Outage and Throughput Analysis of Cognitive Users in Underlay Cognitive Radio Networks With Handover

Interference characterization in cognitive networks with handover has received less attention in stochastic geometry-based interference management and control, especially in cognitive radio networks, because of the possibility of complicating the analysis of various performance metrics of interest, such as outage probability and throughput. However, because of the possible mobility that is observed in real practical systems, some of the receivers may be located outside the coverage regions of their paired transmitters. In order to ensure that any receiver located outside the coverage region of its paired transmitter continues to receive its required service from its paired transmitter while still achieving tractable analysis for various performance metrics of interest, we adopted multiuser diversity via packet relaying. With this approach, any secondary nodes waiting to transmit can be used to sustain coverage between any typical active transmitter and receiver pair, while reducing their own waiting period in the process. We obtained tractable analysis for outage probability, spectral efficiency and throughput and showed the effect of handover rate over the network performance. The outcomes of the numerical results show that the proposed approach is capable of improving the overall network performance by improving coverage and throughput among network users in the cognitive radio networks.


I. INTRODUCTION
The continuous demand for an effective and efficient data transmission service has necessitated the exploration of the benefits of the cognitive radio network (CRN) in tackling the issue of spectrum scarcity. CRN provides an opportunity to accommodate more users on the network by allowing unlicensed users, called secondary users (SUs), to make use of channels assigned to licensed users, also known as primary users (PUs), through channel sensing. With more users now expected on the network, interference becomes an important factor, which if not properly managed can derail the essence of CRN. Hence, interference management and control have been receiving great attention in the last two decades.
The associate editor coordinating the review of this manuscript and approving it for publication was Wei Wang .
One useful approach to characterize interference in a wireless communications network is the adoption of stochastic geometry (SG), because of its ability to produce tractable analysis, especially when nodes are assumed to be independently and identically distributed following the Poisson point process (PPP) [1], [2]. Subsequent efforts have considered the introduction of exclusion regions around active PUs in which no SU is allowed to transmit. A similar concept is now being applied at secondary networks to enhance SUs' quality of service (QoS). With the introduction of exclusion regions in CRN, coverage in both primary and secondary networks can be improved at the expense of SUs' spectral usage efficiency, albeit under the assumption that all users are static. In order to capture the possible handover in the system modeling while still achieving an acceptable level of coverage and spectral efficiency for low-mobility users, multiuser diversity can be adopted in the domain of CRN. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Multiuser diversity not only allows efficient usage of spectral resources, but can also improve network performance through the creation of diverse independent transmission paths between any typical transmitter and receiver pair. Mobility can result in dynamic network topology over time, which may improve network throughput [3]. While the introduction of mobility is expected to improve overall network performance, its adoption in SG-driven interference management and control techniques has received less attention in CRN because of the possibility of complicating the analysis of various performance metrics of interest, such as outage probability and throughput. However, in practical systems, some users may be located outside the coverage regions of their paired transmitters owing to possible mobility. Investigating the impact of such a scenario on the network performance in CRN is hence very important. The coverage region is defined as the area within which the transmission signal power of any test transmitter is strong enough to ensure successful reception at its paired receiver. In this paper, this coverage region is the same as the protection region.
Users located outside the coverage region because of mobility can now take advantage of multiuser diversity to improve spectral usage efficiency. With mobility, several issues, such as spectrum handoff and channel selections, can complicate analysis [4], while characterizing interference in both primary and secondary networks is also more difficult, especially when the assumption of independence among users is relaxed. Various mobility issues in CR cellular networks were considered in [4], [5]. As a typical mobile user moves away from its source, there is a possibility that such a user traverses among diverse networks with different transmission frequencies, leading to frequent handoff -an occurrence that heavily deteriorates users' QoS [6]. Management of handover in dense cellular networks [7], [8] and dense 5G networks [9] was considered, using coverage probability and throughput to identify the effects of handover on the network, while the two-tier cellular network velocity aware handover management scheme considered in [10] is capable of sacrificing the best signal-to-noise plus interference ratio (SINR) association so as to reduce the handover rate and its effect. In these scenarios, SINR was derived based on stationary PPP analysis [7], [8], [10] to ensure tractability and was verified through simulations in order to demonstrate its accuracy, with the test user assumed to move at a constant velocity [8], [10]. It is worth noting that these efforts are based on the cellular network in which a typical mobile user is able to change its association whenever it moves away from the serving base station (BS) towards another BS. This idea allows the mobile user to initiate an association with a target BS if it is closer than the serving BS and terminate its existing association with the serving BS. Such a method means any typical mobile user can get its required service or connection from any active BS, an approach that is not suitable in a typical cognitive network considered in this work, where a tagged receiver can only get its required service through either direct or indirect association with its corresponding tagged transmitter. In such a case, a typical receiver cannot be disconnected from its paired transmitter.
