Declaration 2 Publication
4.6 Results and Discussions
4.6.3 Performance of the Proposed Queue Disciplines
The performance of a queue system is governed by the steady state behaviour of its traffic which includes the assessment of queue characteristics and variations. The steady state
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
75
conditions are derived from the preliminary solution of global balance rate equations for a typical queue system. From literature, there are four major performance indices required to be investigated for any queue discipline. They are provided in literature and given as [Kleinrock, 1975; Bolch et al., 1998; Hillier and Lieberman, 2001]:
(i) The steady state number of instances in the queue designated as Lq
(ii) The steady state waiting time in the queue designated as Wq
(iii) The steady state number of instances in the system designated as L (iv) The steady state waiting time in the system designated as W
These indices are different as the queue disciplines changes and multiple servers exist in the system [Kleinrock, 1975]. Therefore, the performance metrics of any queue discipline changes with increment in number of servers and the changes in queue parameters. Because of this, the set of descriptors for computing these metrics are different for both M/M/s/∞/FCFS and M/Ek/s/∞/FCFS queue disciplines. In this investigation, the behaviour of both disciplines will be examined. It follows that for a typical M/M/s/∞/FCFS discipline, the set of performance descriptors are given by [Hillier and Lieberman, 2001]:
ß = \&j1Þ
R! 1 − Þ)# 4.24A)
where, á = ¸3&ja
`! + &j1 R! 1 − Þ)
1Y ad
¼
Y
4.24p)
âß = ß
& [(`E£BR] 4.25) = ß+ &
j 4.26)
â =
& 4.27)
And the descriptors of the M/Ek/s/∞/FCFS discipline are given as:
ß = 1 + 2 &#
RjRj − &) 4.28)
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
76 âß = ß
& [(`E£BR] 4.29)
= &â 4.30) â = âß+ 1
Rj [(`E£BR] 4.31)
where Lq is the number of spikes in the queue, Wq is the waiting time of a spike in the queue in minutes, W is the overall waiting time of a spike in the queue system and L is the overall number of spikes in the system.
By applying equations (4.24a) - (4.31), the performance indices of the system (Lq, Wq, W and L) are computed via simulation using MATLAB® for servers between 3 and 20. The results generated show that the M/Ek/3/∞/FCFS queue discipline gives a better and convergent solution corresponding to increment in server number. Table 4-4 shows a summary of the performance metrics at s = 3 for the compared queue systems for different rain regimes. Observations from this table indicate that the computed metrics (in bolded) are generally lower for the semi- Markovian queue discipline than seen for the Markovian discipline. This is mainly due to the effects of the Erlang-k stages which are totally absent in the Markovian model. For instance, the values for the number of spikes and their corresponding waiting time in queue (Lq and Wq) in any rain regime are higher for a typical M/M/s. Since the default number of natural servers in the rain process is about 3, spike numbers (L) of 2.6, 1.68, 2.88 and 1.34 are obtained in M/Ek/s for drizzle, widespread, shower and thunderstorm regimes respectively. This is compared to 4.78, 3.26, 5.13 3.51 respectively for M/M/s. Similarly, for the same number of servers, the system waiting times (W) of 13.57, 11, 19.44 and 14 minutes are obtained respectively in M/Ek/s.
Again, the respective figures of 24.87, 21.27, 34.53 and 24.65 minutes are obtained in M/M/s.
This reveals that lower values of performance indices are indeed obtained for M/M/s disciplines across all rain regimes. To further understand their performances, the disciplines are examined as the servers in their systems are increased.
As seen from the compared results in Figures 4-9 and 4-10, there is an obvious difference in the performance ‘curve’ of the M/M/s discipline. In addition to the earlier observation, the step-wise zero-convergence of the Markovian discipline is rather slow for the L and W indices. It is also observed that constant values are attained for these performance pair when a minimum of 10 servers are active in the system. Beyond this, the addition of more servers to the system has zero effect on the convergence rate; this is atypical of proper queue system. The performance of the M/Ek/s queue discipline however shows that the zero-convergence of L and W is continuous as
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
77
Table 4-4: System performance of M/Ek/3/∞/FCFS and M/M/3//∞FCFS for different rainfall regimes in Durban
REGIME DISCIPLINE
SERVICE TIME
INTERARRIVAL TIME
PERFORMANCE METRICS
j k & Lq Wq L W
DRIZZLE
M/Ek/3 0.0809 5 0.1921 9.45 1.82 13.58 2.61
M/M/3 0.0809 X 0.1921 12.52 2.41 24.88 4.78
WIDESPREAD
M/Ek/3 0.0729 5 0.1533 6.43 0.99 11.00 1.69
M/M/3 0.0729 X 0.1533 7.55 1.16 21.27 3.26
SHOWER
M/Ek/3 0.0615 4 0.1486 14.02 2.08 19.44 2.89
M/M/3 0.0615 X 0.1486 18.27 2.72 34.53 5.13
T/STORM
M/Ek/3 0.0489 3 0.0922 7.16 0.71 14.51 1.34
M/M/3 0.0489 X 0.0922 7.19 0.66 27.65 2.55
X means that this parameter has no equivalent value for exponential distribution
(a)
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
3 5 7 9 11 13 15 17 19
Steady state waiting time in the system (minutes)
Number of servers
Drizzle Widespread Shower Thunderstorm
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
78 (b)
Figure 4-9: The effect of increasing the number of servers on the overall system waiting time (a) M/M/s/FCFS/∞ (b) M/Ek/s/FCFS/∞
(a)
1 10
3 5 7 9 11 13 15 17 19
Steady State Waiting Time in the System (Minutes)
Number of Servers
Drizzle (k = 5) Widespread (k = 5) Shower (k = 4) Thunderstorm (k = 3)
10 15 20 25 30 35
3 5 7 9 11 13 15 17 19
Steady state number of spikes in the system
Number of servers
Drizzle Widespread Shower Thunderstorm
Queueing Theory Approach to Rain Fade Analysis at Microwave and Millimeter Bands in Tropical Africa
79
(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.