The probabilistic law of the distribution of the duration of a satellite session with a ground object

     Part II of .
     Brief explanations to PART II of the work. An example of the design and subsequent analysis of the operation of the Iridium satellite system of civilian cellular communications through an artificial satellite covering 100% of the planet Earth’s surface gives us reason to think about mistakes and miscalculations of owners and designers.

     The orbital part of the system is conceived of from 77 satellites, which is equal to the number of electrons of the chemical element Iridium orbiting like satellites around an atomic nucleus. The system was put into operation on 01.11.1998, but already on 13.8.1999 the company began bankruptcy proceedings. The desired effect of the system was not achieved, and the cause of bankruptcy was called "the difficulty of attracting subscribers." The service was re-launched in 2001. Iridium Satellite LLC, which acquired 6 billionth property for just 25 million.
     Currently, the system is composed of 66 satellites in 11 circular orbits with a height of H≈780 km, moving at a speed of 27,000 km / h. The satellite’s orbital period T ≈100 m , the satellite’s residence time over the horizon of the user's standing point ≈10 m. Ground stations (4 in total) are located: Tempe, Arizona; Wahiaawa, Hawaii - owned by 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 reasons for past failures of the project should be attributed to the weak elaboration of the project in the direction of modeling the processes of functioning of the system as a whole. Such a forecast and evaluation of effectiveness is a very expensive part of any project of this kind, and the owners, apparently, were not ready to conduct such studies. It is clear that the real functioning of the system for the owners seemed somewhat unexpected. We pay attention to the characteristic - the satellite’s stay time over the horizon of the user's standing point ≈10 m. Firstly, not every satellite behaves above the phone’s owner’s horizon in this way (part of the satellite doesn’t have time to rise above the horizon already), and secondly, in such a situation switching (messages, channels, packets) becomes very complicated, hence the bulkiness and high the cost of satellite phones and other equipment.
     To obtain a high-quality forecast and an objective picture of the system’s behavior, modeling of the basic processes of the system as a whole is required, and for this, qualitative models of the functioning of subsystems, aggregates, units and individual elements are required. The system designers, apparently, did not have not only finances, time, but also solid models. One model that is very essential for a qualitative forecast and the success of the upcoming functioning is considered in this paper.
     Publications known to the author contain prototypes of models (in which the final result is given without a detailed consideration of the conclusion of its receipt), similar to those considered in the author's work, but contain gross errors, which is what readers pay attention to.
     In this paper, a detailed derivation of the analytical expression for the distribution law of a random variable of mixed type is given. Among the known classical probabilistic laws this law is not, therefore, the work contains an element of novelty, and the list of laws is replenished with another new one. In the author’s previous work (Part I), the equivalence of two random phenomena is established. The first ℬ is the satellite’s hit in the MLO IP during one revolution of the satellite around the Earth and the second is the fact that the longitude of the ascending node of the satellite’s orbit belongs to one of the intervals at the Earth’s equator at any time within this revolution. In addition to this (in Part I), the rationale for using (in determining the probability of the first event) the uniform distribution of the longitude of the ascending node of the satellite’s orbit is given.
     The mathematical side of the problem is reduced to obtaining a general expression for the probability of a random event ℬ in the form of 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 integration limits must be determined by solving spherical triangles on the sphere obtained as a result of "drawing" the situation of the position of the MLO boundary and the boundary position (touching) of the satellite paths with this boundary. In other words, the geometric picture 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.
Conclusion of the probabilistic law of the distribution of the duration of an information exchange session

     The formulated assumptions of the model ultimately provide an analytical expression for the probability of the occurrence of a random event ℬ, as a function of three deterministic and one random variables. We will determine the probabilistic law in the form F t (t) - the distribution function of the random variable t c - the chord length of the boundary circle L of the segment - zone. For convenience and simplification, we will transform the interval of variation of the values ​​of m with to one.
     The conversion to a unit interval is carried out by normalizing the possible values ​​of t with t max , i.e. t = t s / t max. Thus, the normalized variable varies in the interval [0, 1]. Therefore, each value of the length of the chord m uniquely corresponds to a certain value of the angular distance ζc.

     Figure 1 - The geometry of the MLO segment and the chord equal to the segment of the random turn trace

     Obviously, the values ​​of this chord m s lie in the interval [0, m max ], where m max is the value of the chord length provided that the plane ((i, λ) passes through the center of the segment. The length m of the chord with a random section of the segment by the plane ((i, λ) is the greater, the closer the plane goes to the center of the segment, i.e.

where ζc is the current value of the angular distance from the plane П to the center of the segment (it corresponds to the linear distance r) and m = t is the current value of the segment chord length.
      An analysis of the behavior of the random variable m shows that it refers to quantities of a mixed type. In the range of its possible values, there are many discrete values ​​of t d , the probability of which being taken by a random value of m differs from zero, i.e. P (m = m d )> 0. At the set of such points, the random variable m s behaves like a discrete variable, at the other points of the admissible set of values ​​the behavior of the random variable m s is described as the behavior of a continuous random variable. The probability of taking any value from such a set is zero P (m = mn ) = 0.
In our case, the set {m d} discrete values ​​formed by a single point m d = {0}. Indeed, the probability that the satellite does not pass through the area of ​​the IP service area is nonzero at some random turn.
     From a right-angled spherical triangle (Fig. 1), we can write ζ = arccos (cosζc / cos т ζc) . A chord on a sphere of length m is tangent to the boundary of the reduced segment with the angle of angle ζ defined by the expression for ζ depending on the value of m.
     If we now indicate in some way the length of the chord and determine the probability that the value of m will be less than m ass , then the distribution function m is defined.
      In probability theory, it is known that the sum of the probabilities of opposing incompatible events forming a complete group is always equal to unity, 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 that the orbital plane P (i, λ) hits a segment of a certain radius ζc, on the other hand, the formula ζ = arccos (cosζc / cos т ζc)for a given m determines the radius of the segment ζ. Consequently, for a segment of radius ζc, one can determine from m ass the radius of the concentric segment ζ corresponding to some m ass . From here you can determine the probability of getting into this inner segment. As a result, we see that the random value of the chord length of a segment of radius ζc in this case will be no less than m ass , i.e., m assm; , and the probability of the plane P (i, λ) getting into the inner segment will be equal to the probability of the last inequality P (m ass ≤ m). The probability of the opposite event just describes the distribution law in integral form.

     The expression (analytical) for the found probability distribution law (form F t (t) of the distribution function) has a graphical representation in the following form (Fig. 2). Below in Fig. 2 are plots of F t (t) obtained for different angles of inclination of the orbital planes with a fixed flight height h and a fixed position and IP characteristics.

     Figure 2 - Graphs of the distribution function of a random value of the satellite stay time in the service area of ​​the measuring point
     With the above results, the researcher is able to form a stochastic model of the satellite system and explore information flows, including information security issues, starting from the opening / closing of the side, where digital signature is used, message encryption / decryption, etc. functions.

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