Mobility-Aware Uplink Interference Model for 5G Heterogeneous Networks

# Mobility-Aware Uplink Interference Model for 5G Heterogeneous Networks

Yunquan Dong, , Zhi Chen, ,
Pingyi Fan, , and Khaled Ben Letaief,
Y. Dong is with the Department of Electrical and Computer Engineering, Seoul National University, Seoul, Korea 151744. Y. Dong was with the Department of Electrical Engineering, Tsinghua University, Beijing, China, 100084. Email: ydong@snu.ac.kr. Z. Chen is with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada, N2L3G1, Email: z335chen@uwaterloo.ca. P. Fan is with the Department of Electrical Engineering, Tsinghua University, Beijing, China, 100084. Email: fpy@tsinghua.edu.cn. Khaled B. Letaief is with the Hamad bin Khalifa University, Qatar (kletaief@hbku.edu.qa). He is also with the Department of Electrical and Computer Engineering, HKUST, Clear Water Bay, Kowloon, Hong Kong (eekhaled@ust.hk)
###### Abstract

To meet the surging demand for throughput, 5G cellular networks need to be more heterogeneous and much denser, by deploying more and more small cells. In particular, the number of users in each small cell can change dramatically due to users’ mobility, resulting in random and time varying uplink interference. This paper considers the uplink interference in a 5G heterogeneous network which is jointly covered by one macro cell and several small cells. Based on the Lévy flight moving model, a mobility-aware interference model is proposed to characterize the uplink interference from macro cell users to small cell users. In this model, the total uplink interference is characterized by its moment generating function, for both closed subscriber group (CSG) and open subscriber group (CSG) femto cells. In addition, the proposed interference model is a function of basic step length, which is a key velocity parameter of Lévy flights. It is shown by both theoretical analysis and simulation results that the proposed interference model provides a flexible way of evaluating the system performance in terms of success probability and average rate.

{keywords}

interference modeling, heterogeneous networks, 5G, user mobility, Lévy flights.

## I Introduction

As the long term evolution/advanced (LTE/LTE-A) cellular system has been deployed all over the world and is reaching maturity, the standards bodies and industry are now organizing a timeframe to standardize the fifth generation (5G) technology, which is expected to be between 2016 and 2018, followed by initial deployments around 2020. As is expected, the network aggregate data rate will be increased by roughly 1000x from 4G to 5G [1, 2]. To achieve this ambition, 5G communication systems need more nodes per unit area besides more Hz and more bit/s/Hz per node [1]. Therefore, more and more small cells such as pico/femto/relay cells are being added to the existing network [3]. In this context, it may not be surprising to expect that in the not too distant future, the number of base stations may exceed the number of cell phone subscribers [3]. A network that consists of a mix of macro cells and small cells is often referred to as a heterogeneous network (HetNet), or DenseNets [4].

Research on HetNets dates back to the discussion on femto cells in 2008 [5] and was admitted by 3GPP LTE-A standard in 2011 [6]. HetNets is also believed to be an important part of the next generation cellular networks. By adding more and more low power small cells, the reuse of spectrum across the space is improved. At the same time, the number of users competing for resources at each base station is reduced. Note that, the spectral efficiency of modern access technologies such as LTE is already very close to the Shannon’s limit [4]. Therefore, enhancing the network efficiency by densifying the network in the spatial domain rather than user efficiency in the frequency domain would be one important step towards 5G communications.

Due to the scarcity of spectrum, lower power base stations are preferred to be deployed in the same band as macro base stations. Naturally, the interference management in HetNets becomes an unavoidable issue. In the literature, this problem has been discussed from various viewpoints. In the physical layer, the downlink co-channel interference can be modeled as an interference channel. Based on this observation, an interference canceling block modulation scheme was proposed in [7], in which interference can be canceled successively since the covariance matrix of the interference is designed to be rank deficient at each receiver. In [8, 9], joint detection algorithms and maximum-likelihood based local detections were proposed. In the MAC layer or above, related issues include: 1) frequency reuse techniques such as fractional frequency reuse [10] or soft frequency reuse [11] and optimum combining [12], 2) load balancing and power control schemes such as range extension technique in 3GPP LTE Rel-10 systems, user association schemes in [14, 15], and the proportional optimal power control in [16], and 3) fundamental research on interference modeling [19, 18, 17] that will facilitate interference management.

