CC BY
23
УДК 681.3
Svetlov1 M.S., Lvov2 A.A., Svetlov3 I.M., Mishchenko2 D.A., Vagarina2 N.S., Svetlova2 M.K.
1Institute of Precision Mechanics and Control of RAS, Saratov, Russia
2Yuri Gagarin State Technical University of Saratov, Saratov, Russia
3Deutsche Telekom (DTAG) Branch in Darmstadt, Hessen, Darmstadt, Germany
MATHEMATICAL MODEL OF THE CONTROL SYSTEM FOR NETWORK WITH RECOVERY
The paper presents the mathematical model of control system for information network recovery based on the theory of queuing systems and semi-Markov processes. Basic functions of the network recovery system are the processing of messages about failures of the main and backup devices, recovery of devices after failures rehabilitation, switching on of devices after recovery of their working capacity. The algorithm of the control system for network with recovery is given. Sequentially options of a failure of switching points for system from several devices are considered. The transition graphs showing a system behavior in case ofa failure of one or several terminals are constructed. Calculation formulas of key parameters of a network, including the average time of duration of stay of system in different statuses, recovery time of a network are synthesized.
Keywords:
INFORMATION NETWORK, QUEUING SYSTEMS, SYSTEM CONTROL FOR INFORMATION NETWORK RECOVERY, SEMI-MARKOV PROCESSES
INTRODUCTION
The tendency of development of the modern information and computer technologies is characterized by the active implementation practically in all fields of activity of the network structures representing multi-channel information distributed systems of the most different functional purpose. One of the most important purposes of creation of distributed systems to which computer networks belong is achievement of high reliability [1].
The various character and complexity of software and hardware tools of information and communication networks require creation of qualitative algorithms of their functioning and control of them for the purpose of support of high rates of working capacity. These indices include data capacity and volume of the broadcast and processed information, noise immunity, degree of protection of information, including from illegal access, high-speed performance and reliability of the long and trouble-free operation, ability of self-recovery.
A specific place in structure of managing directors of a network of systems is held by the control system for the network with recovery (CSNR) which is an important and most significant element of support of reliable and smooth operation of information and communication networks. Each information network contains the mains and standby functional devices, the systems of a network shall exercise control of these devices and create error messages in their operation, provide switching of devices and their recovery.
Despite numerous options of the models of information networks [2-6] described in literature, the universal mathematical model which is adequately describing all modes, features and subtleties of functioning of information networks is not offered so far and there are no qualitative models of separate structural elements of information networks considering specific features of operation of network management systems. In particular, it concerns also CSNR.
In this report the possible principles of simulation of networks are considered and the option of a mathematical model with detail accounting of specifics of CSNR is offered. For determinacy we will stop on the principles of operation of CSNR in relation to switches (switching points) of a network on the example of one switching point.
Such indices of reliability as probability of a failure, failure density are considered. The difficult systems consisting of many elements except statuses of working capacity and nonservice ability can have also other intermediate statuses which these characteristics do not consider. In this regard another is applied to reliability assessment of difficult systems additional characteristics, for example, availability.
THE CONTROL SYSTEM FOR THE NETWORK WITH RECOVERY
In case of a research of difficult computing systems the mathematical apparatus of the theory of the queuing systems (QS) allowing solving successfully problems of the analysis and synthesis of such systems is traditionally used [7]. However, models of queuing systems in case of their relative simplicity have some and very essential shortcomings. It reduces the value of such models of assumption about the exponential distribution law of density of probabilities in relation to key parameters of system and independence of functioning of its elements. Besides, if we receive days off of characteristics of system as estimated results of simulation only of mean values, then it also restricts use of such models [8]. In most cases process of recovery is considered as Markov process when some important time response characteristics of system are not considered [9].
The conducted researches directed to extension of opportunities of mathematical models of a class of the modelled systems allow offering more complex structure of model with use of accidental processes of semi-Markov type.
We will consider the mode of a failure of one switching point. We will read that the system consists of two devices: one (main) device works, and another - in a reserve status. Let Ai and A2 - failure density of the main and reserve terminals, respectively, and Ai > A2.
If in some time point the main working terminal failed, then the reserve terminal instantaneous gets into gear. At the same time it is considered that reliability of switching devices conforms to all necessary requirements and is conditionally absolute.
