# Erlang's Fixed-Point Approximation for Performance Analysis of HetNets

Authors: Guozhi Song; Jigang Wu; John Schormans; Laurie Cuthbert
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We consider the analytic modelling of wireless systems with multiple access technologies in the perspective of teletraffic engineering and provide a framework for the performance analysis and evaluation of a wireless HetNet (heterogeneous network) system with both cellular and WLAN access technologies. In particular, an approach with Erlang's fixed-point approximation to calculate the new call blocking and handover call dropping probabilities in such systems is introduced. The model is versatile enough to cover not only cellular/WLAN HetNet systems but other wireless HetNets with difference access technologies in general.


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## Body

## 1. Introduction

The term HetNet (heterogeneous network) used in wireless networks indicates the usage of multiple types of access technologies for a wireless system, and these radio access networks (RAN) can form a motley of macrocells, picocells, and femtocells in order to offer wireless service in an environment with a wide variety of wireless coverage zones, ranging from an open outdoor environment to office buildings, homes, and underground spaces. Therefore it has a hierarchical structure with overlapped and overlaid wireless service areas and complex interoperation between these macrocells and smaller cells (sometimes in the form of WLAN) working together to provide a mosaic of coverage. Currently the extensive research of HetNets can be basically categorized into three groups. The first group focuses on the new architecture and implementation of HetNet system such as the work Li et al. did in [1]. The second group is trying to find the good and optimized RAN selection algorithms like what Wang et al. did in [2] and Mapp et al. in [3]. The last group works on the system performance analysis of HetNets, for instance, the work of  Xie and Baras on throughput in [4] and Wang and Du on routing in [5]. In this paper, we will discuss the application of Erlang’s fixed-point approximation in the performance analysis of HetNet systems which is modelled as a product-form loss network.Ross indicated in [6] that the efficient algorithms for exact blocking probabilities and other performance measures do not exist for all product-form loss networks. The reduced fixed-point approximation is a method of approximating blocking probability in product-form loss networks. And loss networks are a special kind of queueing networks (or network of queues) without any queueing (or buffer) at each node. Traditionally for a single loss system which is actually an M/M/n/n queue such as a telephony system, the loss probability is estimated using Erlang’s function that is discussed in Erlang’s classical works [7, 8] which also laid the foundation for teletraffic engineering. It is applicable to a single-call-type system. But for our HetNet systems we adopt the loss network model with the help of reduced fixed-point approximation that is being used in the studies of TCP behaviour [9], optical burst switched networks [10] and also routing [11], and so forth.What is often referred to as reduced load approximation is a fairly broad class of approximation techniques. The general approach is to diminish the arrival rates of offered traffic to a subnetwork (which may be a single link) by a factor equal to the probability that a new call on that route would not be blocked on the other link of its path. Typically this method leads to a set of fixed-point equations for which exists a (not necessarily unique) solution. The underlying assumption is that of independent blocking between the individual subnetworks and, although invalid in most nontrivial situations, does yield particularly good results when traffic correlations are small.The famous Erlang’s fixed-point approximation (EFPA) is a member of the reduced load class, one which analyses each link as a separate subnetwork. The EFPA performs well asymptotically. Kelly in [12] proved that the estimates for a network with fixed routing and no controls tend towards the exact probabilities (i) when the link capacities and arrival rates are increased simultaneously keeping the network topology fixed (Kelly limiting regime), and (ii) when the number of links and routes are increased while the link loads are held constant (diverse routing limit) proved by Ziedins and Kelly in [13].The EFPA is a solution to the set of fixed-point equations:(1.1)Bj=E(ρj,Cj),j=1,…,J,ρj=∑r∈Raj,rνr∏i∈r∖{j}(1-Bi),j=1,…,J,
where Erlang’s formula, E(ν,C), gives the probability that the Erlang loss model is fully utilised and is given by
(1.2)E(ν,C)=νCC!(∑n=0Cνnn!)-1.
The interpretation is that Bj is the probability that linkj is full given its offered traffic load is ρj·ρj is an approximation obtained by considering the carried traffic on link j: the throughput of link j is (1-Bj)ρj and ∑r∈Raj,rνr(1-Lr) is the sum of the contributions made by each route r to j’s carried load. Applying the independent blocking assumption yields
(1.3)Lr=1-Pr(call accepted on each linki∈r)=1-∏i∈rPr(call accepted on linki)=1-∏i∈r(1-Bi),
and the expression (1.3) follows by equating the two expressions for carried traffic on j and substituting for Lr. Kelly in [14] proved that there is a unique vector (B1,…,BJ)∈[0,1]J satisfying (1.1).


