A number of assumptions in the derivations of the round trip time expression have been discussed in previous sections. These assumptions were concerned with the nonlinear effects of the building and lift system and also the static population distributions. No account has been taken of the way in which passengers arrive in a building or the randomness of their destinations. In practice passengers do not conveniently arrive at a building such that 80% of the rated car capacity of a lift is available to step into a lift and be transported to their destinations. Also passengers do not request the same number of destinations on each trip. The effect of random arrival rates and destinations is to cause cars to become unequally spaced and, in extreme circumstances, to bunch. If large numbers of passengers arrive, queues can also develop. Throughout this chapter values for H and S have been described as average values. It is using these values that the average RTT is calculated. Obviously, if the RTT has wide deviations from the average, a poor quality can result. How then does the RTT vary?
7.7.2 Effect of Randomness of Passenger Destinations
Returning to statistical terminology,1 in Section 5.3 it has been indicated that the probability of a variable is defined by a numeric value between zero and unity. Probability then is simply a measure of the likelihood of some occurrence. For example, the probability of stopping at a floor may be 0.1. However, the actual number of stops is called the expected number of stops. The expectation is the mean or average value of some random variable, often termed the expectance.
It is often of interest to know the spread of possible outcomes about the mean. This variation is essentially measured by the statistical characteristic termed variance. If the square root of a value of variance is taken, the value then obtained is termed standard deviation. If large values of variance occur with respect to values obtained for expectance of some variable then that variable has wide deviations of value from its mean.
A mathematical study has been made by Gaver and Powell (1971). They considered an uppeak situation, with no interfloor traffic, and assumed that all lifts depart from the main terminal with a constant number of passengers. No analysis was made with car load variations. The expression they deduced allows for lifts to be expressed from the main terminal to a certain floor and then to service that floor and the floors above. They also made some assumptions to try to overcome the problem that the rated speed may not be reached for a flight of one floor. Both cases of equal and unequal floor population are considered. The expression is a linear function of the number of stops and the reversal floor, so the evaluation of the variance of the round trip time was reduced to the
1 Statistical analysis and probability theory are highly mathematical fields and are best dealt with in specialised texts (Papoulis, 1965)
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calculation of the variance of the two variables (number of stops and reversal floor). The final expression is complex and is best illustrated by examples.
Consider Table 7.1 for a group of lifts serving floors 15 to 26 (12 floor zone) in a building, the lift rated speed being 4.0 m/s.
Table 7.1 Statistical data
Passengers in lift 15 17 19 21
Expectance of RTT(s) 149 157 164 170
Variance of RTT(s2) 68 66 62 57
Standard deviation(s) 8.25 8.15 7.89 7.55
The variance decreases when the number of passengers increases, since there is less randomness in the number of stops. However, two points must be made: the number of floors served is modest and
smaller than the number of passengers in the lift; and the group serves an express zone, which means that a significant part of the round trip time, corresponding to the express flights in both directions, is not subjected to variance of round trip time. If a group of lifts with the same characteristics but serving the 12 floors next to the main terminal is considered, the variance will be the same but the round trip time will be decreased by the travelling time of two flights of 15 floors. This, for an average interfloor distance and a rated speed of 4.0 m/s, is about 25 s. For such a group of lifts next to the main terminal the variance becomes more significant.
The figures above assume equal floor populations; for unequal floor populations the variance increases. An example given by Gaver and Powell with slightly uneven populations gives an RTT expectance of 148 s and a variance of 74 s2 for 15 passengers in a lift (compare with Table 7.1).
Thus significant variance of RTT does exist for equal and slightly unequal floor populations. Gaver and Powell conclude “if the distribution varies considerably from uniform we must formulate a new
representation of round trip time”. How these variations affect a lift system is not quantified. In the long term little effect on the carrying capacity will be noticed, but in the short term queues can build-up reducing the quality of service. It would appear, therefore, that as far as conventional calculations are concerned randomness of destination need not be considered.
7.7.3 Effect of Variations in Arrival Rate
Passengers do not arrive uniformly in time. It is generally accepted that the arrival of people at a
building obeys a Poisson process, ie: the probability of n calls being registered in the time interval T for an average rate of arrivals λ is:
(7.14)
The assumption of passenger arrival by the Poisson process is not completely proven, and its validation would require extensive data logging on a wide range of lift systems (see Section 5.6.1).
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Petigny (1972) and Tregenza (1972), following different methods of analysis but both considering a Poisson arrival distribution, have related the system behaviour with the arrival of passengers. Petigny’s study is purely mathematical and considers a single lift during an uppeak situation. By using probability theory he deduces Equations (7.6) and (7.13) for S and H for the case of defined floor populations. Also by setting up a complex mathematical study involving Markov chain theory, he deduces an expression for the average number of passengers per trip. Although an interesting study, Petigny’s work relates to a single lift only.
Tregenza uses a more practical approach and produces an expression similar to Equation (4.11) for RTT, which he writes as:
(7.15) where t1=2tv and t3=2tp and E symbolises expectance.
Rewriting Equation (7.15) in the style of Equation (4.11):
(7.16) [The subscript p indicates Poisson pdf]
Instead of taking PT as the car capacity or a fraction of it, as in conventional design, he relates the parameters HT, ST and PT as universal functions of an arrival rate parameter pro. He defines pro as the probability of no calls being registered from the main terminal floor to any floor above during the period of one interval. For equal floor populations he produces four equations. These have already been
derived as Equations (5.14), (5.15), (5.16) and (5.17).
Section 5.6.3 showed that if the first two terms of the expansion of the equations for Hp and Sp were taken, then H, using a rectangular probability function and Equation (5.12), was identical to Hp and likewise for S using Equation (5.5) and Sp.
Figures 7.3 and 7.4 show the equations for H and S plotted for both the rectangular and Poisson pdfs. These graphs confirm that if the rectangular pdf is used to derive the H and S equations, their values will always be larger than if the Poisson pdf is used. Thus the resulting value for the RTT will always be larger and a design based on it will be slightly conservative. As the equations using the rectangular pdf are simpler to calculate, they are generally used in design calculations.
The above analysis assumes equal floor populations/demand. Using statistical techniques, expressions can be obtained (Alexandris, 1976) for unequal floor populations. They are:
(7.15) (7.16) (The mathematical symbol means “multiply all terms like…”.)
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Figure 7.3 Mean highest reversal floor H for different numbers of floors and car capacities solid line: conventional calculations, rectangular probability distribution
line with x: calculation using Poisson probability distribution
Figure 7.4 Mean number of stops S for different numbers of floors and car capacities solid line: conventional calculations, rectangular probability distribution
line with x: calculation using Poisson probability distribution
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7.8 COMPARISON OF NONLINEARITIES AND THEIR TOTAL EFFECT ON THE VALUE OF THE