JCCJournal of Computer and Communications2327-5219Scientific Research Publishing10.4236/jcc.2021.911009JCC-113620ArticlesComputer Science&Communications An Analytical Model for a Sectored Cellular Network with Embedded Small Cells Tsang-LingSheu1*Yan-JingWu2Yi-HsunLin1Department of Information Technology and Communication, Shih Chien University, Kaohsiung Campus, KaohsiungDepartment of Electrical Engineering, National Sun Yat-sen University, Kaohsiung05112021091112814914, October 202122, November 2021 25, November 2021© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Sectored Cell Small Cell New-Call Blocking Handoff-Call Dropping Markov Chain1. 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 (mm-wave), 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].

2) A macro cell using a multi-directional antenna consists of multiple sectors [18] [19] [20] [21].

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 co-channel 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 small-cell 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.

2. The Proposed SCNSC2.1. 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 nf frequency bands, where nf 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

Symbols and denotations of the SCNSC
SymbolsDenotations
nthe number of sectors for a macro cell
Rthe radius of a macro cell (in the unit meters)
rthe radius of a small cell (in the unit meters)
Ara small cell coverage divided by to a sector region
nfthe number of frequency bands (either 2 or 3)
Nthe total number of channels in the system
Nsrthe total number of available channels for a sector
Nscthe total number of available channels for a small cell
NCsrnew calls initiating in a sector region
NCscnew calls initiating in a small cell coverage
HCsr_srhandoff calls of MSs moving between two sectors
HCsr_schandoff calls of MSs moving from a sector to its embedded small cell
HCsc_srhandoff calls of MSs moving from a small cell to its associated sector

denote the total number of channels, i.e., time slots, frequency carriers, or time- and-frequency blocks, in the system. Nsr (Nsc) is the total number of available channels in the same frequency band for a sector (small cell) region. There are two kinds of new calls (NCs), initiating in a sector region or a small cell coverage, designated by NCsr and NCsc, respectively. Three kinds of handoff calls (HCs) are generated. One is from those MSs moving from a sector region to its embedded small cell, denoted as HCsr_sc. The second one is from those MSs moving in the reverse direction, denoted as HCsc_sr. The third one is from those MSs moving between two sectors, denoted as HCsr_sr.

A r = π r 2 π R 2 / n = n r 2 R 2 (1)

2.2. Handoff Rate

As elaborated in Section 2.1, there are three kinds of handoff calls (designated by HCsc_sr, HCsr_sc, and HCsr_sr) in the SCNSC. Referring to the works [28] [29], Equation (2) expresses the probability of an MS moving from a small cell area to its associated sector, and vice versa.

E [ V ] × L π × A (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.

1) Let ωsc_sr denote the handoff rate of HCsc_sr. L and A in Equation (2) are equal to 2πr and πr2, respectively. Then,

ω s c _ s r = E [ V ] × ( 2 π r ) π × ( π r 2 ) = 2 × E [ V ] π × r (3)

2) Let ωsr_sc denote the handoff rate of HCsr_sc. Similarly, L and A in Equation (2) are equal to 2πr and π R 2 / n − π r 2 , respectively. Then,

ω s r _ s c = E [ V ] × ( 2 π r ) π × ( π R 2 / n − π r 2 ) = 2 × E [ V ] × r × n π × ( R 2 − n × r 2 ) (4)

3) Let ωsr_sr denote the handoff rate of HCsr_sr. A in Equation (2) is the same as that in Equation (4). L represents the boundary of a sector, which is equal to 2 R + 2 π R / n . However, HCsr_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).

L = ( 2 R + 2 π R / n ) × ( R / ( 2 R + 2 π R / n ) ) (5)

ω s r _ s r = E [ V ] × ( 2 R + 2 π R / n ) π × ( π R 2 / n − π r 2 ) × R 2 R + 2 π R / n = E [ V ] × R × n π 2 × ( R 2 − n × r 2 ) (6)

3. 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.

3.1. 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 Nsc, 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 Nsr, is expressed as Equation (8).