## 2. System Model

Assume there are in totalNc cells and Nw WLANs in our HetNet system. Let us firstly define some symbols to facilitate the further analysis. Let Cia be the set of cells adjacent to cell i, Wio the set of WLANs inside the coverage of cell i,Wka the set of WLANs adjacent to WLAN k, and Cko the set containing the overlaying cell of WLAN k that is, a dual-coverage area as show in Figure 1. This deployment was used in [15] to evaluate and compare performance between a cellular-only network and a cellular-WLAN interworking network.Figure 1
A HetNet (cellular/WLAN) deployment and its converted network topological representation.The use of reservation channels or guard channels is analogous to the use of trunk reservation in fixed-wire circuit-switched networks to give priority to fresh traffic over overflow traffic [16]. So each node can be modelled as an Erlang loss system. The service time at a node would be the channel holding time in the corresponding network the node represented.We use the logical topology abstraction method that was proposed and developed in [17, 18] to derive the network topology of the system by mapping each cellular network and WLAN as nodes and handover traffic as links.


### 2.1. Node Traffic Flow Analysis

Traffic analysis of each node is based on the conservation of traffic. Traffic flows in and out of the node will be equal. The arriving traffic can be divided into four parts: new calls, horizontal handover calls, vertical handover calls, and overflow calls. Traffic either leaves the cell or WLAN normally by completion or handover, or due to the lack of resources is blocked or dropped.The total arrival rate at celli is the sum of the arrival rate of new calls, handover calls (including horizontal handover calls from adjacent cells and vertical handover calls from the WLANs inside the cell), and overflow calls from WLANs within the cell.Similarly, the total arrival rate at WLANk is the sum of the arrival rate of new calls, handover calls (including horizontal handover calls from adjacent WLANs and vertical handover calls from the overlaid cellular cell), and overflow calls from the overlaid cell.The new connection arrival processes to celli and WLAN k are Poisson with rates λi(cn) andλk(wn), respectively, which are independent of other arrival processes. The channel holding time of a connection in cell i (i.e., the time that a user is using resources in cell i) is an exponential distributed random variable with mean 1/μic. The channel holding time in WLAN k is exponentially distributed with mean 1/μkw. Both are independent of earlier arrival times and connection holding times.At the end of a holding time, a connection in celli of the cellular system may terminate and leave the system with probabilityTi(c) or move within the system and continue in an adjacent cell or WLAN with probability 1-Ti(c). The probability that a connection continues and moves to an adjacent cell of cell i or WLAN k inside cell i is 1-Ti(c)=∑j∈CiaHij(cc)+∑k∈WioHik(cw), where Hij(cc) is the probability of attempting a horizontal handover to adjacent cell j and Hik(cw) is the probability of attempting a vertical handover to WLAN k inside cell i.Similarly, at the end of a holding time of a connection in WLANk, a call may terminate and leave the system with probability Tk(w) or move within the system and continue in an adjacent WLAN or an overlay cell of the cellular system with probability 1-Tk(w). The probability that a connection continues and moves to an adjacent WLAN to WLAN k or overlaying cell i is 1-Tk(w)=∑l∈WkaHkl(ww)+∑i∈CkoHki(wc), where Hkl(ww) is the probability of attempting a horizontal handover to adjacent WLAN l, and Hki(wc) is the probability of attempting a vertical handover to overlaying cell i.Each celli of the cellular system has a capacity of Cic units of bandwidth. While each WLAN k has a capacity of Ckw units of bandwidth. I define Ric and Rkw as the reservation parameters of cell i and WLAN k for handover calls, respectively. They act as the admission policy to provide handover connections priority over new connections. Two admission schemes are considered: the cut-off priority and the fractional guard channel.Letni be the number of connections in cell i. When cell i is in any of the states ni≤Cic-Ric, it accepts new and handover calls under both admission policies. When cell i is in any of the states ni>Cic-Ric, only handover requests are accepted for the cut-off priority scheme. For the fractional guard channel scheme, besides handover requests, each new connection request is also accepted with probability ωic. The same admission policies apply to WLANs with probability ωkw. Note that cut-off priority is a particular case of the fractional guard channel with ωkw=0. If a handover request is not accepted in one network, then the request is transferred to the other network. In the cellular system, only when the user is inside the dual-coverage area can the connection be transferred to the WLAN. Connections that are within the WLAN can always be attempted to be transferred to the corresponding overlaying cell in the cellular system.The occupancy of a cell evolves according to a birth-death process independent of other cells as shown in Figure2(a). The process for cell i evolves with birth rate ρic (2.1)(a) for the unreserved states and αic (2.1)(b) for the reserved states. The death rate of cell i in state ni is niμic. The total traffic offered to cell i in state ni is
(2.1)λic(kic)={λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo),kic≤Cic-Ric(a),λi(cn)ωic+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo),kic>Cic-Ric(b),
with ωic=0 for the cut-off priority scheme, and 0≤ωic≤1 for the fractional guard channel scheme.  The term λji(cc) is the horizontal handover rate of cell j offered to cell i, for adjacent cells i and j, and λki(wc) is the vertical handover rate of WLAN k offered to overlaid cell i, λzk(co) is the proportion of all handover traffic that is not accepted in cell z due to the resource depletion and thence overflowed to WLAN k.State transition diagrams for each cellular cell (a) and WLAN (b).
(a)(b)Similarly, as shown in Figure2(b) the occupancy of WLAN k evolves with birth rates ρkw (2.2)(a) and αkw (2.2)(b) based on the state mk, and death rate kkw(μkw+υkw). The total traffic offered to WLAN k in state kkw is
(2.2)λkw(miw)={λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co),mk≤Ckw-Rkw(a),λk(wn)ωkw+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co),mk>Ckw-Rkiwc(b),


