**1.1.3 Battlefield networks**

Future battlefield networks will consist of various heterogeneous networking systems and tiers with disparate capabilities and characteristics, ranging from ground ad hoc mobile, sensor networks, and airborne-rich sky networks to satellite networks. It is an enormous challenge to create a suite of novel networking technologies that efficiently integrate these disparate systems (Ryu et al., 2003).

The key result of this chapter is the application part (Section 5) with the extension of the Equivalent Random Traffic method for estimation of throughput for networks with traffic splitting and correlated streams. The excellent accuracy (relative error less than 1%) is shown by numerical examples. The ERT-method has been developed for planning of alternate routing in telephone systems by many authors: (Wilkinson, 1956); (Bretschneider, 1973); (Fredericks, 1980) and others. In this chapter we propose an extension of the ERT-method

The average revenue *H* equals:

takes the form:

We get:

*<sup>H</sup>* <sup>=</sup> *<sup>A</sup>*(*P*<sup>0</sup> <sup>+</sup> *<sup>P</sup>*1) · *<sup>K</sup>* <sup>+</sup> *<sup>B</sup>*(*p*0*P*<sup>0</sup> <sup>+</sup> *<sup>p</sup>*1*P*1) · <sup>1</sup> *P*<sup>0</sup> + *P*<sup>1</sup> + *P*<sup>2</sup> <sup>=</sup> <sup>2</sup>(*AK* <sup>+</sup> *Bp*<sup>0</sup> + (*<sup>A</sup>* <sup>+</sup> *Bp*0)(*AK* <sup>+</sup> *Bp*1))

Call Admission Control in Cellular Networks 115

This expression is the ratio of two polynomials, each one with probabilities *p*<sup>0</sup> and *p*<sup>1</sup> included in the first power. Consider the expression (1) as a function of one of the probabilities *p*. After multiplying by the relevant constant we can get it into the form (*p* + *a*)/(*bp* + *c*). The derivative of this expression has the form (*ac*)/(*bp* + *c*)2. Consequently, the expression (1) has invariable sign in the range of values (0, 1), and its extreme values are located at the ends of the interval, i.e. probabilities *p*<sup>0</sup> and *p*<sup>1</sup> can only take values 0 or 1. Therefore, the dynamic

This model (strategy 1) is a special case of the previous one: you can reserve 0, 1 or 2 channels, corresponding to choice of probability (*p*0, *p*1) in the form of (1,1) (1,0) or (0,0), which, in own order, corresponds to the values of *R* of 0, 1 or 2. Accordingly, the revenue from formula (1)

*<sup>H</sup>*<sup>0</sup> <sup>=</sup> <sup>2</sup>(*AK* <sup>+</sup> *<sup>B</sup>*)(<sup>1</sup> <sup>+</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*)

*<sup>H</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup>(*<sup>B</sup>* <sup>+</sup> *AK*(<sup>1</sup> <sup>+</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*))

How many channels should be reserved? This depends on the parameters *K*, *A* and *B*. To find the optimal value of *R*, one should solve two equations pointing to the boundary of *K*:

*H*<sup>0</sup> = *H*<sup>1</sup> and *H*<sup>1</sup> = *H*<sup>2</sup> .

2 + *A* + *B*

*<sup>A</sup>*2(<sup>1</sup> <sup>+</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*) .

2(*A* + 1) *A*<sup>2</sup>

*<sup>H</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>*AK*(<sup>1</sup> <sup>+</sup> *<sup>A</sup>*) 2 + 2*A* + *A*<sup>2</sup>

*K*<sup>1</sup> = 1 +

*K*<sup>2</sup> = 1 +

It is easy to verify that for any values of *A* and *B*, the inequality *K*<sup>1</sup> *< K*<sup>2</sup> is true since:

Hence we have the following solution for optimal reservation *R* at *N* = 2 channels:

*R* = 0 if *K < K*<sup>1</sup> *R* = 1 if *K*<sup>1</sup> *< K < K*<sup>2</sup> *R* = 2 if *K*<sup>2</sup> *< K*

*<sup>K</sup>*<sup>2</sup> <sup>−</sup> *<sup>K</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>A</sup>* <sup>+</sup> <sup>2</sup>*<sup>B</sup>* <sup>+</sup> *<sup>A</sup>*<sup>2</sup> <sup>+</sup> *AB*

reservation strategy has advantage over fractional dynamic reservation strategy.

**2.2 Dynamic reservation strategy is better than static reservation strategy**

<sup>2</sup> + (*<sup>A</sup>* <sup>+</sup> *Bp*0)(<sup>2</sup> <sup>+</sup> *<sup>A</sup>* <sup>+</sup> *Bp*1) (1)

<sup>2</sup>(<sup>1</sup> <sup>+</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*)+(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*)<sup>2</sup> (2)

<sup>2</sup> + (*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*)(<sup>2</sup> <sup>+</sup> *<sup>A</sup>*) (3)

*<sup>A</sup>*(<sup>1</sup> <sup>+</sup> *<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*) (4)

to take account of correlated streams. Sections 5.7 and 5.8 contain next step in ERT-method extension, namely the application of Neal's theory (Neal, 1971), and his formulas for covariance of correlated streams.