Among them, the authors of [17] investigated the difference as well as the equivalence among some commonly used interference models for adhoc/sensor networks, such as the additive interference model, the capture threshold model, the protocol model and the interference range model. As pointed in [17], different interference models can produce significantly different results. The uplink intercell interference modeling for HetNets was investigated in [18], in which the distribution of the location of scheduled users and the moment generating function (MGF) of their interference were found. Most recently, [19] studied the downlink interference in a HetNet using stochastic geometry theory, in which every interfering base station, locating outside of a guard zone, follows the Poisson point process (PPP). A dominant interferer was also assumed to locate at the edge of the guard zone. Together with the Gamma approximation method, the Laplace transform of total interference was given, which can be used to evaluate users’ success probability and average rate.

Although very important, previous works focused on static adhoc networks or HetNets, where the interferes are fixed. However, the uplink interference in 5G HetNets are produced by users with mobility. In addition, among those works considering user mobility in HetNets, most of them were investigating how user mobility affected handover performances [22, 23]. As a result, it is still not clear whether users’ mobility will change uplink interference model or not, which is the motivation of this paper.

Particularly, since more and more base stations are deployed in the network, each cell becomes smaller and smaller. As a result, the number of users in a cell or an interfering area is very limited. In this case, users’ mobility will have more impact on the number of users in a cell, which determines their uplink interferences to other type of users in the same area.

This paper focuses on characterizing the uplink interference in 5G HetNets based on the Lévy flights [24] moving model. In this model, each user moves one step in every time interval . Formally, when a user moves from one location to another without a directional change or pause, a flight is defined as the longest straight-line distance between its starting point and end point. Some recent studies on human mobility show that the flight length distributions have a heavy-tail tendency [25, 24]. By normalizing the flight length with a basic step length , the flight length turns to be the number of basic steps in each flight. Therefore, can be seen as an indicator of user velocity, i.e., a user with a larger moves more quickly on average.

Under the Lévy flight mobility model, the number of interferers in an interfering region varies from time to time. Particularly, the number of users in the interfering region can be modeled by a Markov process, in which the state transition probability is a function of user mobility. It is also seen that users may sometimes move out of the macro cell due to mobility. To eliminate this kind of boundary effect, we proposed a modified reflection model, which can simulate the network using a single macro cell. In this model, a user is assumed to re-enter the macro cell from the opposite edge when it reaches the cell edge, without changing its moving direction. In addition, this paper considers the uplink interference for both closed subscriber group (CSG) femto cells which serve some authenticated users only, and the open subscriber group (OSG) femto cells which admit any user coming into its coverage, by deriving their moment generating functions, mean values and variations. It will be seen from simulation results that the uplink interference is actually a constant, regardless how fast users move. It is also shown that the proposed interference model is useful in evaluating the system performance such as the probability of successful transmission and the average transmission rate.

The rest of this paper is organized as follows. The system model is presented in Section II. User’s average probability of coming into or gonging out of a small cell is presented in Section III, based on which the number of users in a small cell is formulated as a Markov Chain in Section IV. After that, the uplink interference is presented in terms of its statistics in Section V. As its two applications, the interference model will be used to evaluate user’s success probability and average transmission rate in Section VI. The obtained result will also be presented via numerical and Monte Carlo simulation results in Section VII. Finally, we will conclude this work in section VIII.

