The proposed CAC schemes can be modeled by Markov chain. The Markov chain for the queuing analysis of a femtocell layer is shown in Figure 12, where the states of the system represent the number of calls in the system. The maximum number of calls that can be accommodated in a femtocell system is *K*. As the call arrival rate in a femtocell is normally very low and the data rate of a femtocellular network is high, there is no need for a handover priority scheme for the femtocellular networks. The calls that have arrived in a femtocellular network are new originating calls, macrocell-to-femtocell handover calls, and femtocell-to-femtocell handover calls. Femtocell-to-femtocell handover calls are divided into two types. The first type of call is when the received SNIR of the T-FAP is greater than or equal to Γ_{2}. The second type of call is when the received SNIR of the T-FAP is between Γ_{1} and Γ_{2}, and these calls are rejected by the macrocellular BS. We define *μ*_{
m
} (*μ*_{
f
}) as the channel release rate of the macrocell (femtocell).

Figure 13 shows the Markov chain for the queuing analysis of the overlaid macrocell layer, where the states of the system represent the number of calls in the system. In Figures 12 and 13, symbols *λ*_{
o,f
} and *λ*_{
o,m
} denote the total originating call arrival rates considering all *n* number of femtocells within a macrocell coverage area and only the macrocell coverage area, respectively. *λ*_{h, mm}, *λ*_{h, ff}, *λ*_{h, fm}, and *λ*_{h, mf} denote the total macrocell-to-macrocell, femtocell-to-femtocell, femtocell-to-macrocell, and macrocell-to-femtocell handover call arrival rates within the macrocell coverage area, respectively. *P*_{
B,m
} (*P*_{
B,f
}) is the new originating call blocking probability in the macrocell (femtocell) system. *P*_{
D,m
} (*P*_{
D,f
}) is the handover call dropping probability in the macrocell (femtocell) system. We assume that for a femtocell-to-femtocell handover, the probability that the received SNIR of the T-FAP is greater Γ_{2} and is represented by *α*, and the received SNIR of the T-FAP is between Γ_{2} and Γ_{2} and is represented by *β.* Figure 13 also shows that the macrocell system provides *S* number of additional states to support handover calls by the proposed adaptive QoS policy. State *N* is the maximum number of calls that can be accommodated by the macrocell system without a QoS adaptation policy. Hence, the system provides a QoS adaptation policy only to accept handover calls in the macrocell system. These handover calls include macrocell-to-macrocell and femtocell-to-macrocell handover calls. Femtocell-to-macrocell handover calls are divided into two types. The first type of call is for those that have directly arrived to the macrocell system. The second type of call is those for which the calls have first arrived to femtocells, but are not accepted to the femtocells owing to lagging of resources or poor SNIR level.

The average channel release rate for the macrocell layer increases as the number of deployed femtocells increases. Because of the increasing number of femtocells, more macrocell users are handed over to femtocell networks. The average channel release rates [29] for the femtocell layer and the macrocell layer are calculated as follows.

For the macrocell layer, the average channel release rate is

{\mu}_{m}={\eta}_{m}\left(\sqrt{n}+1\right)+\mu ,

(11)

and for the femtocell layer, it is

{\mu}_{f}={\eta}_{f}+\mu ,

(12)

where 1/*μ*, 1/*η*_{
m
}, and 1/*η*_{
f
} are the average call duration (exponentially distributed), average cell dwell time for the macrocell (exponentially distributed), and the average cell dwell time for the femtocell (exponentially distributed), respectively.

Equating the net rate of calls entering a cell and requiring handover to those leaving the cell, the handover call arrival rates are calculated as follows [29].

The macrocell-to-macrocell handover call arrival rate is

{\lambda}_{h,\mathit{mm}}={P}_{h,\mathit{mm}}\frac{\left(1-{P}_{B,m}\right)\left({\lambda}_{m,o}+{\lambda}_{f,o}{P}_{B,f}\right)+\left(1-{P}_{D,m}\right)\left\{{\lambda}_{h,\mathit{fm}}+{\lambda}_{h,\mathit{ff}}\left(1-\alpha +\alpha {P}_{D,f}\right)\right\}}{1-{P}_{h,\mathit{mm}}\left(1-{P}_{D,m}\right)},

(13)

the macrocell-to-femtocell handover call arrival rate is

{\lambda}_{h,\mathit{mf}}={P}_{h,\mathit{mf}}\frac{\left(1-{P}_{B,m}\right)\left({\lambda}_{m,o}+{\lambda}_{f,o}{P}_{B,f}\right)+\left(1-{P}_{D,m}\right)\left\{{\lambda}_{h,\mathit{fm}}+{\lambda}_{h,\mathit{ff}}\left(1-\alpha +\alpha {P}_{D,f}\right)\right\}}{1-{P}_{h,\mathit{mm}}\left(1-{P}_{D,m}\right)},

(14)

the femtocell-to-femtocell handover call arrival rate is

{\lambda}_{h,\mathit{ff}}={P}_{h,\mathit{ff}}\frac{{\lambda}_{f,o}\left(1-{P}_{B,f}\right)+{\lambda}_{h,\mathit{mf}}\left(1-{P}_{D,f}\right)}{1-{P}_{h,\mathit{ff}}\left(1-{P}_{D,f}\right)\left\{\alpha +\left(1-\alpha \right){P}_{D,m}\right\}},