Let /Ji and ^2 are the recovery intensities of the main and reserve terminals, respectively, the recovery being started at once in case of failure occurrence. We assume that all the random variables that determine the functioning of the system are distributed according to exponential law. So, ¡(t) u rj(t) are random functions of changing the durations of fail-safe operation and recovery terminals, respectively that are fair expressions:
PH< t} = 1 - e~x; P{r< t} = 1 - e
For simplicity in formulas the argument of t is lowered in designations of functions of time. As criterion of quality of operation of CSNR we will select the average time of trouble-free operation. We introduce the set E of states of the system. For this case, two terminals: E = (eo, ei, e2, e3, et].
Let e0 is a status when both terminals are serviceable (the first is in operation mode and the second is in reserve one) ; ei is a status when one terminal failed, being in operation, and another, being in a reserve, joined the work; e2 is a status when the terminal which is in a reserve failed, and other terminal is operational and put into operation; e3 is a status
when both terminals are recovered after failures: the first is in operation mode and the second is in reserve one; e4 is a status when both terminals are recovered after failures during an operating time.
We will break space E statuses into not crossed subspaces (subsets) corresponding to working (E1) and not working (E2) to system statuses: Ei = {eo, ei, e2}; E2 = {e3, e4}.
The transition graph of system is shown in the figure 1.
For i=0, 1, 2 we will receive:
^ = »00 + fmT + P02T2 ; = »01 + Pi0r0 ; = »02 + P20V
We will add values of probabilities in a system of equations and solving it concerning r0 :
- 1
Г" = Л
(Л +' )(Л +') Л (Л+')+Л(Л +' ).
+
In that specific case,
when
'i='2=':
— 1
r"= Л
2Л +Л + ' Л+Л
For a case of Ài=À, X2-2Л+' Л1 '
:0 and '1=' we have:
Figure 1 - Transition graph of semi-Markov model of an information network
We will define the following important model parameters #.(#2) and 71.(72). These are random functions of changing the durations of failure-free operation and recovery of the main (reserve) terminal, respectively, with distribution functions:
F(#) = 1- eA = P{# < t} ; F(#) = 1 - e'* = P{#2 < t} ;
F(t) = 1 - e-»-1 = P{»< t} ; F(t) = 1 - e-n' = P{»< t} .
Moreover, stay duration do ^ O4 of a network in statuses eo+ e4 are defined either. The system which is able e0 is to first failure the main or reserve terminals, i.e. do = min {#1, #2}. Time of stay in a status of e1 is defined by error-free running time of the working terminal and duration of recovery of the failed working terminal, more precisely, the smallest of them, i.e. 01 = min{#1, TJ1}. We will similarly receive: d-2 = min{#1, T2}; 03 = min{T1, 72}; 04 = min{T1, 71}. Owing to independence of random variables #1 and #2 it is had:
- (A+A2)t
P{min {#, #2} > t} = P{# > t}P{#2 > t} = 1 - e y1 2J _
We will define distribution functions of random variables do ^ 04:
- (*+A2) t
G0(t) = P {d0 < t} = P{min{#1, #2} < t} = e 1 2' ;
G(t) = 1 - e ^^t ; G2(t) = 1 - e-(A+»2)t ; G(t) = 1 -e c"1+»2)' ; G4(t) = 1 -e~2f1 .
The mean values of stay of a network in different statuses:
M[00] = —!— ; Md = —!— ; M0] = —!—;
\+» \+»l
M[0] = M[04] = -L .
»+» 2»
We will defi ne probabil ities of transitions of a network from one status in another:
P01 = p#<#} = * ; P02 = P{#2 <#1} = ;
A+A A+A
P0 = P{T<#}=»; P20 = PT2 <#1}
A+» A+»
P23 = p{#<T2}=yA—; P4 = P{#<T}— •
Average time zi stay of system in some subset of statuses E* will be defined as
*
r,= M[0] +Z P,'T,, jeE .
We received a formula of determination of average time of trouble-free operation of switching point. Now we will consider the mode of failure in the functioning of all networks from three subsystems: access networks, backbone network and information center (command center).
Let A and » be the failure rates and the recovery rate of each subsystem. The main characteristic in this case is the probability of failure-free operation of the network KT.
The availability quotient of a network depends on number of the recovering devices. We will consider cases when there is one, two or three the recovering devices. We will enter the following state space of system:
E = {e0, e1, e2, e3} ,
where e0 is the state when all three subsystems are operable; e1 is the state when two subsystems are operable, and one is faulty; e2 is the state when one subsystem is operable, and two are faulty; e3 is the state where all subsystems are faulty.
