on reliability of binary systems in a random environment

ISSN 0146�4116, Automatic Control and Computer Sciences, 2013, Vol. 47, No. 6, pp. 342–351. © Allerton Press, Inc., 2013.Original Russian Text © V.V. Rykov, Tran Anh Nghia, 2013, published in Avtomatika i Vychislitel’naya Tekhnika, 2013, No. 6, pp. 73–85.

342

1. INTRODUCTION

Most technical systems and biological objects operate in a changing environment, the changes beingof both regular (seasons) and random nature; their frequency can be as well as commensurable with failurerates of the system or to be greater or less than it. The way that reliability of the system depends on thesefactors is of significant interest, given rapidly changing technical capabilities of the modern world. In thepaper the influence of the environment variability to the stationary characteristics and the reliability func�tion is studied.

There are a number of papers devoted to studying the behavior of queuing systems operating in a ran�dom environment. The paper of M. Eisen and M. Tainiter [1] was one of the first publications on this sub�ject, where they studied the M/M/1(ME) system (here and in the following ME denotes Markov environ�ment and means that the respective system operates in a random Markov environment), assuming that theenvironment can take two states only. The same system was studied in [2] and then generalized in [3] foran arbitrary finite number of states of the environment. M. Newts [4] used the matrix�analytical approachto study the behavior of single and multi server systems in a random environment. Then, the M/M/1(ME)and M/M/∞ (ME) models have been studied in [5, 6]. As for further research on the subject, the behaviorof queuing systems has been spread out in different directions involving generalization of the input flowmodels, the queuing scheme, and the structure of the random environment. One can find a rather detailedoverview of up�to�date publications in, for instance, [7, 8]. However, the reliability of systems operatingin a random environment has received insufficient attention thus far. In this paper, we study the reliabilityof the binary Markov system operating in a random Markov environment.

To shade the structural side of the issue and to avoid superfluous technical difficulties, we consider thesimplest binary model of a cold redundancy system with only one repair device that operates in a randomMarkov environment with a finite number of states. Note that generalization for hot and warm redun�dancy system with several recovery devices does not entail any principal difficulties and only involveschanging parameters of the model. Studying heterogeneous reliability systems operating in a randomMarkov environment implies extending significantly the phase space that describes the behavior of thesystem of random processes and, hence, applying more complicated computational algorithms and pro�cedures. Examples of how the model involved can be used to study a hybrid communication system aregiven in [9, 10].

The paper is structured as follows. The next section describes the model of the system involved and therandom process of its operation. In the third and fourth sections, we give the Kolmogorov equations forprobabilities of system states and the procedure of calculating stationary probabilities. The fifth sectiondeals with calculating the reliability function for the system. Finally, the last section gives the results ofnumerical analysis of the model for a system of two elements n = 2 that operates in a random environmenttaking two possible states m = 2. We complete the work with the conclusions and references.

On Reliability of Binary Systems in a Random Environment V. V. Rykov and Tran Anh Nghia

Peoples’ Friendship University, ul. Miklukho�Maklaya 6, Moscow, 117198 Russiae�mail: [email protected], [email protected]

Received April 29, 2013; in final form, July 15, 2013

Abstract—The influence of random environments to technical systems reliability is studied. A Markovmodel of reliability of the system that operates in a random Markov environment is proposed. Generalrelations for stationary and non�stationary Quality of Service (QoS) characteristics of such system aregiven. Numerical study and comparison for a cold backup system operating in stable and random two�state environments are performed.

Keywords: reliability of the system, random environment, Markov process

DOI: 10.3103/S0146411613060096

AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 47 No. 6 2013

ON RELIABILITY OF BINARY SYSTEMS IN A RANDOM ENVIRONMENT 343

2. THE MODEL

First, we consider a model of a general cold redundancy system that consists of n heterogeneous ele�ments and operates in a random environment taking m values. The states of such a system can be describedby (n + 1)�dimensional vectors x = (i, j) = (i, j1, …, jn), the first component i of which describes the states

of the environment and takes m values (i = while the binary components jk (k =

indicate the states of the system’s elements.We denote the set of states of such a system by

E = {x = (i, j) = (i, j1, … , jn): jk = {0, 1},

The total number of the system states is finite and equals N = m × 2n. We designate the sets of up anddown states by E0 and E1, respectively.