### 2.2. Link Traffic Flow (Handover Traffic) Analysis

Let us now focus on the traffic flow on the links, that is, handover traffic and overflow traffic to derive the handover rates in (2.1) ((a), (b)) and (2.2) ((a), (b)). It will be found that the handover rates and the blocking rate and dropping rate are interdependent with each other.


#### 2.2.1. Handover Calls to Cell

The horizontal handover rateλji(cc)of cell j offered to cell i, for adjacent cells i and j, is all the nonblocked (dropped) traffic in cell j multiply with a handover probability factor and is shown in the Figure 3. Blue lines are the traffic that can go through to the next stage, and red lines are the blocked or dropped traffic.Figure 3
Horizontal handover traffic between cellular cells.So the formula for horizontal handover is given by (2.3)
(2.3)λji(cc)=Hji(cc)⋅[λj(cn)⋅(1-Bj(c))+(∑x∈Cjaλxj(cc)+∑y∈Wjoλyj(wc)+∑z∈Wjoλzj(wo))⋅(1-Dj(c))],
where Bj(c) and Dj(c) are the new connection blocking and handover dropping probabilities in cell j, respectively.In (2.3), λzj(wo) corresponds to all handover traffic that is not accepted in WLAN z and hence transferred to cell j, so
(2.4)λzj(wo)=Dz(w)⋅(λjz(cw)+∑l∈Wzaλlz(ww)).
For λki(wc), the vertical handover rate of WLAN k offered to overlay cell i is all the non-blocked (dropped) traffic in WLAN k multiplied with a handover probability factor as shown in the Figure 4. Blue lines are the traffic that can go through to the next stage and red lines are the blocked or dropped traffic. So the formula for vertical handover is given by (2.5)(2.5)λki(wc)=Hki(wc)⋅[λk(wn)⋅(1-Bk(w))+(∑x∈Wkaλxk(ww)+∑y∈Ckoλyk(cw)+∑z∈Ckoλzkco)⋅(1-Dk(w))],
where Bk(w) and Dk(w) are the new connection blocking and handover dropping probabilities in WLAN k, respectively.Figure 4
Vertical handover traffic from WLAN to cellular cell.In (2.6), λzk(co) is the proportion of all handover traffic that is not accepted in cell z and thence transferred to WLAN k, then
(2.6)λzk(co)=Dz(c)⋅(λkz(wc)+∑l∈Czaλlz(cc))⋅γzk,
where γzk is the coverage factor between WLAN k and overlay cell z, that is, the ratio between the radio coverage area of WLAN k and the radio coverage area of cell z with 0<γzk≤1.


