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1.4 The simplest call flow and its properties

Lecture



The simplest call flow is called a stationary ordinary flow without aftereffect. The simplest call flow is completely determined and given by the probability of the receipt of exactly K calls in the time [0, t). Denote this probability by Pk (t) with K = 0.1, 2.3, ..., and t> 0. Find the expression for Pk (t):

  1.4 The simplest call flow and its properties

Consider the small time duration Δ t and calculate the probability that at least one call will be received during this time interval. By definition, the flow parameter we called the relationship limit:

  1.4 The simplest call flow and its properties

Consequently, up to infinitesimally higher order, as Δ t → 0, it can be considered the probability that in the time Δ t there will be at least one call equal to:

  1.4 The simplest call flow and its properties

and the probability that there will not be a single call equal to :

  1.4 The simplest call flow and its properties

Since, by definition, the simplest flow is a flow without aftereffect, the probabilities of incoming calls in non-overlapping periods of time are independent. Therefore, n intervals of time can be considered as n independent experiments, in each of which the time interval Δ t can be “occupied” with probability λ⋅t / n.

The probability that among n intervals there will be exactly K "occupied" can be determined by the theorem on the repetition of experiments (according to the Bernoulli formula) from the expression

  1.4 The simplest call flow and its properties

If n is sufficiently large, this probability is approximately equal to the probability of exactly K arriving at the time interval [0, t), since the probability of two or more calls arriving at the interval Δ t has a negligible probability (the simplest stream is ordinary!). To find the exact value

P k (t), you need to go to the limit as n → ∞:

  1.4 The simplest call flow and its properties

The probability distribution PK (t) is called the Poisson distribution. To verify that the sequence of probabilities PK (t) is a series of distributions, it is necessary to show that the sum of all probabilities PK (t) is equal to one. Indeed, based on the Maclaurin series

  1.4 The simplest call flow and its properties

, we get:

  1.4 The simplest call flow and its properties

To construct a Poisson distribution, it is necessary for all K to calculate PK (t). This is a distribution of a discrete random variable. When λ⋅t = 4, the distribution is as follows:

  1.4 The simplest call flow and its properties

The envelope Poisson distributions for different λ⋅t have the following form:

  1.4 The simplest call flow and its properties

As can be seen from the figure, as the envelope grows, it becomes more and more symmetrical. When λ⋅t⩾10, there is a good agreement between the envelope of the Poisson distribution law and the normal distribution law (which is the distribution law of a continuous random variable), the formula and graph of which is:

  1.4 The simplest call flow and its properties


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Teletraffic Theory

Terms: Teletraffic Theory