**5.9.3 Variance**

By analogy of (13) we get the variance:

$$V\_1 = M\_1 \left( 1 - M\_1 + \frac{M\_{1+}}{M\_1 - M\_{1+}} \right) \tag{25}$$

where

$$M\_{1+} = A\,\beta (D+1,0)$$

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.

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

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 traffic for the following scheme (Fig. 24).

Fig. 24. Scheme with correlated streams.

Using formulas (24) – (25) we find:

*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: Cov= 0.0282596 .

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

Beigy, H. & Meybodi, M.R. (2003). User based call admission control policies for cellular

Call Admission Control in Cellular Networks 135

Beigy, H. & Meybodi, M.R. (2004). A new fractional channel policy. *Journal of High Speed*

Beigy, H. & Meybodi, M.R. (2005). A general call admission policy for next generation wireless

Beneš, V.E. (1966). Some examples of comparisons of connecting networks. *Bell System Techn.*

Bretschneider, G. (1973). Extension of the equivalent random method to smooth traffics.

Chang, C.; Chang, C.J & Lo, K.R. (1999). Analysis of hierarchical cellular systems with

Cruz-Pérez, F.A.; Toledo-Marín, T. & Hernández-Valdez, G. (2011). Approximated

Descloux, A. (1962). On the components of overflow traffic. *Internal Memorandum. Bell*

Fang, Y. & Zhang, Y. Call admission control schemes and performance analysis in wireless mobile networks. *IEEE Transactions on Vehicular Technology*, Vol. 51 (2002), 371–382. Fredericks, A.A. (1980). Congestion in blocking systems – a simple approximation technique.

Ghaderi; M. & Boutaba, R. (2006). Call admission control in mobile cellular networks: a

Guerin, R. (1988). Queueing-blocking system with two arrival streams and guard channels.

Haring, G.; Marie, R.; Puigjaner, R. & Trivedi, K. (2001) Loss formulas and their application to

Hedberg, I. (1981). A simple extension of the Erlang loss formula with continuous first order partial derivatives. *L.M. Ericsson, Internal Report* XF/Sy 81 171 (1981), 4 pp. Hong, D. & Rappaport, S. (1986). Traffic modelling and performance analysis for cellular

procedure. *IEEE Transactions on Vehicular Technology*, Vol. 35 (1986), 77–92. Huang, Q.; Ko, K.-T.; Chan, S. & Iversen, V.B. (2008). Loss performance evaluation

Kosten, L. (1937). Über Sperrungswahrscheinlichkeiten bei Staffelschaltungen.

Leong, C.W. & Zhuang, W. (2001). Call admission control for voice and data traffic in wireless communications. *Computer Communications*, Vol. 25, (2002), 972–979.

comprehensive survey. *Wireless Communications and Mobile Computing*, Vol. 6 (2006),

optimization for cellular networks. *IEEE Transactions on Vehicular Technology*, Vol. 50

mobile radio telephone systems with prioritized and nonprioritized handover

in heterogeneous hierarchical networks. *Mobility Conference 2008*, Vol. 16,

*Proceedings of Seventh International Teletraffic Congress*, Stockholm, June 1973, paper

reneging and dropping for waiting new calls and handover calls. *IEEE Transactions*

Mathematical Analysis Methods of Guard-Channel-Based Call Admission Control in Cellular Networks. *Cellular Networks - Positioning, Performance Analysis, Reliability*.

networks. *Computer Communications*, Vol. 28 (2005), 1798–1813.

Edited by: A. Melikov, ISBN 978-953-307-246-3, InTech, 2011.

*IEEE Transactions on Communications*, Vol. 36 (1988), 153–163.

Iversen, V.B. (2011). *Teletraffic Engineering and Network Planning*. 382 pp. 2011

http://www.com.dtu.dk/education/34340/.

*Elktr. Nachr.-Techn.*, Vol. 14, No. 1, 1937, 5–12.

(2003), 45–58.

411. 9 pp.

69–93.

(2001), 664–673.

doi*>*10.1145/1506270.1506291.

*Networks*, Vol. 13 (2004), 25–36.

*J.*, Vol. 45 (1966), No. 10, 1829–1935.

*on Vehicular Technology*, Vol. 48 (1999), 1080–1091.

*Telephone Laboratories Inc.*, December 1962. 6 pp.

*The Bell System Tech. J.*, Vol. 59 (1980), No. 6, 805–827.

mobile systems: a survey. *CSI Journal on Computer Science and Engineering*, Vol. 1

channels:

*M*<sup>∗</sup> = *M*<sup>1</sup> + *M*<sup>2</sup> = 1.053239, *V*<sup>∗</sup> = *V*<sup>1</sup> + *V*<sup>2</sup> + 2 Cov = 1.56229 . Using the extended ERT-method we get the equivalent group:

*L*∗ = 3.33306, *N*∗ = 3.44900 . Therefore, using Erlang-B formula

*M* = *L*<sup>∗</sup> · *E*(*N*<sup>∗</sup> + *K*, *L*∗).

We can obtain mean intensity *M* of overflow stream for various values of *K* as shown in Table 4. The results of calculations show the excellent accuracy of the extended ERT-method.


Table 4. Accuracy of the Extended ERT-method for correlated streams.

However, such accuracy is not preserved when the number of channels *K* in the final group increases. Table 4 shows values of the loss for different *K* values. It is obvious that for increasing *K* the accuracy drops. For the value *K* = 5 the relative error is greater than 5%. The same effect one observes in Table 1 and Table 2. For decreasing (very small) blocking probabilities the accuracy increases, but the absolute error decreases.