N s c = N n f × A r (7)

N s r = N n f × ( 1 − A r ) (8)

3.2. Analytical Model3.2.1. 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 NCsc and NCsr, are denoted as λsc and λsr, respectively. The arrival rates from HCs, consisting of HCsr_sr, HCsr_sc, and HCsc_sr, are denoted by λsr_sr, λsr_sc, and λsc_sr, respectively. Similarly, the departure rates caused by call handoff, consisting of HCsr_sr, HCsr_sc, and HCsc_sr, are denoted as μsr_sr, μsr_sc, and μsc_sr, respectively. The departure rates caused by call termination include μsc and μsr, which stand for the calls terminating in a small cell and a sector region, respectively. μsc contains two parts: one, denoted as μsc_t, is for NCc originally initiating in a small cell and the other, denoted as μsr_sc_t, is for HCsr_sc terminating in a small cell. μsr contains three parts: one, denoted as μsr_t, is for NCs originally initiating in a sector and the other two, denoted as μsc_sr_t, and μsr_sr_t, respectively for HCsc_sr and HCsr_sr terminating in a sector. Table 2 summarizes the abovementioned transition rates in the Markov model.

Therefore, λsc and λsr 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, λsr_sr, λsr_sc, and λsc_sr are equal to μsr_sr, μsr_sc, and μsc_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 (μsc) is the sum of μsc_t and μsr_sc_t, which are expressed by Equations (11) and (12), respectively. Similarly, the departure rate caused by call termination in a sector region (μsr) is the sum of μsr_t, μsc_sr_t, and μsr_sr_t, which are expressed in Equations (13) to (15).

λ s c = λ × A r (9)

λ s r = λ × ( 1 − A r ) (10)

μ s c _ t = μ × A r × λ s c λ s c + λ s r _ s c (11)

Arrival rates and departure rates in the Markov model
SymbolsDescriptions
λthe total arrival rate
λsrthe arrival rate of NCsr
λscthe arrival rate of NCsc
λsr_srthe arrival rate of HCsr_sr
λsr_scthe arrival rate of HCsr_sr
λsc_srthe arrival rate of HCsc_sr
μthe total departure rate
μsr_srthe departure rate of HCsr_sr
μsr_scthe departure rate of HCsr_sc
μsc_srthe departure rate of HCsc_sr
μscthe departure rate for call termination in a small cell
μsrthe departure rate for call termination in a sector region
μsc_tthe departure rate of NCsc that terminates in a small cell
μsr_tthe departure rate of NCsr that terminates in a sector region
μsr_sc_tthe departure rate of HCsr_sc that terminates in a small cell
μsc_sr_tthe departure rate of HCsc_sr that terminates in a sector region
μsc_sr_tthe departure rate of HCsr_sr that terminates in a sector region

μ s r _ s c _ t = μ × A r × ( 1 − λ s c λ s c + λ s r _ s c ) (12)

μ s r _ t = μ × ( 1 − A r ) × λ s r λ s r + λ s c _ s r + 2 × λ s r _ s r (13)

μ s c _ s r _ t = μ × ( 1 − A r ) × λ s c _ s r λ s r + λ s c _ s r + 2 × λ s r _ s r (14)

μ s r _ s r _ t = μ × ( 1 − A r ) × 2 × λ s r _ s r λ s r + λ s c _ s r + 2 × λ s r _ s r (15)

3.2.2. 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 Nsr (Nsc) and there are nf frequency bands. Therefore, the total number of states in this portion, denoted as N1, is calculated as below.

N 1 = ( N s r + 1 ) n f × ( N s c + 1 ) n f (16)

2) NC in a small cell being blocked: when the number of calls in a small cell reaches the maximum value, i.e., Nsc, an NC will be blocked. The total number of states in the portion, denoted as N2, is calculated as Equation (17).

N 2 = ( N s r + 1 ) n f × ( N s c + 1 ) n f − 1 × n f (17)

3) NC in a sector being blocked: when the number of calls in a sector reaches the maximum value, i.e., Nsr, an NC will be blocked. The total number of states in the portion, denoted as N3, is calculated as Equation (18).

N 3 = ( N s r + 1 ) n f − 1 × ( N s c + 1 ) n f × n f (18)

4) HC moving into a small cell being dropped: when the number of calls in a small cell reaches the maximum value (Nsc), an HC will be dropped. The total number of states in the portion, denoted as N4, is calculated as Equation (19).

