**3. Comparison of four strategies in single tier network: Common case**

## **3.1 Fractional dynamic reservation**

Theorem. For a *N*-channel loss system where *B*-calls are accepted with probability *pi*, depending on the number of busy channels *i* (*i* = 0, 1, . . . , *N*), the optimal fractional dynamic reservation is limited to probabilities *pi* equal to 0 or 1.

Fig. 5. Dependence of revenue on the size of the reservation *R* for the dynamic reservation *D*

Call Admission Control in Cellular Networks 119

Fig. 6. Dependence of revenue on the size of the reservation for three strategies: *D* = dynamic reservation, *F* = static (fixed) reservation, and *L* = restriction on the number of

and fixed reservation *F*. The cost of handover call is *K* = 2.

admitted *B*-calls.

Fig. 4. Fractional dynamic reservation: the common case.

The stationary state probabilities *Fi* are defined by equations (up to normalization factor):

$$\begin{aligned} F\_0 &= 1, \\ F\_1 &= A + B p\_0, \\ F\_2 &= (A + B p\_0)(A + B p\_1) / 2, \\ \dots &\dots \\ F\_N &= (A + B p\_0)(A + B p\_1) \dots (A + B p\_{N-1}) / N! \end{aligned}$$

The lost revenue is equal to:

$$\mathcal{C} = \frac{B \cdot (F\_0 p\_0 + F\_1 p\_1 + \dots + F\_{N-1} p\_{N-1}) + (AK + B) \cdot F\_N}{F\_0 + F\_1 + \dots + F\_N}$$

We should maximize the average revenue:

$$H = A \cdot K + B \cdot 1 - \mathbb{C} \,. \tag{5}$$

Note that this expression is the general case of formula (1). As above, the expression (5) is the ratio of two polynomials, each of which includes the probability *pi* in first power. Consider the expression (5) as a function of the probability *pi* for any *i*. By the same argument as above we prove the Theorem.

Problem 2. The previous theorem does not imply which of the probabilities *pi* are equal to 0 or 1. Common sense is that *pi* = 1 for *i* = 1, ..., *R*, and *pi* = 0 for *i* = *R* + 1, ..., *N*. How to prove this mathematically?

### **3.2 Dynamic reservation as maximum revenue strategy**

We compare two strategies: dynamic and static reservation. On the basis of numerical results we have shown that with the optimal reservation *R* the expected revenue is always higher for the model with the dynamic reservation (Fig. 5). Naturally, when the handover call cost increases, then the number of reserved channels will increase. For *K* = 2 the optimal dynamic reservation is *R* = 2, and for *K* = 4 it is *R* = 4. The curves, of course. coincide at the ends of the definition interval when *R* is equal to 0 or *N*.

Numerical calculations (Fig. 6) show that strategy 4, restriction of number of *B*-calls admitted, is similar to strategy 3. For large values of *R* these strategies are almost identical, but even with the optimal value of *R*, strategy 4 has only a slight advantage over strategy 3.

Conclusion. The results of numerical analysis confirm that the optimal strategy is dynamic reservation. This statement is strictly proved in the case of a two-channel system.

8 Will-be-set-by-IN-TECH

i − 1 i i + 1

··· ···

The stationary state probabilities *Fi* are defined by equations (up to normalization factor):

*FN* = (*<sup>A</sup>* + *Bp*0)(*<sup>A</sup>* + *Bp*1)...(*<sup>A</sup>* + *BpN*−1)/*N*!

