Declaration 2 Publication
7.2 Conclusions
This section hereby discusses the conclusions drawn from chapters three, four, five and six of this thesis.
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
139
7.2.1 Chapter Three – Comparison and Analysis of Rainfall Microstructures in Tropical Africa
The variation of rainfall microstructures over Stratiform-Convective (SC) bounds is an important indices to investigate rainfall at different climatic regions. Over tropical Africa, the location of Durban (subtropical) and Butare (equatorial) have shown that the delineation of SC bounds is separated by at least 2.5 mm/h. Considering that ?. at both locations varies, then this is very much expected. From the results, Butare will experience early transition into convective rainfall types as an equatorial location quickly than Durban, a subtropical location.
equatorial locations are noted to have significant occurrence of showers and thunderstorm rainfalls compared to subtropical locations. For stratiform rainfalls, there is an obvious difference in the probability distribution profiles of rainfall microstructres at both climatic locations. The rainfall rate, rainfall DSD and radar reflectivity distributions all seem to have different profiles at both locations over stratiform rain regime. This is mainly due to the significantly higher percentage of drizzle rainfalls in Durban, resulting in the preponderance of smaller rain droplets in this category. Thus, rainfall rates below 10 mm/h, during rain events, have been reported to follow temperate characteristics [Fashuyi et al., 2006]. However, convective rainfalls at both location have the closer probability profiles for the same category of rain microstructures. This is further proven by the close model parameters of both rainfall DSD and radar distributions as seen in the results. Also, the proponderance of small droplets at both locations is clearly opposite to what is obtained under convective. This suggests that shower and thunderstorm characteristics of rainfall are a little similar under convective conditions. One deviation to this, is however in the monthly variation of radar reflectivities, which is clearly dependent on seasonal cycles. An investigation of SC thresholds at other global locations also shows that a rainfall rate of roughly about 10 mm/h, as suggested by literatures, is the valid transition parameter.
7.2.2 Chapter Four - Queueing Theory of Rain Spikes over Radio Links
The Markov theory of rainfall queues was developed empirically from the rainfall time series data in Durban. The appearances of rainfall spikes, which are linked to rain cell movement over a radio link, tend to appear randomly during rain events. Their service and inter-arrival times follow a Birth-Death scenario which qualifies them as instances of rainfall traffic. The service time distribution clearly exhibited exponential or Erlang-k characteristics, while inter-arrival time distribution is strictly an exponential process. Therefore, results from the data showed that the typical distribution may either follow a Markovian (M/M/s/∞/ FCFS) or semi-Markovian
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
140
(M/Ek/s/∞/ FCFS) queue discipline. However, the semi-Markovian discipline showed better error fitting results and realistic convergence under steady state conditions. The natural server number present in rainfall processes was found to be closer to three (3) for most rain regimes.
Therefore, it is concluded that the rainfall process in Durban is naturally derived from a M/Ek/3/∞/ FCFS queue discipline. Further results also show that the ‘jump’ probability from one spike to another within a rainfall time series can be represented by a ` × `) matrix with different steady state vectors for each regime. The service times of rainfall spike within the queue are also found to be positively correlated and dependent on the maximum rainfall rates attained by the spike. This shows that spikes in Durban with high rainfall rates tend to spend more time in any typical rain event suggesting higher outage periods.
7.2.3 Chapter Five - Comparison of Rainfall Queue Characteristics in Tropical Africa
The semi-Markovian queue description is extended to Butare, an equatorial location in this chapter. Rainfall queues in subtropical and equatorial Africa tend to exhibit similar queue characteristics when examined over annual cycles. However, this is not the case when both locations are compared on the basis of rainfall regimes. Compared to Durban, Butare is found to have increasing service times (with increasing k) as the maximum rain rate of spike increases.
However, there is no distinct trend observed for the inter-arrival times in relation to any spike’s maximum rain rate. Clearly, the queue dynamics of spikes generated at shower and thunderstorm events at Butare are obviously different from observations in Durban. These results have clear implications for rain attenuation analysis as this could lead to significant differences in outage frequencies due to rainfall. The geography and climatic characteristics are factors that might influence this. The jump probabilities at both locations are similar for different regimes and even for the overall data at steady state conditions. This suggests that dynamics of Markov jump probability at both locations, have similar meteorological characteristics, as present at both locations. Queue generated spikes can be applied in the simulation of rainfall conditions, provided the queue parameters are known. From this, it is conclusive to know that rainfall event durations tend to be different given an equivalent random number of spikes generated as a queue at any regime. Thus, it is possible to simulate different rainfall conditions at both locations and apply this to radio propagation studies in the future.
7.2.4 Chapter Six - Queueing Theory Application in Rain Cell Estimation
An extension of the queueing theory approach is employed in the development of standard probability functions representing rain spikes. The spike instances of the semi-Markovian queue
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
141
are investigated by considering them as a time-varying PDF entity with distribution parameters strongly dependent on the maximum rain rate attained by the spike. The Erlang-k distribution was found useful in the description of rainfall PDFs for different rainfall regimes in subtropical and equatorial Africa. It was observed that individual spikes generated during rain events are unique and have varying mean service times over different regimes. An approximate rain rate function, obtained from the underlying characteristics of spikes, is found to exist as a scaled version of the PDF. The rain rate function is representative of circular rain cells, under queue- generated conditions. These queued cells are assumed to move over radio links, propelled by an advection velocity, required to translate its time series to distance. The results of RCDs for different rain regimes in Durban and Butare show spatial variation of cell areas at the two sites.
Butare, being closer to the equator, is expected to experience more convective rainfalls resulting in smaller rain cells. This is confirmed in the results obtained under thunderstorm regimes for scenarios where rainfall rates are greater than 40 mm/h. While computations of length factors from single units of circular rain cells may be valid for terrestrial links with path lengths less than 10 km. Further computations beyond 10 km, show rapid decline of length factors as a result of absent multiple rain cells. Finally, it is seen that the the path attenuation due to rain at 19.5 GHz at Butare tend to give higher figures for rain rates < 30 mm/h compared to Durban.
The DSD formation process and geographical factors as mentioned in Chapter three is a major influence on the predicted rain attenuation at Butare.