N 4 = N s r × ( N s r + 1 ) n f − 1 × ( N s c + 1 ) n f − 1 × n f (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 N5, is calculated as Equation (20) depending on the number of sectors (denoted by n), is even or odd.

N 5 = { ( N s r + 1 ) n f − 1 × ( N s c + 1 ) n f − 1 × N s c × n f + N s r × ( N s c + 1 ) n f × n f − ( N s c + 1 ) × N s c × N s r × n f , if n iseven ( N s r + 1 ) n f − 1 × ( N s c + 1 ) n f − 1 × N s c × n f + ( N s r × 2 ( N s r + 1 ) − N s r 2 ) × ( N s c + 1 ) n f × n f − ( N s c + 1 ) n f − 1 × ( ( N s r + 1 ) n f − 1 − 1 ) × n f , if n isodd (20)

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 Nt × Nt.

N t = N 1 + N 2 + N 3 + N 4 + N 5 (21)

3.3. Performance Measures3.3.1. 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 Nsc and one more NC makes the element become Nsc + 1. Thus, the NC blocking probability in a small cell, denoted as Pbsc, 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 Nsr. Thus, the NC blocking probability in a sector, denoted as Pbsr, is expressed by Equation (23).

P b s c = { ∑ i = 0 N s r ∑ j = 0 N s r ∑ y = 0 N s c P ( i , j , N s c +1 , y ) × n f ,if n iseven ∑ i = 0 N s r ∑ j = 0 N s r ∑ k = 0 N s r ∑ y = 0 N s c ∑ z = 0 N s c P ( i , j , k , N s c +1 , y , z ) × n f ,if n isodd (22)

P b s r = { ∑ j = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c P ( N s r +1 , j , x , y ) × n f , if n iseven ∑ j = 0 N s r ∑ k = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c ∑ z = 0 N s c P ( N s r +1 , j , k , x , y , z ) × n f , if n isodd (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 Nsc. one more HC makes the element become (Nsc + 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 Pdsr_sc, is calculated as Equation (24).

P d s r _ s c = { ∑ i = 0 N s r ∑ j = 0 N s r ∑ y = 0 N s c P ( i , j , ( N s c + 1 ) ' , y ) × n f ,  if  n  is even ∑ i = 0 N s r ∑ j = 0 N s r ∑ k = 0 N s r ∑ y = 0 N s c ∑ z = 0 N s c P ( i , j , k , ( N s c + 1 ) ' , y , z ) × n f ,  if  n  is odd (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 Pdsc_sr and Pdsr_sr be the HC dropping probabilities in the two cases. Equations (25) and (26) express the two cases, respectively.

P d s c _ s r = { ∑ x = 0 N s c ∑ y = 0 N s c − 1 [ ∑ j = 0 N s r − 1 P ( ( N s r + 1 ) ' , j , x , y ) × λ s c _ s r 2 λ s r _ s r + λ s c _ s r     + P ( ( N s r + 1 ) ' , N s r , x , y ) ] × n f , if n iseven ∑ x = 0 N s c ∑ y = 0 N s c ∑ z = 0 N s c − 1 [ ∑ j = 0 N s r − 1 ∑ k = 0 N s r − 1 P ( ( N s r + 1 ) ' , j , k , x , y , z ) × λ s c _ s r 2 λ s r _ s r + λ s c _ s r     + ∑ k = 0 N s r − 1 P ( ( N s r + 1 ) ' , N s r , k , x , y , z ) × 2 λ s c _ s r λ s r _ s r + λ s c _ s r     + P ( ( N s r + 1 ) ' , N s r , N s r , x , y , z ) ] × n f , if n isodd (25)

where λ s c _ s r 2 × λ s r _ s r + λ s c _ s r stands for the proportion of HCs moving into a sector

from its embedded cell because an HC in a sector may arrive from an embedded cell or two adjacent sectors if n is even.

As to the metric Pdsr_sr, it is straightforward equal to the HC dropping probability in a sector excluding Pdsc_sr.

