**2.1 Definition of time interval variables**

In this section the different time interval variables involved in the analytical model of a mobile cellular network are defined.

First, the *unencumbered service time* per call *xs* (also known as the *requested call holding time* (Alfa and Li, 2002) or *call holding duration* (Rahman & Alfa, 2009)) is the amount of time that the call would remain in progress if it experiences no forced termination. It has been widely accepted in the literature that the unencumbered service time can adequately be modeled by a negative exponentially distributed random variable (RV) (Lin et al., 1994; Hong & Rappaport, 1986). The RV used to represent this time is **X***s* and its mean value is ����� � 1��.

Now, c*ell dwell time* or *cell residence time xd*(*<sup>j</sup>*) is defined as the time interval that a mobile station (MS) spends in the *j-*th (for *j* = 0, 1, …) handed off cell irrespective of whether it is engaged in a call (or session) or not. The random variables (RVs) used to represent this time are **X***d*(*<sup>j</sup>*) (for *j* = 0, 1, …) and are assumed to be independent and identically generally phasetype distributed. For homogeneous cellular systems, this assumption has been widely accepted in the literature (Lin et al., 1994; Hong & Rappaport, 1986; Orlik & Rappaport, 1998; Fang & Chlamtac, 1999; Alfa & Li, 2002; Rahman & Alfa, 2009).

In this Chapter, cell dwell time is modeled as a general phase-type distributed RV with the probability distribution function (pdf) *<sup>d</sup> f t* **<sup>X</sup>** , the cumulative distribution function (CDF) *<sup>d</sup> F t* **<sup>X</sup>** , and the mean ����� � 1��.

The *residual cell dwell time xr* is defined as the time interval between the instant that a new call is initiated and the instant that the user is handed off to another cell. Notice that residual cell dwell time is only defined for new calls. The RV used to represent this time is **X***r*. Thus, the probability density function (pdf) of **X***r*, *<sup>r</sup> f t* **<sup>X</sup>** , can be calculated in terms of **X***d* using the excess life theorem (Lin et al., 1994)

$$f\_{\mathbf{X}\_r}(t) = \frac{1}{E[\mathbf{X}\_d]} \left[ 1 - F\_{\mathbf{X}\_d}(t) \right] \tag{1}$$

where *E*[**X***d*] and *<sup>d</sup> F t* **<sup>X</sup>** are, respectively, the mean value and cumulative probability distribution function (CDF) of **X***d*.

Finally, we define *channel holding time* as the amount of time that a call holds a channel in a particular cell. In this Chapter we distinguish between channel holding times for handed off (CHTh) and channel holding time for new calls (CHTn). CHTh (CHTn) is represented by the random variable �� ��� ( ( ) *<sup>N</sup>* **<sup>X</sup>***<sup>c</sup>* ).


$$\mathcal{L}\{f\_{\mathbf{X}\_{r}}(t)\} = \mathcal{L}\left\{\frac{1}{E\{\mathbf{X}\_{d}\}}\right\} - \mathcal{L}\left\{\frac{1}{E\{\mathbf{X}\_{d}\}} F\_{\mathbf{X}\_{d}}(t)\right\} \tag{2}$$

$$f\_{\mathbf{X}\_r}^\*(\mathbf{s}) = \begin{bmatrix} \frac{1}{s} \end{bmatrix} \frac{1}{E\{\mathbf{X}\_d\}} \begin{bmatrix} 1 - f\_{\mathbf{X}\_d}^\*(\mathbf{s}) \end{bmatrix} \tag{3}$$

$$E\{ (\mathbf{X}\_r)^n \} = \frac{E\{ (\mathbf{X}\_d)^{n+1} \}}{(n+1)E\{ \mathbf{X}\_d \}} \tag{4}$$

$$E\{\mathbf{X}\_r\} = \frac{E\{\mathbf{X}\_d\}}{2} + \frac{VAR(\mathbf{X}\_d)}{2E\{\mathbf{X}\_d\}}\tag{5}$$

$$\frac{E\{\mathbf{X}\_d\}}{2} + \frac{VAR\{\mathbf{X}\_d\}}{2E\{\mathbf{X}\_d\}} > E\{\mathbf{X}\_d\}$$

$$\text{Cov}\,V\{\mathbf{X}\_d\} > 1\tag{6}$$

$$F\_{\mathbf{X}\_c^{(h)}}(t) = 1 - \left[1 - F\_{\mathbf{X}\_s}(t)\right] \left[1 - F\_{\mathbf{X}\_d}(t)\right] \tag{7}$$