Multiuser diversity has been demonstrated to be aided by packet relaying in wireless networks [3], [11]- [22]. In these works, the source and destination were demonstrated to be aided by a full duplex multiple input multiple output (MIMO) relaying node in [11], [22], time switching based half duplex relaying techniques [12], vertical cooperative relay [13], [16], [17], incremental relaying [21], cooperative relay [23], and full duplex based two-way amplify and forward relaying [24]. With the introduction of relaying techniques, several metrics of interest, such as success probability and ergodic capacity [11], outage probability [12], [13], [24], symbol error probability and channel capacity [24], can be used to evaluate the performance of the network. The relaying technique can be implemented as half duplex radios [12], [13], [18], full duplex radios [24], decode and forward nodes [12], [18], amplify and forward nodes [24], time switching relaying [16] etc, depending on the interests of the authors. Although full duplex relaying is expected to be more efficient than the half duplex technique, there is a possibility of self-interference in the former. The introduction of relaying techniques also enhances the idea of cooperative communication known as virtual MIMO techniques [24], which can improve network performance even when users are mobile. This cooperative communication can be in the form of horizontal cooperation or vertical cooperation [13]. In horizontal cooperation, PUs or SUs cooperate among one another to increase transmission efficiency and throughput, while vertical cooperation ensures cooperation between PUs and SUs so as to aid the transmission opportunity probability for SUs.
In this work, we modelled interference in underlay CRN with handover using the stochastic geometry approach, considering the case where primary and secondary receivers may be located outside the coverage region of their tagged transmitters owing to mobility. To the best of our knowledge, such an area has received less attention, especially when characterizing interference using SG in the domain of CRN. The closest efforts to our work are the work of [3], where multiuser diversity was considered in mobile ad hoc networks, [13], [14] where vertical cooperation in static CRN was considered and [11], where relay nodes aid source and destination communications in a static MIMO network. In [25], we characterized interference at the mobile receivers in CRN. The analysis, however, did not capture important mobility factors such as spectral efficiency, handover rate and handover cost.
For any communication link in a mobile network, the time-variant nature of the channel strength is an important characteristic. Such variation can be well accommodated through the use of diversity [3]. In this paper, we adopted multiuser diversity in order to keep a typical receiver with low mobility within the reach of its corresponding transmitter when such a receiver moves away from the coverage region of its transmitter. The main contributions of this paper are thus summarized as follows; • We modelled the association between primary transmitters (PTs) and primary receivers (PRs) as well as secondary transmitters (STs) and secondary receivers (SRs), taking into consideration that PRs and SRs may be located within or outside the coverage area of their respective paired PTs and STs. Hence, when a tagged receiver is located within the coverage region of its corresponding transmitter, communication between the pair is achieved following the direct transmission method without the help of the relaying technique. However, when the tagged receiver is located outside the coverage region of its paired transmitter owing to mobility, communication between the transmitter-receiver pair is achieved through the packet relaying technique. In such a case, a tagged transmitter is considered to select any waiting secondary node that produces the highest signal at the tagged receiver through the multiuser diversity technique -a technique that can lead to handoffs. Multiuser diversity via packet relaying for mobile nodes was first proposed in [3].
• With the adoption of multiuser diversity, we obtained expressions for SINR for two cases: when multiuser diversity is adopted in a primary network (via vertical cooperation) for the purpose of reducing outage and when it is adopted in a secondary network (via horizontal cooperation) to improve spectral efficiency.
