Declaration 2 Publication
4.3 The Mathematics of Queueing Theory of Rainfall Process
4.3.4 The Physical Manifestation of Spike Generation in Rain Traffic
The rainfall formation cycle involves the generation of rain clouds from the residues of the natural hydrological processes [Rodriguez et al., 2012]. Over time within a rain event, this traffic process naturally results in a spontaneous and random variation of rainfall rates. Thus, the process itself is an infinite process which exists as a reaction of nature and environment to natural climatic-related variables.
INPUT
SOURCE QUEUE SERVER Served
instances
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This rain cloud generation process and consequent precipitation, when observed from a stationary point of view (or a reference time) can be seen as a natural and self-regulated traffic as seen in Fig. 4-3. Also, the arrival of rain clouds can be seen as FCFS traffic process with different rainfall ‘services’ being offered. These offered services being the remote cause of the variation in rain rates, can be seen as a train of rain spikes parallel to cloud motion. The cloud motion is sustained by a certain quantity called the advection velocity, which varies with different types of rainfall structure and cell sizes [Pawlina, 2002; Begum et al., 2006]. It should be noted that rainfall clouds are dissipative as they travel along the direction of the prevailing advection velocity. This indicates that their area of influence and density diminishes as they travel due to the production of rain droplets; this is essentially a process of mobile energy transfer. It could therefore be assumed that the peak of a spike roughly coincides with the cloud portion of a rain cell at its highest density. On rain cell (or cloud) mobility, several authors have proposed different values of advection velocities for stratiform and convective rains [Pawlina, 2002; Begum et al., 2006]. For example, Pawlina [2002] proposed that advection velocity values be lower for stratiform rains and, higher values for convective rains.
Usually, the appearance of an individual spike determines the temporal service being offered by the passing cloud. The period of service (or service time) in this case, roughly multiplied by the number of spike appearances determines the length of the rainfall duration. From Figs. 4-3 and 4-4, we can observe three major concepts describing the appearance of a spike (or the arrival of a cloud) - they are the inter-arrival, service and overlap times. The inter-arrival time, ta, is defined as the time difference between the arrivals of two consecutive rain clouds/spikes at the reference point. The service time, ts, may be defined as the actual duration of the spike. A third
Figure 4-3: The concept of Rainfall Traffic Generation as a Queueing Model
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Figure 4-4: Identification of some queue parameters from rain spike profiles for a rainfall event on the 24th of January 2009 between 15:15 and 15:56 hrs.
parameter, to, known as the overlap time represents the intercept time between a ‘dying’ spike and arriving spike. Overlapping spikes can be seen as a major factor accommodating the regeneration of detectable rainfall rates between two clouds, with one arriving and the other departing. Also, regular overlaps in the successive arrival of spikes are evidences of queues in the system.
The approximate parameter estimates serving as probability density predictors to a Markovian queue distribution can be used to infer the behaviour(s) of such an array. For this study, the set of expressions for both the mean service time, £Á1, mean inter-arrival time, £Á%, and mean overlap time, £Á\, for successive instances of spikes in rainfall events of N sampled population may be given as [Hillier and Lieberman, 2001]:
£Á = 11 V13 ∆£1, cÂ
d
=1
jà [(`E£BR] ∀£1∈ ℝ 4.10)
£Á = 1% V%3 £%, cÄ d
=1
&Ã [(`E£BR] ∀£%∈ ℝ 4.11)
£Á = 1\ V\3 £\,
cÄ d
=1
f [(`E£BR] ∀£\∈ ℝ 4.12)
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where Ns, Na and No are the maximum number of samples for the service time, inter-arrival time and overlap time respectively. Other notations representing the mean (or average) service, mean inter-arrival and mean overlap times are given as their reciprocals. They are: the mean service rate, jà , mean arrival rate, &Ã, and mean overlap rate, f, respectively. Again, by inspection, if the service time has a maximum number of N sampled population, then the number of arrival samples will be lesser than N.
Broadly speaking, the overall number of arrived clouds represents the total sampled population of the rainfall queueing system under investigation. For easy understanding, this elaborate process of cloud/spike arrivals, as well as departures, is assumed to bear a similitude to the Birth-Death (BD) process. Firstly, for discrete systems, the inter-arrival process signifies the commencement (or birth) of a spike/cloud within a rainfall event. Then, the service time determines the generation, decomposition and the consequent death of the arriving spikes/clouds. Hence, rainfall spike queueing process can be regarded as a special case of the Discrete Markov Chain Process (DMCP). In this study, we will neglect the complicated derivations related to generic BD processes as they can be seen in related literature by Kleinrock [1975], Bolch et. al.[1998] and Hillier and Lieberman [2001]. We shall mainly apply the knowledge from the literature to acquire an understanding of rainfall queueing problems and its consequent application.