## Ii System Model

Fig. 1 presents one macro cell of a heterogeneous network. The area is covered by a macro-eNB (M-eNB), as well as some low power pico-eNBs, femto-eNBs and relays (collectively referred to as home-eNBs, H-eNBs) [20, 21]. The small cells served by these H-eNBs can either be OSG cells or CSG cells. Denote the radius of the macro cell as , the radius of a small cell of interest (a pico/femto/relay cell, OSG or CSG) as . In general, is much larger than .

Users served by H-eNBs and M-eNBs are referred to as the home users (H-UEs) and macro users (M-UEs), respectively. Since pico/femto/relay cells are much smaller than macro cells, the transmit power of H-UEs’ () will be much smaller than that of M-UEs. As a result, the uplink interference from M-UEs to H-UEs is very strong. Due to large scale attenuation and small scale fading, the received signal power at an e-NB can be given by , where is the distance between the user and the e-NB, is the pathloss exponent, and is the random channel power gain. Without loss of generality, Rayleigh fading model will be used in out simulations.

Due to large scale attenuation, both the desired signal and the interference will be attenuated greatly. Similar to the interference range model in [17], this paper assumes that only the interference from M-UEs within an interfering circle will be considered, as shown by Fig. 1. Denote the radius of the interfering circle as interfering radius , which is usually larger than the cell radius . In addition, it is assumed that users in the same small cell will access to the H-eNB in a time division multiple access (TDMA) manner, so that interference among them is avoided. Since the transmit power of H-UEs is low and the attenuation is high, interference from other small cells are also neglected.

Assume that there are users distributed uniformly in the macro cell. Assume that all the incoming traffic from the users can be absorbed by the network. Due to mobility, each user will move to a new location in every time interval , according to the Lévy flight model. In this model, each move of a user is defined as a flight. The direction of a flight is uniformly distributed among and the flight length follows the power law distribution. Particularly, the probability density function (pdf) of is given by

 fX(x)=αΔαxα+1,x∈[Δ,+∞) (1)

where falls in between 0.53 and 1.81, as shown by many human mobility traces [25, 24]. By normalizing the flight length using a basic step length , one will get with .

It is clear that users tend to take longer flights if is larger. Therefore, the general moving velocity is determined by the basic step length , for any given . In this sense, this paper will show whether user mobility will affect uplink interference, by investigating the functional relationships between the statistics of uplink interference and .

To simulate the whole network using a single macro cell, this paper proposed a modified reflection model, as shown in Fig. 2. Assume that a user moves from point along the direction . Suppose that the flight length is so large that the user tends to leave the macro cell from point . Under the modified reflection model, the user will enter the macro cell again from its opposite point on the cell edge, i.e., point , along the same direction. If the flight is so large that the user can leave the macro cell once again from point , then it will re-enter the macro cell from point , and so on. Under this model, it is noted that the number of users in the macro cell will not change.

Actually, the number of users in a small cell is a random variable. As a result, the uplink interference is also random. Let be an arbitrary chosen small cell with radius , and be the number of users in at the beginning of time interval , it can be seen that the process is a Markov chain. In order to show this, the average probability that a user moves into and goes out of a small cell will be discussed first.

## Iii Average Incoming/Outgoing Probability

Since each flight can take any length no shorter than in any direction, every user outside of a small cell of interest has the chance to come into the cell. Likewise, any user in may move out with some probability. For a user who is outside of , define the probability that it comes into after one move as its incoming probability. For a user who lies in , define the probability that it goes out of after one move as its outgoing probability. By taking average over all possible user locations, the average incoming probability and average outgoing probability can be obtained, where is the radius of and is the basic move length.

Assume that the initial location of each user is uniformly distributed in the macro cell. It will be shown by the following lemma that, the location of each user is uniformly distributed throughout the operation. In fact, this can be readily understood since each user moves in a pure random way.

###### Lemma 1.

The end point of a random flight will be uniformly distributed in the macro cell, if its start point follows the uniform distribution.

Proof: See Appendix A.