• Based on the SINR expressions derived for each of the cases considered, we carried out outage probability analysis in both primary and secondary networks. Our analysis takes into consideration the dependence between the distribution of PUs and SUs.
• Following the analysis of outage probability in both primary and secondary networks, we derived expressions for spectral efficiency and throughput while considering the effects of vertical handoff [26], [27] and horizontal handoff [26] through analysis of the handover rate and handover cost in both networks. The remainder of this paper is structured as follows: In Section II, we present the details of the system model, while the analysis of SINR and outage probability are presented in Section III. In Section IV, an analysis of the network throughput is presented. Section V presents numerical results obtained through simulations, while Section VI concludes the paper and offers suggestions for future research. The summary of the notations used in this paper is given in Table 1.

II. SYSTEM MODEL
In a typical CRN where SUs are allowed to make use of PUs' channels as long as their transmissions do not cause excessive interference at the primary network, some of the users are expected to be mobile when a practical system is considered. As a result of the mobility involved, there is constant change in the network topology, though the distribution of users is expected to remain unchanged. Since PU transmitters in CRN are normally radars, TV stations and cellular base stations licensed to use their respective assigned channels, we considered such devices to be located in a fixed location (or at least restricted to within their protection regions when mobile in order not to violate channel allocation policy), while any typical PR located outside the coverage of its paired PT is likely to experience loss of connection from its paired PT. Such a PR can take advantage of vertical cooperation to sustain its association or connection with its corresponding transmitter. However, STs make use of the spectrum in an opportunistic manner and a typical ST is not expected to move off its protection region, though communication can be sustained with the mobile SR located outside the coverage of its paired transmitter via horizontal cooperation.
Without loss of generality, we considered a transmittercentric scenario as shown in Fig. 1, in which a typical PT is located at the center of a disk (known as the protection region) VOLUME 8, 2020 of radius D (see Fig. 1a and Fig. 1b) and a typical ST is located at the center of a disk of radius d (see Fig. 1c and Fig. 1d). It is worth noting that transmitters (either PTs or STs) are not necessarily located at the center of their respective protection zones, but this has been assumed for simplicity. This, however, is not a limitation on the distributions of users. We considered the protection zone of each transmitter to be the same as the coverage area of such transmitter [28], [29]. This implies that the signal generated by a typical transmitter is not strong enough outside its protection zone to satisfy the SINR requirement.
The introduction of a protection zone of radius D around each active PT means the distribution of active PTs can be best represented as the Matern hard core process. However, because its probability generating function is not available [2], [30], the distribution of active PTs was approximated as a PPP p of intensity λ p , obtained through an approximation using equidense PPP, while each mobile PR y p i was initially considered to be located at a random distance r p On the other hand, STs are distributed according to independent PPP s of intensity λ s and are not allowed to be active inside any PU's protection zone, or inside any currently transmitting ST's protection zone, in order to avoid interference at the primary network while improving SUs' QoS. Hence, the locations of active STs follow a modified version of the Poisson hole process m s of intensity λ m s while each active mobile SR is similarly considered to be initially located at a random distance r s following the uniform distribution given as f r s (l) = 2l d 2 , ∀l ≤ d around the location of its corresponding transmitter.
When a typical ST x s k ∈ s is not eligible to transmit owing to its location, such an ST can serve as a decode and forward relaying node in either vertical or horizontal cooperation in order to improve channel efficiency while reducing its own waiting period in the process. As a tagged mobile receiver moves away from the coverage area of its pair tagged transmitter, the probability of outage increases. In order to sustain communication between the tagged transmitter and the tagged mobile receiver, the tagged transmitter through the multiuser diversity technique selects any waiting secondary node within its coverage region that generates the highest signal power 1 w max i is the signal power generated by any ST x s i ∈ s . The tagged transmitter hence sends its intended message to the selected relaying node(s) in the first hop. These selected nodes are known to be closer to the tagged receiver and provide higher capacity than the tagged transmitter under the Rayleigh fading of unit mean. During the second hop, each selected relay node (RN) sends the received packet to the intended receiver. Since the RNs are expected to be located closer to the tagged receiver (PR or SR) than the tagged transmitter (PT or ST) in this case, a better connection is achieved, provided that Rayleigh fading of unit mean is assumed. We further assumed that all RNs are capable of re-routing the received signal following the time switching based half-duplex technique, as in [12].