#### 2.2.2. Handover Calls to WLAN

Similarly, we can derive the termλlk(ww), the horizontal handover rate of WLAN l offered to WLAN k, for adjacent WLAN l and k, which is all the non-blocked (dropped) traffic in WLAN multiplied with a handover probability factor as shown in the Figure 5. Blue lines are the traffic that can go through to the next stage and red lines are the blocked or dropped traffic. So the formula for horizontal handover is given by (2.7)(2.7)λlk(ww)=Hlk(ww)⋅[λl(wn)(1-Blw)+(∑x∈Wlaλxl(ww)+∑y∈Cloλyl(cw)+∑z∈Cloλzl(co))⋅(1-Dlw)].
The term λjk(cw) is the vertical handover rate of cell j offered to WLAN k, for overlaying cell j and k, which is all the non-blocked (dropped) traffic in cell j multiplied with a handover probability factor as shown in the Figure 6. Blue lines are the traffic that can go through to the next stage and red lines are the blocked or dropped traffic. So the formula for vertical handover is given by (2.8)(2.8)λjk(cw)=Hjk(cw)⋅[λj(cn)(1-Bjc)γjk+(∑x∈Cjaλxj(cc)γjk+∑y∈Wjoυyj(wc)γjk+∑z∈Wjoλzj(wo))⋅(1-Djc)].Figure 5
Horizontal handover traffic between WLANs.Figure 6
Vertical handover traffic from cellular cell to WLAN.


### 2.3. Balance Equations

From the analysis of the birth-death process in celli, the detailed balance equations are
(2.9)πic(kic-1)⋅(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))=πic(kic)kic(μic+υic),0<kic≤Cic-Ric,πic(kic-1)⋅(λi(cn)ωic+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))=πic(kic)kic(μic+υic),Cic-R<kic≤Cic,
where πic(kic) is the steady state probability that there arekic calls is in cell i and is derived as
(2.10)πic(kic)=(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))kicπic(0)kic!(μic+υic)kic,ni≤Cic-Ric,πic(kic)=(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))Cic-Ricπic(0)kic!(μic+υic)kic×(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))kic-Cic+Ricπic(0)kic!(μic+υic)kic,ni>Cic-Ric,
where πic(0) is the normalisation constant, obtained as:
(2.11)πic(0)=∑kic=0Cic-Ric1kic!(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo)μic+υic)kic+∑ni=Cic-Ric+1Cic(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))Cic-Rickic!(μic+υic)ni×(λi(cn)ωic+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))kic-Cic+Rickic!(μic+υic)ni.
From the process WLAN k, the detailed balance equations are
(2.12)πkw(kkw-1)(λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))=πkw(kkw)kkw(μkw+υkw),0<kkw≤Ckw-Rkw,πkw(kkw-1)(λk(wn)ωkw+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))=πkw(kkw)kkw(μkw+υkw),Ckw-R<kkw≤Ckw,
where πkw(kkw) is the steady-state probability that kkw calls is in WLAN k, and is derived as(2.13)πkw(kkw)=(λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))kkwπkw(0)kkw!(μkw+υkw)kkw,kkw≤Ckw-Rkw,πkw(kkw)=(λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))Ckw-Rkwπkw(0)kkw!(μkw+υkw)kkw×(λk(wn)ωkw+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))kkw-Ckw+Rkwπkw(0)kkw!(μkw+υkw)kkw,kkw>Ckw-Rkw,
where πk’w(0) is the normalisation constant, obtained as
(2.14)πkw(0)=∑kkw=0Ckw-Rkw1kkw!(λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co)μkw+υkw)kkw+∑kkw=Ckw-Rkw+1Ckw(λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))Ckw-Rkwkkw!(μkw+υkw)kkw×(λk(wn)ωkw+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))kkw-Ckw+Rkwkkw!(μkw+υkw)kkw.


