**5.9.2 The modified Erlang formula**

Formula (21) in the form

22 Will-be-set-by-IN-TECH

*<sup>i</sup>* , *<sup>i</sup> <sup>&</sup>gt;* 0 , where *<sup>ν</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup>

*F*+1 ∏ *k*=*j*+1

� *F*+1 ∏ *k*=*j*+1

*D*+1 ∏ *k*=*j*+1

> � *D*+1 ∏ *k*=*j*+1

> > � *D i* − 1

*νi*+*k*(*D* + 1 − *i* − *k*) *A*

�

, (21)

⎞

*β*(*D*, 0). (20)

⎠ , 0 *< i < D* (22)

*B k ν<sup>k</sup>*

*A k ν<sup>k</sup>* ⎞ ⎠

*B k ν<sup>k</sup>*

> ⎞ ⎠

*A k ν<sup>k</sup>* ⎞ ⎠

⎞ ⎠

*M*<sup>1</sup> = *A* · *β*(*D*, 0) (18)

<sup>2</sup> <sup>−</sup> *<sup>M</sup>*<sup>1</sup> *<sup>M</sup>*<sup>2</sup> (19)

*<sup>E</sup>*(*N*, *<sup>L</sup>*) . (17)

with the following recurrent formulas for coefficients *νi*:

+ 1 +

Cov

*Q*<sup>1</sup> =

*Q*<sup>2</sup> =

In the following discuss the applicability of Neal's results.

*D*! (*<sup>D</sup>* − *<sup>i</sup>*)! *<sup>A</sup><sup>i</sup>*

> �*D i*

� <sup>⎛</sup> ⎝1 +

*N* − *L*

*M*<sup>2</sup> = *B* · *β*(0, *F*)

*F* ∑ *j*=0

1 + *F* ∑ *j*=1

1 + *D* ∑ *j*=1

*D* ∑ *j*=0

variance *V*. From (18) follows that loss probability of the first stream is *β*(*D*, 0).

*i ν<sup>i</sup> β*(*i*, 0) = *A β*(*i* − 1, 0) − *A*

By solving this system of linear equations we get expressions for *β*(*i*, 0):

*i* ∏ *j*=0 *νj*

together with statement (15) indeed satisfies the system of equations (20).

*D*−*i* ∑ *j*=1

*j* ∏ *k*=1

Values *ν<sup>j</sup>* are obtained by formula (17). Using direct test we can ascertain that (21) and (22)

⎛

<sup>=</sup> *A Q*<sup>1</sup> <sup>+</sup> *B Q*<sup>2</sup>

⎝*β*(*D*, *j*)

⎝*β*(*j*, *F*)

Based on Neal's formulae (15) – (19 we get the lost stream intensities *M*<sup>1</sup> and *M*<sup>2</sup> and the

We extract the equations which have *j* = 0 from the equation system (16). Eliminating members with zero coefficients and considering that *β*(0, 0) is known, we obtain the system

⎛ ⎝ � *D j* − 1

⎛ ⎝ � *F j* − 1

⎛

*<sup>ν</sup><sup>i</sup>* <sup>=</sup> *<sup>L</sup> <sup>i</sup>* · *<sup>ν</sup>i*−<sup>1</sup>

**5.9 Comments on Neal's results**

with *D* equations referring to *β*(*i*, 0):

*<sup>β</sup>*(*D*, 0) = <sup>1</sup> *D* ∑ *i*=0

*β*(*i*, 0) = *β*(*D*, 0)

**5.9.1 Algorithm**

Then

where

$$\beta(D,0) = \frac{\frac{A^D}{D!}}{\sum\_{i=0}^D \frac{A^{D-i}}{(D-i)!} \prod\_{j=0}^i \nu\_j} \tag{23}$$

is an obvious analogy to the Erlang formula for the scheme shown in Fig. 23. This formula is applicable to the ERT-method. It allows for non-integral number of channels.

From this the mean intensity *M*<sup>1</sup> follows:

$$M\_1 = \frac{\frac{A^{D+1}}{D!}}{\sum\_{i=0}^{D} \frac{A^{D-i}}{(D-i)!} \prod\_{j=0}^{i} \nu\_j} \tag{24}$$

Formula (24) is easily implemented and allows for non integer values of *N*.