$$F\_{\mathbf{x}\_r^{(N)}}(t) = 1 - \left[1 - F\_{\mathbf{X}\_s}(t)\right] \left[1 - F\_{\mathbf{X}\_r}(t)\right] \tag{8}$$

$$f\_{\mathbf{x}\_{\mathbf{c}}^{(N)}}^{\*}(\mathbf{s}) = f\_{\mathbf{X}\_{\mathbf{S}}}^{\*}(\mathbf{s}) + \mathbf{s} \, \Sigma\_{p \in \Omega\_{\mathbf{X}\_{\mathbf{S}}}} \, \xi = p + \mathbf{s} \, \frac{f\_{\mathbf{X}\_{\mathbf{r}}}^{\*}(\boldsymbol{\xi})}{\xi} \frac{f\_{\mathbf{X}\_{\mathbf{S}}}^{\*}(\mathbf{s} - \boldsymbol{\xi})}{\varepsilon - \xi} \tag{9}$$



$$f\_{\mathbf{X}\_d}(t) = \sum\_{j=1}^{n} P\_j \,\eta\_j \, e^{-\eta\_j t} \tag{10}$$

$$F\_{\mathbf{x}\_c^{(h)}}(t) = 1 - [e^{-\mu t}] \left[ \sum\_{i=1}^{n} P\_i e^{-\eta\_i t} \right]$$

$$F\_{\mathbf{x}\_c^{(h)}}(t) = 1 - \sum\_{i=1}^{n} P\_i e^{-(\mu + \eta\_i)t} \tag{11}$$

$$P\_l^{\{N\}} = \frac{{}^{P\_l \prod\_{f=1}^n \eta\_l}}{\sum\_{l=1 \atop f \neq l}^n {}^{P\_l \prod\_{f=1}^n \eta\_f}} \tag{12}$$

$$f\_{\mathbf{X}\_d}^\*(\mathbf{s}) = \sum\_{j=1}^m P\_j \prod\_{l=1}^j \frac{\eta\_l}{(s+\eta\_l)} \tag{13}$$

$$P\_j = a\_j \prod\_{l=1}^{j-1} (1 - a\_l) \tag{14}$$

$$f^\*\_{\mathbf{x}^{(h)}\_{\mathcal{L}}}(\mathbf{s}) = \frac{\mu}{\mathbf{s} + \mu} + \frac{s}{\mathbf{s} + \mu} [f^\*\_{\mathbf{X}\_d}(\mathbf{s} + \mu)] \tag{15}$$

$$f^\*\_{\mathbf{X}\_c^{(N)}}(\mathbf{s}) = \frac{\mu}{\mathbf{s} + \mu} + \frac{s}{\mathbf{s} + \mu} [f^\*\_{\mathbf{X}\_r}(\mathbf{s} + \mu)] \tag{16}$$

$$f\_{\mathbf{x}\_{\mathcal{L}}^{(h)}}^{\*}\text{ (s)} = \sum\_{j=1}^{m} P\_j^{O(h)} \prod\_{l=1}^{j} \frac{\eta\_l}{(s+\mu+\eta\_l)}\tag{17}$$

$$P\_j^{O(h)} = \left[ \prod\_{l=1}^{j-1} \frac{\eta\_l}{\mu + \eta\_l} \right] \left[ P\_j + \sum\_{k=j+1}^m P\_k \left( \frac{\mu}{\mu + \eta\_j} \right) \right] \tag{18}$$

$$f\_{\mathbf{X}\_{\mathbf{r}}}^{\*}\left(\mathbf{s}\right) = \sum\_{j=1}^{\frac{m(m+1)}{2}} P\_{j}^{(N)}\left(\prod\_{k=h(j)}^{f(j)} \frac{\eta\_{k}}{s + \eta\_{k}}\right) \tag{19}$$

$$P\_j^{(N)} = \frac{P\_{f(j)} \prod\_{k=1 \atop k \neq l(j)}^m \eta\_k}{\sum\_{l=1}^m \left[ \left( \prod\_{k=1 \atop k \neq l}^m \eta\_k \right) \left( \Sigma\_{l=1}^m P\_l \right) \right]} \tag{20}$$

$$f(j) = \begin{vmatrix} \frac{1 \pm \sqrt{1 + 8(j-1)}}{2} \end{vmatrix} \tag{21}$$

