Measurement Of Mean Waiting Time

Abstract

The mean value of waiting time, Wm, for an elevator system on a certain floor can be obtained by detecting intervals of the elevator arrival T.sub.1, T.sub.2, T.sub.3, . . . , T.sub.i at the floor and carrying out the operation: Wm = (1/2) (T.sub.1.sup.2 + T.sub.2.sup.2 + T.sub.3.sup.2 + . . . T.sub.i.sup.2)/(T.sub.1 + T.sub.2 + T.sub.3 + . . . T.sub.i) An analog integrator AIN begins integration of a constant voltage as each elevator signals his arrival and resets at the next elevator arrival so as to provide an output of a sawtooth waveform. A filter F flattens this sawtooth output to supply a quantity proportional to the mean waiting time. In a digital embodiment, counters are employed to perform the summing operations.

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to a method and a system for measuring the mean waiting time.

2. Description of the Prior Art

In transportation systems or the like (which will hereinafter be generally referred to as subjects), it is natural that the number of subjects is usually smaller than the number of users. Therefore, a user should wait until a subject becomes available. And, this time interval is generally called the waiting time. Although this invention is applicable to various systems, the case of an elevator will be taken as an example to help understanding.

In an elevator system, a plurality of elevators may be installed side by side and controlled systematically related with one another for improving the operation efficiency. The average or mean waiting time is an index for expressing the degree of operation (e.g. travel) service of the elevator system. It is the average or mean value of time lengths (waiting time) during which users wait on a floor from their arrival at this floor until they get on an elevator. Thus, the waiting time differs from user to user. However, it can be said that the shorter the average waiting time is, the better are the conditions of transportation.

The mean waiting time Wm of each passenger is represented by

where W1, W2 . . . Wi . . . WN indicate the waiting times respectively taken by N passengers before they get on the respective elevators at a floor after their arrivals at the floor.

Conventional methods for measuring the mean waiting time comprise such a method in which researchers on respective floors measure the waiting time of users with a stop watch, and such a method in which the lapse of time from the registration of a hall call from respective floors until an elevator stops at the floor is automatically measured, but they cannot be considered satisfactory. Namely, when researchers are distributed on respective floors, researchers whose number is equal to at least the number of service floors are necessary, provided that one researcher measures the waiting time on one floor. When the numbers of elevators and users are large, the waiting times of respective users can hardly be measured even if the number of researchers is increased.

Therefore, there is usually adopted a method in which a researcher measures the length of time from the registration of a hall call at the floor under measurement till the arrival of an elevator. According to this method, however, only the waiting time of users who first arrived at the floor can be measured and hence the reliability of the measurement is naturally low.

Even when the measurement of time intervals from the registration of a hall call till the erasing of the registration may be automated for cutting down manpower, the above-mentioned drawback can not be eliminated by such a method and the real mean of the waiting time can never be measured when users successively arrive at the floor.

Further, according to these methods, it is troublesome to obtain the mean waiting time from the measurement data, and rather impossible for practical purposes to determine an exact value of the mean waiting time

by these methods.

This invention intends to provide a method and apparatus for measuring the mean waiting time with high reliability and a simple structure.

As mentioned above, it is practically impossible to measure the waiting time of each passenger individually and, accordingly, alternatively, the present invention resolves this problem by measuring the mean waiting time

per passenger under the application of the queuing theory. The principle for measuring the mean waiting time will be described hereinafter, taking an elevator system as an example.

Assuming that:

Qm : the mean value of the number of persons who are left behind at a floor and who are intending to go in the direction under consideration;

.lambda. : the rate of arrival of passengers at the floor who are intending to go in the direction under consideration (persons/second);

M : the mean value of intervals between elevator arrivals at the floor in the direction under consideration (seconds) (where, the passing of the floor in the direction under measurement because of no registration of the cage or hall call is deemed as an arrival, and the passing of the floor in the direction under measurement because of full capacity is deemed as no arrival); and

Lm : the mean value of the number of persons who are waiting on the floor for an elevator in the direction under measurement, when an elevator arrives at the floor in that direction,

There holds the relation:

Lm = Qm + .lambda..sup.. M (1)

letting arbitrary successive intervals between elevator arrivals be t (seconds), the total waiting time of all the users who have arrived at the floor during t (seconds) can be expressed by

Qm.sup.. t + 1/2.sup. . .lambda. t.sup.. t (2)

based on the assumption that all the left-behind passengers can get on the next elevator.