## 2.1. Node Traffic Flow Analysis

Traffic analysis of each node is based on the conservation of traffic. Traffic flows in and out of the node will be equal. The arriving traffic can be divided into four parts: new calls, horizontal handover calls, vertical handover calls, and overflow calls. Traffic either leaves the cell or WLAN normally by completion or handover, or due to the lack of resources is blocked or dropped.The total arrival rate at celli is the sum of the arrival rate of new calls, handover calls (including horizontal handover calls from adjacent cells and vertical handover calls from the WLANs inside the cell), and overflow calls from WLANs within the cell.Similarly, the total arrival rate at WLANk is the sum of the arrival rate of new calls, handover calls (including horizontal handover calls from adjacent WLANs and vertical handover calls from the overlaid cellular cell), and overflow calls from the overlaid cell.The new connection arrival processes to celli and WLAN k are Poisson with rates λi(cn) andλk(wn), respectively, which are independent of other arrival processes. The channel holding time of a connection in cell i (i.e., the time that a user is using resources in cell i) is an exponential distributed random variable with mean 1/μic. The channel holding time in WLAN k is exponentially distributed with mean 1/μkw. Both are independent of earlier arrival times and connection holding times.At the end of a holding time, a connection in celli of the cellular system may terminate and leave the system with probabilityTi(c) or move within the system and continue in an adjacent cell or WLAN with probability 1-Ti(c). The probability that a connection continues and moves to an adjacent cell of cell i or WLAN k inside cell i is 1-Ti(c)=∑j∈CiaHij(cc)+∑k∈WioHik(cw), where Hij(cc) is the probability of attempting a horizontal handover to adjacent cell j and Hik(cw) is the probability of attempting a vertical handover to WLAN k inside cell i.Similarly, at the end of a holding time of a connection in WLANk, a call may terminate and leave the system with probability Tk(w) or move within the system and continue in an adjacent WLAN or an overlay cell of the cellular system with probability 1-Tk(w). The probability that a connection continues and moves to an adjacent WLAN to WLAN k or overlaying cell i is 1-Tk(w)=∑l∈WkaHkl(ww)+∑i∈CkoHki(wc), where Hkl(ww) is the probability of attempting a horizontal handover to adjacent WLAN l, and Hki(wc) is the probability of attempting a vertical handover to overlaying cell i.Each celli of the cellular system has a capacity of Cic units of bandwidth. While each WLAN k has a capacity of Ckw units of bandwidth. I define Ric and Rkw as the reservation parameters of cell i and WLAN k for handover calls, respectively. They act as the admission policy to provide handover connections priority over new connections. Two admission schemes are considered: the cut-off priority and the fractional guard channel.Letni be the number of connections in cell i. When cell i is in any of the states ni≤Cic-Ric, it accepts new and handover calls under both admission policies. When cell i is in any of the states ni>Cic-Ric, only handover requests are accepted for the cut-off priority scheme. For the fractional guard channel scheme, besides handover requests, each new connection request is also accepted with probability ωic. The same admission policies apply to WLANs with probability ωkw. Note that cut-off priority is a particular case of the fractional guard channel with ωkw=0. If a handover request is not accepted in one network, then the request is transferred to the other network. In the cellular system, only when the user is inside the dual-coverage area can the connection be transferred to the WLAN. Connections that are within the WLAN can always be attempted to be transferred to the corresponding overlaying cell in the cellular system.The occupancy of a cell evolves according to a birth-death process independent of other cells as shown in Figure2(a). The process for cell i evolves with birth rate ρic (2.1)(a) for the unreserved states and αic (2.1)(b) for the reserved states. The death rate of cell i in state ni is niμic. The total traffic offered to cell i in state ni is
(2.1)λic(kic)={λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo),kic≤Cic-Ric(a),λi(cn)ωic+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo),kic>Cic-Ric(b),
with ωic=0 for the cut-off priority scheme, and 0≤ωic≤1 for the fractional guard channel scheme.  The term λji(cc) is the horizontal handover rate of cell j offered to cell i, for adjacent cells i and j, and λki(wc) is the vertical handover rate of WLAN k offered to overlaid cell i, λzk(co) is the proportion of all handover traffic that is not accepted in cell z due to the resource depletion and thence overflowed to WLAN k.State transition diagrams for each cellular cell (a) and WLAN (b).
(a)(b)Similarly, as shown in Figure2(b) the occupancy of WLAN k evolves with birth rates ρkw (2.2)(a) and αkw (2.2)(b) based on the state mk, and death rate kkw(μkw+υkw). The total traffic offered to WLAN k in state kkw is
(2.2)λkw(miw)={λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co),mk≤Ckw-Rkw(a),λk(wn)ωkw+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co),mk>Ckw-Rkiwc(b),


