Declaration 2 Publication
4.6 Results and Discussions
4.6.4 Jump State Probabilities of Rainfall Spikes
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(b)
Figure 4-10: The effect of increasing the number of servers on the expected number of spikes for different rain regime (a) M/M/s/FCFS/∞ (b) M/Ek/s/FCFS/∞
more servers are added to the system. The effect of the stages for different rain regime also ensures that lower values of these performance indices are attained irrespective of the server numbers. An extensive summary of the simulated performance results for both queue disciplines are presented in the tables in Appendix F. As seen from the investigations and of the performance indices, it is conclusive that the M/Ek/s discipline is the most appropriate to describe the spike traffic in Durban. This is because it offers the advantages of lower system waiting times, shorter queue times and better zero-convergence of performance metrics. In addition, it can be seen from the L and W metrics for all regimes that the M/Ek/s queue applied in this study is twice as efficient as M/M/s queue.
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identification of generated spike by their peak rain rates can be best distinguished by a known probability process. If the prevailing rain regime classification is applied (i.e. drizzle, widespread, shower and thunderstorm) to group the peak rain rates, it would be paramount to know, for example, the probability of a drizzle spike being succeeded by a thunderstorm spike and so on. This poses a genuine question as thus: what is the probability that the regime state of a rain spike jumps (or changes) to another regime state on arrival of another spike in the queue? Based on this question, it is expedient to develop state transition matrices to estimate the spike jump from one regime state to another. Since the rainfall process itself is of Markovian scheduling, it is logical to depict the probability of jumping from one current transition state, Mo
to any state, Mn. This is described as:
_a|_\) = D ´ for (, ∈ ℝ ∀` 4.32)
where n is the number of possible states of system transitions as spikes are generated in the queue process. Invariably, (4.32) can be represented as a simple matrix as below:
_a|_\) = ãD … D a
Da⋮
…⋱ ⋮
Daa
ä 4.33)
By examining the processed rainfall queue data, it is observed that spikes from different rain regimes have a defined number of states for probability transition. Rain spikes generated in drizzle regime for instance have only one state because drizzle events are peak spike threshold of rain rates below 5 mm/h. For widespread events, generated spikes can only transit between two states, i.e. drizzle and widespread, with threshold below 10 mm/h. For shower events, generated spikes have a three-state transition probability of jumping randomly through drizzle, widespread and shower with threshold of 40 mm/h. Finally, thunderstorm storm events have four complete transition states where the spikes can transit randomly through drizzle, widespread, shower and thunderstorm.
By physically examining the processed data from rainfall measurements in Durban, it is possible to identify the transitions of a generated spike in one regime state to another. By following the state diagrams, it can be deduced that a two-state system for example has four possible or 22 transition states. A special case of steady state occurs in the system when the transition probability matrix, P, jumps continuously until it attains an infinite number of transitions [Bolch et al., 1998]. This condition leads to a steady state problem which must satisfy the condition that follows:
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å¶ = å for å ≠ 0 4.34)
where q is the system vector at steady state such that its elements represent constant values of row elements of P under this condition. The solution of q can be solved by applying the eigenvalue approach so that:
¶e− &ç)åe = 0 4.35)
where I is an n x n identity matrix dependent on the number of states.
Following this arrangement, qT becomes the eigenvector of P and can be easily resolved. The results that follow hereafter present the state transitions matrices of transiting spikes in Durban for the rainfall regimes of widespread, shower and thunderstorm.
4.6.4.1 Jump State Transition Matrices for Rain Regimes
There jumping state transition matrices obtained in this subsection are so classified according to their rainfall event bound. Therefore, it is expected that the number of states increase from drizzle to thunderstorm events. A drizzle event can only generate drizzle spikes and hence, will continually maintain one state which is drizzle. Widespread events can generate drizzle and widespread spikes, hence, only two states are possible. Following this trend, it is obvious that shower and thunderstorm events can only transit spikes in three and four states respectively.
Transition state diagrams are presented in Figures 4-11 to show the transition possibilities of between distinct spikes in similar events. Figures 4-11a, 4-11b and 4-11c show the state transition states available for widespread events, shower and thunderstorm events. Since the drizzle events are perpetually stuck in one state, the state diagram is excluded. The elements pertaining to each of these transitions are presented as transition matrices. For any widespread event, the state transition matrix is given as:
¶è é|1|%é = êD Dë
Dë Dëëì 4.36)
where PDD, PDW, PWD and PWW are the transition probabilities of drizzle (D) and widespread (W) spikes.
The state transition matrix available for shower events is thus given:
¶1¤\è| = ¨D
Dë
Dª
Dë
Dëë
Dªë
Dª
Dëª
Dªª© 4.37)
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(b)
(c)
Figure 4-11: The state transition diagrams for spikes generated in different rainfall regimes (a) Widespread (b) Shower (c) Thunderstorm .
where PDD, PDW, PDS, PWD, PWW, PWS, PSD, PSW, PSS are the transition probabilities of drizzle (D), widespread (W) and shower (S) spikes.
Finally, thunderstorm events have transitional matrix as follows:
D
T W S
W S
D
D W
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¶¤íaé|1\, = î
D Dë Dª De
Dë
Dª
De
Dëë
Dªë
Deë
Dëª
Dªª
Deª
Dëe
Dªe
Dee
ï 4.38)
where PDD, PDW, PDS, PDT, PWD, PWW, PWS, PWT, PSD, PSW, PSS, PST, PTD, PTW, PTS, PTT are the transition probabilities of drizzle (D), widespread (W), shower (S) and thunderstorm (T) spikes.
Table 4-5 presents the populated transition matrices for each of the rainfall events as given in (4.36), (4.37) and (4.38). The number of product operation or stages required to attain steady- state conditions, n, is also presented. The results from the steady state vector are hereby discussed in the context of probability, the vector itself being a stochastic vector. It is important to note that the sum of all the elements contained in this is approximately equal to one.
Therefore, it follows at steady state that in a widespread event, there is 73.7% chance of drizzle spikes compared to 26.3% chance of widespread spikes. In shower events at steady state, the probabilities of drizzle, widespread and shower spikes are 51.5%, 19.5% and 29% respectively.
Finally, for spikes under thunderstorm events, the occurrence probabilities of 42.7%, 9.9%, 21.4% and 25.8% are observed for drizzle, widespread, shower and thunderstorm spikes respectively. In all these regimes, drizzle spikes are all obviously dominant as seen from the discussion in comparison to other spike types.
Table 4-5: State Transition Matrix and Steady State Vectors for Spike Generation in Durban
REGIME TRANSITION MATRIX n STEADY STATE VECTOR
DRIZZLE [1] [1]
WIDESPREAD ê0.7528 0.24720.6935 0.3065ì 4 [0.7372 0.2628]
SHOWER ¨0.6839
0.3621 0.3165
0.1782 0.2069 0.2152
0.1379 0.4310 0.4684©
12 [0.5146 0.1947 0.2913]
T/STORM î
0.6071 0.0714 0.0714 0.2500 0.3750
0.3333 0.2308
0.2500 0.1111 0.0769
0.125 0.3333 0.3846
0.2500 0.2222 0.3077
ï 10 [0.4276 0.0989 0.2136 0.2587]
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