According to the random waypoint model [31], [32] in a discrete-time based system, each typical low mobility receiver uniformly selects its destination point D k randomly within the deployment region, and moves to such destination by uniformly selecting a velocity v, ∀v ∈ [V min , V max ]. At D k , such a mobile receiver remains static for a predefined time t p , before proceeding to the next destination following the same rule. Each receiver is considered to move independently, while updating its position at the beginning of each time slot. While the initial location of any tagged receiver can be assumed to follow a uniform distribution within the coverage region of its paired transmitter, the distribution of mobile receivers following a random waypoint model is non-uniform at the steady state [31]. The implication of non-uniformity of such a model is reduced in our proposed method, since the tagged receiver is not always located outside the coverage region of its paired tagged transmitter and can in fact be static during its transmission period.
The proposed approach can be summarized as follows: • When a typical mobile PR is located at a distance r p ≤ D from its paired PT, direct transmission is maintained between the primary transmitter-receiver pair.
However, when such a tagged PR is located outside the coverage region of its paired PT, the SINR threshold requirement for coverage at the tagged PR may not be satisfied via direct transmission. Hence, the decode and forward ST nodes are used to sustain communication between the tagged PT and PR. The process is the same in the case of an ST and SR.
• To avoid interference resulting from multiple movements of packets from one node to another, the hop limit is restricted to two. A similar restriction was imposed in [33]. We further assumed that any selected relaying node would only be able to decode the destination address of the packet, but not the message itself, so as to ensure data security. 2 In such a case, a typical transmitter will transmit to the RN during the first hop, while each selected relay will transmit to the intended receiver during the second hop, provided there is at least one ST waiting to transmit within the coverage of such a typical transmitter and along the trajectory of the typical mobile receiver. We assumed that these RNs are capable of re-routing the received packet under vertical and horizontal cooperation by a distance of radius d.
• In a case where multiple potential RNs are located in the direction of a typical mobile receiver, the near destination relay selection rule [13] through the multiuser diversity technique is followed.
• If there is no ST waiting to transmit within a tagged transmitter's coverage and along the trajectory of a tagged mobile receiver, the transmitter maintains direct association with the receiver. It is worth noting that higher ST intensity will produce higher spectrum opportunities, though with higher handoff rates. This multiuser diversity based approach can be represented in terms of the presence of RN within the coverage region of any active transmitter as: where By neglecting possible overlapping of protection regions in the primary network, a typical ST waiting to transmit within the coverage of a tagged PT cannot be located in the coverage region of another PT. Hence, any RN can serve at most one mobile PR at a time. Similarly, we assumed that a typical RN can only serve at most one mobile SR at a time. Hence, For primary transmission, the packet either moves from the tagged PT to the tagged PR under direct transmission or from any typical PT through the selected RNs to the paired PR following vertical cooperation. Similarly, for secondary transmission, the packet moves from any typical ST to its intended SR under direct transmission and from any typical ST to its intended SR via an RN following horizontal cooperation. Although any tagged low mobility receiver (PR or SR) is expected to move with a velocity v, ∀v ∈ [V min , V max ], we considered V min ≥ 0 in order to capture the situations where such a receiver may be static, such that the probability density function of V [34] is given as From this, the time in which any tagged low mobility PR resided in the coverage region of its paired PT and the time such PR spent in the coverage region of any selected RN can be known. Similarly, the time spent in the coverage region of its tagged ST and any selected RN by any tagged low mobility SR can be determined. From [34]- [36], we know that the expected mean time spent by the low mobility receiver in the coverage region of its tagged transmitter expressed as and the expected mean time spent by such a receiver in the coverage region of the selected RN expressed as depend only on E[ 1 V ] and E[V ] respectively [36]. Next, we derive the analysis of outage probability in both primary and secondary networks.
Assumption 1: We assumed stationary point processes to model the distribution of users. Hence, the expressions for the outage probability are independent of velocity v.