### Iii-a Average Outgoing Probability

As shown in Fig. 3, a certain user UE locates at the origin of the polar coordinate system, i.e., point . The center of the macro cell is point , . There is also a small cell (femto/pico/relay cells, CSG or OSG) centered at point . The radius of cell is . Due to the isotropic property of the circular cell, the relative position of a user to depends only on its distance to the cell center. Therefore, it is sufficient to consider users at different locations by changing the cell center of , i.e., changing .

For any given , suppose that UE moves along direction , which will intersect at points . It is seen that satisfies the following equation

 (r0+rθcosθ)2+(rθsinθ)2=R2. (2)

From (2), we have

 rθ=√R2−r20sin2θ−r0cosθ. (3)

Next, the outgoing probability can be solved case by case. In cases 1) to 3), it is assumed that the flight length is relatively small so that UE will not move out of the macro cell. The outgoing probability of going out for large flights is considered in case 4).

#### Iii-A1 Δ<R

If , it is seen that for every . Then UE can move out of if the flight length satisfies . Then we have

 Po(r0|0rθ}dθ

where represents the outgoing probability as a function of when it is conditioned on event .

If , UE will be very close to the edge. In this case, UE will move out of directly in some scenarios, since every flight is not shorter than .

Define as the angle which enables UE to reach the cell edge when the flight length is exactly . Then point lies on the curve defined by (2). Thus, can be solved as

 θ1=arccosR2−r20−Δ22Δr0,θ1∈(0,π).

It is clear that holds true if , which means that UE will certainly move out of . For any , the user can move out only if the flight length satisfies . Then the conditional outgoing probability is

 Po(r0|R−Δrθ}dθ.

#### Iii-A2 R≤Δ<2R

In this situation, UE will move out of directly in most directions, except that and . We have

 Po(r0|0rθ}dθ.

#### Iii-A3 Δ≥2R

In this situation, UE will move out of with probability 1, regardless of its moving direction and location, i.e., .

#### Iii-A4 The case when flight length is very large

Finally, It is noted that if the flight length is very large, it is possible that UE will move out of the macro cell from point and re-enter from point . In fact, UE may move out of the macro cell and re-enter for many times. The probability that UE will come back to is

 Preo(r0)=Pr{ return to Ck}=∞∑m=1Pr{rθ+2ml1−2l2

where and are the half chord length within the macro cell and small cell , respectively.

By Lemma1, the location of UE is uniformly distributed in the macro cell. Thus the probability that is smaller than is . Then we can get the pdf of as , which is independent from the angle.

According to taking average over and following the analysis above, the proposition below summarizes the average outgoing probability.

###### Proposition 1.

The average outgoing probability that a user in will move out of the cell is given by (5), as shown on the top of next page,

where , , and , .

### Iii-B Average Incoming Probability

As shown in Fig. 4, UE locates at the origin (point ) of the polar coordinate system. Its distance to the center of the macro cell (point , also the center of a chosen small cell () is . To evaluate the probability that UE comes into when UE locates at different locations, it is equivalent to fix the position of UE while changing the position of the cell center (point ). In addition, since their relative position depends only on the distance between them, only needs to be changed.

Any point locates on the edge of must satisfy

 (d0−ρθcosθ)2+(ρθsinθ)2=R2. (6)

Solving from this equation, we have two roots of

 ρ1=d0cosθ−√R2−d20sin2θ~{}~{}~{}andρ2=d0cosθ+√R2−d20sin2θ

which corresponds to segment and in Fig. 4, respectively.

Define as the angular coordinate of the intersection point , i.e., . It is seen that satisfies equation (6). Then we have by solving the equation.

Define as the angular coordinate of the tangent line of circle which passes the origin, i.e., . We have .

It is noted that holds true for any . Particularly, solving from , i.e.,

 arccosΔ2+d20−R22Δd0=arcsinRd0

we know that satisfies .

As shown in Fig. 4, the incoming probability is also the probability that the end point of a flight falls into . In the following part, we will discuss the incoming probability case by case.

First, assume that the flight length is relatively small and UE will come into directly.

#### Iii-B1 Δ<2R

In this case, UE has non-zero probability to enter if .

