivision multiple-access (CDMA), and wideband CDMA projects and has experience in product development, design, deployments, testing, and integration. Since July 2004, he has been with the Department of Electrical and Computer Engineering, Texas A&M University at Qatar, Doha, Qatar, where he is currently a Professor. His research interests include communication theory and its application to design and performance, analysis of cellular systems, indoor communication systems, mobile networks, broadband wireless access systems, cooperative networks, cognitive radio, diversity techniques, and beyond fourthgeneration systems.