# CONTINUOUS-STOCHASTIC MODELS (0-SCHEMES), Basic relations. - Modeling of systems

## CONTINUOUS-STOCHASTIC MODELS (0-SCHEMES)

The features of the continuous-stochastic approach will be considered using the example of using as standard mathematical schemes queuing systems (English, queueing system), which we will call ( 1-schemes . mass service are a class of mathematical schemes developed in the theory of mass service and various applications for the formalization of the processes of the functioning of systems that are inherently processes of service [6, 13, 33, 37, 51].

## Basic relations.

As a process of service, various processes of functioning of economic, production, technical and other systems, for example, streams of supplying products to some enterprise, streams of parts and components on the assembly conveyor of the shop, applications for processing computer information from remote terminals, etc. At the same time, characteristic for the operation of such objects is the random appearance of requests (requirements) for maintenance and termination of service at random moments strap, t. e. the stochastic nature of their operation process. Let us dwell on the basic concepts of queuing required for the use of O-schemes, both in the analytical and in the simulation.

In any elementary act of service, two main components can be distinguished: the expectation of service by the application and the actual maintenance of the application. This can be represented in the form of some 1st service device F, (Figure 2.6), consisting of the accumulator of orders R ", in which L = 0, of applications, where T, ® is the capacity of the/th storage, and channel service requests (or just a channel) A ,. For each element of service device/7, event streams come: to the drive I/- the stream of requests for the channel A/- the flow of services

A stream of events is a sequence of events that occur one after another at some random time. There are streams of homogeneous and inhomogeneous events. The event flow is called homogeneous if it is characterized only by the moments of arrival of these events (calling moments) and is given by the sequence {/ "} = {0 1 ^/ 2 ^ where/ i - the moment when the nth event occurred is a nonnegative real number. The homogeneous flow of events can also be specified as a sequence of time intervals between the nth and (n-1) th events {m i }, which is uniquely related to the sequence of the triggering moments {/ "}, where m i • л &> 1, о 0, that is, T |/|.

The flow of non-uniform events is the sequence {/ ",/ i ), where/ i - calling moments; g i is a set of event tags. For example, with respect to the maintenance process for a heterogeneous flow of applications can be Fig. 2.6. Application Maintenance Device

given membership to a particular source of requests, the presence of priority, the ability to service a particular type of channel, etc.

Consider a thread in which events are separated by time intervals T |, m 2> ... which are generally random variables. Suppose that the intervals m, m 3 , ... are independent of each other. Then the flow of events is called a thread with a limited aftereffect.

An example of an event flow is shown in Fig. 2.7, where T, is the interval between events (random variable); T n is the observation time, t s is the time of the event.

The intensity of the flow can be calculated experimentally using the formula where N is the number of events that occurred during the observation time T i . If Gu = const or is defined by some formula 2y /<7> _ |), then the flow is called deterministic. Otherwise, the stream is called random.

Random streams happen:

- ordinary, when the probability of simultaneous occurrence of 2 or more events is zero;

- stationary, when the frequency of occurrence of events is constant;

- without aftereffect, when the probability does not depend on the moment of the previous events.

The flow of events is called ordinary, if the probability that for a small time interval A, the adjacent I, falls more than one events P & gt; | (/, A /) is negligibly small in comparison with the probability that exactly one event F (i, A /) falls on the same time interval Ar, that is, P x (I, AO P & gt ;, (I, AO) • If for any interval A/the event as the sum of the probabilities of events forming a complete group and inconsistent, then for an ordinary event flow where O (R0 is a quantity whose order of smallness is higher than A /, that is, A stationary flow of events is a flow for which the probability of occurrence of a particular number of events in the time interval t depends only on the length of this section and does not depend on where on the time axis 0/.

Consider the ordinary flow of events on the time axis 0 ( and find the average number of events occurring on the time interval A /, which is adjacent to the time t. We obtain Then the average number of events occurring on the time interval A ( per unit time will be (P, (I, A))/A. Let us consider the limit of this expression as A/- »0. If this limit exists, then it is called the intensity (density) of the ordinary flow of events Rn [P r (r,

A/- * 0 Fig. 2.7. Graphical image of P-scheme

Д /)/Д /] A (/). The flux intensity can be any non-negative function of time having a dimension inverse to the dimension of time. For a stationary flow, its intensity does not depend on time and is a constant value equal to the average number of events occurring per unit time X (O const.

## thematic pictures

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