Fractal analysis of rainfall event duration for microwave and millimetre networks: rain queueing theory approach

Abstract

Rain fade in radio networks is generated from random fluctuations of rainfall rates, within rain events of spatiotemporal dimensions. These events can be represented as a catenation of single rain spikes occurring as a possible three-stage processbirth, overlap and death. Using the queueing theory approach, the birth-death characteristics of single spikes are investigated as inter-arrival and service time distributions. A total of 548 spike samples from rainfall events in Durban (29°52'S, 30°58'E), South Africa are examined based on distrometer measurements. Rainfall regime analysis of drizzle, widespread, shower and thunderstorm bounds is applied to determine the queue pattern. It is found that the queue patterns in Durban exhibit an Erlang-k distribution (E k ) for both the service and overlap times, while exponential distribution (M) is suitable for inter-arrival time. The mean error statistics for the regimes give root-mean-square errors of 0.64, 1.3 and 2.02% for the service, inter-arrival and overlap distribution, respectively, with acceptable Chi-Squared (χ 2 ) statistics. The M/E k /s/∞ steadystate analysis is later undertaken to investigate the performance of the proposed queue system. Based on the overall data, a power-law relationship is found to exist between the service time and peak rain rate per spike.

Approximate queue scheduling for rainfall synthesis over radio links in subtropical regions

“…Because of these unique extremities in the nature of typical queues, there exist certain parameters which govern the incidence of queue characteristics. In rainfall queues, these parameters in a First-Come, FirstServed (FCFS) process over Equatorial and subtropical locations, are described by Alonge and Afullo [12], [13] as service time, inter-arrival time and overlap time. The service time distribution, , of these arriving spikes is described as having several number of stages, k, thereby following an Erlangian process.…”

“…From studies of Alonge and Afullo [12], [13] on this subject, steady state analysis of rainfall queues tend to therefore follow a semi-Markovian FCFS M/E k /s queue discipline over subtropical and tropical locations. The equivalent number of servers, s, which is the minimum ratio of service time to inter-arrival time, converges at 3.…”

“…Earlier researchers [9]- [11] have applied the Markov Chain (MC) approach, however, these methods have locationspecific transition matrices; making this approach seemingly incompatible elsewhere. This current work of rainfall synthesis over radio links is based on Queueing Theory Technique (QTT) proposed by Alonge and Afullo in [12], [13]. This technique applies Birth-Death (BD) process and queueing concepts from rainfall rate time series data collected over Durban between January 2009 and December 2010.…”

Rainfall time series synthesis from queue scheduling of rain event fractals over radio links

“…The interarrival time parameter for any rain queue follows an exponentially distributed process, i.e., t a~e xp(λ), given by Alonge and Afullo [2015]…”

“…This is largely the consequence of the location-specific characteristics of rainfall across the world. Another approach that could be applied to grasp the underlying characteristics of time-varying rainfall is the consideration of this process as a queue-governed system [Alonge and Afullo, 2014;Alonge and Afullo, 2015]. Queueing systems are generally representative of processes where activities over a period of time possess fractal identities with self-similar behavior(s), in the counting domain [Bolch et al, 1998;Kleinrock, 1975;Hillier and Lieberman, 2001].…”