**4.3 On optimal channel distribution (future study)**

Fig. 13 shows two arrangements each of 18 radio channels for use by 4 call streams. Fig. 13.a shows a two-tier network with 3 individual channels per stream in the first tier and 6 common 12 Will-be-set-by-IN-TECH

b) Static reservation: Divide all macro-cell channels allocated to a cell into two groups: one for the common use by all calls and the other for high-mobility calls only (the rigid

(a) (b)

networks: (a) Three micro-cells and one macro-cell, (b) Seven micro-cells and one macro-cell. Numerical results for two two-tier network examples are obtained. They are not qualitatively different from the results of the one-tier model discussed above. Fig. 10.a (three micro-cells and one macro-cell) and Fig. 10.b (seven micro-cells and one macro-cell) show that the dynamic reservation strategy gives the higher maximum revenue in both cases if the reserved number of channels *R* is properly chosen. The parameters are as follows: *A* = high-mobility call flow, *B* = low-mobility call flow, *N*<sup>1</sup> = number of micro-cell channels for each cell, *N*<sup>2</sup> =

**Conclusion:** In case of two-tier network, the results of numerical analysis confirm that the

In hierarchical overlaying cellular networks, traffic overflow between the overlaying tiers is used to increase the utilization of the available capacity. The arrival process of overflow traffic has been verified to be correlated and bursty. This characteristic has brought great challenges to performance evaluation of hierarchical networks. In most published works, the discussion is focussed on traffic loss analysis in homogenous hierarchical networks, e.g. micro/macro cellular phone systems as in the numerical analysis below. In the paper (Huang et al., 2008), the authors address the problems of performance evaluation in more complicated scenarios by taking account of heterogeneity and user mobility in hierarchical networks. They present an approximate analytical loss model. The loss performance obtained by our approximated analytical model is validated by simulation in a heterogeneous multi-tier overlaying system. Fig. 12 shows the dependence of revenue on channel rearrangement from macro-cell to micro-cell. We are looking for maximum revenue when low-mobility calls cost one unit and

Fig. 13 shows two arrangements each of 18 radio channels for use by 4 call streams. Fig. 13.a shows a two-tier network with 3 individual channels per stream in the first tier and 6 common

Fig. 10. Dependence of the revenue on the reserved number of channels for two-tier

division-based CAC scheme).

number of micro-cell channels.

**4.2 Channel rearrangement effect**

high-mobility calls cost *K* = 3 units.

**4.3 On optimal channel distribution (future study)**

optimal strategy is dynamic reservation.

Fig. 11. An illustration of channel rearrangement.

Fig. 12. Dependence of revenue on channel rearrangement from macro-cell to micro-cell.

channels in the second tier. In Fig. 13.b some kind of a homogeneous single tier network is depicted: each call has access to 9 channels equally distributed between streams. Such kind of arrangement could be implemented by modern DSP techniques.

Fig. 14 depicts the loss probability curves for these two schemes. Case (a) relates to pure loss system, case (b) relates to scheme with one waiting positions per stream. What is surprising? In case (a), beginning with a loss probability as low as 0.56% (less than 1%), it is advantageous to use the equally distributed scheme. Therefore, the traditional two-tier network could be recommended here at a very low call rates. Table 1 contains more detailed data on loss probability. When a single waiting position is added, the advantage of the equally distributed scheme increases even more and the cross point of curves occurs at the loss probability equal to 0.025%.


.......................................................................................................................................................................................... ........

V

*M* = *L* · *E*(*N*, *L*) (10)

M<sup>1</sup> ....................................................................................................................................................... ....... .......................... (11)

(13)

(14)

≥ 1 (12)

*L N* + 1 − *L* + *M*

*L N* + 1 − *L* + *M*

M

.......................... ................................................................................................................... ........ .......................... L N ∞

**5.3 Kosten's model and its new interpretation**

(reference is usually made to Riordan's paper (Riordan, 1956)):

*V* = *M* ·

*<sup>Z</sup>* <sup>=</sup> *<sup>V</sup> <sup>M</sup>* <sup>=</sup>

Fig. 17. An illustration of the ERT-method extension.

After substitution of (*M* + *N* + 1) into (11) we get:

1 − *M* +

 1 *M* − *M*<sup>1</sup>

*V* = *M* ·

We can reduce this expression to a simpler form:

*V* = *M*<sup>2</sup>

formula (7) follows:

or

probability a traffic stream with given mean value and variance is subject to.

.......................... ................................................................................................................... ....... .......................... L N 1

1 − *M* +

From these two parameters one introduce a new parameter *Z*, the so-called peakedness:

Experience shows that peakedness *Z* is a very good measure for the relative blocking

We offer a new interpretation of Kosten's results. We consider the scheme in Fig. 17. There are *N* common channels and one separate channel. From (10 and (11 we get a new formula for the variance *V* when both mean *M* and mean *M*<sup>1</sup> = *L* · *E*(*N* + 1, *L*) are known. From recurrence

.......................................................................................................................................................................................... .......

