Blog_Post_2, Saturday_4_3:

    Module 2 has really stirred up some thoughtful analysis in my mind, especially with regards to probability and the occurrence of defined/random events. Systems Engineers (especially within the Space Industry) are constantly dealing with probability mathematics, as they are routinely accessing data sets and capabilities which reside on large information grids, which are in turn, heavily dependent upon Satellite Communication networks (SATCOMS) and Wireless Communication networks. Within these realms, the applications for probability calculations appears endless, and some very exciting solutions have been achieved as a result of the ability to calculate probability for certain events within sets of data. In particular, Stochastic Geometry, which is a specialized branch of probability mathematics dealing with quantifying random special patterns, time averages for data packet delays, applied probability, random phenomena, and spatial placement/orientation. As I dug deeper into the world of SATCOM and Stochastic Geometry, I quickly became familiar with a reoccurring term: Signal to Interference Noise Ratio (SINR). This ratios is of particular importance to scientists and engineers because the geometry of signal/communication nodes in satellite constellations and wireless network grids effectively determines the level of SINR encountered when attempting to relay data via links (Baccelli & Blaszczyszyn, 2009, p. 250-255). Consider Key Performance Indicators (KPIs) such as User Coverage Probability, Down-link Coverage Probability, and data download rates by satellite spatial placement. Stochastic Geometric calculations are leveraged to derive downlink coverage probabilities for satellites in Low-Earth Orbit (LEO) to users on the earth. The KPI for Probability of Coverage can be defined as: Pc(T) = P(SINR>T). In this case, T represents a minimum threshold for successful data transmission (Okati et al., 2020, p. 4-5). Turning the conversation back to the text, consider a satellite with uplink/downlink A and B, designed to deliver continuous data between points 1 and 2. During any given day in the week, the probability of failure due to weather events, solar flares, or other global phenomena for either Links A or B is 0.15. Additionally, if Link A fails there is a 0.05 probability that Link B will also fail. Given this data, we can now compute the probability of no data service between points 1 and 2 during any given point throughout the week.

Solution:

Let:                     

·         P(A) = Probability of Link A failure

·         P(B) = Probability of Link B failure

·         P(C) = Probability of no data service between points 1 and 2

Thus, Event C will be the intersection of Events A and B, and P(A) = P(B) =0.15, P(B│A) =0.05. Therefore, P(C) =P(B∩A) = P(B│A) P(A) = 0.15 x 0.05 = 13.5075 probability of no data service (Khisty et al., 2012, p. 127) I have included what I thought was a very representative model for satellite constellations with regards to Stochastic Geometry and Probability of data service. Also, while I did not use any of his text as reference material,  I highly recommend the book, Understanding satellite navigation by Acharya (2014) as a great foundation for understanding SATCOM activity.

Figure 1:Stochastic Geometric Constellation. Okati, N., Riihonen, T., Korpi, D., Angervuori, I., & Wichman, R. (2020). Downlink Coverage and Rate Analysis of Low Earth Orbit Satellite Constellations Using Stochastic Geometry. IEEE. https://arxiv.org/pdf/2004.13378.pdf

 

           

 

 

Acharya, R. (2014). Understanding satellite navigation (1st ed.). Academic Press. https://doi.org/10.1016/B978-0-12-799949-4.12001-9

Baccelli, F., & Blaszczyszyn, B. (2009). Stochastic Geometry and Wireless Networks: Volume I Theory. Foundations and Trends in Networking, 3(3-4), 249-449. https://www.stat.berkeley.edu/~aldous/206-SNET/Papers/baccelli-1.pdf

Khisty, C. J., Mohammadi, J., & Amedkudzi, A. A. (2012). Systems engineering with economics, probability, and statistics (2nd ed.). J. Ross Publishing.

Liu, D. (2018). Systems engineering: Design principles and models (1st ed.). CRC Press.

Okati, N., Riihonen, T., Korpi, D., Angervuori, I., & Wichman, R. (2020). Downlink Coverage and Rate Analysis of Low Earth Orbit Satellite Constellations Using Stochastic Geometry. IEEE. https://arxiv.org/pdf/2004.13378.pdf