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

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 parameters:


..................................................................................... ........ .......................... .................................................................................... ....... .......................... .................................................................................... ....... .......................... .................................................................................... ....... .......................... .................................................................................... ....... .......................... .................................................................................... ....... .......................... .................................................................................... ....... .......................... N A B L − A − B F D ∞ M<sup>1</sup> ∞ M<sup>2</sup>

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

After rather sophistical derivations of two-dimensional binomial moment generating functions and using Kosten's approach, Neal obtains linear equations for the two-dimensional probabilities *β*(*i*, *j*),(0 ≤ *i* ≤ *D* , 0 ≤ *j* ≤ *F*). The initial value is:

$$\beta(0,0) = E(N,L) \tag{15}$$

and other probabilities are defined by linear balance equations:

$$
\langle (i+j)v\_{i+j}\beta(i,j) \rangle = A\,\beta(i-1,j) + B\cdot\beta(i,j-1) - A\begin{pmatrix} D\\i-1 \end{pmatrix} \beta(D,j) - B\begin{pmatrix} F\\j-1 \end{pmatrix} \beta(i,F),
$$

$$
\text{where}
\quad 0 \le i \le D, \quad 0 \le j \le F, \quad i+j > 0,\tag{16}
$$

**5.9.2 The modified Erlang formula**

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

By analogy of (13) we get the variance:

**5.10 Extended ERT-method. Numerical example**

traffic for the following scheme (Fig. 24).

2

3

Using formulas (24) – (25) we find:

= 0.0282596 .

Cov

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

Fig. 24. Scheme with correlated streams.

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

*β*(*D*, 0) =

applicable to the ERT-method. It allows for non-integral number of channels.

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

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

Substituting *A* by *B* and *D* by *F* in (24) and (25) we get *M*<sup>2</sup> and *V*<sup>2</sup> in similar way.

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

3

2

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

*M*<sup>1</sup> = 0.180018, *M*<sup>2</sup> = 0.873221, *V*<sup>1</sup> = 0.252930, *V*<sup>2</sup> = 1.25284 .

Using (18) - (19) we calculate the covariance of the two streams:

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

1 − *M*<sup>1</sup> +

*M*1<sup>+</sup> = *A β*(*D* + 1, 0)

Consider an example where the extended ERT-method can be used and the covariance obtained by using formulas (18) - (19). Let us calculate the mean value *M* of the overflow

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

We now can calculate the intensity of the flow which is overflowing to the group with *K*

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

*D* ∑ *i*=0

*D* ∑ *i*=0

is an obvious analogy to the Erlang formula for the scheme shown in Fig. 23. This formula is

Call Admission Control in Cellular Networks 133

*A<sup>D</sup> D*!

> *i* ∏ *j*=0 *νj*

(23)

(24)

(25)

*AD*−*<sup>i</sup>* (*D* − *i*)!

*AD*+<sup>1</sup> *D*!

> *i* ∏ *j*=0 *νj*

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

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

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

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

M2, V<sup>2</sup>

2 K M

M1, V<sup>1</sup>

*AD*−*<sup>i</sup>* (*D* − *i*)!

Formula (21) in the form

**5.9.3 Variance**

where

with the following recurrent formulas for coefficients *νi*:

$$\nu\_i = \frac{L}{i \cdot \nu\_{i-1}} + 1 + \frac{N - L}{i}, \quad i > 0, \quad \text{where} \quad \nu\_0 = \frac{1}{E(N, L)}. \tag{17}$$

Then

$$M\_1 = A \cdot \beta(D, 0) \tag{18}$$

$$\begin{aligned} M\_2 &= B \cdot \beta(0, F) \\ \text{Cov} &= \frac{A \, Q\_1 + B \, Q\_2}{2} - M\_1 \, M\_2 \end{aligned} \tag{19}$$

where

$$\begin{aligned} Q\_1 &= \frac{\sum\_{j=0}^{F} \left( \beta(D\_r j) \prod\_{k=j+1}^{F+1} \frac{B}{k \,\nu\_k} \right)}{1 + \sum\_{j=1}^{F} \left( \binom{F}{j-1} \prod\_{k=j+1}^{F+1} \frac{B}{k \,\nu\_k} \right)} \\ Q\_2 &= \frac{\sum\_{j=0}^{D} \left( \beta(j, F) \prod\_{k=j+1}^{D+1} \frac{A}{k \,\nu\_k} \right)}{1 + \sum\_{j=1}^{D} \left( \binom{D}{j-1} \prod\_{k=j+1}^{D+1} \frac{A}{k \,\nu\_k} \right)} \end{aligned}$$

Based on Neal's formulae (15) – (19 we get the lost stream intensities *M*<sup>1</sup> and *M*<sup>2</sup> and the variance *V*. From (18) follows that loss probability of the first stream is *β*(*D*, 0).

## **5.9 Comments on Neal's results**

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

## **5.9.1 Algorithm**

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

$$\dot{\mathbf{a}}\,\nu\_{i}\,\beta(\mathbf{i},0) = A\,\beta(\mathbf{i}-1,0) - A\begin{pmatrix} D\\ \mathbf{i}-1 \end{pmatrix}\,\beta(D,0) \,. \tag{20}$$

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

$$\beta(D,0) = \frac{1}{\sum\_{i=0}^{D} \frac{D!}{(D-i)! \cdot A^i} \prod\_{j=0}^{i} \nu\_j} \tag{21}$$

$$\beta(i,0) = \beta(D,0) \binom{D}{i} \left( 1 + \sum\_{j=1}^{D-i} \prod\_{k=1}^{j} \frac{\nu\_{i+k}(D+1-i-k)}{A} \right), \qquad 0 < i < D \tag{22}$$

Values *ν<sup>j</sup>* are obtained by formula (17). Using direct test we can ascertain that (21) and (22) together with statement (15) indeed satisfies the system of equations (20).
