Numerical Values

The round trip time equation. Equation (4.11), is:

(4.11)

It will be noted that the equation comprises three parts. Each part comprises a time independent variable (H, S, P) and a time dependant variable (tv, ts, tp). It is these six parameters which must be evaluated in order to determine the round trip time equation. Each variable will now be taken in turn in the following sections and typical values indicated. Installed systems may have different values.

5.2 DETERMINATION OF P

To evaluate Equation (4.11) a value for the average number of passengers (P) carried during each trip needs to be determined. If passengers arrived efficiently then as each lift arrived there would be a number of passengers waiting (P) equal to the rated car capacity (CC) of the lift ready to board. Unfortunately life is not like that and passengers arrive in a random fashion.

To avoid passengers being left behind (a queue) to wait for the next lift, it is necessary to assume a lower than 100% utilisation factor for car occupancy. This assumption arises as statistical theory implies that, as the utilisation of a facility increases towards its maximum, the probability of immediate use of that facility reduces. Thus to achieve maximum utilisation of a facility it is necessary to have a queue of applicants waiting (like an airport). This is not considered satisfactory for a lift system. Therefore the design utilisation has to be lower to allow for statistical variations to be accommodated. How much lower than 100% should P be set below CC?

The probability of the immediate use of a facility is shown diagrammatically in Figure 5.1 with respect to system utilisation. As system utilisation increases, then the probability of a passenger being left behind increases, until at 100% utilisation there is

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a high probability of being left behind to queue. The shape of the curve has been shown to apply to such diverse facilities, access to a telephone line, availability of a lavatory, a free bank teller, etc.

Figure 5.1 System utilisation

Looking at the curve of Figure 5.1 again, it can be seen that above 70% utilisation the change in slope increases significantly and at 90% and above it is very rapidly increasing. The ratio of the slope at 90% compared to the slope at 50% is some 25:1. Usually the 80% point is considered to be the “knee” of the curve for most system utilisation judgements and this value is selected for lifts also. Therefore, for lift systems it is reasonable to consider:

(5.1)

Early explanations justified the 80% value by saying that passengers never loaded a car above 80% car capacity even when queues existed, thus showing remarkable restraint. Other theories to explain the 80% figure proposed are either: circulation difficulties (passengers at the back of the car always want to get out at the first stop); or operational problems (passengers obstructing door closing), both of which have the effect of increasing the round trip time. The 80% derating factor appears to have been arrived at by intuition and experience, rather than theory.

5.3 DETERMINATION OF S

During a round trip a lift car stops at a number of floors for passengers to alight. The round trip time is affected by the number of stops made. Each trip will be different, but what is the average number of stops that can be expected? The term “probable number of stops” has been avoided as statisticians understand1 a parameter labelled “probable”

1 It is not sensible to proceed further without defining some statistical terms. 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 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.

Statistical analysis and probability theory are highly mathematical fields and are best dealt with in specialised texts.

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will yield a number between zero and unity. Thus the more acceptable term which will be used is

expected number of stops. Basset Jones in 1923 published a method of calculating the expected number of stops for floors with equal populations. Consider a building with N floors above the main terminal. Assume that each floor is equally likely as a destination for passengers.

The probability that one passenger will leave the lift at any particular floor is 1/N.

The probability that one passenger will NOT leave the lift at any particular floor is the complement of this probability, viz:

(5.2)

Since each passenger is assumed to be independent of all others, the product law of probability gives the probability that NO passengers from a lift containing P passengers will leave the lift at any particular floor as:

(5.3) Note there are P terms. Hence the probability that a stop will be made at any particular floor is:

(5.4) The expected or average number of stops (S) for N floors will then be:

(5.5)

Values for S can be calculated each time using the formulae. As there are a finite set of rated capacities offered by lift manufacturers, eg: those indicated in BS EN81 and BS ISO4190, it is possible to tabulate these values as shown in Table 5.1.

5.4 DETERMINATION OF H

Some design procedures assume H to be N, or in tall buildings, N−1. Arbitrary rules are offered

(Strakosch, 19981). It is possible, however, to deduce an expression for H with respect to N and P using probability theory. It is not clear who first derived an expression for H or when, although Schroeder (1955) writing in German could have been the first. Using the same assumptions and definitions as in Section 5.3, assume a passenger is equally likely to travel to any floor.