*<sup>M</sup>*<sup>1</sup> <sup>=</sup> *L M*

*L L M <sup>M</sup>*<sup>1</sup> − *L*

> − 1 = *M*

Note that *M* − *M*<sup>1</sup> is the load carried by a single channel and therefore it is always less than one. Formula (14) is useful for applications of the ERT-method in case of traffic splitting.

*<sup>M</sup>* <sup>+</sup> *<sup>N</sup>* <sup>+</sup> <sup>1</sup> <sup>=</sup> *L M*

= *M* 

*N* + 1 + *M*

*M*<sup>1</sup>

1 − *M* +

 *M M* − *M*<sup>1</sup>

*M*<sup>1</sup> *M* − *M*<sup>1</sup>

− *M*  M V

1 − *M* +

Fig. 16. Kosten's model: *N* fully accessible channels and a channel group of infinite capacity.

Call Admission Control in Cellular Networks 127

The basic idea the Equivalent Random Traffic (ERT) method is to use Erlang-B formula for overflow traffic offered to a secondary channel group of infinite capacity (Fig. 16), the so-called Kosten model (Kosten, 1937). Kosten's paper contains formulae for all binomial moments of number of busy channels in secondary overflow group. In practice, only the two first moments are used for characterization of the overflow traffic: mean traffic intensity *M* and variance *V*

### **5.2 Erlang formula and its generalization for non-integer number of channels**

We consider a system of *N* identical fully accessible channels (servers, trunks, slots, call center agents, pool of wavelengths in the optical network, etc) offered Poisson traffic *L* and operating as a loss system (blocked calls cleared). The probability that all *N* channels are busy at a random point of time is equal to:

$$B = E(N, L) = \frac{\frac{L^N}{N!}}{\sum\_{i=0}^{N} \frac{L^i}{i!}} \tag{6}$$

This is the famous Erlang-B formula (1917) (Iversen, 2011). For numerical analysis of (6) we use the well-known recurrent formula:

$$E(N+1,L) = \frac{L \cdot E(N,L)}{N+1+E(N,L)}\tag{7}$$

with initial value*E*(0, *L*) = 1. For the ERT-method we need Erlang-B formula for non-integral number of channels. How to get solution for a non-integral *N*? The traditional approach is based on the incomplete gamma function using:

$$E(N,L) = \frac{L^N \cdot e^{-L}}{\Gamma(N+1,L)}\tag{8}$$

where

$$
\Gamma(N+1, L) = \int\_L^{\infty} t^N \, e^{-t} \mathbf{d}t
$$

We propose a new approach for engineering applications. Let the value *N* be from the interval (0,1). We introduce a parabolic approximation for ln *R* = *lnE*(*N*, *L*) at points *N* = 0, *N* = 1, and *N* = 2:

$$\begin{aligned} \ln E(0, L) &= \ln(1) = 0 \\ B &= \ln E(1, L) = \ln \frac{L}{L + 1} \\ \text{C} &= \ln E(2, L) = \ln \frac{L^2}{L^2 + 2L + 2} \end{aligned}$$

Then we get the requested approximation:

$$
\ln E(\mathbf{N}, L) = \left(\frac{\mathbf{C}}{2} - B\right) \mathbf{N}^2 + \left(2B - \frac{\mathbf{C}}{2}\right) \mathbf{N}
$$

or in a more convenient form

$$E(N,L) = L^N \left(L+1\right)^{N^2-2N} \left(L^2+2L+2\right)^{\frac{N-N^2}{2}}\tag{9}$$

Thus we have an initial value of *E*(*N*, *L*) for *N* inside the interval (0,1) and we may calculate *E*(*N*, *L*) at any *N* by means of recurrent formula (7). In (Schneps-Schneppe & Sedols, 2010) the proposed approximation (6) is compared numerically with earlier known Erlang-B formula approximations (Rapp, 1964); (Szybicki, 1967); (Hedberg, 1981) and it is shown to be more accurate.

