Probabilistic law of the distribution of the duration of the session of the satellite with a ground object
PART II .
Brief explanations to PART II of the work. An example of the design and subsequent analysis of the functioning of the satellite system "IRIDIUM" of civil cellular communications through the satellite, covering 100% of the surface of planet Earth, gives reason to think about the mistakes and miscalculations of the owners and designers.
The orbital part of the system is designed from 77 satellites, which is equal to the number of electrons of the chemical element Iridium, which circulate like satellites around the atomic nucleus. The system was put into operation on 11/01/1998, but on 13/8/1999, the company began bankruptcy proceedings. The desired effect of the system could not be achieved, and the reason for bankruptcy was called "the difficulty of attracting subscribers." The service was re-launched in 2001. Iridium Satellite LLC, which acquired a $ 6 billion property in just 25 million.
Currently, the system is formed from 66 satellites in 11 circular orbits with a height of H≈780 km, moving at a speed of 27,000 km / h. The orbital period of the satellite T ≈100 m , the residence time of the satellite over the horizon of the user's standing point is ≈10 m . Ground stations (4 items total) are located: Tempe, Arizona; Wahiava, Hawaii - belong to the US Department of Defense Information Systems Protection Agency; Avezzano, Italy. The issues of information security and cryptographic protection of information flows are the prerogative of the Agency.
Among the causes of past failures of the project should include a weak study of the project in the direction of modeling the functioning of the system as a whole. Such a forecast and evaluation of the effectiveness of a very expensive part of any project of this kind and the owners, apparently, were not ready to conduct such research. It is clear that the actual functioning of the system for the owners seemed somewhat unexpected. Let's pay attention to the characteristic - the time of stay of the satellite over the horizon of the user's standing point ≈10 m . First, not every AES behaves over the horizon of the owner of the phone in this way (some of the AES, not having time to ascend over the horizon, already comes), and secondly, in such a situation, switching (messages, channels, packages) becomes more complicated, hence the bulkiness and high the cost of satellite phones and other equipment.
To obtain a qualitative forecast and an objective picture of the behavior of the system, modeling of the main processes of the system as a whole is required, and this requires high-quality models of the functioning of subsystems, units, nodes and individual elements. The system designers, apparently, did not have not only finances, time, but also good-quality models. One very essential for a qualitative forecast and the success of the upcoming operation of the model is considered in the present work.
The publications known to the author contain prototypes of models (in which the final result is presented without a detailed consideration of the conclusion of its receipt), similar to those considered in the author’s work, but contain gross errors, and the attention of readers is drawn to this.
This paper presents a detailed derivation of the analytical expression for the distribution law of a random variable of mixed type. Among the well-known classical probabilistic laws of this law there is no, therefore, the work contains an element of novelty, and the list of laws is supplemented by another new one. In the previous work of the author (Part I), the equivalence of two random phenomena is established. The first is the hit of the satellite in the MLA IP at one revolution of the satellite around the Earth and the second is the fact that the longitude of the ascending node of the satellite orbit belongs to one of the intervals at the earth equator at an arbitrary time within this revolution. In addition to this (in Part I), the rationale for using (when determining the probability of the first event) the uniform distribution law of the ascending node of the satellite orbit is given.
The mathematical side of the problem is reduced to obtaining a general expression for the probability of a random event ℬ as the sum of two integrals of the distribution density of a random variable, which is described by a uniform distribution law. In this case, the limits of integration must be determined by solving spherical triangles on a sphere, obtained as a result of the situation of the position of the MLO boundary and the boundary position (tangency) of the routes of the satellite with this boundary. In other words, the geometric pattern of the “frozen” joint motion of the satellite and the measuring point together with the Earth is considered and taken as the basis of the model.
Derivation of the probability distribution law of the duration of an information exchange session
The formulated assumptions of the model ultimately provide an analytical expression for the probability of occurrence of a random event ℬ, as a function of three deterministic and one random variable. We will determine the probability law in the form of F t (t) - the distribution function of a random variable τ s - the chord length of the boundary circle L of the segment - zone. For convenience and simplification, we transform the interval of change of values of m with into one.
Conversion to a unit interval is performed by normalizing the possible values of m with over m max , i.e. t = t s / t max . Thus, the normalized variable varies in the interval [0, 1]. Consequently, each value of the chord length t uniquely corresponds to some value of the angular distance c.
Figure 1 - Geometry of the MLO segment and the chord, equal to the segment of the random turn track
Obviously, the values of this chord t c lie in the interval [0, t max ], where t max is the value of the chord length, provided that the plane P (i, λ) passes through the center of the segment. The length m t of the chord with a random section of a segment by the plane P (i, λ) is the greater, the closer to the center of the segment is the plane, i.e
where ζc is the current value of the angular distance from the plane P to the center of the segment (it corresponds to the linear distance r ) and t = t is the current value of the chord length of the segment.