Remark: Incorporating velocity v into the analysis of outage probability is very difficult and can complicate the analysis. Hence, the network topology was assumed to remain unchanged in each time slot and independent of the other time slots. The same assumption was made in [7], [8], [10] for cellular networks and was verified to be accurate through simulations.

III. ANALYSIS OF OUTAGE PROBABILITY
The outage probability is derived by determining the probability that the signal power received at a typical receiver is not above the predefined SINR threshold θ (i.e. P(SINR ≤ θ)). We derive the outage probability for PU and SU in the following subsections.

A. PRIMARY USER OUTAGE PROBABILITY
In order to derive an expression for the PT outage probability, two cases are considered -direct transmission between PT and PR and transmission from PT to PR via the selected relay nodes, depending on the required capacity, i.e. the number of packets n at any transmission time of PT. Generally, the PU outage probability can be given as where p z is the probability that there is at least one ST waiting to transmit within the coverage area of the typical PT and along the trajectory of the typical mobile PR, ε p is the PT outage probability under direct transmission and E(ε ind ) is the expected PT outage probability when a tagged PT transmits to its intending PR via RN(s). Although p z ∈ [0 1], its derivation can likewise be obtained from the Choquet capacity of a PPP [37] as p z = 1 − exp(−λ s πD 2 ), where the definition of D is provided in [38] as where θ p is the SINR threshold requirement for PUs' transmissions, ζ is a design factor [38] and η is the path loss exponent.
where I pp = x s i ∈ s P r c i,k ||y p i || −η is the interference from active RNs at the tagged PR. P p is the transmit power of PT, c k,k is the fading coefficient between the tagged transmitter and its intended receiver, ||y p k || is the Euclidean distance between the tagged PT and its intended PR, and σ 2 represents the Gaussian noise. Also, c i,k is the fading coefficient between other active transmitters and the tagged PR, ||y p i || is the Euclidean distance between other active transmitters and the tagged PR, P s is the transmit power of ST and P r ≈ P s is the RN transmit power. We assumed that each channel multi-path fading is Rayleigh distributed with the unit mean (i.e E[c] = 1). The outage probability in this case is given in the following proposition.
Proposition 1: The outage probability at any tagged PR, given that the location of such a tagged PR is within the coverage area of its pair PT, can be expressed as where f (v) =  [7], [10]. The derivation presented in Proposition 1 is straightforward from (7). The proof is summarized in Appendix A for brevity.
At η = 4, the ε p can be simplified to 2) INDIRECT TRANSMISSION BETWEEN PRIMARY TRANSMITTER-RECEIVER PAIR VIA VERTICAL COOPERATION As a result of its ability to be mobile, a tagged PR may be located outside the coverage region of its paired PT. In such a case, the PT transmits to the PR through a relaying node or set of relays using a message splitting technique [15]. When message splitting is used, the PT message is divided into segments in order to meet the selected relay's capacity requirements. Any typical transmitter is able to generate the distance through signal strength from its paired receiver similar to the reference signal received power and reference signal received quality techniques in 4G long-term evolution. Assumption 2: We assumed that the PT's message is divided into n number of segments depending on the n number of packets required to be transmitted. If packets do not arrive sequentially at the intending receiver, they simply wait to be re-assembled.
When the primary transmitter and receiver pair communicate via RN(s), outage is possible in two conditions. The first is the case in which the message is lost during the first hop (i.e. SINR I R k ≤ θ p ) at any selected RN. This means that any of the selected RN(s) cannot decode the message from the PT, hence the message is lost during the first hop, i.e. N n=1 SINR I R k ≤ θ p , where n = 1..N is the number of selected RNs. This implies that the intended PR cannot receive the PT message during the second hop. In such a case, the SINR received at the intended PR located outside the coverage of its paired PT in each hop is SINR I y p k θ p . The other case is when the message is lost during the second hop, (i.e. SINR II y p k ≤ θ p ), which means all the selected RN(s) can decode the PT message (i.e. N n=1 SINR I R k > θ p ); however, the intended PR cannot decode the received PT message during the second hop. For simplicity and following Assumption 2, we assumed that each typical PT selects only one RN (i.e. has only one packet to send at any time, n = 1). The joint outage probability can then be obtained following the information theory formula [13], [21] as P(I c ≤ R), where θ p = 2 R − 1, R being the rate at which the information is being sent [13] and I c being given as From (10), the expected outage probability under indirect transmission between any primary transmitter-receiver pair 208050 VOLUME 8, 2020 via SU relay ε ind is obtained as The assumption of no correlation between the two hops as in [13] means (11) can be solved by deriving the outage probability expression for each of the segments involved. Hence, the SINR at any selected relay during the first hop is given as where ||y R k || is the Euclidean distance between the tagged PT and the selected RN, I pr , I RN and I rr are the interference from other active PTs, interference from active STs and interference from active RNs at any typical selected RN (R k ) respectively. Note that the location of the RN is actually within the coverage area of the tagged PT. This implies that the SINR at R k is the same as SINR y p k presented in (7). Hence, P(SINR I R k ≤ θ p ) = ε p . The next step is to derive the outage probability analysis for P(2SINR I y p k ≤ θ p ) = ε 1 .
