A` = Ax· Enx+`(Ax) . (9.10)
As the offered traffic is m, the traffic congestion of the system becomes: C = A`
m . (9.11)
Notice: the blocking probability is not Enx+`(Ax). We should remember the last step (9.11), where we relate the lost traffic to the originally offered traffic, which in this case is given by m (9.8).
We notice that if the overflow traffic is from a single primary group with PCT–I traffic, then the method is exact. In the general case with more traffic streams the method is approximate, and it does not yield the exact mean blocking probability.
Example 9.2.1: Paradox
In Sec. 6.3 we derived Palm’s theorem, which states that by superposition of many independent arrival processes, we locally get a Poisson process. This is not contradictory with (9.8) and (9.9),
because these formulæ are valid globally. 2
9.2.2
Numerical aspects
When applying the ERT–method we need to calculate (m, v) for given values of (A, n) and vice versa. It is easy to obtain (m, v) for given (A, n) by using (9.4) & (9.5). To obtain (A, n) for given (m, v), we have to solve two equations with two unknown. It requires an iterative procedure, since En(A) cannot be solved explicitly with respect to neither n nor A (Sec.7.5). However, we can solve (9.7) with respect to n:
n = A · m + v m
m +mv − 1 − m − 1 , (9.12)
so that we know n for given A. Thus A is the only independent variable. We can use Newton- Raphson’s iteration method to solve the remaining equation, introducing the function:
f (A) = m − A · En(A) = 0 .
For a proper starting value A0 we improve this iteratively until the resulting values of m and v/m become close enough to the known values.
Yngv´e Rapp (1965 [86]) has proposed a good approximate solution for A, which can be used as initial value A0 in the iteration:
A ≈ v + 3 · v m · nv m − 1 o . (9.13)
174 CHAPTER 9. OVERFLOW THEORY From A we get n, using (9.12). Rapp’s approximation is sufficient accurate for practical applications, except when Ax is very small. The peakedness Z = v/m has a maximum, obtained when n is little larger than A (Fig. 9.4). For some combinations of m and v/m the convergence is critical, but when using computers we can always find the correct solution. Using computers we operate with non-integral number of channels, and only at the end of calculations we choose an integral number of channels greater than or equal to the obtained results (typical a module of a certain number of channels (8 in GSM, 30 in PCM, etc.). When using tables of Erlang’s B–formula, we should in every step choose the number of channels in a conservative way so that the blocking probability aimed at becomes worst case.
The above-mentioned method presupposes that v/m is larger than one, and so it is only valid for bursty traffic. Individual traffic stream in Fig. 9.5 are allowed to have vi/mi < 1, provided the total aggregated traffic stream is bursty. Bretschneider ([9], 1973) has extended the method to include a negative number of channels during the calculations. In this way it is possible to deal with smooth traffic (EERT-method = Extended ERT method).
9.2.3
Parcel blocking probabilities
The individual traffic streams in Fig. 9.5do not have the same mean value and variance, and therefore they do not experience equal blocking probabilities in the common overflow group with ` channels. From the above we calculate the mean blocking (9.11) for all traffic streams aggregated. Experiences show that the blocking probability experienced is proportional to the peakedness Z = v/m. We can split the total lost traffic into individual lost traffic parcels by assuming that the traffic lost for stream i is proportional to the mean value mi and to the peakedness Zi = vi/mi of the stream. We obtain:
A` = g X i=1 A`,i = c · A`· g X i=1 m1,i· vi m1,i = c · A`· v ,
from which we find the constant c = 1/v.
The (traffic) blocking probability for traffic stream i, which is called the parcel blocking probability for stream i, then becomes:
Ci = A`,i
mi = vi
9.2. EQUIVALENT RANDOM TRAFFIC METHOD 175 Furthermore, we can divide the blocking among the individual groups (primary, secondary, etc.). Consider the equivalent group at the bottom of Fig. 9.5 with nx primary channels and ` secondary (overflow) channels, we can calculate the blocking probability due to the nx primary channels, and the blocking probability due to the ` secondary channels. The probability that the traffic is lost by the ` channels is equal to the probability that the traffic is lost by the nx+ ` channels, under the condition that the traffic is offered to the ` channels:
H(l) = A · Enx+l(A) A · Enx(A)
= Enx+l(A) Enx(A)
. (9.15)
The total loss probability can therefore be related to the two groups: Enx+l(A) = Enx(A) ·
Enx+l(A) Enx(A)
. (9.16)
By using this expression, we can find the blocking for each channel group and then for example obtain information about which group should be increased by adding more channels.
Example 9.2.2: Example 9.1.1 continued
In example 9.1.1 the blocking probability of the primary group of 8 channels is E8(10) = 0.3383. The blocking of the overflow group is
H(8) = E16(10) E8(10)
= 0.02231
0.3383 = 0.06592 . The total blocking of the system is:
E16(10) = E8(10) · H(8) = 0.3383 · 0.06592 = 0.02231 .
2
Example 9.2.3: Hierarchical cellular system
We consider a cellular system HCS covering three areas. The traffic offered in the areas are 12, 8 and 4 erlang, respectively. In the first two cells we introduce micro-cells with 16, respectively 8 channels, and a common macro-cell covering all three areas is allocated 8 channels. We allow overflow from micro-cells to macro-cells, but do not rearrange the calls from macro- to micro-cells when a channel becomes idle. Furthermore, we look away from hand-over traffic. Using (9.6) & (9.7) we find the mean value and the variance of the traffic offered to the macro-cell:
Cell Offered Number of Overflow Overflow Peakedness traffic channels mean variance
i Ai ni(j) m1,i vi Zi
1 12 16 0.7250 1.7190 2.3711
2 8 8 1.8846 3.5596 1.8888
3 4 0 4.0000 4.0000 1.0000
176 CHAPTER 9. OVERFLOW THEORY
The total traffic offered to the macro-cell has mean value 6.61 erlang and variance 9.28. This corresponds to the overflow traffic from an equivalent system with 10.78 erlang offered to 4.72 channels. Thus we end up with a system of 12.72 channels offered 10.78 erlang. Using the Erlang-B formula, we find the lost traffic 1.3049 erlang. Originally we offered 24 erlang, so the total traffic blocking probability becomes B = 5.437%.
The three areas have individual blocking probabilities. Using (9.14) we find the approximate lost traffic from the areas to be 0.2434 erlang, 0.5042 erlang, and 0.5664 erlang, respectively. Thus the traffic blocking probabilities become 2.03%, 6.30% and 14.16%, respectively. A computer simulation with 100 million calls yields the blocking probabilities 1.77%, 5.72%, and 15.05%, respectively. This corresponds to a total lost traffic equal to 1.273 erlang and a blocking probability 5.30%. The accuracy of the method of this chapter is sufficient for real applications. 2