P d s r _ s r = { ∑ j = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c P ( ( N s r + 1 ) ' , j , x , y ) × n f − P d s c _ s r , if n iseven ∑ j = 0 N s r ∑ k = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c ∑ z = 0 N s c P ( ( N s r + 1 ) ' , j , k , x , y , z ) × n f − P d s c _ s r , if n isodd (26)

3.3.2. 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 Qnsr, is the product of Nsr + 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 Qhsr, by multiplying Nsr + 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.

Q n s r = { ∑ j = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c ( N s r + 1 ) × P ( N s r + 1 , j , x , y )     , if n iseven ∑ j = 0 N s r ∑ k = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c ∑ z = 0 N s c ( N s r + 1 ) × P ( N s r + 1 , j , k , x , y , z )     ,if n isodd (27)

Q h s r = { ∑ j = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c ( N s r + 1 ) × P ( ( N s r + 1 ) ' , j , x , y )     ,  if  n  is even ∑ j = 0 N s r ∑ k = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c ∑ z = 0 N s c ( N s r + 1 ) × P ( ( N s r + 1 ) ' , j , k , x , y , z )     ,  if  n  is odd (28)

Q n s c = { ∑ i = 0 N s r ∑ j = 0 N s r ∑ y = 0 N s c ( N s c + 1 ) × P ( i , j , N s c + 1 , y )     , if n iseven ∑ i = 0 N s r ∑ j = 0 N s r ∑ k = 0 N s r ∑ y = 0 N s c ∑ z = 0 N s c ( N s c + 1 ) × P ( i , j , k , N s c + 1 , y , z )     , if n isodd (29)

Q h s c = { ∑ i = 0 N s r ∑ j = 0 N s r ∑ y = 0 N s c ( N s c + 1 ) × P ( i , j , ( N s c + 1 ) ' , y )     , if n iseven ∑ i = 0 N s r ∑ j = 0 N s r ∑ k = 0 N s r ∑ y = 0 N s c ∑ z = 0 N s c ( N s c + 1 ) × P ( i , j , k , ( N s c + 1 ) ' , y , z )     , if n isodd (30)

Referring to Equation (27), the state probability P ( N s r + 1 , j , x , y ) in Qnsr contains two sets: Nsr ongoing calls that are composed of λsr, λsc_sr, and λsr_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 Wnsr, can be expressed by Equation (31). On the other hand, the average waiting time (AWT) of HCs in a sector, denoted as Whsr, is derived as Equation (32) by referring to Equation (28). The state probability P ( N s r + 1 , j , x , y ) in Qhsr contains two sets: Nsr ongoing calls that are composed of λsr, λsc_sr, and λsr_sr, and one HC generating from λsc_sr, and λsr_sr that may get dropped. Similarly, we can derive the AWT of NCs and HCs in a small cell, respectively denoted as Wnsc and Whsc, by referring to Equations (29) and (30).

W n s r = { ∑ j = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c P ( N s r + 1 , j , x , y ) λ s r + ∑ j = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c N s r × P ( N s r + 1 , j , x , y ) λ s r + λ s c _ s r + 2 λ s r _ s r , if n iseven ∑ j = 0 N s r ∑ k = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c ∑ z = 0 N s c P ( N s r + 1 , j , k , x , y , z ) λ s r + ∑ j = 0 N s r ∑ k = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c ∑ z = 0 N s c N s r × P ( N s r + 1 , j , k , x , y , z ) λ s r + λ s c _ s r + 2 λ s r _ s r , if n isodd (31)

W h s r = { ∑ j = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c P ( ( N s r + 1 ) ' , j , x , y ) λ s c _ s r + 2 λ s r _ s r + ∑ j = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c N s r × P ( ( N s r + 1 ) ' , j , x , y ) λ s r + λ s c _ s r + 2 λ s r _ s r , if n iseven ∑ j = 0 N s r ∑ k = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c ∑ z = 0 N s c P ( ( N s r + 1 ) ' , j , k , x , y , z ) λ s c _ s r + 2 λ s r _ s r + ∑ j = 0 N s r ∑ k = 0 N s r ∑ x = 0 N s c ∑ y = 0 N s c ∑ z = 0 N s c N s r × P ( ( N s r + 1 ) ' , j , k , x , y , z ) λ s r + λ s c _ s r + 2 λ s r _ s r , if n isodd (32)

