V2

no e

Fig. 8.4. The Gaussian density function.

The central limit theorem states that the probability density function of the sum of a large number of independent variables tends towards n(|j., o : x) as 'n' increases. Substituting, t = (x - |j.)/o in equation 8.37, and neglecting o p(t) = -¡L= e~t2/2 ...(8.38)

Equation 8.38 is called standard normal distribution. 8.3.4. Statistical Parameters

The random process may be discrete or continuous. Similarly the time index of random variables can be discrete or continuous. Thus, there are four different types of process namely (a) continuous time continuous state (b) continuous time discrete state (c) discrete time continuous state and (d) discrete time discrete state. In telecommunication switching sytstem, our interest is discrete random process and therefore for modelling a switching system, we use discrete state stochastic process. A discrete state stochastic process is often called a chain.

A statistical properties of a random process may be obtained in two ways :

(i) Observing the behaviour of the system to be modelled over a period of time repeatedly. The data obtained is called a single sample. The average determined by measurements on a single sample function at successive times will yield a time average.

(ii) Simultaneous measurements of the output of a large number of statistically identical random sources. Such a collection of sources is called an ensemble and the individual noise waveforms is called the sample function. The statistical average made at some fixed time t = t1 on all the sample functions of the ensample is the ensemble average.

The above two ways are analogous to obtaining the statistics from tossing a die repeatedly (large number) or tossing one time the large number of dice.

In general, time average and ensemble average are not the same due to various reasons. When the statistical characteristics of the sample functions do not change with time, the random process is described as being stationary. The random process which have identical time and ensemble average are known as ergodic processes. An ergodic process is stationary, but a stationary process is not necessarily ergodic.

Telephone traffic is nonstationary. But the traffic obtained during busy hour may be considered as stationary (which is important for modelling) as modelling non-stationary is difficult.

8.3.5. Pure Chance Traffic

Here, the call arrivals and call terminations are independent random events. If call arrivals are independent random events, their occurrence is not affected by previous calls. This traffic is therefore sometimes called memory less traffic. A.A. Markov in 1907, defined properties and proposed a simple and highly useful form of dependency. This class of processes is of great interest to our modelling of switching systems. A discrete time Markov chain i.e. discrete time discrete state Markov process is defined as one which has the following property.

= P [fX(tn+1) = Xn+1l/iX(tn) = X^l where t1 < t2 < tn <tn+1 and xt is the ith discrete state space value.

Equation 8.39 states that the propability that the random variable X takes on the value xn+1 at time step n + 1 is entirely determined by its state value in the previous time step n and is independent of its state values in earlier time steps ; n - 1, n- 2, n- 3 etc.

8.3.6. The Birth and Death Process

The birth and death process is a special case of the discrete state continuous time Markov process, which is often called a continuous-time Markov chain. The number of calls in progress is always between 0 and N. It thus has N + 1 states. If the Markov chain can occur only to adjacent states (i.e. probability change from each state to the one above and one below it) the process is known as birth-death (B-D) process. The basic feature of the method of Markov chains is the kolmogorov differential-difference equation, for the limiting case, can provide a solution to the state probability distribution for the Erlang systems and Engset systems.

Let N(t) be a random variable specifying the size of the population at time t. For a complete description of a birth and death process, we assume that N(t) is in state k at time t and has the following properties :

1. P(k) is the probability of state k and P(k + 1) is the probability of state k + 1.

2. The probability of transition from state k to state k + 1 in short duration At is Xk At, where Xk is called the birth rate in state k.

3. The probability of transition from state k to state k - 1 in the time interval At is |j.k At, where |j.k is called the death rate in state k.

4. The probability of no change of state in the time interval At is equal to 1 - (kk + |j.k) At.

5. The probability in At, from state k to a state other than k + 1 or k - 1 is zero.

Based on the above properties, birht and death process of N(t) and state transition rate diagram are shown in Fig. 8.5.

At statistical equilibrium (i.e. stationary), let Pjk is the conditional probability, that is the probability of state increases from j to k. Similarly Pkj is the probability of state decrease from k to j.

The probabilities P(0), P(1), P(N) are called the state probabilities and the conditional probabilities Pjk, Pkj are called transition probabilities. The transition probabilities satisfy the following condition :

No change

Call end

Call end t

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