First, if , it is seen that if and if . Then UE will be in if the direction of the flight satisfies and the flight length satisfies . The corresponding conditional incoming probability is

 Pi(d0|R

Second, if , it is seen that holds true if . It is also seen that holds true if . Therefore, UE will come into if and the flight length satisfies , or and the flight length satisfies . The corresponding probability is,

 Pi(d0|√Δ2+R2

Third, if , then holds true for any and the conditional incoming probability is

 Pi(d0|R+Δ≤d0)=2∫θ3012πPr{ρ1

#### Iii-B2 Δ≥2R

In this case, UE will certainly move across unless is larger than . Likewise, the incoming probability can also be obtained, by replacing the lower limit of integrals on with .

In addition to what discussed above, UE may also come into indirectly. For example, UE starts from point along direction . If the flight is very large, UE may leave the macro cell from point and re-enter at point according to the modified reflection model. It can even leave the macro cell again from point and re-enter at point , and so on. In this case, the probability that UE will come into is given by

 Pr{come~{}in~{}indirectly}=∞∑m=1Pr{2ml1+ρ1

where is the half chord length.

If the user moves along direction instead, this probability turns to be

 Pr{come~{}in~{}indirectly}=∞∑m=1Pr{2ml1−ρ2

Finally, by taking the average over and , we obtain the incoming probability as follows.

###### Proposition 2.

The average probability that UE will come into the cell of interest is given by (9), as shown on the top of next page.

## Iv The Number of Users in Ck

Let be an arbitrary small cell of interest and be its radius. Denote the number of users in at the beginning of time interval as . Due to user mobility, will be a random variable. In fact, dominates the number of interferers in the uplink. In this section, the stochastic characteristics of will be investigated.

### Iv-a Queueing Model Formulation

Assume that the users leave at the beginning of each time interval, which is denoted by , and arrive at at the end of each interval, i.e., . Specifically, and .

By its definition, is the number of users in at time , where users arriving at between are included, and users leaving between are not included.

As shown in the previous section, is the average incoming probability and is the outgoing probabilities. Although the probability of coming into or moving out of a cell is different for different users and different locations, we assume that each user outside of may come into it with probability , and each user in will leave with probability , in the average sense. In the following part, we will denoted and by and for notation simplicity.

If , we will have users outside of . Define the probability that there will be users coming into the cell at time as

 ν(j,N−k)=CjN−kPji(1−Pji)N−k−j

where is the combination function and .

Likewise, define the probability that there are users leaving the cell at time as

 μ(j,k)=CjkPjo(1−Po)k−j

where .

Therefore, the transition probability of the Markov chain is

 pkj=Pr{ξn=j|ξn−1=k}=min(k,N−j)∑r=max(0,k−j)μ(r,k)ν(j+r−k,N−k)

where and .

Note that the upper limit of the summation can also be expressed by: . That is, is the gap to the goal of users on condition that all of other users will coming in, and can only be filled by users who will not leave . Therefore, the maximum users can leave is at most .

With , all the statistics of are hence determined.

### Iv-B Stationary Distribution of ξn

Since both and are positive and smaller than , and the Markov chain has finite states, one can readily show that the Markov chain considered here has a stationary distribution .

Define the probability generating function (PGF) of as , which will be given by the following theorem.

###### Theorem 1.

The PGF of the stationary distribution of , i.e., the number of users in , is given by

 ξ(z)=(PizPi+Po+PoPi+Po)N. (10)
###### Proof.

Firstly, the stationary distribution satisfies the following equations.

 πP=π,πe=1

where e is a row vector of ones.

For the -th element of stationary distribution , we have

 πj=N∑k=0πkpkj,j=0,1,⋯,N.

By multiplying on both sides and take the summation from to , we have

where the order of the summations is changed in (a) and variable substitution is used in (b).

Then the theorem is established by solving from the above equation. ∎

###### Remark 1.

Using the polynomial expansion to we will have

It is clear that , which means that the number of users in , i.e., is a Binomial distributed random number in the limit sense.

