Loss systems

4.8.4.1 Grade of Service parameters

We distinguish between several performance parameters depending on the system and strategy considered.

For loss systems the main performance parameter is the blocking or congestion probability. This can be defined in several ways:

• The time congestion E denotes the proportion of time the system is blocked

• The traffic congestion C denotes the proportion of the offered traffic which is not carried.

The call congestion B is typically observed by the user who initiates call attempts. For traffic engineering the relevant measure is the traffic congestion C.

4.8.4.2 Erlang's loss systems

The most successful and simple model is Erlang's loss system where the blocking probability is given by Erlang's B-formula. The traffic is described by the offered traffic A, the system (only one link) by the number of channels, and the strategy is full accessibility with lost calls cleared.

The above-mentioned three elements of the model are each described by only one parameter. Only single-channel calls are considered. The network performance is described by the blocking probability E1,n(A), i.e. the probability that a call attempts is blocked because all n

channels are busy.

For Erlang's loss system time, call, and traffic congestions are equal. The state probabilities are given by the truncated Poisson distribution, and when the number of channels is very large this becomes a Poisson distribution.

This model has been very successful for traffic engineering. The background for this success is that the traffic is very well modeled by one parameter only. The underlying mathematical assumption is a Poisson arrival process. This is fulfilled when the traffic is generated by many independent users, which is the case for telephony. If the arrival process is a Poisson process, then the model is insensitive to the holding time distribution, which means that only the mean holding time is of importance. So the model is very robust to the traffic and models the real world extremely well.

Improvement function:

This denotes the increase in carried traffic when the number of channels is increased by one from n to n + 1:

4.8.4.3 Engset's loss system

The Poisson arrival process is the most random process, and the calls are generated by a very large number of independent sources, each having an infinitesimal calling rate. In many real systems the number of users is limited, and the arrival process is more regular or smooth than random traffic.

This is modeled by Engset's loss system where we have a finite number S of users (traffic sources) which alternates between the states off (= idle) and on (= busy). When a source is idle it generates γ calls per time unit (mean inter-arrival time = 1/γ. It is on during a mean holding s. When it is on it generates no new calls. If we let β = γ . s and consider the strategy lost calls cleared, then the blocking probability is given by Engset's formula:

The state probabilities are given by the truncated Binomial distribution, and when S ≥ n this becomes the Binomial distribution. For the same number of channels and the same offered traffic, the Engset system will have lower blocking probability than the Erlang system because the offered traffic is more smooth. For Engset's loss system we have E ≥ B ≥ C. It can be shown that the call congestion is the time congestion when the number of users S is reduced by one:

The traffic congestion can be obtained by:

4.8.4.4 Peakedness

We characterize variations of traffic by peakedness. Even if the traffic is stationary, i.e. there are no variations of the parameters describing the traffic process, then the traffic intensity is fluctuating around the mean value m (measured in channels) because we can only describe the traffic by a statistical distribution. The fluctuations around the mean value are described by the variance v.

The peakedness Z is defined as Z = v/m and has the unit [channel]. For the offered traffic in Erlang's loss system the peakedness is Z = 1, because the Poisson distribution has m = v. The offered traffic is the carried traffic when number of channels is unlimited and the carried traffic is the mean value of number of busy channels (m = A). For the Engset model (and Binomial distribution) Z < 1. In fact, we have Z = 1−A/S and number of sources S must always be greater than the offered traffic A.

For Z = 1 the traffic is random, whereas for Z < 1 the traffic is smooth. Below we consider overflow traffic with Z > 1 which is peaked or bursty traffic. The traffic congestion will be almost proportional with Z. We will characterize a traffic stream by mean and variance or peakedness.

It is noticed that peakedness has the dimension [channels]. Therefore, it is proper for circuit- switched networks, whereas for packet-switched network the coefficient of variation v/m2 is more appropriate.

Above we have used the parameters (S, β ) to characterize the traffic streams. Alternatively we may also use (A,Z) related to (S, β ) by the formulae:

In addition to Erlang and Engset model we also have the Pascal model which has peakednes Z > 1.

If we let S and β be negative in the above formulae, then we get the Pascal model. Another model with Z > 1 is the Interrupted Poisson process [4.1].

4.8.4.5 Overflow traffic

For planning circuit switched networks with e.g. alternate routing we have to be able to characterize the traffic which is blocked from one link and routed via another link. The basic methods for this problem is the Equivalent Random Traffic (ERT) method by Wilkinson and the equivalence method of Fredericks-Hayward.

Given an Erlang loss system with n channels and offered traffic A we are able to derive the mean value and peakedness of the blocked traffic:

We may also for given mean value m and peakedness Z solve the two equations and _find (A, n) which is called the equivalent group. The idea of the ERT-method is to find the total mean value and variance of all traffic streams offered to a group, and then replace this system by an equivalent Erlang loss system.

The method of Fredericks-Hayward is easier to apply. This method proposes that a system with n channels, which are offered A erlang with peakedness Z, has the same blocking probability as an Erlang loss system with n=Z channels, offered traffic A=Z (and thus peakedness Z=1):

There are several other methods to deal with overflow traffic. Using the above Erlang-Engset- Pascal models (BPP-traffic models) and traffic congestion we get results similar to the above methods. Also Interrupted Poisson processes are used to model bursty traffic processes.

4.8.4.6 Principles of dimensioning

When dimensioning service systems we have to balance grade-of-service requirements against economic restrictions.

In telecommunication systems there are several measures to characterise the service provided. The most extensive measure is Quality-of-Service (QoS), comprising all aspects of a

connection as voice quality, delay, loss, reliability etc. We consider a subset of these, Grade- of-Service (GoS) or network performance, which only includes aspects related to the capacity of the network.

For proper operation, a loss system should be dimensioned for a low blocking probability. In practice the number of channels n should be chosen so that En(A) is 1-5% to avoid overload due to many non-completed and repeated call attempts which both load the system and are a nuisance to subscribers.

If Erlang's B-formula is applied with a fixed blocking probability for dimensioning trunk groups, then we will observe that

a. The utilisation per channel is, for a given blocking probability, highest in large trunk groups, but very low in small groups At a blocking probability E = 1 % a single channel can at most be used 36 seconds per hour! See Fig. 4.8.3.

b. Large trunk groups are more sensitive to a given overload than small trunk groups. This is explained by the low utilisation of small groups, which therefore have a higher spare capacity.

Thus two conflicting factors are of importance when dimensioning trunk groups: we may choose among a high sensitivity to overload or a low utilisation of the channels.

4.8.4.6.1 Improvement principle (Moe's principle)

If we replace the requirement of a fixed blocking probability with an economic requirement, then the improvement function Fn(A) should take a fixed value so that the extension of a trunk group with one additional channel increases the carried traffic by the same amount for all groups.

We will then notice that the utilisation of small groups becomes better corresponding to a high increase of the blocking probability. On the other hand the congestion in large groups decreases to a smaller value.

FB is called the improvement value.