$$h(f) = f - \frac{f(f)(f(f) - 1)}{2} \tag{22}$$

$$f\_{\mathbf{x}\_c^{\{N\}}}(\mathbf{s}) = \Sigma\_{j=1}^{\frac{m(m+1)}{2}} P\_j^{O(N)} \left( \prod\_{k=h(j)}^{f(j)} \frac{\eta\_k}{s + \mu + \eta\_k} \right) \tag{23}$$

$$P\_j^{O(N)} = \left[ \prod\_{l=h(f)}^{f(f)-1} \frac{\eta\_l}{\mu + \eta\_l} \right] \left[ P\_j^{(N)} + \sum\_{k=f(f)+1}^m \frac{P\_k^{(N)}}{\frac{\mu^2 - k + 2}{2} \left( \mu + \eta\_{f(f)} \right)} \right] \tag{24}$$

$$E\left\{\mathbf{X}\_c^{(N)}\right\} = \frac{1}{\mu} \left[1 - \frac{\eta}{\mu} \left[1 - f\_{\mathbf{X}\_d}^\*(\mu)\right]\right] \tag{25}$$

$$E\left\{\mathbf{X}\_c^{(h)}\right\} = \frac{1}{\mu} \left[1 - f\_{\mathbf{X}\_d}^\*(\mu)\right] \tag{26}$$

$$E\left\{\mathbf{X}\_c^{(N)}\right\} > E\left\{\mathbf{X}\_c^{(h)}\right\} \tag{27}$$

$$f\_{\mathbf{X}\_d}^\*(\mu) > \frac{\eta}{\mu + \eta} \tag{28}$$

$$\mathbb{E}\,\mathrm{Cov}^2\left(\mathbf{X}\_c^{(N)}\right) = \frac{-4\eta \,\mathrm{E}\left\{\mathbf{X}\_c^{(h)}\right\} + 2\left[1 - \eta \frac{d\left.f\_{\mathbf{X}\_d}^\*(\mu)}{d\mu}\right]}{\left[\mathrm{E}\{\mathbf{x}\_c^{(N)}\}\mu\right]^2} - \mathbf{1} \tag{29}$$

$$\mathcal{L}oV^{2}\{\mathbf{X}\_{c}^{(h)}\} = \frac{2}{\left(\varepsilon\{\mathbf{x}\_{c}^{(h)}\}\right)^{2}\mu} \left[\frac{d f\_{\mathbf{X}\_{d}}^{\star}(\mu)}{d\mu} + E\{\mathbf{X}\_{c}^{(h)}\}\right] - 1\tag{30}$$

$$E\left\{ \left( \mathbf{X}\_c^{(N)} \right)^n \right\} = \frac{1}{\mu} \left[ nE\left\{ \left( \mathbf{X}\_c^{(N)} \right)^{n-1} \right\} - \eta E\left\{ \left( \mathbf{X}\_c^{(h)} \right)^n \right\} \right] \tag{31}$$

$$E\left\{ \left( \mathbf{X}\_c^{(h)} \right)^n \right\} = \frac{n}{\mu} \left( \left( -1 \right)^n \frac{d^n \left[ f\_{\mathbf{x}\_d}^\*(\mu) \right]}{d\mu^n} + E\left\{ \left( \mathbf{X}\_c^{(h)} \right)^{n-1} \right\} \right) \tag{32}$$

(i.e., when finite variance is considered), log-normal, gamma, hyper-Erlang of order (2,2), hyper-exponential of order 2, and Coxian of order 2. Three different mobility scenarios for the numerical evaluation are assumed: *E*{**X***d*}=5*E*{**X***s*} (low mobility), *E*{**X***d*}=*E*{**X***s*} (moderate mobility), and *E*{**X***d*}=0.2*E*{**X***s*} (high mobility). In the plots of this section we use *E*{**X***s*}=180 s. In our numerical results, the effect of CoV and skewness of CDT on CHT characteristics is investigated. In the plots presented in this section, "HC" and "NC" stand for channel holding time for handoff calls (CHTh) and channel holding time for new calls (CHTn), respectively.