###### Remark 2.

Denote and . It is well known that Binomial distribution can be approximated by Poisson distribution when is very large and is very small, which will make further analysis easier.

It is known that the mean and variance of a random variable are related with its PGF through following equations

 E[X]=G′X(z)|z=1D[X]=G′′X(z)−(G′X(z))2+G′X(z)|z=1.

Then the statistics of the number of users in are given by the following proposition.

###### Proposition 3.

The average and the variance of number of users in are given by, respectively

 E[ξn]=NPiPi+PoD[ξn]=NPiPo(Pi+Po)2. (11)
###### Remark 3.

Note that both and are functions of basic step length , which is an index of moving velocity. Therefore, both and are also functions of user velocity.

## V The Randomness of Uplink Interference

Usually, M-UEs transmit power of is much higher than that of H-UEs since M-UEs are very far from the M-eNB. As a result, M-UEs’ uplink signals will be a great interference to the H-UEs nearby. In addition, this uplink interference will change randomly along time due to the following reasons.

First, each interferer is located randomly and moves randomly. Therefore, their distances to the interfered H-eNB are also random, which introduces uncertainty to the interference. Second, interfering signals suffer from small scale fading, which vary quickly along time. Last but not the least, the number of interferers is random due to user mobility. Particularly, its fluctuation is further accelerated by the miniaturization of cells. As a result, the uplink interference also has a ‘fading’ property.

However, it should be noted that the ‘fading’ of the uplink interference caused by users’ mobility is a kind of large scale fading and a slow fading. Generally speaking, the velocity of a user is 3 km/h for pedestrians and about 120 km/h if the user is in a vehicle. Therefore, the flight time is relatively large, which makes the fluctuation of the uplink interference much slower than small scale fading.

In the following part, the fading property of uplink interference will be characterized in terms of distribution and statistic moments, based on which the impact of user mobility on uplink interference can be revealed.

### V-a Uplink Interference to CSG Femto cells

In a CSG femto cell with radius , only some authenticated users within its cell coverage are allowed to communicate with the H-eNB. Those unauthenticated UEs have to be linked to the M-eNB, even if it is in . As shown in Fig. 1, each UE within the circle of interfering radius , which is referred to as , is an interferer to femto UEs in .

Let be the number of M-UEs in the interfering circle in the -th time interval. Thus it is a Binomial distributed random variable with its PGF given by (10), Theorem 1.

Denote the distance between M-UE and the femto e-NB as where m is assumed. Let be the small scale fading power gain. Let be its cumulative distribution function (CDF), is the average power gain and is the second order moment. Denote M-UEs’ transmit power as , it is seen that the instantaneous interference can be expressed as , with its moments given by following proposition.

###### Proposition 4.

The first and second order moments of the interference from a uniformly distributed M-UE within the interfering circle are

 μc=E[Icj]=2PmtPγ(Rβ−2I−1)(β−2)Rβ−2I(R2I−1)μ(2)c=E[I2cj]=(Pmt)2P(2)γ(R2β−2I−1)(β−1)R2β−2I(R2I−1). (12)
###### Proof.

Since M-UE is uniformly distributed within the interference circle, the probability that its distance to the H-eNB is less than is . It is readily obtained that the pdf of is . Next, the first and second order moment of the interference can be obtained readily by taking its average over and . ∎

###### Remark 4.

In the case of , (12) holds in the limitation sense. That is,

 μc=limβ→22PmtPγ(Rβ−2I−1)(β−2)Rβ−2I(R2I−1)=2PmtPγR2I−1limβ→2Rβ−2I−1(β−2)1Rβ−2I=2PmtPγR2I−1lnRI.
###### Remark 5.

Actually, both and are decreasing with pathloss exponent , which can be proved by checking their derivatives versus .

Since there are M-UEs within the interfering circle, the total interference will be

 Ic=ξiii∑j=1Icj=ξiii∑j=1γjPmtdβj.