Whereas, letting the distribution of t, i.e. the probability density function of intervals between elevator arrivals, be x(t), the mean waiting time for one user can be obtained as follows:

(see Journal of Royal Statistical Society Series B, Vol. 16, page 86)

= 1/.lambda.M (Lm - .lambda.M) M + 1/2M (M.sup.2 + .sigma..sup.2)

= Lm/.lambda. - M/2 (1 - .alpha..sup.2)

where .alpha. = .sigma./M ;

.alpha. : coefficient of variation (relative dispersion) of the interval t between succeeding elevator arrivals; and

.sigma..sup.2 : variance of the interval t between succeeding elevator arrivals.

In the case of no overflow of passengers, there holds a relation Lm = .lambda. M and the equation (3) can be transformed into

Wm = M/2 (1 + .alpha..sup.2) (4)

now, consider a case in which an elevator in one direction under consideration (e.g. upwards) arrives at a floor under consideration (e.g. the second floor) with time intervals of T.sub.1, T.sub.2, T.sub.3, . . . , T.sub.n. The mean value of intervals between arrivals M and the variance .sigma..sup.2 in this case are expressed by the following equations: ##SPC1##

Hence, the coefficient of variation is:

Substituting the equations (5) and (7) into the equation (4), the mean waiting time in the case of no overflow of passengers is given by:

The present invention is based on the above-described analysis. More particularly, the mean waiting time in a system in which at least one subject becomes available at time intervals, is measured by taking a quantity corresponding to the ratio of the total sum of squares of said time intervals to the total sum of said time intervals. Said quantity is apparently proportional to the mean waiting time.

SUMMARY OF THE INVENTION

An object of this invention is to provide a method of estimating or approximately determining the mean waiting time of a user for at least one subject which becomes available at time intervals by:

measuring a first value representative of a sum of said time intervals

measuring a second value representative of a sum of the squares of said time intervals

and

calculating the ratio

of the first value to said second value.

Another object of this invention is to provide a system for estimating mean waiting time and, in particular, the mean waiting times for users of an elevator system, including a means for producing time interval signals, the intervals of which relate to time intervals of travel service provided by the elevator system, an integrator circuit for integrating a constant input during each interval of the time interval signals, and a filter for smoothing an output of the integrator. The integrator may be an analog form and, in addition, digital circuitry may be employed for counting and summing, subsequent to storage, of a plurality of pulses representative of a plurality of intervals of waiting times.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a graphic representation of elevator arrivals at a floor;

FIG. 2 is a circuit diagram of an embodiment of the invention;

FIG. 3 is a graphical illustration of the operation of the circuit of FIG. 2; and

FIG. 4 is a block diagram of an embodiment of the invention employing a digital method.

DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 1 shows intervals T.sub.1, T.sub.2, . . . T.sub.n on a time axis at each end of which upward elevators arrive at the second floor. FIG. 2 shows a system for measuring the average waiting time which comprises an analog integrator AIN and a filter F. The analog integrator includes an operational amplifier OAI, a feedback capacitor CI connected to both ends of the operational amplifier OAI, a switch SWI1 and a resistor RI1 for the resetting operation, and an input resistor RI2. The switch SWI1 closes to establish a closed loop including the resistor RI1 and the feedback capacitor CI upon the arrival of an elevator and allows the charge stored in the capacitor CI to discharge swiftly to reset the analog integrator AIN. Then, the switch SWI1 opens immediately after resetting. Thus, the analog integrator AIN begins to integrate the input again. With the input resistor RT12 is connected another switch SWI2 in series therewith which opens when the switch SWI1 is closed. This makes the input of the analog integrator AIN zero and assures the resetting operation by the switch SWI1. Accordingly, practically, no serious problem arises, even if the switch SWI2 is dispensed with. To the input resistor RI2 is applied an input voltage E.sub.o through the switch SWI2.

The output of the analog integrator AIN is supplied to a filter F including an operational amplifier, a feedback capacitor CF, a feedback resistor RF1 and an input resistor RF2. Thus, the filter F provides an output signal which is the average of the output of the analog integrator AIN.

FIG. 3 illustrates the operation of the average waiting time measuring system shown in FIG. 2 with the assumption that the switch SWI1 is closed and the switch SWI2 is open at the time t = 0. Provided that an elevator comes up to arrive at the floor under consideration at the time t = 0, the switch SWI1 will open and SWI2 will close. Then, a constant input E.sub.o is given to the analog integrator AIN which then provides a linearly increasing output as shown in FIG. 3. The integration constant giving the gradient of this line can be expressed by:

K = e.sub.o /r.sub.2 C.sub.1

where e.sub.o, r.sub.2 and C.sub.1 represent the magnitude of the input voltage E.sub.o, the resistor RI2 and the capacitor CI, respectively.

