**5.4 ERT-method**

The ERT-method has been developed for planning of alternate routing in telephone networks by several authors: (Wilkinson, 1956);(Bretschneider, 1973);(Fredericks, 1980); and others. Fig. 18 explains the essence of the method. In (Fig. 18.a) *g* traffic streams which may for example be overflow traffic from other exchanges are offered to a transit exchange. As it is non-Poisson traffic, it cannot be described by classical traffic models. We do not know the distributions (state probabilities) of the traffic streams, but we are satisfied (most often the case in applications of statistics) by describing the i-th traffic stream by its mean value *Mi* and variance *Vi*. The aggregated overflow process of the *g* traffic streams is said to be equivalent to

$$L\_1 \xrightarrow{L\_1} \xrightarrow{M\_1} \xrightarrow{M\_1} \begin{matrix} M\_1 \\ \hline \hline \hline \hline \hline \hline \hline \hline \end{matrix} \xrightarrow{M\_1} \begin{matrix} M\_1 \\ \hline \hline \hline \hline \hline \hline \hline \end{matrix} \xrightarrow{M\_1} \begin{matrix} L \\ \hline \hline \end{matrix} \xrightarrow{M} \begin{matrix} N \\ \hline \hline \hline \end{matrix} \xrightarrow{N} \begin{matrix} K \\ \hline \hline \hline \end{matrix} \xrightarrow{N} \begin{matrix} L \\ \hline \hline \end{matrix} \xrightarrow{N} \begin{matrix} N \\ \hline \hline \end{matrix} \begin{matrix} N \\ \hline \hline \end{matrix} \xrightarrow{N} \begin{matrix} N \\ \hline \hline \end{matrix} \begin{matrix} L \\ \hline \end{matrix} \xrightarrow{N} \begin{matrix} N \\ \hline \hline \end{matrix} \begin{matrix} N \\ \hline \end{matrix} \xrightarrow{N} \begin{matrix} N \\ \hline \end{matrix} \begin{matrix} N \\ \hline \end{matrix} \xrightarrow{N} L \begin{matrix} L \\ \hline \end{matrix} \begin{matrix} N \\ \hline \end{matrix} \begin{matrix} N \\ \hline \end{matrix}$$

Fig. 18. Application of the ERT-method: (a) *g* independent traffic streams offered to a common group of *K* channels, (b) equivalent group, (c) Erlang-B formula applied to a common group with *N* + *K* channels.

the traffic overflowing from a single full accessible group with the same mean and variance as the total overflow traffic. The total traffic offered to the group with *K* channels has the mean value:

$$M = \sum\_{i=1}^{\mathcal{S}} M\_i$$

We assume that the traffic streams are independent (non-correlated), and thus the variance of the total traffic stream becomes:

$$V = \sum\_{i=1}^{\mathcal{S}} V\_i$$

Therefore, the total traffic is described by *M* and *V*. We now consider this traffic to be equivalent to a traffic flow which is lost from a full accessible group and has same mean value *M* and variance *V* (Fig. 18.b). For given values of *M* and *V*, we therefore solve equations (10) and (11) with respect to *N* and *L*. Then it is replaced by the equivalent system (Fig. 18.c) which is a full accessible system with (*N* + *K*) channels offered the traffic *L*.

## **5.5 On accuracy of the ERT-method**

Let us give a computational analysis of the classical ERT-method by a three-tier network shown in Fig. 19. There are four streams each offering a traffic equal to 5 erlang traffic. On first tier there are two servers per stream, on second tier there are three servers, and on third tier two servers. Application of the ERT-method to dimension the alternate routing networks consists of three steps.

18 Will-be-set-by-IN-TECH

The ERT-method has been developed for planning of alternate routing in telephone networks by several authors: (Wilkinson, 1956);(Bretschneider, 1973);(Fredericks, 1980); and others. Fig. 18 explains the essence of the method. In (Fig. 18.a) *g* traffic streams which may for example be overflow traffic from other exchanges are offered to a transit exchange. As it is non-Poisson traffic, it cannot be described by classical traffic models. We do not know the distributions (state probabilities) of the traffic streams, but we are satisfied (most often the case in applications of statistics) by describing the i-th traffic stream by its mean value *Mi* and variance *Vi*. The aggregated overflow process of the *g* traffic streams is said to be equivalent to

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

(a) (b)

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

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

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

......................... LN K M V

.......................... L N +K L · E(L, N +K)

(c)

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

**5.4 ERT-method**

···

N*<sup>g</sup>*

common group with *N* + *K* channels.

the total traffic stream becomes:

**5.5 On accuracy of the ERT-method**

consists of three steps.