16 Will-be-set-by-IN-TECH

We consider a system of *N* identical fully accessible channels (servers, trunks, slots, call center agents, pool of wavelengths in the optical network, etc) offered Poisson traffic *L* and operating as a loss system (blocked calls cleared). The probability that all *N* channels are busy at a

This is the famous Erlang-B formula (1917) (Iversen, 2011). For numerical analysis of (6) we

*<sup>E</sup>*(*<sup>N</sup>* <sup>+</sup> 1, *<sup>L</sup>*) = *<sup>L</sup>* · *<sup>E</sup>*(*N*, *<sup>L</sup>*)

with initial value*E*(0, *L*) = 1. For the ERT-method we need Erlang-B formula for non-integral number of channels. How to get solution for a non-integral *N*? The traditional approach is

*<sup>E</sup>*(*N*, *<sup>L</sup>*) = *<sup>L</sup><sup>N</sup>* · *<sup>e</sup>*−*<sup>L</sup>*

We propose a new approach for engineering applications. Let the value *N* be from the interval (0,1). We introduce a parabolic approximation for ln *R* = *lnE*(*N*, *L*) at points *N* = 0, *N* = 1,

*<sup>B</sup>* <sup>=</sup> ln *<sup>E</sup>*(1, *<sup>L</sup>*) = ln *<sup>L</sup>*

*<sup>C</sup>* <sup>=</sup> ln *<sup>E</sup>*(2, *<sup>L</sup>*) = ln *<sup>L</sup>*<sup>2</sup>

*<sup>L</sup>* <sup>+</sup> <sup>1</sup> ,

*<sup>L</sup>*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>L</sup>* <sup>+</sup> <sup>2</sup> .

*<sup>N</sup>*−*N*<sup>2</sup>

<sup>2</sup> (9)

 ∞ *L t <sup>N</sup> e* −*t* d*t*

Γ(*N* + 1, *L*) =

ln *E*(0, *L*) = ln(1) = 0 ,

 *C* <sup>2</sup> <sup>−</sup> *<sup>B</sup> N*<sup>2</sup> + <sup>2</sup>*<sup>B</sup>* <sup>−</sup> *<sup>C</sup>* 2 *N*

*E*(*N*, *L*) = *L<sup>N</sup>* (*L* + 1)*N*<sup>2</sup>−2*N*(*L*<sup>2</sup> + 2*L* + 2)

Thus we have an initial value of *E*(*N*, *L*) for *N* inside the interval (0,1) and we may calculate *E*(*N*, *L*) at any *N* by means of recurrent formula (7). In (Schneps-Schneppe & Sedols, 2010) the proposed approximation (6) is compared numerically with earlier known Erlang-B formula approximations (Rapp, 1964); (Szybicki, 1967); (Hedberg, 1981) and it is shown to be more

ln *E*(*N*, *L*) =

*L<sup>N</sup> N*! ∑*<sup>N</sup> i*=0 *Li i*!

*<sup>N</sup>* <sup>+</sup> <sup>1</sup> <sup>+</sup> *<sup>E</sup>*(*N*, *<sup>L</sup>*) (7)

<sup>Γ</sup>(*<sup>N</sup>* <sup>+</sup> 1, *<sup>L</sup>*) (8)

(6)

**5.2 Erlang formula and its generalization for non-integer number of channels**

*B* = *E*(*N*, *L*) =

random point of time is equal to:

use the well-known recurrent formula:

where

and *N* = 2:

accurate.

based on the incomplete gamma function using:

Then we get the requested approximation:

or in a more convenient form

Fig. 16. Kosten's model: *N* fully accessible channels and a channel group of infinite capacity.

## **5.3 Kosten's model and its new interpretation**

The basic idea the Equivalent Random Traffic (ERT) method is to use Erlang-B formula for overflow traffic offered to a secondary channel group of infinite capacity (Fig. 16), the so-called Kosten model (Kosten, 1937). Kosten's paper contains formulae for all binomial moments of number of busy channels in secondary overflow group. In practice, only the two first moments are used for characterization of the overflow traffic: mean traffic intensity *M* and variance *V* (reference is usually made to Riordan's paper (Riordan, 1956)):

$$M = L \cdot E(N, L) \tag{10}$$

$$V = M \cdot \left(1 - M + \frac{L}{N + 1 - L + M}\right) \tag{11}$$

From these two parameters one introduce a new parameter *Z*, the so-called peakedness:

$$Z = \frac{V}{M} = \left(1 - M + \frac{L}{N + 1 - L + M}\right) \ge 1\tag{12}$$

Experience shows that peakedness *Z* is a very good measure for the relative blocking probability a traffic stream with given mean value and variance is subject to.

We offer a new interpretation of Kosten's results. We consider the scheme in Fig. 17. There are *N* common channels and one separate channel. From (10 and (11 we get a new formula for the variance *V* when both mean *M* and mean *M*<sup>1</sup> = *L* · *E*(*N* + 1, *L*) are known. From recurrence