Define as the Moment generating function (MGF) of each individual interference. Then the MGF of the total interference and its average and variance are summarized by the following theorem.

###### Theorem 2.

The MGF of the uplink interference to CSG femto cell UEs is

 GIc(s)=(PiGIcj(s)Pi+Po+PoPi+Po)N. (13)

Its average and variance are given by, respectively

 E[Ic]=NPiPi+PoμcD[Ic]=NPiPi+Poμ(2)c−NP2i(Pi+Po)2μ2c. (14)
###### Proof.

By its definition, one has

 GIc(s)=E[esIc]=E[es∑ξiiij=1Icj]=N∑k=0Pr{ξiii=k}(E[esIcj])k=ξ(GIcj(s))

where was given by (10). This proves (13).

Then the average uplink interference will be

 E[Ic]=G′Ic(s)|s=0=NPiPi+Poμc.

Similarly, its second moment is

 E[I2c]=G′′Ic(s)|s=0=N(N−1)P2i(Pi+Po)2μ2c+NPiPi+Poμ(2)c

where and are given by Proposition 4.

Therefore, the variance of uplink interference is

 D[Ic]=E[I2c]−E2[Ic]=NPiPi+Poμ(2)c−NP2i(Pi+Po)2μ2c.

###### Remark 6.

It is known that the pdf of a random variable is completely determined by its MGF [27]. Thus Theorem 2 gives a full characterization of the uplink interference to a CSG femto cell. Explicit expressions for can also be obtained for any given .

###### Remark 7.

Two key parameters for the results in Theorem 2 are and . While specifies the interfering area, indicates the mobility of users. Therefore, this theorem has presented how user mobility affects the uplink interference.

### V-B Uplink Interference to OSG Femto cells

OSG femto or pico/relay cells will admit every users coming into their coverage. Therefore, only M-UEs outside the cell but within the interfering radius will cause interference.

Assume there are users in all within the circular area of radius , in which users locates within . Thus the number of interferers in the interfering ring is .

Let be the distance between an interferer and the H-eNB. Its interference to H-UEs is and the total interference is

 Io=ξii∑j=1Ioj=ξii∑j=1γjPmtdβj.

First, the first and second moments of are given by the following proposition.

###### Proposition 5.

The first and second order moments of the interference from a uniformly located M-UE within the interfering ring are

 νo=E[Ioj]=2PmtPγ(Rβ−2I−Rβ−2)(β−2)Rβ−2IRβ−2(R2I−R2)ν(2)o=E[I2oj]=(Pmt)2P(2)γ(R2β−2I−R2β−2)(β−1)R2β−2IR2β−2(R2I−R2). (15)

The proof of Proposition 5 is similar to that of Proposition 4 and is omitted here. It can be proved that and are also decreasing with .

Define for , it is clear that is decreasing with and . By comparing (12) and (15), we have and .

Define as the MGF of the instant interference from M-UE, the MGF of and its average and variance are given by the following theorem.

###### Theorem 3.

The MGF of the uplink interference to OSG femto cell UEs is

 GIo(s)=(Pi(qGIoj(s)+1−q)Pi+Po+PoPi+Po)N. (16)

The average and variance are given by, respectively

 E[Io]=NPiqPi+PoνoD[Io]=NPiqPi+Poν(2)c−NP2iq2(Pi+Po)2ν2c (17)

where , and are calculated with and .

###### Proof.

For any user who has moved into the interfering circle , its location is uniformly distributed in the area by Lemma 1. Thus its probability of lying in the interfere ring is . Then the probability that there are users in the interfering ring is

 Pr{ξii=k}=N−k∑i=0Pr{ξi=i,ξiii=k+i}=N−k∑i=0Pr{ξiii=k+i}Ckk+iqk(1−q)i.

Next, the MGF of is

 Gξii(z)=N∑k=0zkPr{ξii=k}=N∑k=0zkN−k∑i=0Pr{ξiii=k+i}