The network is operable if all subsystems are in order. Consequently, the subset of states is operable E = {e0} and subset of failure states
E2 = {e1 > e2> e3} •
Transition graph of semi-Markov model of system with consecutive connection is shown in the
Figure 2.
Figure 2 - Transition graph of semi-Markov model of system with consecutive connection
We will define time of stay of information network in different states. Let i = 1,2,3 is a random variable equal to a period of a time between failures «i» systems, 77 is the random
variable equal to restoration duration by means of «i» device.
Functions of distribution of random variables ¿¡i and t]j have the form:
F (Ç) = P[4i< t} = 1 - e-
F (ц) = Р{ц < t} = 1 - e-'
The network is in the eo to first failure one of three subsystems: 6a= min{H2,.
Time of stay in the state ei is defined by time of a time between failures of two subsystems and time of recovery of one subsystem:
0i = mn{Çi¿27I} •
The same way we will receive that time of stay in a condition of e2 is defined by duration of a time between failures and recovery of one subsystem (at one recovery device) or duration of a time between failures of one and recovery of two subsystems (at two recovery devices):
r„ =
We will consider that expression is fair:
<= iFi(t)dF(2(t) = JF(i(t)f (f)dt. 0 0
probabilities of transitions take a
2 \min {41,V1,V2}-
Time of stay in a condition of e3 is defined by time of recovery of one (at one recovery device), or two (at two recovery devices) , or three (at three recovery devices):
K
6^= •min {^ } 1 tnm {ri^nT}}-
We will define functions of distribution of duration of stay in conditions of eo and ei:
G0(t)=P{min{fi, ?3}<t} = 1-e-3At; Gi(t) = 1-
e-(2At +p)t_
Functions of distribution of duration of stay in a condition e2 at one and at two or three recovery devices:
[l _e~(^+M)t. (t) = [
2 [1 _e~a+2M)t .
Functions of distribution of duration of stay in a condition of e3 at one, either two, or three recovery devices:
4.
Then form:
P + ; P12 = 2A/(2A + rf) .
The case of one or two and three recovering devices is considered:
2p/(A+2p). 23 ~\A/(A+2v).
The availability quotient will be determined by a formula:
T
Kr =-T-T +T
T0 +T В
where To=pi0o= 1/3A
or = T„ /(T„ +Tt) ,
time per fault of a net-
Оз(г) =
1-e-
1-e-2И ; 1-e-3*.
work; - time of recovery of a network (Tb),
equal time of stay in a subset of statuses of failures E'2.
Now we will define ^ :
Solving system in case of one recovering device we will receive:
Average values of duration of stay in conditions of eo and ei will register in a look:
M\Qa] = 1/3A ; M[] = 1/(2A + M) .
Average values of duration of stay in a condition of e2 at one or two and three recovery devices:
2 A2 + 2Ац+цг
И
M [02] =
\1/(А+И),
~\l/(A+2m)
Average values of stay duration in a condition of e3 at one, either two, or three recovery devices are M[&] = 1/m, M[&] = 1/2m, M[&] = 1/3m respectively.
Now we will define probabilities of transitions:
Pm = P32 = 1 ; P = <min{,
P2
It is probability that the refusal of the working subsystems will happen before recovery of the refused subsystems. We will receive in cases of one or two and three restoring devices:
= 1- P = P{mini^} <H ,
is probability that
Pi Ч
[P{min{ni,V2}<^}.
P = 1 - P =
-}1 A 1 11
IPH <z},
lP{ê <min{H h}}.
B rf 6A3 +6A2H+3Afj.2
In case of two recovering devices:
T = A2 + 2Arf+. K = _
B ; r 3A3+6Ap2 +6A2^+2fi '
In case of three recovering devices:
T =aa+A+3IL. k J-eJ
B ; r \A+p) '
CONCLUSIONS
The provided mathematical model allows defining all required characteristics of an information network taking into account possible operation modes of CSNR. Similar approach can be applied also to other structural systems and subsystems of an information network.
It is necessary to tell that the experimental data obtained in case of a research of program model on the basis of the offered mathematical model almost entirely matched calculated data for the switching center of an information network in case of support of probability fault-free (or taking into account recovery) operations at least 0,999999. The other applications of the proposed techniques are given in [10-12].
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Т.=цв.+ P, г
Ï2 =И02 + P21Г2 + P23r3
Т3 =И0з + P32r2