By Rayleigh fading assumption, (13) can be expressed as where D 1 = D + d. The derivations of L I pp , L I sp and L I rp are similar to the ones in Proposition 1 except that the Laplace transforms (LTs) were taken at s = θ p 2P p l −η . The last part of (11) is P(SINR I y p k Let the distance between the tagged PT and the selected RN during the first hop be r p2 and the distance between the selected RN and the tagged PR during the second hop be r R in any typical two consecutive time slots; ε 2 is given as {exp(−s 1 σ 2 )L I pp (s 1 )L I sp (s 1 )  (15), which is given as P(P p c k,k r −η p2 + P r c k,k r −η R ≤ ), where = θ p (σ 2 + I pp + I sp + I rp ). By substituting (8), (14) and (16) into (11), (5) can be obtained.

B. SECONDARY USER OUTAGE PROBABILITY
The same process as in the analysis of PT outage probability was followed to derive ST outage probability under direct and indirect associations. For direct association, a typical ST transmits directly to its paired SR, while the transmission from ST to SR is directed via RN during indirect association. SU outage probability is thus given as where p f is the probability that there is at least one ST waiting to transmit within the protection region of the typical ST and along the trajectory of the typical mobile SR, ε s is the outage probability under direct transmission between the secondary transmitter-receiver pair and E(ε relay ) is the expected outage probability when a tagged ST transmits to its intended SR via the selected RN.

1) DIRECT TRANSMISSION BETWEEN SECONDARY TRANSMITTER-RECEIVER PAIR
In this case, a typical ST x s k ∈ m s sends a direct message to its intended SR y s k considered to be located within its paired ST coverage area, otherwise known as the protection region of radius d. The SINR at the SR can be expressed as where I ss = x s i ∈ m s \x s k P s c i,k ||y s i || −η is the interference from other active STs at the tagged SR, I ps = x p i ∈ p P p c i,k ||y s i || −η is the interference from active PTs at the tagged SR and I rs = x s i ∈ s P r c i,k ||y s i || −η is the interference from active RNs at the tagged SR. ||y s k || is the Euclidean distance between the tagged ST and its intended SR and ||y s i || is the Euclidean distance between any transmitter and the tagged SR. The outage probability in this case is given in Proposition 2.
Proposition 2: The outage probability at any tagged SR, given that the location of such a tagged SR is within the coverage area of its pair ST, can be expressed as where θ s is the SINR threshold for SU and z = θ s P s r −η s . Proof: The proof is presented in Appendix B. VOLUME 8, 2020 At η = 4, the close form of (19) can be obtained as

2) INDIRECT TRANSMISSION VIA HORIZONTAL COOPERATION
A tagged SR may be located outside the coverage region of its paired ST owing to mobility. In such a case, the ST transmits to the SR through any selected RN, hence outage is possible under two scenarios: if outage occurs during the first hop (i.e. SINR R k ≤ θ s ) and if outage occurs during the second hop (i.e. SINR R k > θ s ). The joint outage probability can then be similarly obtained following the information theory formula given as The outage probability ε relay is thus given as From (22), we can obtain the SINR at any selected RN during the first hop as where ||y R k || represents the Euclidean distance between the tagged ST and the selected RN, while I RN = x s i ∈ m s \x s k P s c i,k ||y R i || −η , I pr = x p i ∈ p P p c i,k ||y R i || −η and I rr = x s i ∈ s P r c i,k ||y R i || −η are the interference from other active STs, interference from active PTs and interference from other active RN at the selected RN R k respectively. Similarly, the location of the RN is actually within the exclusion region of the tagged ST, hence the SINR at R k is the same as the one presented in (18). Hence, P(SINR I R k ≤ θ s ) = ε s . Next, we derive the analysis for P(2SINR I y s k ≤ θ s ) = ε 3 .