## 2.2. Link Traffic Flow (Handover Traffic) Analysis

Let us now focus on the traffic flow on the links, that is, handover traffic and overflow traffic to derive the handover rates in (2.1) ((a), (b)) and (2.2) ((a), (b)). It will be found that the handover rates and the blocking rate and dropping rate are interdependent with each other.


### 2.2.1. Handover Calls to Cell

The horizontal handover rateλji(cc)of cell j offered to cell i, for adjacent cells i and j, is all the nonblocked (dropped) traffic in cell j multiply with a handover probability factor and is shown in the Figure 3. Blue lines are the traffic that can go through to the next stage, and red lines are the blocked or dropped traffic.Figure 3
Horizontal handover traffic between cellular cells.So the formula for horizontal handover is given by (2.3)
(2.3)λji(cc)=Hji(cc)⋅[λj(cn)⋅(1-Bj(c))+(∑x∈Cjaλxj(cc)+∑y∈Wjoλyj(wc)+∑z∈Wjoλzj(wo))⋅(1-Dj(c))],
where Bj(c) and Dj(c) are the new connection blocking and handover dropping probabilities in cell j, respectively.In (2.3), λzj(wo) corresponds to all handover traffic that is not accepted in WLAN z and hence transferred to cell j, so
(2.4)λzj(wo)=Dz(w)⋅(λjz(cw)+∑l∈Wzaλlz(ww)).
For λki(wc), the vertical handover rate of WLAN k offered to overlay cell i is all the non-blocked (dropped) traffic in WLAN k multiplied with a handover probability factor as shown in the Figure 4. Blue lines are the traffic that can go through to the next stage and red lines are the blocked or dropped traffic. So the formula for vertical handover is given by (2.5)(2.5)λki(wc)=Hki(wc)⋅[λk(wn)⋅(1-Bk(w))+(∑x∈Wkaλxk(ww)+∑y∈Ckoλyk(cw)+∑z∈Ckoλzkco)⋅(1-Dk(w))],
where Bk(w) and Dk(w) are the new connection blocking and handover dropping probabilities in WLAN k, respectively.Figure 4
Vertical handover traffic from WLAN to cellular cell.In (2.6), λzk(co) is the proportion of all handover traffic that is not accepted in cell z and thence transferred to WLAN k, then
(2.6)λzk(co)=Dz(c)⋅(λkz(wc)+∑l∈Czaλlz(cc))⋅γzk,
where γzk is the coverage factor between WLAN k and overlay cell z, that is, the ratio between the radio coverage area of WLAN k and the radio coverage area of cell z with 0<γzk≤1.


### 2.2.2. Handover Calls to WLAN

Similarly, we can derive the termλlk(ww), the horizontal handover rate of WLAN l offered to WLAN k, for adjacent WLAN l and k, which is all the non-blocked (dropped) traffic in WLAN multiplied with a handover probability factor as shown in the Figure 5. Blue lines are the traffic that can go through to the next stage and red lines are the blocked or dropped traffic. So the formula for horizontal handover is given by (2.7)(2.7)λlk(ww)=Hlk(ww)⋅[λl(wn)(1-Blw)+(∑x∈Wlaλxl(ww)+∑y∈Cloλyl(cw)+∑z∈Cloλzl(co))⋅(1-Dlw)].
The term λjk(cw) is the vertical handover rate of cell j offered to WLAN k, for overlaying cell j and k, which is all the non-blocked (dropped) traffic in cell j multiplied with a handover probability factor as shown in the Figure 6. Blue lines are the traffic that can go through to the next stage and red lines are the blocked or dropped traffic. So the formula for vertical handover is given by (2.8)(2.8)λjk(cw)=Hjk(cw)⋅[λj(cn)(1-Bjc)γjk+(∑x∈Cjaλxj(cc)γjk+∑y∈Wjoυyj(wc)γjk+∑z∈Wjoλzj(wo))⋅(1-Djc)].Figure 5
Horizontal handover traffic between WLANs.Figure 6
Vertical handover traffic from cellular cell to WLAN.