When the next upward elevator arrives at the floor after a time interval of T.sub.1, the switch SWI1 will close and SWI2 will open. The analog integrator AIN is reset immediately after that time, the time required for resetting being not more than about 100 microseconds. Whereas, the intervals between elevator arrivals T.sub.i (i = 1, 2, 3, . . . , n) seldom go below 20 seconds. Thus, the resetting time of said order can almost be neglected. Therefore, the output of the analog integrator AIN will have a sawtooth waveform. Immediately after the resetting operation, the switch SWI1 will open and SWI2 will close to begin the integration again.

The resetting and integration will be carried out upon the arrival of elevators; therefore, the output of the analog integrator AIN will have a sawtooth waveform as shown in FIG. 3, in which the bases are equal to the intervals of elevator arrivals T.sub.1, T.sub.2, T.sub.3, . . . , T.sub.n with a common gradient and hence different height.

Then, such an output is averaged at the filter F. Letting an arbitrary interval of arrival be T.sub.i, the area of the sawtooth waveform is:

(T.sub.i .sup.. KT.sub.i)/2 = K/2 .sup.. T.sub.i.sup.2

where K is a gradient (the gain of the integrator). Thus, the total area of sawtooth waveforms for N elevator arrivals can be expressed by

Letting the average of the sawtooth waveform, i.e. the output of the filter, be X', the relation:

holds, and hence

Thus, it is apparent that the output of the filter F is proportional to the mean waiting time.

The mean waiting time is the average or mean value of waiting times in an arbitrarily chosen time period T (T will hereinafter be referred to as the averaging time). So, the larger the time constant of the filter F is, the longer becomes said averaging time. If the time constant of the filter F is selected to be too large, variations in the output of the filter F become so small that the measurement of the desired waiting time becomes impossible. Therefore, the time constant of the filter F should be selected to be appropriate for measuring the mean waiting time in the necessary averaging time T.

Provided that the time constant of the filter F is selected to have an averaging time T.sub.o, the output X' of the filter F represents the average or mean waiting time in the preceding time period T.sub.o at any time.

In the above embodiment, the time interval T.sub.i is detected every time an elevator arrives at the floor, but with respect to an elevator which is the subject of the embodiment, a time interval between any two adjacent time periods in which the elevator is available on the floor considered is not represented only by the arrival intervals.

Namely, departure intervals, intervals of the door-opening operation and intervals of the door-closing operation, etc. also show indirectly the arrival interval of an elevator, i.e., the time interval T.sub.i is related to the time interval between the time periods in which an elevator is available.

Thus, in alternative embodiments, the time intervals under measurement can be arranged to be the time intervals from the departure of an elevator to the arrival of the next elevator, or from the time an elevator closes its door to the time the next elevator opens its door.

Further, the overall mean waiting time on a floor can be measured by averaging the mean upward and downward waiting times.

FIG. 4 shows a block diagram of another embodiment of the invention employing a digital computation device.

A pulse oscillator POS constantly generates pulses of a fixed frequency and supplies its output to AND gates A1 and A2. A flip-flop circuit FF inverts the outputs from output terminals 01 and 02 every time an arrival signal RS of an elevator enters the circuit. Counters CU1 and CU2 alternately perform counting operations. More particularly, while the counter CU1 counts an input quantity, the counter CU2 supplies a quantity counted in the preceding interval to a squaring device SD to square it and the result is then stored in a register REG. The output of the square device SD is sequentially stored in P1 to PN of the register REG and the stored contents are added by an adder ADD. A divider DV divides the output of said adder ADD by the output of an average time device ATD. Thus, the mean waiting time Wm can be calculated.

When an elevator reaches the floor at time t = 0 in FIG. 3, a starting signal SS is given to an AND gate A3 so as to allow a pulse train to be supplied to the AND gate A1. An arrival signal RS is sent to the flip-flop FF to provide an output signal on the output terminal 02.

The AND gate A1 remains open during an arrival interval T.sub.1 and the counter CU1 counts the pulse train during the time interval T.sub.1. Thus, the calculated value of the counter CU1 becomes equal to the arrival interval T1.