Likewise, the derivations of L I ss , L I ps and L I rs are similar to the ones in Proposition 2 except that each LT was taken at z = θ s 2P s l −η . To obtain analysis for P(SINR I y s k + SINR II y s k ≤ θ s ) = ε 4 , let the distance between the tagged ST and the selected RN during the first hop be r s2 and the distance between the selected RN and the tagged PR during the second hop be r R in any typical two consecutive time slots; ε 4 is given as {exp(−z 1 σ 2 )L I ss (z 1 )L I ps (z 1 ) where (26), which is given as P(P s c k,k r −η s2 + P r c k,k r −η R ≤ s ), where s = θ s (σ 2 + I ss + I ps + I rs ).

IV. ANALYSIS OF THROUGHPUT
Following the analysis of outage probability presented in the previous section, we obtained the analysis of average throughput in this section. According to [7], the average throughput is given as where B, S E and D H are the overall bandwidth of the channel, average spectral efficiency, and handover cost (defined as the fraction of time in which no data are transmitted to the tagged receiver owing to the handover process) respectively. Note that handover occurs when a typical transmitter initiates a connection with any selected RN in order to sustain communication with its paired receiver. The average spectral efficiency is given as [2] where (a) is obtained by making variables x = e z − 1. At the primary network, the average spectral efficiency can be expressed as Similarly, the average spectral efficiency at the secondary network is given as As a typical mobile receiver moves along its trajectory, any selected RN may be used to sustain its association with its typical transmitter. This results in vertical handoff (when used in the primary network) and horizontal handoff (when used in the secondary network). It is worth noting that the proposed approach only makes use of ST as relaying nodes. Hence, D H is simplified as given in [16] as where H o is the handover rate and d e is the handover delay. At constant velocity v with no pause time assumptions, H o [39] is given as The average throughput in the primary network can be obtained by substituting (30) into (28), hence, At η = 4, the average throughput in the primary network can be expressed as Similarly, the average throughput in the secondary network can be obtained by substituting (31) into (28). This is expressed as This, at η = 4, can be obtained as

V. NUMERICAL RESULTS
We now carry out the numerical simulations of the analytical approach presented in the previous sections in order to demonstrate the performance of the analyses. We carried out the Monte Carlo simulation averaged over 50 000 channel realizations in order to validate the presented analytical approach. Except when stated otherwise, the following parameters were used in the simulations: P p = 0 dB, P s = −32 dB, η = 4, σ 2 = −180 dB, λ s = 0.3, λ p = 0.03, θ s = θ p = 3, r p = 0.5, r s = 0.1, m = 81, d ≈ 0.295 and the channel bandwidth was set at B = 1 MHz.
The effects of the SINR threshold on outage probability as presented in Fig. 2 shows that the SU outage probability is directly proportional to the SINR threshold required to guarantee coverage between any typical transmitter and its paired receiver. The relationship is the same for PUs. Interestingly, the proposed model is able to further reduce the outage probability, hence improving the user's coverage when the SINR threshold increases while other parameters remain constant. With an increase in the intensity of RNs inside the  coverage region of the tagged receiver, the analysis showed that coverage can be improved. Depending on the user's QoS requirement, the proposed model can help a typical transmitter to meet the SINR requirement at the intended receiver especially when the intended receiver is located outside the coverage region of such typical transmitter. Although the analysis of the outage probability for each network was based on the use of stationary point process analysis, simulation results show that such analysis accurately captures the performance of mobile users.