A homogeneous system with the same failure and recovery characteristics for all elements is the sim�plest example of this general model. In this case, states can be enlarged by combining the states with the

same number j = of failure elements; the set of states then takes the form E = {(i, j): i = j = with the total number of states N = m × (n + 1).

Now suppose that changes in the environment are described by a homogeneous Markov process with thefinite number m of states and the generator (Infinitesimal Matrix – IM) Λ = To avoid superfluoustechnical difficulties, we also assume all elements of the system to be homogeneous being exponentiallydistributed times in up and down states with parameters α and β, while appropriate parameters for up anddown time distributions of the elements that operate in the state i�th environment are denoted by αi andβi. It is supposed that when the environment changes its state, the elements of the system instantly changetheir failure and recovery intensities. The behavior of such a system is described by the two�dimensionalMarkov process

X(t) = (I(t), J(t)),

with its first component taking m values and describing the states of the environment, while its secondcomponent J(t) describes the states of the system (the number of down elements) and takes n + 1 valuesfrom the set {0, 1, …, n}. By our assumptions, the system can move only to “neighboring” states; i.e., itssecond component is a birth�and�death process (BDP).

In the following, we use matrix designations and understand vectors, as usual, as vector�columns anduse a prime for the transpose operation and an overdot for the derivative. We introduce the following des�ignations:

• Λ = is the IM (generator) of the environment process;

• is the row vector of transition intensities of the environment from the state i

(i =

• λi = is the intensity of change of the i�th state of the environment;

• diag is the diagonal matrix that has the components of the vector on its main diagonal;

• Ak = [ai, j(k)] is the IM (generator) of the process when the system operates in the k�th environment,where

• x = (i, j) are the states of the system that are arranged in the following lexicographic order:

E = {(1, 0), (1, 1), … (1, n), … (m, 0), (m, 1), …, (m, n)}.

In the following the notation i(x) and j(x) for the components i and j of the vector x = (i, j) is used.

,1, )m ,1, )n

if the �th element is up,

if the �th element is down,

0,

1,k

kj

k⎧

= ⎨⎩

1, ,i m= .1, }k n=

1

n

kkj

=

∑ 1, ,m 0, }n

.,[ ]i kλ

,[ ]i kλ

( )λ = λ λ λ�

,1 ,2 ,' , ,..., 'i i i i m

;1, )m

,1 i kk m≤ ≤λ∑

'iλ�

'iλ�

{ },, 0,...,i j n∈

for

, for

for,

, 1,

( ) ( ) ,

, 1;

k

i j k k

k

j i

a k j i

j i

α = +⎧⎪

= − α + β =⎨⎪β = −⎩

344


RYKOV, TRAN ANH NGHIA

Under these assumptions, the process X(t) = (I(t), J(t)) is a two�dimensional Markov process with thestate space E and the block�wise IM Q = [Qx, y], its diagonal blocks Qx, x for i(y) = i(x) having the form ofthe three�diagonal matrices Qx, x = Ai(x) – λi(x) I and its off�diagonal blocks Qx, y having the form Qx, y =λj(x), j(y) I, where I is the identity matrix,

(1)

We give some comments on the above. (1) For an unchanging (stable) environment, the respective process becomes an ordinary BDP that

describes operation of a homogeneous cold reservation system.(2) For a heterogeneous system with elements of different failure and recovery intensities, one needs to

expand the state space of the system, while leaving the IM the same structure.

