CARRO BOMBA FORESTAL C5 413.- PREGUNTA
499.- PREGUNTA Equipo de Bombeo
By a stochastic sum we understand the sum of a stochastic number of random variables (Feller, 1950 [27]). Let us consider a trunk group without congestion, where the arrival process and the holding times are stochastically independent. If we consider a fixed time interval T , then the number of arrivals is a random variable N . In the following N is characterised by:
N : density function p(i) ,
mean value m1,n, (3.40)
variance σn2,
Arriving call number i has the holding time Ti. All Ti have the same distribution, and each arrival (request) will contribute with a certain number of time units (the holding times) which
3.3. STOCHASTIC SUM 73 is a random variable characterised by:
T : density function f (t) ,
mean value m1,t, (3.41)
variance σt2,
The total traffic volume generated by all arrivals (requests) arriving within the considered time interval T is then a random variable itself:
ST = T1+ T2+ · · · + TN. (3.42) T p p p p 1 2 3 1 2 T1 1 i i 1 T T T 2 T T 2 T 3 T
Figure 3.3: A stochastic sum may be interpreted as a series/parallel combination of random variable.
In the following we assume that Ti and N are stochastically independent. This will be fulfilled when the congestion is zero.
The following derivations are valid for both discrete and continuous random variables (sum- mation is replaced by integration or vice versa). The stochastic sum becomes a combination of random variables in series and parallel as shown in Fig.3.3 and dealt with in Sec.3.2. For a given branch i we find (Fig. 3.3):
m1,i = i · m1,t, (3.43)
σi2 = i · σt2, (3.44)
74 CHAPTER 3. PROBABILITY THEORY AND STATISTICS By summation over all possible values (branches) i we get:
m1,s = ∞ X i=1 p(i) · m1,i = ∞ X i=1 p(i) · i · m1,t, m1,s = m1,t · m1,n, (3.46) m2,s = ∞ X i=1 p(i) · m2,i = ∞ X i=1 p(i) · {i · σt2+ (i · m1,t)2} , m2,s = m1,n· σt2+ m21,t· m2,n, (3.47) σs2 = m1,n· σt2+ m 2 1,t· (m2,n− m21,n) , σs2 = m1,n· σt2+ m 2 1,t· σ 2 n. (3.48)
We notice there are two contributions to the total variance: one term because the number of calls is a random variable (σn2), and a term because the duration of the calls is a random variable (σ2t).
Example 3.3.1: Special case 1: N = n = constant (mn = n) m1,s = n · m1,t,
σ2s = σt2· n . (3.49)
This corresponds to counting the number of calls at the same time as we measure the traffic volume
so that we can estimate the mean holding time. 2
Example 3.3.2: Special case 2: T = t = constant (mt = t) m1,s = m1,n· t ,
σs2 = t2· σn2. (3.50) If we change the scale from 1 to m1,t, then the mean value has to be multiplied by m1,t and the variance by m21,t. The mean value m1,t = 1 corresponds to counting the number of calls, i.e. a
3.3. STOCHASTIC SUM 75
Example 3.3.3: Stochastic sum
As a non-teletraffic example N may denote the number of rain showers during one month and Timay denote the precipitation due to the i’th shower. ST is then a random variable describing the total precipitation during a month. N may also for a given time interval denote the number of accidents registered by an insurance company and Ti denotes the compensation for the i’th accident. ST then is the total amount paid by the company for the considered period. 2
Chapter 4
Time Interval Distributions
The exponential distribution is the most important time distribution within teletraffic theory. This time distribution is dealt with in Sec. 4.1.
Combining exponential distributed time intervals in series, we get a class of distributions called Erlang distributions (Sec.4.2). Combining them in parallel, we obtain hyper–exponen- tial distribution (Sec. 4.3). Combining exponential distributions both in series and in par- allel, possibly with feedback, we obtain phase-type distributions, which is a class of general distributions. One important sub–class of phase-type distributions is Coxian-distributions (Sec. 4.4). We note that an arbitrary distribution can be expressed by a Cox–distribution which can be used in analytical models in a relatively simple way. Finally, we also deal with other time distributions which are employed in teletraffic theory (Sec. 4.5). Some examples of observations of life times are presented in Sec.4.6.
