An Analytical Model for a Sectored Cellular Network with Embedded Small Cells

In order to handle the huge amounts of data transmitted through hot spots and to increase cell coverage in buildings, this paper proposes a sectored cellular network with small cells (SCNSC) embedded in each sector. Two region types, sectors and variable-radius small cells are classified in an SCNSC macro cell. Because the system capacity is enhanced by using fractional frequency reuse (FFR), the overall channel bandwidth is divided into two or three frequency bands depending on whether the number of sectors in a macro cell is even or odd. For the purpose of topological evaluation on the proposed SCNSC, we build an analytical model using Markov chain theory. The performance measures include new-call blocking and handoff-call dropping probabilities, average waiting times, and normalized throughputs in each sector and its embedded small cell by varying mobile stations’ moving speeds, the number of sectors, and the radius size of an embedded small cell. From the evaluation results, it is demonstrated that adjusting either the number of sectors or the coverage of an embedded cell can satisfy different service demands.


Introduction
Advancements in mobile wireless communication technology have led to a tremendous growth in the number of wireless devices and the volume of data transmitted, respectively. One of the primary goals for fifth generation (5G) communication systems is to provide up to 1 Gbps data rate anywhere and anytime. This can be achieved by combining several novel communication technologies, such as massive multiple-input multiple-output (MIMO), millimeter wave (mmwave), and small cell deployment [1] [2]. In recent decades, several schemes [3] [4] [5] have been presented to alleviate the problem of high link interruption probability due to insufficient radio resource by delaying or advancing the instant at the beginning of handoff procedure. On the other hand, the authors in [6] [7] [8] [9] focused on selecting the target base station (BS) for a mobile station (MS) to reduce handoff times based on certain criteria related to MS signal strength. Further, several works [10] [11] [12] intended to lower link interruption rate with their proposed channel allocation mechanisms.
However, the previously-proposed research works have not considered the topology of a sectored macro cell with small cells. To enhance cell coverage and data rate for a large number of users, the coverage performance of small cells has been well studied. Multiple classes of BSs, macro BSs, hotspot BSs, and femtocell BSs, are installed in a heterogeneous cellular network. On the other hand, sectored antennas can be deployed to support fractional frequency reuse (FFR) in a cell to increase subscriber capacity [13]. The works in the literature [14]- [27] are divided into three categories as below.
1) A macro cell using an omnidirectional antenna is installed with multiple small cells, which are referred to as either femto cells or pico cells [14] [15] [16] [17].
3) A macro cell using a multi-directional antenna consists of multiple sectors and one or more small cells are installed in each sector [22]- [27].
As to the third category, Wu et al. [22] [23] presented a three-sectored macro cell architecture with a femtocell inside each sector and a femtocell exclusion region (FER) around a macro BS. The expression of FER radius as a function of the azimuth angle relative to the macro BS was derived, which can be directly used to divide the macro cell coverage area into inner and outer regions for cochannel assignments to the macro cell and femtocells. Further from the proceeding works, Kalbkhani et al. [24] divided a macro cell into six sectors with multiple femtocells randomly installed in some sectors. According to the amount of received interferences, they presented an algorithm for channel allocation in femtocells and a closed-form formula for femtocell downlink outage probability. Tsai [25] proposed that installing three-dimension directional antennas on small cells can mitigate two-tier interference of the ultra-dense heterogeneous smallcell network and investigated the impacts of the three-dimension directional antennas and the resource block usage ratio on the system capacity and link reliability in an apartment building. Dividing a macro cell into a center zone and an edge zone (the cell coverage beyond the center zone) of three sectors, the authors in [26] aimed to mitigate interference between a macro cell and multiple femtocells in the center and edge zones by using fractional frequency reuse (FFR) under the macro cell allocating a frequency band and the femtocells selecting the sub-bands.
Although the aforementioned schemes [22] [23] [24] [25] [26] have improved the outage probability, including the new-call blocking and handoff-call dropping probabilities, in a macro cell with multiple small cells, they have not considered the influences of MS mobility, the coverage size of a small cell, and the number of sectors on the outage probability of a call moving from adjacent cells.
Our interest in this paper is to design a sectored cellular network with small cells (SCNSC) embedded in each sector. Two region types, sectors and variable-radius small cells are classified in an SCNSC macro cell. The proposed SCNSC can increase the system capacity by using fractional frequency reuse (FFR) since two sectors that are not adjacent can allocate the same frequency band. Each frequency band is further divided into two portions assigned to sectors and small cells, respectively. For performance evaluation, we derive the new-call blocking and handoff-call dropping probabilities, average waiting times, and normalized throughputs in each sector and its embedded small cell in terms of the call generation rate, MS moving speed, the number of sectors, and the radius size of a small cell using Markov chain theory.
The remainder of this paper is organized as follows. Section 2 defines the proposed SCNSC. In Section 3, we build an analytical model for the SCNSC by using a multi-dimensional Markov chain. The analytical results and discussions are presented in Section 4. Finally, concluding remarks are given in Section 5.