#### **4.1 Cell dwell time distribution completely characterized by its mean value**

Fig. 2 plots the mean value of both CHTn and CHTh versus the mean value of CDT when it is modeled by negative-exponential (EX), constant, and Pareto with 1<2 distributions. It is important to remark that all of these distributions are completely characterized by their respective mean values. As expected, Fig. 2 shows that, for the case when CDT is exponentially distributed, mean CHTn is equal to mean CHTh. An interesting observation on the results shown in Fig. 2 is that, irrespective of the mean value of CDT, there exists a significant difference between the mean value of CHTn when CDT is modeled as exponential distributed RV and the corresponding case when it is modeled by a heavytailed Pareto distributed RV (this behavior is especially true for the case when =1.1). Notice, however, that this difference is negligible for the case when =2 and high mobility scenarios (say, *E*{**X**d}<50 s) are considered. Similar behaviors are observed if mean CHTh is considered. Consequently, for high mobility scenarios where CDT can be statistical characterized by a Pareto distribution with shape parameter close to 2, the exponential distribution represents a suitable model for the CDT distribution. Fig. 2 also shows that, for

Fig. 2. Mean new and handoff call channel holding time for deterministic, negative exponentially, and Pareto distributed CDT against the mean CDT.

(i.e., when finite variance is considered), log-normal, gamma, hyper-Erlang of order (2,2), hyper-exponential of order 2, and Coxian of order 2. Three different mobility scenarios for the numerical evaluation are assumed: *E*{**X***d*}=5*E*{**X***s*} (low mobility), *E*{**X***d*}=*E*{**X***s*} (moderate mobility), and *E*{**X***d*}=0.2*E*{**X***s*} (high mobility). In the plots of this section we use *E*{**X***s*}=180 s. In our numerical results, the effect of CoV and skewness of CDT on CHT characteristics is investigated. In the plots presented in this section, "HC" and "NC" stand for channel holding time for handoff calls (CHTh) and channel holding time for new calls (CHTn),

Fig. 2 plots the mean value of both CHTn and CHTh versus the mean value of CDT when it is modeled by negative-exponential (EX), constant, and Pareto with 1<2 distributions. It is important to remark that all of these distributions are completely characterized by their respective mean values. As expected, Fig. 2 shows that, for the case when CDT is exponentially distributed, mean CHTn is equal to mean CHTh. An interesting observation on the results shown in Fig. 2 is that, irrespective of the mean value of CDT, there exists a significant difference between the mean value of CHTn when CDT is modeled as exponential distributed RV and the corresponding case when it is modeled by a heavytailed Pareto distributed RV (this behavior is especially true for the case when =1.1). Notice, however, that this difference is negligible for the case when =2 and high mobility scenarios (say, *E*{**X**d}<50 s) are considered. Similar behaviors are observed if mean CHTh is considered. Consequently, for high mobility scenarios where CDT can be statistical characterized by a Pareto distribution with shape parameter close to 2, the exponential distribution represents a suitable model for the CDT distribution. Fig. 2 also shows that, for

**4.1 Cell dwell time distribution completely characterized by its mean value** 

Fig. 2. Mean new and handoff call channel holding time for deterministic, negative

0 100 200 300 400 500 600 700 800 900

EX HC(Constant) NC(Constant) HC(Pareto =1.1) NC(Pareto =1.1) HC(Pareto =2) NC(Pareto =2)

Mean cell dwell time

exponentially, and Pareto distributed CDT against the mean CDT.

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Mean new and handoff call channel holding time

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respectively.

a given value of the mean CDT and considering the case when CDT is Pareto distributed with =1.1 (=2), mean CHTn always is greater (lower) than mean CHTh. This behavior can be explained by the combined effect of the following two facts. First, as comes closer to 1 (2), the probability that CDT takes higher values increases (decreases). This fact contributes to increase (reduce) the mean CHTh. Second, in general, new calls are more probable to start on cells where users spent more time and, as comes closer to 1, this probability increases. This fact contributes to increase mean CHTn relative to the mean CHTh. Then, the combined effect is dominated by the first (second) fact as comes closer to 2 (1). This leads us to the behavior explained above and illustrated in Fig.2. It may be interesting to derive the condition upon which the mean CHTn is greater than the mean CHTh when CDT is heavy-tailed Pareto distributed. This represents a topic of our current research.

#### **4.2 Cell dwell time distribution completely characterized by its first two moments**

Fig. 3 plots the mean value of both CHTn and CHTh versus the *CoV* of CDT when it is modeled by Pareto with shape parameter >2, lognormal, and Gamma distributions; all of them with mean value equal to 180 s. It is important to remark that all of these distributions are completely characterized by their respective first two moments. Fig. 3 shows that both mean CHTn and mean CHTh are highly sensitive to the type of distribution of CDT; this fact is especially true for *CoV*>2. Notice that, for the particular case when *CoV*=0, the mean values of both CHTn and CHTh are identical to the corresponding values for the case when CDT is deterministic with mean value equals 180 s, as expected. Fig. 3 also shows that, for values of CoV of CDT greater than 1 (1.2), mean CHTn is greater that mean CHTh when CDT is Gamma (log-normal) distributed. On the other hand, when CDT is Pareto distributed and irrespective of the value of its CoV, CHTh always is greater that mean CHTn. This behavior is mainly due to the heavy-tailed characteristics of the Pareto distribution.

