## Possible applications.

Usually in applications when modeling different systems with reference to an elementary service channel * K _{ t } *, we can assume that the flow of applications w, e

*W*, t ie the time intervals between the moments of the appearance of the orders (the triggering moments) at the input

*K*form a subset of the unmanaged variables, and the service flow w, gC /, ie the time intervals between the beginning and the end of the service of the application, forms a subset of the controlled variables.

_{ h } Applications serviced by * K _{ and } * and applications that left the device

*P,*for various reasons unhandled (for example, /), form the output stream

*y*Y, ie, the time intervals between the exit times form a subset of the output variables.

_{ t }e The process of functioning of service device D can be represented as the process of changing the states of its elements in time z, (/). The transition to a new state for * P, * means the change in the number of requests that are in it (in the channel * K, * and in the drive * N. * Thus, the vector of states for H has the form z, = (z®, z *), where * z, * is the state of the accumulator * H _{ t } * (r® = 0 - the drive is empty, z ® = 1 - there is one application in the drive, z ® = L ® - the drive is full),

*L,*is the drive capacity

*H*measured by the number of applications that can fit in it, z * - channel state

_{ h }*K,*(z * = 0 - channel is free, z * = l-channel is busy, etc. .).

In the practice of modeling systems with more complex structural relationships and behavioral algorithms, not separate service devices are used for formalization, but * Q-schemes *, formed by the composition of many elementary maintenance devices L, (queuing networks) . If the channels * K, * of the various service devices are connected in parallel, then multi-channel service (multi-channel * Q-scheme), * occurs, and if devices * n _{ t } * and their parallel compositions are connected in series, then there is a multi-phase service (multi-phase

*Q-scheme).*Thus, to specify the

*Q-scheme*

*R*, reflecting the relationship of the elements of the structure (channels and stores) to each other.

The links between the elements of the * Q-scheme * are represented in the form of arrows (stream lines reflecting the direction of motion of applications). There are open and closed * Q-schemes. * In an open * Q-scheme * the output flow of the served requests can not re-enter an element, that is, there is no feedback, and in closed * O-schemes * there are feedbacks on which orders move in the direction opposite to the input-output movement.

The internal (internal) parameters of the * O-scheme * will be the number of phases

_{ }, the number of channels in each phase

1, & pound ;, the number of drives of each phase * to =, * /, _{ ф }, capacity of the/th storage. It should be noted that in queuing theory depending on capacity the following terminology is used for queuing systems: a system with losses (/, * = 0, that is, there is no accumulator in the device * P, *, but there is only service channel A)), waiting systems ( /, * - oo, that is, the accumulator D has infinite capacity and the queue of applications is not restricted) and a system of mixed type (with a limited storage capacity D). We denote the entire set of eigenfunctions of the * 0-scheme * as a subset of H.

To define * 0-scheme *, it is also necessary to describe the algorithms of its functioning that determine the set of rules of behavior of applications in the system in various ambiguous situations. Depending on the place of occurrence of such situations, the algorithms (disciplines) of waiting for applications in the accumulator H are distinguished, and the services are serviced by the channel * K _{ {} * of each elementary servicer D

*0-scheme.*The heterogeneity of applications, reflecting the process in one or another real system, is accounted for by introducing priority classes.

Depending on the priority dynamics in * 0-schemes *, static and dynamic priorities are distinguished. Static priorities are assigned in advance and do not depend on the states of the * 0-scheme *, ie they are fixed within the solution of a specific simulation problem. Dynamic priorities arise in the simulation, depending on the situations that arise. Based on the rules for selecting applications from the storage drive D for maintenance by the channel * K _{ {}, *, you can select relative and absolute priorities.

*Relative Priority*means that a higher priority application that has arrived at Drive D is waiting for the end of the maintenance of the previous application by the

*K*channel and only then takes up channel.

_{ {}*Absolute priority*means that the application with a higher priority received in drive D, interrupts the service with the

*K,*channel with lower priority, and itself occupies the channel (at the same time it is superseded from A , the application can either leave the system, or can be re-written to some place in D).