System Model
As illustrated in Figure 1, a macro cell in the proposed SCNSC consists of two region types, sectors and variable-radius small cells. The coverage of a macro cell is sectored by using a multi-directional antenna and a variable-radius small cell is installed within each sector. Assuming that the maximum coverage of a small cell is an inscribed circle within a sector, a small cell coverage spanning the boundary of two sectors is not allowable.
We summarize the symbols and denotations for the SCNSC in Table 1. Let n be the number of sectors for a macro cell. R and r, respectively, represent the radiuses of a macro cell and a small cell. Ar is defined as the ratio of a small cell coverage to a sector region, as expressed in Equation (1). The system capacity is enhanced by using FFR. Therefore, the overall channel bandwidth is divided into n f frequency bands, where n f is two (three) if n is even (odd). Figure 2 shows channel allocation for the system when n is even. As shown in Figure 2(a), the system bandwidth is divided into two frequency bands in different colors and each frequency band is composed of two portions, one is for small cells and the other is for sectors. As illustrated in Figure 2(b), in order to reduce co-channel interference, two adjacent sectors allocate different frequency bands and the two embedded small cells use different frequency bands from their respective sectors. Similarly, Figure 3 shows channel allocation for the system when n is odd. Let N    (1)

Handoff Rate
As elaborated in Section 2. [ ] where L is the length of the boundary between a small cell and a sector, E[V] is the average moving speed of an MS, and A is the coverage area of a small cell or sector. As illustrated in Figure 4, three kinds of handoff rate are derived as follows.
3) Let ω sr_sr denote the handoff rate of HC sr_sr . A in Equation (2) is the same as that in Equation (4). L represents the boundary of a sector, which is equal to . However, HC sr_sr is defined as a handoff call when an MS moves from a sector to two adjacent sectors without including an MS moving out of the macro cell. Thus, L in Equation (2) is modified as expressed in Equation (5). Replacing A and L in Equation (2) with the formulas for the kind of handoff, we obtain Equation (6).

Analytical Model of the SCNSC
In order to analyze the system performance of the proposed SCNSC, we build a mathematical model using Markov chain [30]. If n is even, each state of the transition matrix is defined as a four-dimensional vector, denoted as [i, j, x, y], in which i (j) is the number of calls in the odd (even)-indexed sector and x (y) is the number of calls in the odd (even)-indexed small cell. On the other hand, if n is odd, each state of the transition matrix is defined as a six-dimensional vector, denoted as [i, j, k, x, y, z], in which i (j or k) is the number of calls in the sector indexed by a multiple of three plus one (two or zero) and x (y or z) is the number of calls in the small cell indexed by a multiple of three plus one (two or zero). Firstly, we calculate the transition rates, which contain the arrival and departure rates between any two of the states. Then, each state probability is obtained and the performance measures, including the NC blocking and HC dropping probabilities, the average waiting time, and the normalized throughput, are derived in terms of the obtained state probabilities.

