Properties of DTIPs

In order to Technology (IT AC DTIP) could be applied to some real discrete process with the aim of finding an effective strategy of its control, this process must have a certain set of restrictive properties, which are characteristic for discrete technological and information processes (DTIPs). In other words, the properties of any DTIP must satisfy certain restrictions, which all set is divided into the following four groups:
1)
restriction on belonging of the process to the category of controlled ones;
2)
group of restrictions on the description of the process phase state;
3)
group of restrictions on applied controls;
4)
restriction on the process quality criterion.

Fig. 1 shows a general structural scheme of a discrete process. In this scheme a control strategy is separated from the process, although in reality it is usually its constituent part. This is done to focus your attention on the control strategy, from which depends the efficiency of flowing of the process itself. Here, the arrows mark incoming and outgoing flows of objects, relating to this process, and the dotted rectangular frame ‚Äî the limits of its impact. Fig. 1. General structural scheme of a discrete process

We turn now to description of the required restrictions, which allow to represent a discrete process as DTIP. First, we note that DTIP is a controlled process, otherwise it makes no sense to consider it from from the standpoint of its effective control. Controllability of the process should be confirmed by a list of those its parameters, characteristics and procedures, which can be varied (e.g., an order of entry of details to the machine, the speed of its processing, possible denials of processing, etc.).

Definition. Phase state of a discrete process is called such quantitative description of its discrete and analog parameters, that was fixed at some moment of its observation, from the values ‚Äã‚Äãof which, as well as from the current control depends the further behavior of this process.

DTIP can be infinite, limited in time and even instant. However, in case or its infinity, it must be stationary (parameters and characteristics of a stationary process don't depend on the time of its observation). Then DTIP is described at any observing moment by the following vector of its phase state:

Z = (Z1, Z2),   (1)

where Z1 = (z11,‚Ä¶,z1i,‚Ä¶,z1m1) ‚Äî vector of the process discrete parameters and Z2 = (z21,‚Ä¶,z2j,‚Ä¶,z2m2) ‚Äî vector of the process analog parameters. Here are given the following notations:

• m1 ‚Äî number of discrete parameters in the process, describing discrete states of its objects;
• m2 ‚Äî number of analog parameters, describing analog states of this process, as well as used therein restrictions;

Note 1.  Consider a simple example of such one restriction. Suppose you have to process a group of details during limited period of time T. Then, the current time t, that has elapsed since the start of processing details, are specified in the phase state vector of the process as its analog parameter, the value of which varies in the range [0, T].

• z1i ‚Äî value of i-th discrete parameter of the process, i = 1,‚Ä¶,m1;
• z2j ‚Äî value of its j-th analog parameter, j = 1,‚Ä¶,m2.

Note 2.  The stationarity condition for infinite DTIPs allows to describe their current state by the vector (1). If a discrete process was non-stationary, then the vector of its state would depend on the time of its flowing. In this case, it would be impossible to synthesize for it numerically the optimal control strategy.

On the DTIP phase states are imposed the following restrictions: numbers of discrete and analog parameters that completely describe behavior of a discrete process for specified controls are finite, and their values ‚Äî limited. In other words, we should have the following inequalities:

m1 < ‚àû, m2 < ‚àû, z1i < ‚àû for all i –∏ z2j < ‚àû for all j.

The necessity in DTIP of the above restrictions are due to the fact that in the case of applying of Technology to this process, must be performed an operation of its finite-dimensional approximation. As a result, from the initial DTIP is formed a derivative (i.e. secondary) stationary discrete process with a finite number of phase states, which will be optimized numerically. Such an operation can be performed only if there are limited number of parameters and limited values each of them.

From the theory of random processes it is known, that a process, which behavior depends on the limited values ‚Äã‚Äãof some set of analog parameters, is called a process with limited aftereffect. Hence it follows, that DTIP is a discrete process either with limited aftereffect, which is given by a set of analog parameters with their restricted values, or without any aftereffect if analogue parameters are absent.

Let's now consider restrictions on the controls, used in DTIP, which, as known, can be instant and non-instant (see above subsection "What is a discrete process" of this Sec.). These restrictions, which determine the listed below properties, include the following:

• action of a non-instant control remains unchanged during its application (property of permanence of non-instant control);
• action of an instant control spreads only on one object of the process (property of a single-purpose instant control);
• at one and the same object of the process can influence at any moment only one control (property of absence of controls accumulation);
• for any state of DTIP the choice of a control is made ‚Äã‚Äãfrom limited set of possible controls (property of limitation of control variants).

Let's explain the essence of these restrictions and corresponding properties of DTIPs.

Property of permanence of non-instant control means that parameters of such control can't be changed during its application. If, for example, as a control you choose the speed of processing same detail on the machine, then this speed should remain constant during the use of this control.