Analysis of the behavior of the random variable t shows that it refers to the values of the mixed type. In the area of its possible values, there is a set of discrete values m d , the probability of accepting which by a random variable m differs from zero, i.e P (m = md )> 0. In the set of such points, the random variable m with behaves as a discrete variable; at the remaining points of the admissible set of values, the behavior of the random variable m with is described as the behavior of a continuous random variable. The probability of accepting any value from such a set is zero. P (m = m, n ) = 0.
In our case, the set {m d } of discrete values is formed by the single point m d = {0}. Indeed, the probability that the satellite does not pass through the region of the IP service area is non-zero at some random turn.
From a rectangular spherical triangle (Fig. 1) we can writeζ = arccos (cosζc / cos t cosc) . A chord on a sphere of length t is tangent to the boundary of a smaller segment with a solution angle ζ defined by the expression for ζ depending on the value of t.
If we now specify the chord length in some way and determine the probability that the value of m will be less than m butt , then the distribution function m will be determined.
In probability theory it is known that the sum of the probabilities of opposite incompatible events that form the full group is always equal to one, i.e.
P (t <t ass ) + P (t ≥t ass ) = 1. F t (t) = P (t <t ass ) = 1 - P (t ≥t ass ).
In this situation, we just have such a case. The satellite either passes through the zone or not.
The formula obtained earlier ( Here ) allows us to determine the probability of the orbit plane P (i, λ) falling into a segment of a certain radius c, on the other hand, the formulaζ = arccos (cosζc / cos ζc) for a given t determines the radius of the segment ζ. Therefore, for a segment of radius ζc, we can determine from the t backside the radius of its concentric segment ζ corresponding to some t backside . From here you can determine the probability of hitting this internal segment. As a result, we see that the random value of the chord length of a segment of radius ζc in this case will not be less than t ass , that is, t ass ≤ t; and the probability of hitting the plane (i, λ) in the inner segment will be equal to the probability of the last inequality P (t ass ≤ t). The probability of the opposite event describes the distribution law in integral form.
The expression (analytical) for the found law of probability distribution (the form F t (t) of the distribution function) has a graphical representation in the following form (Fig. 2). The figure 2 below shows the graphs F t (t) obtained for different inclination angles of the orbit planes with a fixed flight height h and a fixed position and characteristics of the PI.
Figure 2 - Graphs of the distribution function of the random variable of the time the satellite is in the service area of the measuring point
With these results, the researcher has the opportunity to form a stochastic model of the functioning of the satellite system and explore information flows, including information security issues, starting from opening / closing a side using a digital signature, encrypting / decrypting messages, etc. functions.
Brief explanations to PART II of the work. An example of the design and subsequent analysis of the functioning of the satellite system "IRIDIUM" of civil cellular communications through the satellite, covering 100% of the surface of planet Earth, gives reason to think about the mistakes and miscalculations of the owners and designers.
The orbital part of the system is designed from 77 satellites, which is equal to the number of electrons of the chemical element Iridium, which circulate like satellites around the atomic nucleus. The system was put into operation on 11/01/1998, but on 13/8/1999, the company began bankruptcy proceedings. The desired effect of the system could not be achieved, and the reason for bankruptcy was called "the difficulty of attracting subscribers." The service was re-launched in 2001. Iridium Satellite LLC, which acquired a $ 6 billion property in just 25 million.
Currently, the system is formed from 66 satellites in 11 circular orbits with a height of H≈780 km, moving at a speed of 27,000 km / h. The orbital period of the satellite T ≈100 m , the residence time of the satellite over the horizon of the user's standing point is ≈10 m . Ground stations (4 items total) are located: Tempe, Arizona; Wahiava, Hawaii - belong to the US Department of Defense Information Systems Protection Agency; Avezzano, Italy. The issues of information security and cryptographic protection of information flows are the prerogative of the Agency.
Among the causes of past failures of the project should include a weak study of the project in the direction of modeling the functioning of the system as a whole. Such a forecast and evaluation of the effectiveness of a very expensive part of any project of this kind and the owners, apparently, were not ready to conduct such research. It is clear that the actual functioning of the system for the owners seemed somewhat unexpected. Let's pay attention to the characteristic - the time of stay of the satellite over the horizon of the user's standing point ≈10 m . First, not every AES behaves over the horizon of the owner of the phone in this way (some of the AES, not having time to ascend over the horizon, already comes), and secondly, in such a situation, switching (messages, channels, packages) becomes more complicated, hence the bulkiness and high the cost of satellite phones and other equipment.