(15)

and the femtocell-to-macrocell handover call arrival rate is

{\lambda}_{h,\mathit{fm}}={P}_{h,\mathit{fm}}\frac{{\lambda}_{f,o}\left(1-{P}_{B,f}\right)+{\lambda}_{h,\mathit{mf}}\left(1-{P}_{D,f}\right)}{1-{P}_{h,\mathit{ff}}\left(1-{P}_{D,f}\right)\left\{\alpha +\left(1-\alpha \right){P}_{D,m}\right\}},

(16)

where *P*_{h,mm}, *P*_{h,mf}, *P*_{h,ff}, and *P*_{h,fm} are the macrocell-to-macrocell handover probability, macrocell-to-femtocell handover probability, femtocell-to-femtocell handover probability, and femtocell-to-macrocell handover probability, respectively.

The probability of handover depends on several factors such as the average call duration, cell size, and average user velocity. The handover probabilities from a femtocell and to a femtocell in integrated femtocell/macrocell networks also depend on the density of femtocells and the average size of femtocell coverage areas. Hence, on the basis of the basic derivation for handover probability calculations in [29], we derive the formulas for *P*_{h,mm}, *P*_{h,mf}, *P*_{h,ff}, and *P*_{h,fm} as follows

{P}_{h,\mathit{mm}}=\frac{{\eta}_{m}}{{\eta}_{m}+\mu},

(17)

{P}_{h,\mathit{fm}}=\left[1-n{\left(\frac{{r}_{f}}{{r}_{m}}\right)}^{2}\right]\frac{{\eta}_{f}}{{\eta}_{f}+\mu},

(18)

{P}_{h,\mathit{ff}}=\left(n-1\right){\left(\frac{{r}_{f}}{{r}_{m}}\right)}^{2}\frac{{\eta}_{f}}{{\eta}_{f}+\mu},

(19)

{P}_{h,\mathit{mf}}=n{\left(\frac{{r}_{f}}{{r}_{m}}\right)}^{2}\frac{{\eta}_{m}\sqrt{n}}{{\eta}_{m}\sqrt{n}+\mu}.

(20)

There is no guard channel for the handover calls in the femtocell layer in our proposed scheme. For the femtocell layer, the average call blocking probability *P*_{B,f} and the average call dropping probability *P*_{D,f} can be calculated as [30]

{P}_{D,f}={P}_{B,f}={P}_{f}\left(K\right)=\frac{{\left(\frac{{\lambda}_{T,f}}{n}\right)}^{K}\frac{1}{K!{\mu}_{f}^{K}}}{{\displaystyle \sum _{i=0}^{K}{\left(\frac{{\lambda}_{T,f}}{n}\right)}^{i}\frac{1}{i!{\mu}_{f}^{i}}}},

(21)

\mathrm{where}\phantom{\rule{0.5em}{0ex}}{{\lambda}_{T}}_{,f}={{\lambda}_{f}}_{,o}+{{\lambda}_{h}}_{,\mathit{mf}}+\alpha {{\lambda}_{h}}_{,\mathit{ff}}+{{P}_{D}}_{,m}\beta {{\lambda}_{h}}_{,\mathit{ff}}.

A QoS adaptation/degradation policy is allowed for the handover calls of a macrocell layer in our proposed scheme. For the macrocell layer, the average call blocking probability *P*_{B,m} and the average call dropping probability *P*_{D,m} can be calculated as [30]

\begin{array}{l}{P}_{B,m}={\displaystyle \sum _{i=N}^{N+S}P\left(i\right)}\\ \phantom{\rule{2.7em}{0ex}}={\displaystyle \sum _{i=N}^{N+S}\frac{{\left({\lambda}_{m,0}+{\lambda}_{h,m}\right)}^{N}{\left({\lambda}_{h,m}\right)}^{i-N}}{i!{\mu}_{m}^{i}}}P\left(0\right),\end{array}

(22)

{P}_{D,m}=P\left(N+S\right)=\frac{{\left({\lambda}_{m,0}+{\lambda}_{h,m}\right)}^{N}{\lambda}_{h,m}^{S}}{\left(N+S\right)!{\mu}_{m}^{N+S}}P\left(0\right),

(23)

\text{where}\phantom{\rule{0.5em}{0ex}}{\lambda}_{h,m}={\lambda}_{h,\mathit{mm}}+{\lambda}_{h,\mathit{fm}}+\alpha {P}_{D,f}{\lambda}_{h,\mathit{ff}}+\left(1-\alpha \right){\lambda}_{h,\mathit{ff}}

\begin{array}{l}\mathrm{and}\phantom{\rule{0.5em}{0ex}}P\left(0\right)=\left[{\displaystyle \sum _{i=0}^{N}\frac{{\left({\lambda}_{m,0}+{\lambda}_{m,h}\right)}^{i}}{i!{\mu}_{m}^{i}}}\right.\\ \phantom{\rule{5.5em}{0ex}}{\left(\right)}^{{\displaystyle +{\displaystyle \sum _{i=N+1}^{N+S}\frac{{\left({\lambda}_{m,0}+{\lambda}_{m,h}\right)}^{N}{\left({\lambda}_{m,h}\right)}^{i-N}}{i!{\mu}_{m}^{i}}}}}-1& .\end{array}\n