N<sup>2</sup>

N<sup>1</sup>

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

M*<sup>g</sup>* V*g*

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

M<sup>2</sup> V<sup>2</sup>

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

Fig. 18. Application of the ERT-method: (a) *g* independent traffic streams offered to a common group of *K* channels, (b) equivalent group, (c) Erlang-B formula applied to a

*M* =

*V* =

is a full accessible system with (*N* + *K*) channels offered the traffic *L*.

the traffic overflowing from a single full accessible group with the same mean and variance as the total overflow traffic. The total traffic offered to the group with *K* channels has the mean

> *g* ∑ *i*=1 *Mi*

We assume that the traffic streams are independent (non-correlated), and thus the variance of

Therefore, the total traffic is described by *M* and *V*. We now consider this traffic to be equivalent to a traffic flow which is lost from a full accessible group and has same mean value *M* and variance *V* (Fig. 18.b). For given values of *M* and *V*, we therefore solve equations (10) and (11) with respect to *N* and *L*. Then it is replaced by the equivalent system (Fig. 18.c) which

Let us give a computational analysis of the classical ERT-method by a three-tier network shown in Fig. 19. There are four streams each offering a traffic equal to 5 erlang traffic. On first tier there are two servers per stream, on second tier there are three servers, and on third tier two servers. Application of the ERT-method to dimension the alternate routing networks

*g* ∑ *i*=1 *Vi*

K

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

M<sup>1</sup> V<sup>1</sup>

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

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

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

L<sup>1</sup>

L<sup>2</sup>

L*<sup>g</sup>*

value:

Fig. 19. Three-tier network.


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 probability is very small.


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

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

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).

If *A* = *p*<sup>1</sup> *L* and *B* = *p*<sup>2</sup> *L*, where *p*<sup>1</sup> + *p*<sup>2</sup> = 1, then the peakedness of partial stream *Vi*/*Mi* is

Call Admission Control in Cellular Networks 131

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

 = *p*<sup>1</sup> *p*2(*V* − *M*). The covariance formula is proved according to the theorem of variance for mutually

 ,

Cov.

> .................................................................................... ....... ..........................

 *D i* − 1

where 0 ≤ *i* ≤ *D* , 0 ≤ *j* ≤ *F*, *i* + *j >* 0 , (16)

∞

M<sup>1</sup> ∞

M<sup>2</sup>

*β*(0, 0) = *E*(*N*, *L*) (15)

*β*(*D*, *j*) − *B*

 *F j* − 1

*β*(*i*, *F*),

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

− 1 = *pi*

*V* = *V*<sup>1</sup> + *V*<sup>2</sup> + 2 ·

During early 1970's, Scotty Neal from Bell Labs studied the covariance of correlated streams in alternative routing networks (Neal, 1971). Below we use results from Neal's paper to develop some formulae in notations of Fig. 23. We are looking for covariance between two overflow streams after groups with *D* and *F* channels, respectively. The key to Neal's solutions is the original work (Kosten, 1937). Neal (Neal, 1971) extended the ERT-method to mutually dependent streams. On basis of the extended Kosten' model (Fig. 23) he developed a technique for taking correlation into account when combining dependent streams of overflow traffic. More precisely, Neal has considered a 5-parameter Markov model (Fig. 23) with 5

> .................................................................................... ....... ..........................

After rather sophistical derivations of two-dimensional binomial moment generating functions and using Kosten's approach, Neal obtains linear equations for the two-dimensional

F

D

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

*Vi Mi*

Cov

1. Number of busy channels in the first choice group (up to *N*),

2. Number of busy channels *i* in the first alternate group (0 ≤ *i* ≤ *D*), 3. Number of busy channels *j* in the second alternate group (0 ≤ *j* ≤ *F*), 4. Number of busy channels in the first imaginary infinite channel group, 5. Number of busy channels in the second imaginary infinite channel group.

> ..................................................................................... ........ ..........................

probabilities *β*(*i*, *j*),(0 ≤ *i* ≤ *D* , 0 ≤ *j* ≤ *F*). The initial value is:

and other probabilities are defined by linear balance equations:

(*i* + *j*)*νi*+*jβ*(*i*, *j*) = *A β*(*i* − 1, *j*) + *B* · *β*(*i*, *j* − 1) − *A*

................................................................................... ....... .................. ........ N

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

A

B

L − A − B

Fig. 23. An illustration to Neal's formulas.

defined by total peakedness *V*/*M*:

**5.8 Correlated streams: Neal's formulae**

and covariance

parameters:

dependent variables

$$(M, V) \longrightarrow \begin{array}{c} \hline N \\ \hline \end{array} \longrightarrow M \cdot E\left(\frac{N}{Z}, \frac{M}{Z}\right)$$

Fig. 20. An illustration of Fredericks & Hayward's approach.