Assumptions
1) MSs are homogeneously distributed in a macro cell.
2) Each new/handoff call occupies one channel.
3) The average moving speed of MSs in either of a sector or a small cell is the same.

4)
The system bandwidth is channelized with a maximum number, denoted as N.
Under the above assumptions, the number of available channels in the same frequency band for a small cell, denoted as N sc , can be expressed by Equation (7).
Moreover, the number of available channels in the same frequency band for a sector region excluding its embedded cell coverage, denoted as N sr , is expressed as Equation (8).

Transition Rates
The transition rates in the Markov model include arrival rate, denoted by λ, and departure rate, denoted by μ. The arrival rate is composes of those from NCs and HCs, while the departure rate is generated from call handoff or termination. The arrival rates associated with NCs, consisting of NC sc and NC sr , are denoted as λ sc and λ sr , respectively. The arrival rates from HCs, consisting of HC sr_sr , HC sr_sc , and HC sc_sr , are denoted by λ sr_sr , λ sr_sc , and λ sc_sr , respectively. Similarly, the departure rates caused by call handoff, consisting of HC sr_sr , HC sr_sc , and HC sc_sr , are denoted as μ sr_sr , μ sr_sc , and μ sc_sr , respectively. The departure rates caused by call termination include μ sc and μ sr , which stand for the calls terminating in a small cell and a sector region, respectively. μ sc contains two parts: one, denoted as μ sc_t , is for NCc originally initiating in a small cell and the other, denoted as μ sr_sc_t , is for HC sr_sc terminating in a small cell. μ sr contains three parts: one, denoted as μ sr_t , is for NCs originally initiating in a sector and the other two, denoted as μ sc_sr_t , and μ sr_sr_t , respectively for HC sc_sr and HC sr_sr terminating in a sector. Table 2 summarizes the abovementioned transition rates in the Markov model. Therefore, λ sc and λ sr can be expressed by Equations (9) and (10), respectively.
In the balanced condition of the Markov model, the arrival rate and departure rate caused by call handoff is the same. That's, λ sr_sr , λ sr_sc , and λ sc_sr are equal to μ sr_sr , μ sr_sc , and μ sc_sr , respectively. We can obtain the handoff-call arrival (departure) rate for each state by multiplying the varied element in the state by the associated handoff rate, as derived in Section 2.2. As to the departure rate caused by call termination in a small cell coverage (μ sc ) is the sum of μ sc_t and μ sr_sc_t , which are expressed by Equations (11) and (12), respectively. Similarly, the departure rate caused by call termination in a sector region (μ sr ) is the sum of μ sr_t , μ sc_sr_t , and μ sr_sr_t , which are expressed in Equations (13) to (15).

Markov States
All of the states in the Markov model are composed of the following five portions: 1) The number of calls in either a small cell or a sector does not exceed the upper limit. That's, each sector-related (small-cell-related) element of a state varies from 0 to N sr (N sc ) and there are n f frequency bands. Therefore, the total number of states in this portion, denoted as N 1 , is calculated as below.
2) NC in a small cell being blocked: when the number of calls in a small cell reaches the maximum value, i.e., N sc , an NC will be blocked. The total number of states in the portion, denoted as N 2 , is calculated as Equation (17).
3) NC in a sector being blocked: when the number of calls in a sector reaches the maximum value, i.e., N sr , an NC will be blocked. The total number of states in the portion, denoted as N 3 , is calculated as Equation (18).
4) HC moving into a small cell being dropped: when the number of calls in a small cell reaches the maximum value (N sc ), an HC will be dropped. The total number of states in the portion, denoted as N 4 , is calculated as Equation (19).
5) HC moving into a sector being dropped: there are two situations in the portion of states. That's, an HC is moving from a small cell to a sector and an HC is moving from a sector to two adjacent sectors. The total number of states in the portion, denoted as N 5 , is calculated as Equation (20) depending on the number of sectors (denoted by n), is even or odd.
By summing Equations (16) to (20), we can obtain the total number of the states, N, as expressed in Equation (21). The size of the transition matrix for the Markov model is N t × N t .