When a next elevator signal arrival after a time lapse of T.sub.1, the flip-flop FF is inverted by the arrival signal RS to provide an output signal on the output terminal O1 and to provide no output signal on the output terminal 02. Then, the AND gate A1 is closed and A2 is opened and the counter CU2 counts the pulse train in the arrival interval T.sub.2, and hence the counted value gives the arrival interval T.sub.2. While the counter CU2 counts the pulse number, the counter CU1 supplies the previously counted value T.sub.1 to the squaring device SD to give T.sub.1.sup.2. This value T.sub.1.sup.2 is stored in a portion P.sub.1 of the register REG. The counter CU1 is reset by reset signals RE1 and RE2 to prepare for the next operation. The counters CU1 and CU2 are so arranged that they cannot be reset during the counting operation, i.e. while receiving an input pulse train from the AND gate A1 or A2.

When the arrival interval T.sub.2 elapses, an arrival signal RS is given to the flip-flop FF again to invert the flip-flop, opening the AND gate A1 and closing the AND gate A2. Then, the counter CU1 begins a counting operation again and the square of the counted value T.sub.2 by the counter CU2, that is T.sub.2.sup.2, is stored in a portion P2 of the register REG, as is similar to the foregoing operation. Thus, the values T.sub.1.sup.2, . . . , T.sub.n.sup.2 are stored in portions P1, . . . , PN of the register REG by a sequence of similar operations. The adder ADD calculates the total sum of the numbers stored in the register REG, that is

while the average time device ATD gives

Thus, the divider DV can provide an output:

In obtaining the averaged mean waiting time by sampling, the whole system is reset at respective sampling instances and the calculation is performed in a similar manner as described above.

Further, in the case of providing a renewed mean waiting time at each time when an elevator arrives at the floor under consideration, the stored contents in the portions P.sub.1, . . . , P.sub.N of the register are sequentially replaced with new information and the content of the average time device ATD is changed accordingly.

As described above, in such a case where a subject or subjects (for instance, an elevator) are available with certain time intervals, the mean waiting time can be measured simply by detecting time intervals T.sub.i which are related with the intervals between successive time periods in which said subjects are available.

Thus, the automated detection of an average waiting time can be achieved with high accuracy by a simple apparatus.

Further, in an embodiment in which an analog integrator integrates the input during each of said time intervals and in which the integrated output is flattened by a filter, there is no need for any special calculating device for measuring the mean waiting time and the averaging time T can be arbitrarily altered by an adjustment of the time constant of the filter. Further, the output of the filter may continuously represent an instantaneous mean waiting time.

Claims

We claim:

1. A system for estimating the mean waiting time of the waiting times for the users of an elevator system which provides travel services from one to another of a plurality of floors comprising:

first means, responsive to successive travel services, which are provided by an elevator system and available at a given floor for traveling in a predetermined direction, for producing, successively, time interval signals whose intervals correspond to time intervals of said successive travel services;

second means, responsive to said time interval signals, for integrating an input of a predetermined constant value during each interval of a respective one of said time interval signal, thereby producing, successively output signals each indicative of the integration of said input; and

third means, responsive to the output signals of said second means, for smoothing the output signals of said second means.

2. A system for estimating the mean waiting time according to claim 1, in which said first means includes switch means, which in response to each of said successive travel services, permits said input to be supplied to said second means for each said time interval and resets said second means at the end of each said time interval to clear the integration of said input, said second means includes an analog integrator which integrates said input, and said third means includes a filter circuit for smoothing the output signals of said second means.

3. A system for estimating mean waiting time according to claim 1, in which said switch means includes a first switch connected between an input means which produces an output of said predetermined constant value and said analog integrator through an input resistor and a second switch connected between output and input terminals of said analog integrator through a feedback resistor, said first switch being closed for each said time interval and open at the end of said each time interval, while said second switch is open when said first switch is closed and closed when said first switch is open.

4. A system for estimating the mean waiting time of the waiting times for the users of an elevator system comprising:

means for generating a signal representative of the waiting times including

first means, responsive to a first prescribed condition of an elevator with respect to a given floor, for initiating the generation of a signal representative of the waiting time;

second means, responsive to a second prescribed condition of an elevator with respect to said given floor occurring subsequent to said first condition, for terminating said waiting time signal; and

third means, responsive to a predetermined time period corresponding to a plurality of said waiting time signals, for averaging said waiting time signals over a selected time interval and producing an output signal representative of said average.

5. A system according to claim 4, wherein said first means is responsive to the passage of an elevator, which is available to carry additional passengers, past a prescribed point with respect to a given floor, for initiating the generation of said waiting time signal.

6. A system according to claim 4, wherein said first means is responsive to the arrival of an elevator at said given floor, for initiating the generation of said waiting time signal.