Fig. 3. Mean new and handoff call channel holding time for gamma, log-normal, and Pareto distributed cell dwell time versus CoV of cell dwell time.

#### **4.3 Cell dwell time distribution completely characterized by its first three moments**

Figs. 4, 5, and 6 (7, 8, and 9) plot the mean value (CoV) of both CHTn and CHTh versus both the CoV and skewness of CDT when it is modeled by hyper-Erlang (2,2), hyper-exponential of order 2, and Coxian of order 2 distributions, respectively. It is important to remark that all of these distributions are completely characterized by their respective first three moments. Results of (Johnson & Taaffe, 1989; Telek & Heindl, 2003) are used to calculate the parameters of these distributions as function of their first three moments. In Figs. 4 to 9, two different values for the mean CDT are considered: 36 s (high mobility scenario) and 900 s (low mobility scenario). From Figs. 2, 5 and 6 the following interesting observation can be extracted. Notice that, for the case when CDT is modeled by either hyper-exponential or Coxian distributions and irrespective of the mean value of CDT, the particular scenario where skewness and *CoV* of CDT are, respectively, equal to 2 and 1, corresponds to the case when CDT is exponential distributed (in the exponential case mean CHTn and mean CHTh are identical).

Fig. 4. Mean CHTn and mean CHTh for hyper-Erlang distributed CDT versus CoV and skewness of CDT, with the mean CDT as parameter.

**4.3 Cell dwell time distribution completely characterized by its first three moments**  Figs. 4, 5, and 6 (7, 8, and 9) plot the mean value (CoV) of both CHTn and CHTh versus both the CoV and skewness of CDT when it is modeled by hyper-Erlang (2,2), hyper-exponential of order 2, and Coxian of order 2 distributions, respectively. It is important to remark that all of these distributions are completely characterized by their respective first three moments. Results of (Johnson & Taaffe, 1989; Telek & Heindl, 2003) are used to calculate the parameters of these distributions as function of their first three moments. In Figs. 4 to 9, two different values for the mean CDT are considered: 36 s (high mobility scenario) and 900 s (low mobility scenario). From Figs. 2, 5 and 6 the following interesting observation can be extracted. Notice that, for the case when CDT is modeled by either hyper-exponential or Coxian distributions and irrespective of the mean value of CDT, the particular scenario where skewness and *CoV* of CDT are, respectively, equal to 2 and 1, corresponds to the case when CDT is exponential distributed (in the exponential case mean CHTn and mean CHTh

Fig. 4. Mean CHTn and mean CHTh for hyper-Erlang distributed CDT versus CoV and

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E(Xd)=36(HC) E(Xd)=36(NC) E(Xd)=900(HC) E(Xd)=900(NC)

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skewness of CDT, with the mean CDT as parameter.

are identical).

Fig. 5. Mean CHTn and mean CHTh for hyper-exponentially distributed CDT versus CoV and skewness of CDT, with the mean CDT as parameter.

Fig. 6. Mean CHTn and mean CHTh for Coxian distributed cell dwell time versus CoV and skewness of cell dwell time, with the mean CDT as parameter.

Fig. 7. CoV of CHTn and CHTh for hyper-Erlang distributed CDT versus CoV and skewness of CDT, with the mean CDT as parameter.

Fig. 8. CoV of CHTn and CHTh for hyper-exponential distributed CDT versus CoV and skewness of CDT, with the mean CDT as parameter.

Fig. 7. CoV of CHTn and CHTh for hyper-Erlang distributed CDT versus CoV and skewness

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E(Xd)=36(HC) E(Xd)=36(NC) E(Xd)=900(HC) E(Xd)=900(NC)

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Fig. 8. CoV of CHTn and CHTh for hyper-exponential distributed CDT versus CoV and

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E(Xd)=36(HC) E(Xd)=36(NC) E(Xd)=900(HC) E(Xd)=900(NC)

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skewness of CDT, with the mean CDT as parameter.

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of CDT, with the mean CDT as parameter.