To obtain a qualitative forecast and an objective picture of the behavior of the system, modeling of the main processes of the system as a whole is required, and this requires high-quality models of the functioning of subsystems, units, nodes and individual elements. The system designers, apparently, did not have not only finances, time, but also good-quality models. One very essential for a qualitative forecast and the success of the upcoming operation of the model is considered in the present work.
The publications known to the author contain prototypes of models (in which the final result is presented without a detailed consideration of the conclusion of its receipt), similar to those considered in the author’s work, but contain gross errors, and the attention of readers is drawn to this.
This paper presents a detailed derivation of the analytical expression for the distribution law of a random variable of mixed type. Among the well-known classical probabilistic laws of this law there is no, therefore, the work contains an element of novelty, and the list of laws is supplemented by another new one. In the previous work of the author (Part I), the equivalence of two random phenomena is established. The first is the hit of the satellite in the MLA IP at one revolution of the satellite around the Earth and the second is the fact that the longitude of the ascending node of the satellite orbit belongs to one of the intervals at the earth equator at an arbitrary time within this revolution. In addition to this (in Part I), the rationale for using (when determining the probability of the first event) the uniform distribution law of the ascending node of the satellite orbit is given.
The mathematical side of the problem is reduced to obtaining a general expression for the probability of a random event ℬ as the sum of two integrals of the distribution density of a random variable, which is described by a uniform distribution law. In this case, the limits of integration must be determined by solving spherical triangles on a sphere, obtained as a result of the situation of the position of the MLO boundary and the boundary position (tangency) of the routes of the satellite with this boundary. In other words, the geometric pattern of the “frozen” joint motion of the satellite and the measuring point together with the Earth is considered and taken as the basis of the model.
Derivation of the probability distribution law of the duration of an information exchange session
The formulated assumptions of the model ultimately provide an analytical expression for the probability of occurrence of a random event ℬ, as a function of three deterministic and one random variable. We will determine the probability law in the form of F t (t) - the distribution function of a random variable τ s - the chord length of the boundary circle L of the segment - zone. For convenience and simplification, we transform the interval of change of values of m with into one.
Conversion to a unit interval is performed by normalizing the possible values of m with over m max , i.e. t = t s / t max . Thus, the normalized variable varies in the interval [0, 1]. Consequently, each value of the chord length t uniquely corresponds to some value of the angular distance c.
Figure 1 - Geometry of the MLO segment and the chord, equal to the segment of the random turn track
Obviously, the values of this chord t c lie in the interval [0, t max ], where t max is the value of the chord length, provided that the plane P (i, λ) passes through the center of the segment. The length m t of the chord with a random section of a segment by the plane P (i, λ) is the greater, the closer to the center of the segment is the plane, i.e
where ζc is the current value of the angular distance from the plane P to the center of the segment (it corresponds to the linear distance r ) and t = t is the current value of the chord length of the segment.
Analysis of the behavior of the random variable t shows that it refers to the values of the mixed type. In the area of its possible values, there is a set of discrete values m d , the probability of accepting which by a random variable m differs from zero, i.e P (m = md )> 0. In the set of such points, the random variable m with behaves as a discrete variable; at the remaining points of the admissible set of values, the behavior of the random variable m with is described as the behavior of a continuous random variable. The probability of accepting any value from such a set is zero. P (m = m, n ) = 0.
In our case, the set {m d } of discrete values is formed by the single point m d = {0}. Indeed, the probability that the satellite does not pass through the region of the IP service area is non-zero at some random turn.
From a rectangular spherical triangle (Fig. 1) we can write
If we now specify the chord length in some way and determine the probability that the value of m will be less than m butt , then the distribution function m will be determined.
In probability theory it is known that the sum of the probabilities of opposite incompatible events that form the full group is always equal to one, i.e.
P (t <t ass ) + P (t ≥t ass ) = 1. F t (t) = P (t <t ass ) = 1 - P (t ≥t ass ).
In this situation, we just have such a case. The satellite either passes through the zone or not.
The formula obtained earlier ( Here ) allows us to determine the probability of the orbit plane P (i, λ) falling into a segment of a certain radius c, on the other hand, the formula
The expression (analytical) for the found law of probability distribution (the form F t (t) of the distribution function) has a graphical representation in the following form (Fig. 2). The figure 2 below shows the graphs F t (t) obtained for different inclination angles of the orbit planes with a fixed flight height h and a fixed position and characteristics of the PI.
Figure 2 - Graphs of the distribution function of the random variable of the time the satellite is in the service area of the measuring point
With these results, the researcher has the opportunity to form a stochastic model of the functioning of the satellite system and explore information flows, including information security issues, starting from opening / closing a side using a digital signature, encrypting / decrypting messages, etc. functions.
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Source: https://habr.com/ru/post/238159/
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