The probability that one passenger will leave the lift at any given floor is 1/N and so the probability that one passenger will NOT leave the car at a given floor is:

(5.6) 1 Page 82.

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The probability that none of the P passengers will leave the car at a given floor is: (5.7)

The probability of the car travelling no higher than the ith floor is equal to the probability that no one leaves the lift at the Nth, (N–1)th, (N–2)th,…and ( i+1)th, floors is:

(5.8) Expanding and simplifying produces:

(5.9)

It is now possible to propose that the {probability that i is the highest floor attained} is equal to the {probability that a lift travels no higher than the ith floor} minus the {probability that the lift travels no higher than the (i−1)th floor}, viz:

(5.10) Then the average (or mean) highest floor H is:

(5.11)

Expanding and simplifying, the expected or average highest reversal floor (H) for N floors will then be: (5.12)

Values for H can be calculated each time using the formulae. As there are a finite set of rated capacities offered by lift manufacturers, eg: those indicated in BS EN81 and BS ISO4190, it is possible to tabulate these values as in Table 5.1. This table shows that H is indeed approximately equal to N, where a large capacity lift is installed, but grossly in error, where a small capacity lift is installed, especially where many floors are served (see Section 7.3.3 for discussion).

5.5 EXAMPLE 5.1

Calculate the round trip time using the data below and Equation (4.11). The data above gives:

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Table 5.1 Values of H and S for EN81 rated capacities *

Floors 6 (4.8) 8 (6.4) 10(8.0) 13(10.4) 16(12.8) 21(16.8) 26(20.8) 33 (26.4) N H S H S H S H S H S H S H S H S 5 4.6 3.3 4.7 3.8 4.8 4.2 4.9 4.5 4.9 4.7 5.0 4.9 5.0 5.0 5.0 5.0 6 5.4 3.5 5.6 4.1 5.7 4.6 5.8 5.1 5.9 5.4 6.0 5.7 6.0 5.9 6.0 6.0 7 6.2 3.7 6.5 4.4 6.6 5.0 6.8 5.6 6.8 6.0 6.9 6.5 7.0 6.7 7.0 6.9 8 7.1 3.8 7.4 4.6 7.5 5.3 7.7 6.0 7.8 6.6 7.9 7.2 7.9 7.5 8.0 7.8 9 7.9 3.9 8.2 4.8 8.4 5.5 8.6 6.4 8.7 7.0 8.8 7.8 8.9 8.2 9.0 8.6 10 8.7 4.0 9.1 4.9 9.3 5.7 9.5 6.7 9.7 7.4 9.8 8.3 9.9 8.9 9.9 9.4 11 9.6 4.0 10.0 5.0 10.2 5.9 10.5 6.9 10.6 7.8 10.8 8.8 10.8 9.5 10.9 10.1 12 10.4 4.1 10.8 5.1 11.1 6.0 11.4 7.1 11.5 8.1 11.7 9.2 11.8 10.0 11.9 10.8 13 11.2 4.1 11.7 5.2 12.0 6.1 12.3 7.3 12.5 8.3 12.7 9.6 12.8 10.5 12.9 11.4 14 12.1 4.2 12.6 5.3 12.9 6.3 13.2 7.5 13.4 8.6 13.6 10.0 13.7 11.0 13.8 12.0 15 12.9 4.2 13.4 5.4 13.8 6.4 14.1 7.7 14.3 8.8 14.6 10.3 14.7 11.4 14.8 12.6 16 13.7 4.3 14.3 5.4 14.7 6.5 15.0 7.8 15.3 9.0 15.5 10.6 15.7 11.8 15.8 13.1 17 14.5 4.3 15.3 5.5 15.6 6.5 16.0 8.0 16.2 9.2 16.5 10.9 16.6 12.2 16.8 13.6 18 15.4 4.3 16.0 5.5 16.6 6.6 16.9 8.1 17.1 9.3 17.4 11.1 17.6 12.5 17.7 14.0 19 16.2 4.3 16.9 5.6 17.4 6.7 17.8 8.2 18.1 9.5 18.4 11.3 18.5 12.8 18.7 14.4 20 17.0 4.4 17.8 5.6 18.2 6.7 18.7 8.3 19.0 9.6 19.3 11.6 19.5 13.1 19.7 14.8 21 17.9 4.4 18.6 5.6 19.1 6.8 19.6 8.4 19.9 9.8 20.3 11.7 20.5 13.4 20.6 15.2 22 18.7 4.4 19.5 5.7 20.0 6.8 20.5 8.4 20.9 9.9 21.2 11.9 21.4 13.6 21.6 15.6 23 19.5 4.4 20.4 5.7 20.9 6.9 21.4 8.5 21.8 10.0 22.1 12.1 22.4 13.9 22.6 15.9 24 20.3 4.4 21.2 5.7 21.8 6.9 22.4 8.6 22.7 10.1 23.1 12.3 23.3 14.1 23.5 16.2