3. KOLMOGOROV EQUATIONS

We use = = to denote state probabilities of the process X(t) at thetime t and its initial distribution (there were no failure elements in the very beginning), where δ is the Kro�necker symbol. Suppose is the vector of state probabilities = where the sub�vec�

tors describe probabilities of the system’s states when it operates in the i�th environment. We use =(a1, …, am) to denote the initial distribution of the environment and = (1, 0,…, 0) to denote the vectorof dimension n + 1 that corresponds to the completely fault�free state of the system when all its elementsare up states. Then, taking into account the structure of the matrix Q, we can represent the system of Kol�mogorov differential equations for state probabilities of the process X(t) with the initial condition

(2)

as the system of equations that correspond to operation of the system in various environments

(3)

Multiplying the equations of this system by the vector�column consisting of unities = (1, …, 1)' from

the right, using the designation = and noting that Ak = we obtain the system of equationsfor the environment state probabilities

(4)

In terms of Laplace transforms = the system (2) with appropriate initial condition,

takes the form of the system of algebraic equations

(5)

which solution have the form

(6)

The latter expression has the form of a fractional rational vector function with respect to the variable s;which inversion allows to find the non�stationary distribution of state probabilities. We can use the latterexpression as well as initial system of equations (3) for numerical analysis.

4. STATIONARY CHARACTERISTICS

Steady state (stationary) probabilities of the system πj = (they exist since the process is irre�

ducible and has finite number of states) coincide with the limit probabilities and satisfy to the system of

1 1 1,2 1,

2,1 2 2 2,,

,1 ,2

...

...[ ] .

... ... ... ...

...

m

mx y

m m m m

A I I I

I A I IQ Q

I I A I

− λ λ λ⎡ ⎤⎢ ⎥λ − λ λ⎢ ⎥= =⎢ ⎥⎢ ⎥λ λ − λ⎣ ⎦

, ( )i j tπ { },( ) ( , )X t x i j= =P , (0)i jπ ,0i ja δ

( )tπ

�

'( )tπ

�

,1' '( ( ),..., ( ))mt tπ π

� �

'( )i tπ

� �

'a

0'e�

' ,(0)π

�

' ' , ' 1 0 0' '( ) ( ) (0) ( ,..., )mt t Q a e a eπ = π π =

� � � � ��

., 0' ' ' ' '( ) ( )( ) ( ) , (0) , ( 1, )k k k k i i k k k

i k

t t A I t I a e k m≠

π = π − λ + π λ π = =∑� � � � ��

1�

' ( )1k tπ

�

�

( )kp t 1�

0,�

.,( ) ( ) ( ) , (0) , ( 1, )k k k i i k k k

i k

p t p t p t p a k m≠

= −λ + λ = =∑�

( )j sπ� ( )stje t dt−

π∫

' ' ' ,( ) (0) ( )s s s Qπ − π = π

� � ��

' ' 1( ) (0)( ) .s sI Q −

π = π −

� ��

lim ( )jt

t→∞

π



equilibrium or global balance equations (GBE). For the vector of stationary probabilities = with the additional normalization condition, this system has the form

(7)

Similarly to (3) these equations can be represented as

(8)

Multiplying last equations by the vector�column consisting of unities from the right and using notation

= pk, one obtains the system of equations for stationary state probabilities of the environment

(9)

which jointly with the normalization condition = 1, allows to find its unique solution.

Taking into account the relation λk = – for the numerical solution, it is convenient to represent thesystem of equations (8) in the form

(10)

We use the steady state probabilities in order to calculate various stationary characteristics of the systemconsidered in the numerical analysis section, namely,

• the stationary probability of operation in the i�th environment pi =

• the probability of failure of the system operating in the i�th environment πi, n,

• the probability of failure of the system πfail =

Note that one can use the results of [11] to obtain some explicit expressions.

5. RELIABILITY FUNCTION

In any state of the environment, the system under consideration fails when all its elements fail, i.e., the

set of faulty states of the system E1 has the form E1 = We use T to designate the life time of thesystem that is the destination time of the component J(t) of the process X(t) to the set E1,

T = inf {t: J(t) ∈ E1}.

