# KJA Consultants Inc.

en français## Elevator and Escalator Consulting Engineers

# Probability

In the period from 1925 to 1932 there was a major increase in high rise office building construction in the centre cores of North American cities. The elevator engineers of that era had to establish a basis for the determination of the elevator system. It was a natural application of the standard theories of probability. The basic problem can be stated as follows: given that there are P passengers embarking on an elevator at the main floor and there are N typical floors, what is the probable number of stops that the elevator will make? This is, in effect, a variation of simple high-school problems in probability. It is equivalent to saying: "If we have N boxes and P apples, and we randomly drop the apples in the boxes, how many boxes will have apples in them?".

The resulting number of probable stops S, given P passengers and N stops above the main floor is given by:

Once the number of probable stops S is calculated, it is possible to construct an artificial building consisting of the main floor plus S floors. The distance between each of the S floors will be equal to the travel divided by S. Using this artificial building, the time to travel from one stop to the next can be calculated based on the elevator speed and acceleration chosen. Time is added for the return express trip top to bottom. Door operation time, interlock time, passenger transfer time and dispatching time are calculated and added to the total time. All of this gives the round trip time (RTT). The designer can then derive the number, capacity and speed of the elevators to provide a desired interval (the RTT divided by the number of elevators) and five minute handling capacity (P*300 divided by the interval).

This technique was originated by Westinghouse in the late 1920's and first used by them for design purposes in the early 1930's. Thereafter, it became the standard method in the industry.

There has been no significant change to that approach over the years. Some people prefer minor variations in the assessment of dispatching time penalty for the group (which was taken by Westinghouse originally at 10%), others make minor distinctions in the calculations of the average jump and so on. The arithmetical approach was satisfactory in the early days of simplified dispatching systems. As well, the morning up peak was invariably the design point up until the 1960's since work habits were predicated upon fixed and universal starting times. Moreover, many people took their lunches at their desks with the result that the noon traffic was not significant. Apart from these factors, since all of the design was done directly or indirectly by the elevator companies, there was a natural tendency to design a generous elevator system. Putting all of this together, the mechanical calculations satisfied everyone for a number of years and, indeed, there was no other option until computers came into widespread use. KJA started using computer simulations for design and analysis in 1970 with the MACH1 program. This program has developed over the ensuing years to become the present KJA SIM program. SIM and its predecessor MACH1 to MACH8 programs have been used in several thousand projects.