7. A system according to claim 4, wherein said second means is responsive to the arrival of an elevator at said given floor, for terminating the generation of said waiting time signal.

8. A system according to claim 7, wherein said first means is responsive to the departure of an elevator from said given floor, for initiating the generation of said waiting time signal.

9. A system according to claim 4, wherein said second means is responsive to the passage of an elevator, which is available to carry additional passengers, past a prescribed point with respect to a given floor, for terminating the generation of said waiting time signal.

10. A system according to claim 4, wherein said first means is responsive to the closure of an elevator door at said given floor, for initiating the generation of said waiting time signal.

11. A system according to claim 4, wherein said second means is responsive to the opening of an elevator door at a given floor, for terminating the generation of said waiting time signal.

12. A system according to claim 4, wherein said first means comprises a source of reference potential, a first condition responsive switch connected thereto, and an integrator circuit connected to said first switch, said first switch being coupled to connect said source of reference potential to said integrator in response to the occurrence of said first condition, and a second condition responsive switch connected in the feedback circuit provided in said integrator, and being coupled to be opened to the occurrence of said first condition.

13. A system according to claim 12, wherein said first and second switches of said second means respectively open and close upon the occurrence of said second condition.

14. A system according to claim 13, wherein said third means comprises means for filtering the output of said integrator circuit.

15. A system according to claim 4, wherein said first means comprises

a pulse generator;

a first counter logically coupled to the output of said pulse generator to count pulses supplied therefrom, and

a first gate circuit, for initiating the supply of pulses generated by said pulse generator to said counter, in response to a signal representative of the occurrence of said first condition.

16. A system according to claim 15, wherein said second means comprises means for disabling said first gate circuit from supplying pulses to said first counter in response to a signal representative of the occurrence of said second condition , whereby the number of pulses counted by said first counter will represent said waiting time signal.

17. A system according to claim 16, wherein said third means comprises a squaring circuit connected to the output of said first counter, a storage register connected to said squaring circuit to store successive outputs thereof representative of the squared values of successive waiting times, an adder circuit connected to the output of said register to sum the contents thereof, a divider circuit having one input connected to the output of said adder circuit, and a timer circuit connected to another input of said divider circuit, said timer circuit supplying a signal representative of the time elapsed from the generation of a selected time interval signal, whereby the output of said divider is representative of the mean waiting time over a preselected time interval.

18. A system according to claim 17, wherein said disabling means includes a flip-flop circuit, the state of the outputs of which are controlled by a signal representative of one of said prescribed conditions, a second gate circuit logically coupled to said pulse generator and one of the outputs of said flip-flop, the other output of said flip-flop controlling said first gate, and a second counter circuit connected to the output of said second gate, the output of said second counter being supplied to said squarer circuit.

19. A system for estimating the mean value per user of the waiting times ( W1, W2 . . . Wi . . Wa) of a plurality of users who, at random, reach a position where at least one subject of use becomes available at irregular time intervals ( T1, T2 . . Ti . . . T.sub.n ), before said users utilize said at least one subject, comprising:

first means responsive to successive time interval signals for measuring a first value representative of the sum

of said time intervals;

second means responsive to successive time interval signals for measuring a second value representative of the sum

of the squares of said time intervals; and

third means for calculating the ratio

of said second value to said first value.

20. A system for estimating the mean value per user of the waiting times ( W1, W2, . . . Wi . . . Wn ) of a plurality of users in an elevator system including at least one elevator providing transportation of users among a plurality of floors, wherein the users arrive at random at a given one of said floors and wait at the floors for said waiting times in order to receive transportation in a predetermined direction, which transportation becomes available at said given one of said floors at irregular time intervals ( T1, T2 . . . Ti Tn ), comprising:

first means responsive to successive time interval signals for measuring a first value representative of the sum

second means responsive to successive time interval signals for measuring a second value representative of a sum

of the squares of said time intervals; and

third means for calculating the ratio

of said second value to said first value.

21. A system for estimating the mean interval of time for a plurality of successive signals the time intervals between which may vary, comprising:

a pulse generator;

first means logically coupled to said pulse generator and responsive to each of said successive signals, for generating respective signals representative of the count of the number of pulses between successive ones of said plurality of successive signals;

second means, responsive to the output of said first means, for generating a signal representative of the sum of the squares of said count representative signals; and

third means, responsive to the output of said second means and a signal representative of a prescribed interval of time, for dividing the output of said second means by said time representative signal, whereby said mean interval of time between said plurality of signals may be determined.