The Problem of Rational Control of Service Discipline in Elevator Control Systems

L. V. Perfil'ev

(Presented by the scientific-technical seminar of the Research Institute of Automation and Electromechanics at TPI)

The operation of elevator control systems under conditions of random passenger flow is accompanied by overloads of the system in peak modes. Peak modes are characterized by the fact that the passenger flow is directed predominantly “upward” or “downward.”

In connection with this, the problem arises of controlling the service discipline (the method of organizing elevator operation) depending on the nature of the passenger flow. The essence of the problem is to choose such a service discipline under which the service-quality indicators of the system attain optimal values.

At present there are no universal methods for the optimal solution of problems in such a general formulation.

In practice, three principal modes of elevator operation have been distinguished, conventionally called “daytime,” “morning,” and “evening,” in each of which elevator operation is organized in the corresponding manner. The “daytime” mode assumes a symmetric passenger flow, i.e., the average number of requests directed “upward” and the average number of requests directed “downward” entering the system over a certain interval of time are equal. In the “evening” mode the passenger flow is directed mainly “downward,” and in the “morning” mode, “upward.”

Switching from one mode to another is performed by the dispatcher depending on the nature of the passenger flow, on the basis of signaling indicating the presence of requests in the system. Sometimes switching is performed at established hours of the day on the basis of previous operating experience with the system. In connection with this there arises the problem of automatic switching of modes depending on the nature of the passenger flow, with the aim of obtaining optimal values of the system’s service-quality indicators.

The solution of the stated problem is connected, first of all, with determining the criteria for transition from one mode to another. Each mode \(R_j\) \((j = 1, 2 \ldots k)\) corresponds to a quite definite method of organizing elevator operation.

During operation of the system, situations arise such that, for any value of \(R_j\), any values of the magnitude of passenger-flow asymmetry, \(\alpha_i\), are possible. By passenger-flow asymmetry is meant the magnitude of the ratio between the average number of requests directed “upward” and the average number of requests directed “downward” entering the system over a certain interval of time.

If by \(\alpha_i\) we denote the value of the ratio between requests of opposite directions, which is determined at each discrete instant of time, then the essence of the problem is that, for a known value—

for each value of $\alpha_i$ compute the probability with which the quantity $\bar{\alpha}_i$ assumes one of the possible values.

Having at our disposal mathematical models of the elevator-control system in all operating modes, it is not difficult, for each value of $R_j$ $(j = 1, 2 \ldots k)$, to compute the function

\[ v = f_i(\alpha_i/\bar{\alpha}_i), \tag{1} \]

where $v$ is the probability that the value of the ratio between calls in opposite directions at each instant of time assumes a value equal to $\alpha_i$, under the condition that the asymmetry of the passenger flow is equal to $\bar{\alpha}_i$.

From expression (1) it is not difficult to pass to the expression

\[ w = \varphi_i(\bar{\alpha}_i/\alpha_i), \tag{2} \]

where $w$ is the probability that the value of the asymmetry of the passenger flow assumes a value equal to $\bar{\alpha}_i$, under the condition that the value of the ratio between calls in opposite directions is equal to $\alpha_i$.

Thus, if at each instant of time it is possible to measure the quantity $\alpha_i$ for a known value of $R_j$, this means that the probability is known of what value of asymmetry the passenger flow has.

The problem is complicated by the fact that in real elevator-control systems it is difficult to measure the value of the parameter $\alpha_i$ (it is difficult to determine how many people wish to travel “up” and how many “down”). A way out of this difficulty can be found if, by means of some quite definite quantity $\beta_i$, which is relatively easy to measure, the values of the parameter $\alpha_i$ are estimated.

In this case it is possible to compute the conditional probability density function of the parameter $\alpha_i$

\[ q = \psi_i(\alpha_i/\beta_i). \tag{3} \]

In such a case, on the basis of expressions (2) and (3), for each operating mode of the system it is possible to determine the probability $w'$, with which the asymmetry of the passenger flow assumes a value equal to $\bar{\alpha}_i$, under the condition that the value of the parameter $\beta_i$ is determined.

\[ w' = \varphi'_i(\bar{\alpha}_i/\beta_i). \tag{4} \]

Taking into account all that has been said above, the problem of determining the criterion for switching from one mode to another can be reduced to the problem of solving a rectangular game.

Here the strategy of one player is known.

Player $P_1$ is the device (automatic dispatcher) that determines the mode in which the system must operate. Player $P_2$ is the passenger flow.

The pure strategies of player $P_1$ are the operating modes of the system $R_j$ $(j = 1, 2, \ldots k)$, and those of player $P_2$ are the values of the asymmetry of the passenger flow, $\bar{\alpha}_i$.

The elements of the payoff matrix of the game are the indicators of service quality, computed for all possible combinations of the quantities $R_j$ and $\bar{\alpha}_i$.

Then it is not difficult to compute the entire set of optimal strategies of player $P_1$ for the known set of mixed strategies of player $P_2$, determined by expression (4).

In this case, the automatic dispatcher implementing the optimal mixed strategies of player $P_1$ can be constructed as follows.

By means of a diode decoder, the measured value of the parameter $\beta_i$ is converted into discrete signals $x_j$ $(j = 1, 2 \ldots m)$ for controlling a device that makes it possible to play out events with probabilities

occurrence, equal to \(p_j\). The values of the probabilities \(p_j\) correspond to the probability values of the mixed strategy of player \(P_1\). Fig. 1 gives a functional diagram of such a device. Depending on which of the \(m\) buses the signal \(x_j\) is applied to, the corresponding probability of the mixed strategy of player \(P_1\) is realized. The events in the device are realized sequentially, one after another.

If the event with probability \(p_j\) has occurred, the signal \(y_j\) appears at the output of the device, which means that player \(P_1\) chooses the operating mode \(R_j\).

If the event with probability \(p_j\) has not occurred, the next \((j+1)\)-st signal is applied to the input of the device.

The operation of the device (Fig. 1) is as follows.

Fig. 1.

The random-signal generator \(GS\) issues, at each discrete instant of time, a pulse with probability equal to \(\rho_1\). The discrete instants of time are set by the generator \(G\), which produces pulses at equal time intervals. The signals of the generator \(GS\) are received by counter 1. If the number of pulses of counter 1 and of counter 2 is recorded, into which in each discrete instant of time a pulse from the generator \(G\) is entered, then it is not difficult to calculate \(\rho_2\)—the probability that one pulse will enter counter 1 after one discrete interval, after two discrete intervals, etc. The value of this probability increases as the number of discrete intervals increases.

With the aid of decoder 1 and the coincidence circuits, depending on the values of the signals \(x_j\), one can control the number of discrete intervals in such a way that the probability \(\rho_2\) takes values equal to the probabilities \(p_j\). In this case, if one pulse has been registered in counter 1, the signal \(y_j\) appears at the output of decoder 2, indicating that it is necessary to establish the operating mode of the system \(R_j\).

The output signal of the coincidence circuits switches off the generator \(G\) and reads the information of decoder 2.