4.1
Exponential distribution
In teletraffic theory this distribution is also called the negative exponential distribution. It has already been mentioned in Sec. 3.1.2 and it will appear again in Sec. 6.2.1.
In principle, we may use any distribution function with non–negative values to model a life– time. However, the exponential distribution has some unique characteristics which make this distribution qualified for both analytical and practical uses. The exponential distribution plays a key role among all life–time distributions.
78 CHAPTER 4. TIME INTERVAL DISTRIBUTIONS This distribution is characterised by a single parameter, the intensity or rate λ:
F (t) = 1 − e−λt , λ > 0 , t ≥ 0 , (4.1)
f (t) = λe−λt, λ > 0 , t ≥ 0 . (4.2)
The gamma function is defined by:
Γ(n + 1) = Z ∞
0
tne−tdt = n! . (4.3)
We replace t by λt and get the ν’th moment: mν = ν! λν , (4.4) Mean value m = m1 = 1 λ, Second moment: m2 = 2 λ2, Variance: σ2 = 1 λ2, Form factor: ε = 2 , ... ... λ
Figure 4.1: In phase diagrams an exponentially distributed time interval is shown as a box with the intensity. The box thus means that a customer arriving to the box is delayed an exponentially distributed time interval before leaving the box.
The exponential distribution is very suitable for describing physical time intervals (Fig.6.2). The most fundamental characteristic of the exponential distribution is its lack of memory. The distribution of the residual time of a telephone conversation is independent of the actual duration of the conversation, and it is equal to the distribution of the total life-time (3.11):
f (t + x|x) = λe −(t+x)λ e−λx = λe−λt = f (t) .
If we remove the probability mass of the interval (0, x) from the density function and nor- malise the residual mass in (x, ∞) to unity, then the new density function becomes congruent
4.1. EXPONENTIAL DISTRIBUTION 79 with the original density function. The only continuous distribution function having this property is the exponential distribution, whereas the geometric distribution is the only dis- crete distribution having this property. An example with the Weibull distribution where this property is not valid is shown in Fig. 3.1. For k = 1 the Weibull distribution becomes iden- tical with the exponential distribution. Therefore, the mean value of the residual life-time is m1,r = m, and the probability of observing a life–time in the interval (t, t + dt), given that it occurs after t, is given by
p{t < X ≤ t + dt|X > t} = f (t) dt 1 − F (t)
= λ dt . (4.5)
Thus it depends only upon λ and dt, but it is independent of the actual age t.
4.1.1
Minimum of k exponentially distributed random variables
We assume that two random variables X1 and X2are mutually independent and exponentially distributed with intensities λ1 and λ2, respectively. A new random variable X is defined as:
X = min {X1, X2} . The distribution function of X is:
p{X ≤ t} = 1 − e−(λ1+λ2)t. (4.6)
This distribution function itself is also an exponential distribution with intensity (λ1+ λ2). Under the assumption that the first (smallest) event happens within the time interval t, t+dt, then the probability that the random variable X1 is realized first (i.e. takes places in this interval and the other takes place later) is given by:
p{X1 < X2| t} = P {t < X1 ≤ t + dt} · P {X2 > t} P {t < X ≤ t + dt} = λ1e −λ1tdt · e−λ2t (λ1+ λ2) e−(λ1+λ2)tdt = λ1 λ1+ λ2 , (4.7)
i.e. independent of t. Thus we do not need to integrate over all values of t.
These results can be generalised to k variables and make up the basic principle of the simu- lation technique called the roulette method, a Monte Carlo simulation methodology.
80 CHAPTER 4. TIME INTERVAL DISTRIBUTIONS
4.1.2
Combination of exponential distributions
If one exponential distribution (i.e. one parameter) cannot describe the time intervals in suffi- cient detail, then we may have to use a combination of two or more exponential distributions. Conny Palm introduced two classes of distributions: steep and flat.
A steep distribution corresponds to a set of stochastic independent exponential distributions in series (Fig. 4.2), and a flat distribution corresponds to exponential distributions in par- allel (Fig. 4.4). This structure naturally corresponds to the shaping of traffic processes in telecommunication and data networks.
By the combination of steep and flat distribution, we may obtain an arbitrary good approx- imation for any distribution function (see Fig. 4.7 and Sec. 4.4). The diagrams in Figs. 4.2
& 4.4 are called phase-diagrams.