## 2.2.1. Handover Calls to Cell

The horizontal handover rateλji(cc)of cell j offered to cell i, for adjacent cells i and j, is all the nonblocked (dropped) traffic in cell j multiply with a handover probability factor and is shown in the Figure 3. Blue lines are the traffic that can go through to the next stage, and red lines are the blocked or dropped traffic.Figure 3
Horizontal handover traffic between cellular cells.So the formula for horizontal handover is given by (2.3)
(2.3)λji(cc)=Hji(cc)⋅[λj(cn)⋅(1-Bj(c))+(∑x∈Cjaλxj(cc)+∑y∈Wjoλyj(wc)+∑z∈Wjoλzj(wo))⋅(1-Dj(c))],
where Bj(c) and Dj(c) are the new connection blocking and handover dropping probabilities in cell j, respectively.In (2.3), λzj(wo) corresponds to all handover traffic that is not accepted in WLAN z and hence transferred to cell j, so
(2.4)λzj(wo)=Dz(w)⋅(λjz(cw)+∑l∈Wzaλlz(ww)).
For λki(wc), the vertical handover rate of WLAN k offered to overlay cell i is all the non-blocked (dropped) traffic in WLAN k multiplied with a handover probability factor as shown in the Figure 4. Blue lines are the traffic that can go through to the next stage and red lines are the blocked or dropped traffic. So the formula for vertical handover is given by (2.5)(2.5)λki(wc)=Hki(wc)⋅[λk(wn)⋅(1-Bk(w))+(∑x∈Wkaλxk(ww)+∑y∈Ckoλyk(cw)+∑z∈Ckoλzkco)⋅(1-Dk(w))],
where Bk(w) and Dk(w) are the new connection blocking and handover dropping probabilities in WLAN k, respectively.Figure 4
Vertical handover traffic from WLAN to cellular cell.In (2.6), λzk(co) is the proportion of all handover traffic that is not accepted in cell z and thence transferred to WLAN k, then
(2.6)λzk(co)=Dz(c)⋅(λkz(wc)+∑l∈Czaλlz(cc))⋅γzk,
where γzk is the coverage factor between WLAN k and overlay cell z, that is, the ratio between the radio coverage area of WLAN k and the radio coverage area of cell z with 0<γzk≤1.


## 2.2.2. Handover Calls to WLAN

Similarly, we can derive the termλlk(ww), the horizontal handover rate of WLAN l offered to WLAN k, for adjacent WLAN l and k, which is all the non-blocked (dropped) traffic in WLAN multiplied with a handover probability factor as shown in the Figure 5. Blue lines are the traffic that can go through to the next stage and red lines are the blocked or dropped traffic. So the formula for horizontal handover is given by (2.7)(2.7)λlk(ww)=Hlk(ww)⋅[λl(wn)(1-Blw)+(∑x∈Wlaλxl(ww)+∑y∈Cloλyl(cw)+∑z∈Cloλzl(co))⋅(1-Dlw)].
The term λjk(cw) is the vertical handover rate of cell j offered to WLAN k, for overlaying cell j and k, which is all the non-blocked (dropped) traffic in cell j multiplied with a handover probability factor as shown in the Figure 6. Blue lines are the traffic that can go through to the next stage and red lines are the blocked or dropped traffic. So the formula for vertical handover is given by (2.8)(2.8)λjk(cw)=Hjk(cw)⋅[λj(cn)(1-Bjc)γjk+(∑x∈Cjaλxj(cc)γjk+∑y∈Wjoυyj(wc)γjk+∑z∈Wjoλzj(wo))⋅(1-Djc)].Figure 5
Horizontal handover traffic between WLANs.Figure 6
Vertical handover traffic from cellular cell to WLAN.