The accuracy of Fredericks & Hayward approach is numerically compared with ERT and with exact values. The calculations were performed for different variants of the scheme shown in Fig. 21. In general, its accuracy is comparable to that of the Wilkinson approach (Table 3). However, in our opinion Wilkinson's approach is more reliable and always yields worst-case values.

$$L\_1 \xrightarrow{\quad} \begin{array}{c} \begin{array}{|c|} \hline N\_1 \\ \hline \end{array} \longrightarrow \\ L\_2 \xrightarrow{\quad} \begin{array}{|c|} \hline N\_2 \\ \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{array} \longrightarrow \\ \begin{array}{|c|} \hline \end{$$

Fig. 21. On accuracy of Fredericks & Hayward's approach.


Table 3. On accuracy of Fredericks & Hayward's approach.

### **5.7 Correlation of overflow streams**

As it is, the ERT-method is not applicable to the analysis of multi-tier networks with correlated streams as shown in Fig. 15.c. In 1960's, this type of problem appeared when dimensioning so-called gradings, the basic structural block in step-by-step exchanges. An important result was developed independently by (Descloux, 1962) and (Lotze, 1964). They determined mean *M* and variance *V* of the overflow stream components when split up after a first choice group as shown in Fig. 22.b. On the basis of Kosten's model with two parameters *M* and *V*, they developed a 5-parameter model for the two stream case: mean values *M*<sup>1</sup> and *M*2, variances *V*<sup>1</sup> and *V*2, and covariance Cov.

Fig. 22. (a) Kosten's model, (b) two-stream model.

If *A* = *p*<sup>1</sup> *L* and *B* = *p*<sup>2</sup> *L*, where *p*<sup>1</sup> + *p*<sup>2</sup> = 1, then the peakedness of partial stream *Vi*/*Mi* is defined by total peakedness *V*/*M*:

$$\frac{V\_i}{M\_i} - 1 = p\_i \left(\frac{V}{M} - 1\right) \text{ .}$$

and covariance

20 Will-be-set-by-IN-TECH

The accuracy of Fredericks & Hayward approach is numerically compared with ERT and with exact values. The calculations were performed for different variants of the scheme shown in Fig. 21. In general, its accuracy is comparable to that of the Wilkinson approach (Table 3). However, in our opinion Wilkinson's approach is more reliable and always yields worst-case

 N <sup>Z</sup> , <sup>M</sup> Z 

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

K M =?

.......................... ................................................................................... ....... .................. ........ (M, V ) N M *·* E

> .................................................................................... ....... ..........................

*L***<sup>1</sup>** *N***<sup>1</sup>** *L***<sup>2</sup>** *N***<sup>2</sup>** *K M* (Wilkinson) *M* (Hayward) *M* (exact) 2 2.5 0 1 1.9740 1.9723 1.9721 2 2.5 0 1 5.9635 5.9620 5.9589 3 4 4 5 2.8717 2.8514 2.8498 3 4 4 5 0.2614 0.2268 0.2608 3 4 4 5 0.1122 0.0859 0.1140 3 2 4 5 0.1636 0.1386 0.1640 2.5 7 8.25 4 9 0.30268 0.2884 0.30273

As it is, the ERT-method is not applicable to the analysis of multi-tier networks with correlated streams as shown in Fig. 15.c. In 1960's, this type of problem appeared when dimensioning so-called gradings, the basic structural block in step-by-step exchanges. An important result was developed independently by (Descloux, 1962) and (Lotze, 1964). They determined mean *M* and variance *V* of the overflow stream components when split up after a first choice group as shown in Fig. 22.b. On the basis of Kosten's model with two parameters *M* and *V*, they developed a 5-parameter model for the two stream case: mean values *M*<sup>1</sup> and *M*2, variances

> .................................................................................... ........ .........................

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

M1, V<sup>1</sup>

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

M2, V<sup>2</sup>

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

B

A

N Cov

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

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

Fig. 20. An illustration of Fredericks & Hayward's approach.

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

Fig. 21. On accuracy of Fredericks & Hayward's approach.

Table 3. On accuracy of Fredericks & Hayward's approach.

Cov.

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

Fig. 22. (a) Kosten's model, (b) two-stream model.

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

L N M, V

(a) (b)

**5.7 Correlation of overflow streams**

*V*<sup>1</sup> and *V*2, and covariance

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

L<sup>2</sup> N<sup>2</sup>

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

L<sup>1</sup> N<sup>1</sup>

values.

$$\text{Cov} = p\_1 \, p\_2 (V - M) \,.$$

The covariance formula is proved according to the theorem of variance for mutually dependent variables

$$V = V\_1 + V\_2 + 2 \cdot \text{Cov.}$$