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Fig. 9. CoV of CHTn and CHTh for Coxian distributed CDT versus CoV and skewness of cell dwell time, with the mean cell dwell time as parameter.

On the other hand, Fig. 4 shows that the case when hyper-Erlang distribution with skewness equals 2 and *CoV* equals 1 is used to model CDT does not strictly correspond to the exponential distribution; however, the exponential model represents a suitable approximation for CDT in this particular case. From Figs. 4 to 9, it is observed that the qualitative behavior of mean and *CoV* of both CHTn and CHTh is very similar for all the phase-type distributions under study. The small quantitative difference among them is due to moments higher than the third one. Analyzing the impact of moments of CDT higher than the third one on channel holding time characteristics represents a topic of our current research.

From Fig. 10 is observed that the difference among the mean values of CHTn and CHTh is strongly sensitive to the CoV of the CDT, while it is practically insensitive to the skewness of the CDT. This difference is higher for the case when the CDT is modeled as hyperexponential distributed RV compared with the case when it is modeled as hyper-Erlang distributed RV. Also, it is observed that this difference remains almost constant for the entire range of values of the CoV of the CDT.

Fig. 10. Difference among the mean values of new and handoff call channel holding times for hyper-Erlang and hyper-exponential distributed cell dwell time versus CoV and skewness of cell dwell time, for the moderate-mobility scenario.

Finally, in Fig. 11 the mean channel holding time for new and handoff calls considering the gamma, hyper-Erlang (2,2), hyper-exponential of order 2, and Coxian of order 2 distributions for the cell dwell time are shown for different values of the coefficient of variation. The numerical results shown in Fig. 11 are obtained by equaling the first three moments of the different distributions to those of the gamma distribution. From Fig. 11, it is observed that for the hyper-exponential and Coxian distributions practically the same results are obtained for the mean channel holding time for both new and handoff calls. The differences among the other distributions are due to the fact that they differ on the higher order moments. To show this, the forth standardized moment (i.e., excess kurtosis) of the different distributions is shown in Fig. 12 for different values of the coefficient of variation, equaling the first three moments of the different distributions to those of the gamma distribution. From Fig. 12, it is observed that the hyper-exponential and Coxian distributions practically have the same value of excess kurtosis but this differs for that of the gamma and hyper-Erlang distributions. The gamma distribution shows the more different value of the excess kurtosis and, therefore, for this distribution the more different values of the mean channel holding times in Fig. 11 are obtained. Then, it could be necessary to capture more than three moments, even though the lower order moments dominate in importance. Similar conclusion was drawn in (Gross & Juttijudata, 1997).

Fig. 10. Difference among the mean values of new and handoff call channel holding times for hyper-Erlang and hyper-exponential distributed cell dwell time versus CoV and

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Coefficient of Variation Skewness

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Finally, in Fig. 11 the mean channel holding time for new and handoff calls considering the gamma, hyper-Erlang (2,2), hyper-exponential of order 2, and Coxian of order 2 distributions for the cell dwell time are shown for different values of the coefficient of variation. The numerical results shown in Fig. 11 are obtained by equaling the first three moments of the different distributions to those of the gamma distribution. From Fig. 11, it is observed that for the hyper-exponential and Coxian distributions practically the same results are obtained for the mean channel holding time for both new and handoff calls. The differences among the other distributions are due to the fact that they differ on the higher order moments. To show this, the forth standardized moment (i.e., excess kurtosis) of the different distributions is shown in Fig. 12 for different values of the coefficient of variation, equaling the first three moments of the different distributions to those of the gamma distribution. From Fig. 12, it is observed that the hyper-exponential and Coxian distributions practically have the same value of excess kurtosis but this differs for that of the gamma and hyper-Erlang distributions. The gamma distribution shows the more different value of the excess kurtosis and, therefore, for this distribution the more different values of the mean channel holding times in Fig. 11 are obtained. Then, it could be necessary to capture more than three moments, even though the lower order moments dominate in importance.

skewness of cell dwell time, for the moderate-mobility scenario.

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Similar conclusion was drawn in (Gross & Juttijudata, 1997).

Fig. 11. Mean new and handoff call channel holding time for gamma, hyper-exponential (2), hyper-Erlang (2,2) and Coxian (2) distributed cell dwell time versus CoV of cell dwell time.

Fig. 12. Kurtosis of cell dwell time for gamma, hyper-exponential (2), Coxian (2) and hyper-Erlang (2,2) distributed cell dwell time versus CoV of cell dwell time.