Property of a single-purpose instant control means that an instant control can be applied only to a single object of the process. If you need to apply at some time moment a group of instant controls to several objects of the process, then it can be done by their successive application to each of these objects. The schemes of numerical optimization, which are used in Technology, allow to realize such a procedure of successive applying instant controls.

Property of absence of controls accumulation means that onto some object of the process can't simultaneously effect two or more controls, that have been applied. In other words, if you apply current control, then the effect of previous control is cancelled.

Property of limitation of control variants means that the choice of a control in each state of the process is made ‚Äã‚Äãfrom a limited set of its possible variants. If, for example, a control is the speed of processing details on the machine, then the value of this speed is selected from a limited set of its possible values.

The requirement for fulfillment of all these restrictions concerning controls, being applied to DTIP, is caused by the specificity of numerical optimization schemes, used in Technology.

Consider now the latest restriction imposed on DTIP, which refers to the selection of a quality criterion for this process, that quantitatively characterizes its efficiency. The requirement of Technology is that this criterion should be additive in the in the broadest sense, the essence of which is as follows:

• for a group of successively applied instant controls the additivity means summing values of losses (incomes), each of them refers to any one of these controls and to corresponding object of the process;
• for a group of consistently applied non-instant controls the additivity means summing values of losses or incomes, each of them refers to that period of time, during which this control is applied.

In other words, for instant controls the property of additivity is considered in relation to the process objects, to which they are applied, and for non-instant ‚Äî in relation to time intervals during which this controls have effect.

It should be noted that in many real DTIPs the additivity property spread for non-instant controls not only on the flow time of this process (a compulsory condition for using Technology), but on its objects. For example, the losses of downtime of a batch of details, waiting for their processing on the machine, equal to the sum of losses from delay in processing every of these details. However, there are processes for which the spread of the principle of additivity onto the process objects is absent. Consider an example that demonstrates the abovementioned.

Let DTIP is the process of reforming a military formation (MF), consisting of several military units. The task is to find by using IT AC DTIP of effective strategy of controlling this process. When formulating a quality criterion for this DTIP were taken into account many factors, including the fact that during the process of reforming this MF, the last should perform its main function, consisting in combat duty. It means that a value of losses from MF reforming depends on which elements (military units) are currently in operation, and which ‚Äî no (being reformed). In other words, the additivity property takes place here only for the flow time of the process, and not concerning its objects, what, as known, it is quite possible (see. above).

So, if you are dealing with some discrete process of any physical nature, which satisfies the above restrictions on its behavior (i.e. represents DTIP), you can confidently solve the task of effective controlling this process with the help of Technology.

It should be noted that with such abstract description of a discrete process is not always easy to recognize it in the reality around us. To make this easier to do, we give a generalized physical interpretation of those objects, which can be parts of a discrete process.

First of all note, that virtually any discrete process has two categories of objects. Some of them have any resources, while others require their processing by using resources of the first objects. Here we use the terms of the queuing theory and call objects of the first category as devices of service (or just devices), and of the second ‚Äî demands. In general, represent a discrete process as a process of servicing demands by devices. At that, the entire set of these objects, which functioning we are interested in, call a service system.

Besides demands and devices, which interact with each other, provide the presence in a discrete processes of abstract objects of a third category, called mediums. The purpose of mediums is to describe a process of changing conditions for functioning devices and demands. If demands and devices can enter the service system and leave it, then objects-mediums, if they are provided, are always present there and only change their states according to certain laws, affecting on the behavior of certain devices and demands.

Fig. 2 is a structural scheme of a discrete process in the above interpretation. If for this process are performed the above restrictions, it will be DTIP, to which can be applied Technology. Fig. 2. Structural scheme of a discrete process in interpretation of a service system

Consider a few examples of DTIPs, noting what objects there are demands, devices and mediums.

Example 1. A seller services a queue of buyers. Here, the device is a seller, and demands ‚Äî buyers (mediums are absent).

Example 2. A batch of details are processed on a machine. Here, the device is a machine, and demands ‚Äî details (mediums are absent too).

Example 3. Stationary flow of details enter for processing on an unreliable machine, that periodically and randomly breaks down, and after its recovery is running again. Here we have two devices: obvious one (a running machine) and implicit (a repair team, recovering broken machine). Here, demands are details, being processed on a machine, as well as requests for its restoration, which are generated when it breaks down. This process has also one medium, describing an order of breakdowns of a machine and its subsequent recoveries.

Example 4. A sum of money, allocated by a trader, should be spend for acquiring goods from a wholesaler for their subsequent sale. Here, demands are goods available at a wholesaler, and the device with a group service of demands ‚Äî a trader himself, whose resource is that amount of money.

Example 5. Some building is being constructed by a construction company and its contractors. Here, devices are specialized teams, performing certain construction operations, and the demand ‚Äî a network graph that describes the sequence of different operations on this construction, which represent the stages of servicing this demand.