$$L \longrightarrow \begin{array}{c} \hline N \\ \hline \end{array} \xrightarrow{M} \begin{array}{c} \begin{array}{c} M \\ \hline \end{array} \xrightarrow{M\_1} \begin{array}{c} M\_1 \\ \hline \end{array} \xrightarrow{M\_1} \begin{array}{c} \hline \end{array} \xrightarrow{M\_1} \begin{array}{c} \hline \end{array} \xrightarrow{M\_1} \begin{array}{c} \hline \end{array} \xrightarrow{M\_1} \begin{array}{c} \hline \end{array} \xrightarrow{M\_1} \begin{array}{c} \begin{array}{c} \hline \end{array} \xrightarrow{M\_1} \begin{array}{c} \hline \end{array} \xrightarrow{M\_1} \begin{array}{c} \begin{array}{c} \hline \end{array} \xrightarrow{M\_1} \begin{array}{c} \begin{array}{c} \hline \end{array} \end{array} \xrightarrow{M\_1} \begin{array}{c} \begin{array}{c} \hline \end{array} \xrightarrow{M\_1} \begin{array}{c} \begin{array}{c} \hline \end{array} \end{array} \xrightarrow{M\_1} \begin{array}{c} \begin{array}{c} \begin{array}{c} \hline \end{array} \end{array} \xrightarrow{M\_1} \begin{array}{c} \begin{array}{c} \begin{array}{c} \hline \end{array} \end{array} \xrightarrow{M\_1} \begin{array}{c} \begin{array}{c} \text{ $M\_1$ } \\ \hline \end{array} \end{array} \xrightarrow{}$$

Fig. 17. An illustration of the ERT-method extension.

formula (7) follows:

$$M\_1 = \frac{L\,M}{N + 1 + M}$$

or

$$M + N + 1 = \frac{L \, M}{M\_1}$$

After substitution of (*M* + *N* + 1) into (11) we get:

$$V = M \cdot \left(1 - M + \frac{L}{\frac{LM}{M\_1} - L}\right) = M\left(1 - M + \frac{M\_1}{M - M\_1}\right) \tag{13}$$

We can reduce this expression to a simpler form:

$$V = M^2 \left( \frac{1}{M - M\_1} - 1 \right) = M \left( \frac{M}{M - M\_1} - M \right) \tag{14}$$

Note that *M* − *M*<sup>1</sup> is the load carried by a single channel and therefore it is always less than one. Formula (14) is useful for applications of the ERT-method in case of traffic splitting.

................................................................. ........ ..........................

lines (parameters: *L* = 5, *N* = 2).

.......................................................................................................................... ........ .......................... 3

process, and is equal to 4.30349, i.e. the relative error is less than 1%.

(in brackets) is 6.91011, i.e. the relative error again is less than 1%.

................................................................................................................................................................................................................. ........

1. First step is a direct application of formulae (10) (11) for two streams and two individual

2. The second step, we apply the formulae (10) (11) to a three-channel group. We get the equivalent parameters *L* = 9.265, *N* = 5.83071 and the lost traffic *M* = 4.3224. The exact value given in brackets is obtained by solving the system of equations of the Markov

3. Third step: The two overflow streams are fed into the two lines. We get the equivalent parameters: *L* = 16.2076, *N* = 10.3188, and the lost traffic *M* = 6.97707. The exact value

The results of calculations show the excellent accuracy of the method. However, such accuracy is not preserved when the number of channels in the third step increases. If instead of two channels we have (2 + *g*) channels in the common group, then Table 2 shows values of the loss for different *g* values. It is obvious the accuracy drops when increasing *g*. For a value of *g* = 10, the relative error is bigger than 3%, but always on the safe side, and the absolute loss

> *g* Loss probability Loss probability Relative error Exact ERT-method % 0.4083 0.4105 0.539 0.3239 0.3277 1.173 0.2459 0.2506 1.911 0.1765 0.1812 2.663 0.1181 0.1218 3.133 0.0724 0.0747 3.177

In (Fredericks, 1980) an equivalence method is proposed which is simpler to use than Wilkinson-Bretschneider's method. The motivation for the method was first put forward by W.S. Hayward. For given values of (*M*, *V*) of a non-Poisson flow, Frederick & Hayward's approach implies direct use of Erlang's formula *E*(*N*, *M*), but with scaling of its parameters

as *E*(*N*/*Z*, *M*/*Z*). The scaling parameter *Z* = *V*/*M* is the peakedness (12) (Fig. 20).

................................................................................................................................................................................................................. ........

M = 4.3224 exact 4.30349 L = 9.265 N = 5.83071

Call Admission Control in Cellular Networks 129

..........................

2

................................................................................................................................................................................................................. ........

M = 6.97707 exact 6.91011 L = 16.2076 N = 10.3188

..........................

.........................

3

........................................................................................................................... ........ ..........................

........................................................................................................................... ........ .........................

........................................................................................................................... ........

.................................................................. ........ ..........................

.................................................................. ........ ..........................

5

5

5

Fig. 19. Three-tier network.

probability is very small.

.................................................................. ........ ..........................

.......................... 5

2

2

2

2

Table 2. On accuracy of the classical ERT-method.

**5.6 Fredericks & Hayward's ERT-method**