*<sup>C</sup>* <sup>=</sup> *<sup>B</sup>* · (*F*<sup>0</sup> *<sup>p</sup>*<sup>0</sup> <sup>+</sup> *<sup>F</sup>*<sup>1</sup> *<sup>p</sup>*<sup>1</sup> <sup>+</sup> ... <sup>+</sup> *FN*−<sup>1</sup>*pN*−1)+(*A K* <sup>+</sup> *<sup>B</sup>*) · *FN F*<sup>0</sup> + *F*<sup>1</sup> + ... + *FN*

Note that this expression is the general case of formula (1). As above, the expression (5) is the ratio of two polynomials, each of which includes the probability *pi* in first power. Consider the expression (5) as a function of the probability *pi* for any *i*. By the same argument as above

or 1. Common sense is that *pi* = 1 for *i* = 1, ..., *R*, and *pi* = 0 for *i* = *R* + 1, ..., *N*. How to prove

We compare two strategies: dynamic and static reservation. On the basis of numerical results we have shown that with the optimal reservation *R* the expected revenue is always higher for the model with the dynamic reservation (Fig. 5). Naturally, when the handover call cost increases, then the number of reserved channels will increase. For *K* = 2 the optimal dynamic reservation is *R* = 2, and for *K* = 4 it is *R* = 4. The curves, of course. coincide at the ends of

Numerical calculations (Fig. 6) show that strategy 4, restriction of number of *B*-calls admitted, is similar to strategy 3. For large values of *R* these strategies are almost identical, but even

The results of numerical analysis confirm that the optimal strategy is dynamic

with the optimal value of *R*, strategy 4 has only a slight advantage over strategy 3.

reservation. This statement is strictly proved in the case of a two-channel system.

The previous theorem does not imply which of the probabilities *pi* are equal to 0

❥ A + B p*<sup>i</sup>*−<sup>1</sup>

✧✦

i+1

❨

✧✦

❨

*H* = *A* · *K* + *B* · 1 − *C* . (5)

★✥

❥ A + B p*<sup>i</sup>*

★✥

✧✦

i

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

❨

★✥

❥

Fig. 4. Fractional dynamic reservation: the common case.

*F*<sup>0</sup> = 1,

... ...

**3.2 Dynamic reservation as maximum revenue strategy**

the definition interval when *R* is equal to 0 or *N*.

The lost revenue is equal to:

we prove the Theorem.

 2.

this mathematically?

Problem

Conclusion.

We should maximize the average revenue:

*F*<sup>1</sup> = *A* + *Bp*0,

Fig. 5. Dependence of revenue on the size of the reservation *R* for the dynamic reservation *D* and fixed reservation *F*. The cost of handover call is *K* = 2.

Fig. 6. Dependence of revenue on the size of the reservation for three strategies: *D* = dynamic reservation, *F* = static (fixed) reservation, and *L* = restriction on the number of admitted *B*-calls.

Fig. 8. A macro-cell covering (a) three and (b) seven micro-cells.

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

New calls Low Mobility

Fig. 9. A two-tier cellular network call flow model.

two reservation strategies:

common pool of channels.

Microcells

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

Overflow Calls

calls of intensity *A* are served by macro-cell only. Low-mobility calls of intensity *B* are served by the micro-cells as first choice and, if the reservation strategy admits it, by the macro-cell as second choice. Arriving calls are served as follows. The mobility class of the call is identified. High-mobility calls of intensity *A* are served by macro-cell only. Low-mobility calls of intensity *B* are served by the appropriate micro-cell as first choice, and if reservation strategy allows it by macro-cell as second choice. In both cases, our optimization criteria is the same: to maximize the revenue when each served A-call costs *K* units and each served B-call costs one unit (*K >* 1). When calls reach the macro-cell level, they are no longer

Call Admission Control in Cellular Networks 121

Macrocell

differentiated according to their mobility classes. Therefore, the calls of the high mobility class terminals and the overflowed handover calls from micro-cells are treated identically. New calls from micro-cells may not use the guard (reserved) channels upon their arrival. If no non-guard channel is available, then new calls are blocked. High mobility calls are blocked if all macro-cell channels are busy. Fig. 9 shows schematically how calls are served and what order is followed when serving them. As above in the case of single-tier network we compare

a) Dynamic reservation: The cutoff priority scheme is to reserve a number of channel for high-mobility calls in the macro-cell. Whenever a channel is released, it is returned to the

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

New calls High Mobility

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