NC Blocking and HC Dropping Probabilities
An NC may get rejected in a small cell if one of the small-cell-related elements in the current state is N sc and one more NC makes the element become N sc + 1. Thus, the NC blocking probability in a small cell, denoted as Pb sc , is expressed as Equation (22). Similarly, an NC may get rejected in a sector if one of the sector-related elements in the current state is N sr . Thus, the NC blocking probability in a sector, denoted as Pb sr , is expressed by Equation (23). On the other hand, an HC moving from a sector to a small cell gets no services when one of the small-cell-related elements in the current state reaches N sc .
one more HC makes the element become (N sc + 1)'. We use the symbol prime to indicate the case when the arriving call belongs to HCs. Therefore, the HC dropping probability for MSs moving from a sector to a small cell, denoted by Pd sr_sc , is calculated as Equation (24).
In addition, there are two cases of an HC getting no services in a sector. One is for an HC moving into a sector from its embedded cell, and the other is for an HC moving from either of two adjacent sectors. Let Pd sc_sr and Pd sr_sr be the HC dropping probabilities in the two cases. Equations (25) and (26)

Average Waiting Time
Before deriving the average waiting time for an MS, it is required to obtain the average queue length (AQL) in a sector or a small cell. The AQL of NCs in a sector, denoted as Qn sr , is the product of N sr + 1 and the summation of the states an NC is blocked in the sector, as expressed in Equation (27). Similarly, we can obtain the AQL of HCs in a sector, denoted as Qh sr , by multiplying N sr + 1 by the summation of the states an HC is dropped in the sector, as expressed in Equation (28). Similar to Equations (27) and (28), the AQL of NCs and HCs in a small cell, can be expressed by Equations (29) and (30), respectively.  in Qn sr contains two sets: N sr ongoing calls that are composed of λ sr , λ sc_sr , and λ sr_sr , and one NC initiating in a sector that may get blocked. As a result, the average waiting time (AWT) of NCs in a sector, denoted as Wn sr , can be expressed by Equation (31). On the other hand, the average waiting time (AWT) of HCs in a sector, denoted as Wh sr , is derived as Equation (32) by referring to Equation (28).
The state probability ( ) 1, , , sr P N j x y + in Qh sr contains two sets: N sr ongoing calls that are composed of λ sr , λ sc_sr , and λ sr_sr , and one HC generating from λ sc_sr , and λ sr_sr that may get dropped. Similarly, we can derive the AWT of NCs and HCs in a small cell, respectively denoted as Wn sc and Wh sc , by referring to Equations (29) and (30).

Normalized Throughput
We define the normalized throughput (NT) as the individual channel utilization for a sector, a small cell, or a macro cell. Let Г sr be the NT for a sector. As expressed in Equation (35), Г sr consists of three conditions: one is for the states the channels are available for both in a sector and a small cell (i.e., the first term of Equation (35)); one is for the states the channels are fully occupied in a sector and an NC or HC gets no services in a sector (i.e., the second and third terms of Equation (35)); the other one is for the states the channels are fully occupied in a small cell and an NC or HC gets no services in a small cell (i.e., the fourth and fifth terms of Equation (35)). Similar to the derivation of Г sr , we obtain the NT for a small cell, denoted as Г sc , as expressed in Equation (36). Finally, let Г mc be the NT for the macro cell. Similarly, we can obtain Г mc , as expressed by Equation (37).