## 2.3. Balance Equations

From the analysis of the birth-death process in celli, the detailed balance equations are
(2.9)πic(kic-1)⋅(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))=πic(kic)kic(μic+υic),0<kic≤Cic-Ric,πic(kic-1)⋅(λi(cn)ωic+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))=πic(kic)kic(μic+υic),Cic-R<kic≤Cic,
where πic(kic) is the steady state probability that there arekic calls is in cell i and is derived as
(2.10)πic(kic)=(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))kicπic(0)kic!(μic+υic)kic,ni≤Cic-Ric,πic(kic)=(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))Cic-Ricπic(0)kic!(μic+υic)kic×(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))kic-Cic+Ricπic(0)kic!(μic+υic)kic,ni>Cic-Ric,
where πic(0) is the normalisation constant, obtained as:
(2.11)πic(0)=∑kic=0Cic-Ric1kic!(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo)μic+υic)kic+∑ni=Cic-Ric+1Cic(λi(cn)+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))Cic-Rickic!(μic+υic)ni×(λi(cn)ωic+∑j∈Ciaλji(cc)+∑k∈Wioλki(wc)+∑l∈Wioλli(wo))kic-Cic+Rickic!(μic+υic)ni.
From the process WLAN k, the detailed balance equations are
(2.12)πkw(kkw-1)(λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))=πkw(kkw)kkw(μkw+υkw),0<kkw≤Ckw-Rkw,πkw(kkw-1)(λk(wn)ωkw+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))=πkw(kkw)kkw(μkw+υkw),Ckw-R<kkw≤Ckw,
where πkw(kkw) is the steady-state probability that kkw calls is in WLAN k, and is derived as(2.13)πkw(kkw)=(λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))kkwπkw(0)kkw!(μkw+υkw)kkw,kkw≤Ckw-Rkw,πkw(kkw)=(λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))Ckw-Rkwπkw(0)kkw!(μkw+υkw)kkw×(λk(wn)ωkw+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))kkw-Ckw+Rkwπkw(0)kkw!(μkw+υkw)kkw,kkw>Ckw-Rkw,
where πk’w(0) is the normalisation constant, obtained as
(2.14)πkw(0)=∑kkw=0Ckw-Rkw1kkw!(λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co)μkw+υkw)kkw+∑kkw=Ckw-Rkw+1Ckw(λk(wn)+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))Ckw-Rkwkkw!(μkw+υkw)kkw×(λk(wn)ωkw+∑l∈Wkaλlk(ww)+∑j∈Ckoλjk(cw)+∑g∈Ckoλgk(co))kkw-Ckw+Rkwkkw!(μkw+υkw)kkw.


## 3. QoS Metrics

In the cellular system the new-connection blocking probabilityBi(c) in cell i is Bi(c)=∑kic=Cic-RicCic-1πic(ni)(1-ωic)+πic(Cic), and the handover dropping probability  Di(c)=πic(Cic).Similarly,Bk(w) in WLAN k is Bk(w)=∑Kkw=Ckw-RkwCcw-1πkw(mk)(1-ωkw)+πkw(Ckw), and  Dk(w)=πkw(Ckw).As we can see, blocking probabilitiesBi(c), Bk(w) and dropping probabilities Di(c), Dk(w) and handover rates are interdependent, so they can be calculated numerically using Erlang fixed-point approximation we have discussed.Blocking probabilitiesBi(c), Bk(w)and dropping probabilitiesDi(c), Dk(w) are given an initial value, then the values of the handover rates in each cellular cell and WLAN can be calculated. With handover rates we then get a new set of blocking probabilities Bi(c), Bk(w)and dropping probabilities Di(c), Dk(w) using the Erlang-B formula. Then we compare the new blocking and dropping probabilities with the old ones; if they are less than a given threshold, they can be accepted; otherwise all the above steps will be repeated once again using the obtained blocking and dropping probabilities as the new initial inputs.


## 4. Conclusions

In this paper, we discussed the methods of reduced load approximation especially Erlang fixed-point approximation and its application in the analysis of wireless HetNet. We analysed a single-rate traffic cellular/WLAN integrated wireless network with detailed study of the node traffic flow and link traffic analysis based on a network topology scheme. QoS metrics in terms of new call blocking probability and handover call dropping probability were considered in the model and calculated using the Erlang fixed-point approximation. This transforms the network problem into finally a set of nonlinear equations. By solving this set of equations the performance of the system in terms of new call blocking probability and handover call dropping probability can be easily derived.



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*Source: 862809-2012-05-22.xml*