At the constant SINR threshold, an increase in PT transmit power reduces outage in the primary network since an increase in transmit power at any typical PT is expected to improve coverage at its pair receiver provided that interference received at such a receiver remains constant. An increase in the transmit power of PT hence continues to produce a decrease in PU outage probability until a point at which such an increase has a non-significant impact on the outage owing to other channels' parameters such as interference, shadowing and fading. Improved performance was observed when p z = 1, as shown in Fig. 3. The presence of RN in the coverage area of any typical PT further reduces outage in such a primary network.
The impact of PU transmit power on SU outage probability as presented in Fig. 4 also shows that SU outage probability  tends to increase with an increase in PT transmit power owing to an increase in interference from active PTs received at the tagged SR, while the outage probability at the secondary network reduces with an increase in STs' transmit power, as shown in Figure 5, because of the possibility of meeting the SINR threshold required at the tagged SR when the tagged ST transmits with higher transmit power, provided that channel conditions such as fading and shadowing remain unchanged while P s << P p . Interestingly, the presented analysis shows improved performance with a higher value of p f in the secondary network and p z in the primary network.
Similar analyses were obtained in [13], [14] for the primary network. Since the intra-network interference within the primary network was not captured in [14], the analysis is expected to underestimate the outage probability as presented in Fig. 6. Similarly, the approximate analysis presented in [13] produces a lower outage probability compared to our analysis in Proposition 1 owing to the fact that intra-network interference in the secondary network was not captured in [13]. Likewise, the performance at the secondary network can be significantly improved with both inter-network and intra-network interference controls in the secondary network. Similar observations were made under indirect transmission as shown in Fig. 7. Generally, an increase in D further reduces the outage probability.   The effect of mobility on the performance at the primary network is presented in Fig. 8. When any tagged PR is moving at a velocity more than 2.4 km/h, the infeasible region is reached, given that the handover delay is 2 seconds. Similarly as shown in Fig. 9, the infeasible region is reached when the average velocity at any tagged SR reaches 0.7 km/h, while the handover delay is 2 seconds. This infeasible region is also reached at a velocity of 1.5 km/h when the handover delay is set at 1 second. Generally, the throughput analysis presented in Fig. 8 and Fig. 9 shows improved performance when handover delay is reduced in the network, while users' throughput experience is affected by an increase in velocity. Similar results were obtained in [9], [10].

VI. CONCLUSION
Mobility in wireless communications networks remains an open issue despite its importance in realizing practical system modeling owing to the difficulty in obtaining tractable expressions for various performance metrics of interest. In this paper, we presented tractable system analyses for an orthogonal multiple access scenario by adopting multiuser diversity via packet relaying. Our analyses take into consideration various important parameters such as handover rate and spectral efficiency.
In order to avoid complicated analysis, we restricted the hop limits to two. This implies that a typical mobile receiver will experience outage if located outside the coverage area of the selected RN and its tagged transmitter, hence the mobility is restricted. In future, it will be interesting to allow unrestricted mobility in the network without compromising network performance. It will also be interesting to know whether integration of message splitting techniques will further complicate system analysis.

APPENDIX A PROOF OF PROPOSITION 1
The outage probability at the tagged PR is given as P(SINR y p k ≤ θ p ). where s = θ p P p l −η . The expression for L I pp is given as L I pp (s) = exp − π γ λ p (sP p ) γ π sin(γ ) , since PTs are distributed following PPP. Also, active ST cannot be located inside the coverage area of active PTs as well as inside the coverage area of other active STs. Hence, the expression for L I sp is given as while the distributions of RNs can be said to follow PPP with the expression for L I rp given as L I rp (s) = exp − π γ λ r (sP r ) γ π sin(γ ) .
From the given expressions for L I pp , L I sp and L I rp , the proof of Proposition 1 is derived.

APPENDIX B PROOF OF PROPOSITION 2
The outage probability at the tagged SR is given as P(SINR y s k ≤ θ s ). Hence, ε s = P P s c k,k ||y s k || −η σ 2 + I ss + I ps + I rs ≤ θ s , = 1 − exp −σ 2 θ s P s l −η L I ss θ s P s l −η L I ps θ s P s l −η × L I rs θ s P s l −η . Since the tagged SR is assumed to be distributed within the coverage region d of its paired ST, From these, the proof of Proposition 2 is derived.