When considering the algorithms for the operation of maintenance devices D (channels * K, * and drives D), it is also necessary to specify a set of rules for which applications leave D and A: for D - either overflow rules, depending on the filling of D, leave the system, or the rules of care associated with the expiration of the waiting time for the application in //, for * K _{ 1 } * - the rules for selecting routes or directions of care. In addition, for applications it is necessary to specify the rules by which they remain in the channel

*K*or are not allowed to be serviced by the channel

*K*ie rules channel locks. In this case, the locks

_{ 1y }*K are distinguished, by*the output and the input. Such locks reflect the presence of control relationships in the

*Q-cxeem*regulating the flow of requests depending on the states (

_{ y }*2-schemes*.The entire set of possible algorithms for the behavior of orders in

*0, -circuit*can be represented as some operator of algorithms of behavior of orders

*A.*

Thus, the * () - scheme *, describing the process of the functioning of the queuing system of any complexity, is uniquely specified in the form & pound; /, H, 2, A, * A * & gt ;

With a number of simplifying assumptions about the subsets of the incoming flows * IV * and the service flows * and * (fulfillment of the stationarity, ordinariness and bounded aftereffect conditions) of the conjugation operator of the elements of the structure * I (single-phase * single-channel service in an open system), subsets of eigen parameters * N * (maintenance with an infinite capacity of the drive), an operator of order processing algorithms * A * (uninterrupted service without interrupts and locks) for the estimation of probability-time x The analytic apparatus developed in the theory of queuing can be used. Under the assumed assumptions, in the notation of Kendall, there will be a classical M/M/1 service system (a single-channel system with a Markov incoming application flow and a Markov service flow). Consider, for example, the basic analytic relations for such an elementary * () scheme * [6, 24, 37].

Example 2.6. Assume that the maintenance process starts when there are no applications in the drive. Then the states of the queuing system are described by the following system of equations:

where * P _{ i } * (0 is the probability of finding the system in the state r

_{ i }(/) e2 at time t, when there are

*n*applications in it.

These equations follow from the fact that the probability of finding applications in the system * n * at the moment of time (/ + Д /) is equal to the probability of finding in the system n applications at the moment * I, * multiplied by the probability that during the time D/in the system there will be no applications and no one will be served, plus the probability of finding in the system (n-1) applications at the time * I, * multiplied by the probability the fact that one application will be processed during the time D/*, and no application will be served, plus the probability of finding in the system (n + 1) applications at time t, multiplied by probabilities be that during the time A I one application leaves the system and have not received a single application. The probability that no application will be received during the time A , and no application leaves the system ', is equal to (1 -A * *) (1-M0) The term containing (A /), with the derivation of the differential equation is omitted Hence, we can write 1- (A + q) A /. Concerning the remaining two terms of the first equation, we note that *

Moving * P _{ n } * (/) to the left and letting N approach zero, we obtain a system of differential equations

Let's find the expression for the mathematical expectation of the number of applications in the storage device and the average waiting time for applications in the storage device for the stationary state * p * = X/p <1> 1. Equating the time derivatives and eliminating the time I of the equations, we obtain systems) of algebraic equations *

Suppose that in the first equation n ** 1. Then (1 + p) p, -p _{ a } + pp _{ 0 }. Substituting here the value of p from the second equation, we find * p _{ 2 } PPo - * Repeating these operations,

we get p _{ i } = pp _{ 0 }, and

hack as this is the sum of the probabilities that

there are no applications in the system, there is one application, two applications, etc. The number of these probabilities must be equal to one, since all possible

System state. Therefore

or

where p _{ 0 } & quot; 1 * -P - * Therefore, p "= p & quot; (1-p).

The resulting expression is a geometric distribution. Mathematical expectation of the number of applications in the system (device),

Note that/"is the average value and there may be fluctuations in the number of requests waiting for maintenance, which can be estimated using variance:

In this case

Therefore,

The mathematical expectation of the number of applications in the drive,

Average wait time of requests in the drive

The possibilities for evaluating characteristics using analytical models of queuing theory are very limited in comparison with the requirements of the practice of research and design of systems formalized in the form of*Q-schemes.*Imitational models that allow to investigate < i> Q-scheme, given by

*Q = (N, U, H,*Z, V,

*R*,

*A}*Without restrictions, many simulation languages are oriented to work with

_{ t }*Q-schemes*in machine implementation of models, for example,

*SIMULA, SIMS CRIPT, GPSS*, etc. In detail, the issues related to simulation simulation of g-schemes will be discussed below.

*
*

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