Parameters and Values
We evaluate the system performance of our proposed SCNSC via the analytical model built in Section 3. By setting the parameters as shown in Table 3, the transmission radius of the macro cell is fixed at 1000 meters and the transmission radius of a small cell is varied to change its coverage. The total number of channels in the system is given 60 and the number of sectors is set to be 3 or 6. In order to observe the effect of the average moving speed (E[V]) of MSs on the

Results and Discussions
Fixing the radius of a small cell (r) at a medium size, 150 meters, Figures 5-9 show the performance measures derived in Section 3.3 versus different traffic loads. In Figure 5(a) and Figure 5(b), it is observed that the NC blocking probabilities in a sector and a small cell, Pb sr and Pb sc , both increase with the system traffic load and the average moving speeds of MSs since the increment in traffic load or MS moving speed results in more NCs and HCs. Besides, it is found that the number of sectors (n) has opposing influences on Pb sr and Pb sc . Since the greater n is, the less the sector region, Pb sr increases with n due to more channels being occupied by handoff calls. On the contrary, Pb sc for n = 6 is lower than that for n = 3 because an even number of sectors (e.g., n = 6) has more channels allocated for a small cell than an odd number of sectors (e.g., n = 3). Figure 6 shows the HC dropping probabilities, Pb sr_sr , Pb sc_sr , and Pb sr_sc . From  always declines when the traffic load reaches a certain level. This is because larger traffic load means more NCs initiating in the system but the number of HCs remains the same and may get services more probably due to a higher percentage of NCs getting blocked.
To evaluate the influences of r and n on the AWT and NT, the average moving speed of MSs is fixed at 40 km/hr. Compared to Figure 5 and Figure 6, we can see from Figure 7 and Figure 8 that the same influence of the traffic load on the AWT of NCs and HCs for both a sector and a small cell. From Figure 7(a) and Figure 7(b), both Wn sr and Wh sr for n =3 are larger than those for n = 6 because the system capacity enhanced by using FFR is more obvious when n is even. Besides, no matter what n is, Wn sr increases but Wh sr decreases with the increment in traffic load when r becomes small. This is because smaller embedded-cell coverage makes a sector region larger and handoffs generating from an Journal of Computer and Communications  embedded cell to a sector take place more frequently to get services in a sector region in which less of N sr are available and then, less of NCs can get services.
Consequently, a decreasing number of NCs getting services brings about a decreasing number of HCs. On the other hand, as observed in Figure 8(a) and Figure 8(b), both Wn sc and Wh sc increase with the increment in traffic load when r becomes small. This is simply because a smaller proportion of small cell coverage to sector region has less number of channels in a small cell.
As to the normalized throughput, we can see from Figure 9(a) and Figure   9(b) that both Г sr and Г sc are larger for r = 50 than those for r = 250, since a smaller size of an embedded cell results in a smaller N sc and more HCs generating from a small cell to its associated sector even though N sr is greater. At last, it is demonstrated from Figure 9(c) that using FFR can enhance the system capac-ity because the NT for the macro cell (Г mc ) for n = 6 is larger than that for n = 3 no matter what r is.

Conclusion
We have presented a sectored cellular network with small cells (SCNSC) architecture. In the SCNSC, a multi-dimensional antenna is used for sectoring a macro cell and each sector is installed with a small cell to support the huge amounts of data transmitted through hot spots and to increase cell coverage in buildings.
The overall system capacity is enhanced by using FFR. We have built an analytical model using Markov chain to do topological evaluation on the proposed SCNCS. The performance measures, including new-call blocking and handoffcall dropping probabilities, average waiting times, and normalized throughputs, are derived for a sector and its embedded small cell. From our analytical results, we have demonstrated that the new-call blocking and handoff-call dropping probabilities in a small cell can be decreased by increasing the number of sectors or the radius of a small cell. On the contrary, the handoff-call dropping probability in a sector can be decreased by decreasing the number of sectors or the radius of a small cell. To maximize the normalized throughput of the system, we can increase the number of sectors and decrease the proportion of small cell area to sector region. In summary, adjusting either the number of sectors or the coverage of an embedded cell can satisfy different service demands.