80% capacity shown in parentheses. N is the number of floors above the main terminal.

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To determine S, use Equation (5.5):

To determine H, use Equation (5.12):

Using Equation (4.11):

Compare the calculated values of S and H to those given in Table 5.1. 5.6 EFFECT OF PASSENGER ARRIVAL PROCESS

5.6.1 General

The derivation of the formulae for H and S assumed that passengers arrived at a constant inter-arrival interval (according to the particular level of arrivals existing) and that the lift arrival at a constant interval to take the intending passengers to their destinations. This effect is illustrated in Figure 5.2(a) where there is a lift system with a rated car capacity of 6 persons and an interval of 20 s; and 6 passengers arrive every 20 s.

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 in batches equal to 80% of the rated car capacity nor do they register the same number of destinations during each trip. The effect of this randomness is to cause the lifts to take different times to carry out a round trip and they become unevenly spaced. This effect is called bunching. (Buses in the street seem to do this. Although they may be on a 20 minute timetabled frequency they only appear every hour in threes!). This effect is illustrated in Figure 5.2(b) for the same conditions as Figure 5.2(a). Note the overall passenger average waiting times have increased and queues develop.

Note that these “snapshots” of lift behaviour are a very simple minded representation of an extremely complex process and does not bear close scrutiny. This is why it is best to set up statistical models to represent the process and then to draw general and averaged conclusions from them.

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Figure 5.2 A simple representation of passenger arrival and lift car departures (a) Constant passenger arrivals: constant lift departures.

Overall average passenger waiting time 8.6 s.

(b) Random passenger arrivals: irregular lift departures. Overall average passenger waiting time 12.6 s.

So if passengers do not obey the constant (sometimes called rectangular) arrival process used to derive Equations (5.5) and (5.12) what process do they obey?

It is generally accepted that passengers arrive into a lift system according to the Poisson probability process. This probability distribution function has been used to describe other phenomena such as: the generation of radioactive particles and telephone calls; failures of electronic equipment; and the

demands on digital computer central servers. Although this arrival process is not proven with respect to lift systems, work by Alexandris (1977) did go some way to confirm it. Using observers, Alexandris surveyed three buildings with widely differing lift and other physical characteristics. He came to certain conclusions:

(a) Comparison of the observed and theoretical values calculated for the mean and variance showed a Poisson fit to be reasonable.

(b) The chi-squared goodness-of-fit tests gave evidence that a Poisson arrival rate assumption at least cannot be rejected.

(c) Although there may be other theoretical distributions which might better accommodate the data, the Poisson distribution must be considered as a good approximation to the actual empirical distribution. 5.6.2 Formulae for S and H using the Poisson probability distribution function

Accepting the Poisson probability distribution function (pdf) as the best representation of the passenger arrival process, what effect does this have on the evaluation of the round trip time equation? Assume that the probability of n calls being registered in the time interval T for an average rate of arrivals λ (in calls per second) is:

(5.13)

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Tregenza (1972) used this relationship to derive formulae for H and S (shown below with subscript, p). For a building with equal floor populations, he defined pr0 as the probability of no calls being registered from the main terminal to any floor above during the period of one interval (T):

(5.14)

Then by the same arguments used in developing Equations (5.5) and (5.12):

(5.15) (5.16) The arrivals and departures are equal and can be defined as:

(5.17)