W n s c = { ∑ i = 0 N s r ∑ j = 0 N s r ∑ y = 0 N s c P ( i , j , N s c + 1 , y ) λ s c + ∑ i = 0 N s r ∑ j = 0 N s r ∑ y = 0 N s c N s c × P ( i , j , N s c + 1 , y ) λ s c + λ s r _ s c , if n iseven ∑ i = 0 N s r ∑ j = 0 N s r ∑ k = 0 N s r ∑ y = 0 N s c ∑ z = 0 N s c P ( i , j , k , N s c + 1 , y , z ) λ s c + ∑ i = 0 N s r ∑ j = 0 N s r ∑ k = 0 N s r ∑ y = 0 N s c ∑ z = 0 N s c N s c × P ( i , j , k , N s c + 1 , y , z ) λ s c + λ s r _ s c , if n isodd (33)

W h s c = { ∑ i = 0 N s r ∑ j = 0 N s r ∑ y = 0 N s c P ( i , j , ( N s c + 1 ) ' , y ) λ s r _ s c + ∑ i = 0 N s r ∑ j = 0 N s r ∑ y = 0 N s c N s c × P ( i , j , ( N s c + 1 ) ' , y ) λ s c + λ s r _ s c , if n iseven ∑ i = 0 N s r ∑ j = 0 N s r ∑ k = 0 N s r ∑ y = 0 N s c ∑ z = 0 N s c P ( i , j , k , ( N s c + 1 ) ' , y , z ) λ s r _ s c + ∑ i = 0 N s r ∑ j = 0 N s r ∑ k = 0 N s r ∑ y = 0 N s c ∑ z = 0 N s c N s c × P ( i , j , k , ( N s c + 1 ) ' , y , z ) λ s c + λ s r _ s c , if n isodd (34)

3.3.3. Normalized Throughput

We define the normalized throughput (NT) as the individual channel utilization for a sector, a small cell, or a macro cell. Let Гsr be the NT for a sector. As expressed in Equation (35), Гsr 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 Гsr, we obtain the NT for a small cell, denoted as Гsc, as expressed in Equation (36). Finally, let Гmc be the NT for the macro cell. Similarly, we can obtain Гmc, as expressed by Equation (37).

4. Analytical Results4.1. 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

Parameters and their values
ParametersValues
R1000 meters
r50, 150, 250 meters
N60
n3, 6
E[V]20, 40, 60 km/hr
λ0.001 - 0.006
μ0.005

HCs dropping probability, E[V] are set to be 20, 40, or 60 km/hr. The arrival rate of NCs varies from 0.001 to 0.006 and the departure rate of NCs is fixed at 0.005. The total number of MSs in the macro cell is 200. Then, the corresponding transition matrix and the associated transition rates between any two states can be derived. All of the state probabilities are obtained by running the calculation on MATLAB. The Excel data for the analytical results are included within the supplementary information file and the performance metrics are presented in Figures 5-9.

4.2. 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, Pbsr and Pbsc, 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 Pbsr and Pbsc. Since the greater n is, the less the sector region, Pbsr increases with n due to more channels being occupied by handoff calls. On the contrary, Pbsc 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, Pbsr_sr, Pbsc_sr, and Pbsr_sc. From Figures 6(a)-(c), it is interesting that any one of the three HC dropping probabilities

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 Wnsr and Whsr 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, Wnsr increases but Whsr 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

embedded cell to a sector take place more frequently to get services in a sector region in which less of Nsr 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 Wnsc and Whsc 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 Гsr and Гsc are larger for r = 50 than those for r = 250, since a smaller size of an embedded cell results in a smaller Nsc and more HCs generating from a small cell to its associated sector even though Nsr is greater. At last, it is demonstrated from Figure 9(c) that using FFR can enhance the system capacity because the NT for the macro cell (Гmc) for n = 6 is larger than that for n = 3 no matter what r is.

5. 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 handoff-call 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.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Sheu, T.-L., Wu, Y.-J. and Lin, Y.-H. (2021) An Analytical Model for a Sectored Cellular Network with Embedded Small Cells. Journal of Computer and Communications, 9, 128-149. https://doi.org/10.4236/jcc.2021.911009

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