To calculate its cumulative distribution function (c.d.f.) F(t) = P{T ≤ t}, we study the respective processwith the set of failure states E1 as an absorbing set of states. Let us represent the IM (generator) Q, the vec�tor of state probabilities and the vector of initial states in the block–wise form

(11)

where the blocks of the matrix with the indices 0 and 1 correspond to transitions of the process from the

set of states E0 to the set E1 and back. Setting = = 0, one can reduce the system of equations (3)to the form

(12)

In terms of the Laplace transform, taking into account the initial condition, this system can be repre�sented as

which solution is

'π�

{ , }j j Eπ ∈

' ' .0, 1 1Qπ = π =

�

� �

,,

1

' ' '( ) 0, ( 1, ) 1 1.k k k i i k k

i k k m

A I I k m≠ ≤ ≤

π − λ + π λ = = π =∑ ∑�

� � �

' 1kπ�

�

, 0,k k i i k

i k

p p≠

−λ + λ =∑

1 ii mp

≤ ≤∑

, ,k kλ

,

1

' ' '0, 1 , ( 1, ).k k i i k k k

i m

A p k m≤ ≤

π + π λ = π = =∑�

� � �

'1,iπ�

�

,1.i ni m≤ ≤

π∑

1( , ).

i mi n

≤ ≤∪

' ,( )tπ

�

' , , ,0 1 0 1

0,0 0,10 0, 0,

1,0 1,1

' ' ' ' ', ( ) ( ( ) ( )) ( , )E E E E

Q QQ t t t e e e

Q Q⎡ ⎤

= π = π π =⎢ ⎥⎣ ⎦

� � � � � �

1,0Q10,' Ee

�

0 0 1 00,0 0,1' ' ' '( ) ( ) , ( ) ( ) .E E E Et t Q t t Qπ = π π = π

� � � ��

0 0 0 1 1 00, 0,0 0, 0,1' ' ' ' ' '( ) ( ) , ( ) ( ) ,E E E E E Es s e s Q s s e s Qπ − = π π − = π

� � � � � ��

346



The life time c.d.f. has the form

(13)

while its Laplace transform is =

Hence, we can represent the moment generation function of the system life time = = as

(14)

Taking into account structural properties of the matrix Q, the latter expression allows for a moredetailed representation, which will be illustrated with an example in the next section. Moreover, since ithas the form of a fractional rational function with respect to the variable s, by its inverting one can find thedensity and c.d.f. of the system life time.

6. NUMERICAL ANALYSIS

For numerical analysis, we restrict ourselves to the system with only two elements (n = 2) operating ina random environment with two states (m = 2). It is supposed that the system operate in cold standbyredundancy regime which means that the only one (principal) element of two elements is operating, whilethe second one starts to operate once the first one fails. There is only one recovery device, which meansthat, if both elements fail, the system is in the failure state and only one of the elements is recovered, whilethe second one waits for recovery. To compare it to the system operating in a nonrandom (stable) environ�ment, note that the system’s operation in this case is described by the birth�and�death process, with itsequations for state probabilities having the form

(15)

The solutions of these equations are the non�stationary distribution of state probabilities that convergeto stationary probabilities when t → ∞. These probabilities can be found from the system of equilibriumequations and are well known (see, for instance, [12]):

(16)

where ρ = αβ–1.

The reliability function R(t) = 1 – F(t) can be found using the solution of system (15), in which thestate 2 should be considered as an absorbing state, and in accordance with [12] equals to

(17)

where s1 and s2 are the roots of the characteristic equations of system (15) taken, for the sake of clarity, withthe minus sign

(18)

To study the system’s behavior in a random environment, the model of the two�dimensional Markovprocess X(t) = {I(t), J(t)} is used, the first component of which describes of the environment behavior,while the second one describes the system’s states, i.e., the number of its faulty elements. Figure 1 showsthe transitions graph of the process X(t).

0 0 1 0

1 10, 0,0 0, 0,0 0,1

1' ' ' '( ) ( ) , ( ) ( ) .E E E Es e Is Q s e Is Q Qs

− −

π = − π = −

� � � ��

1

1

'( ) { } ( ) ( )1,j E

j E

F t T t t t∈

= ≤ = π = π∑P�

�

( )F s�

1' ( )1.E sπ

��

( )f s� M sTe − ( )sF s�

1 0

10, 0,0 0,1' '( ) ( )1 ( ) 1.E Ef s s s e Is Q Q−

= π = −

� ��

,

,

.

0 0 1

1 0 1 2

2 1 2

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

t t t

t t t t

t t t

π = −απ + βπ⎧⎪π = απ − α + β π + βπ⎨⎪π = απ − βπ⎩

�

�

�

fail

2

0 1 22 2 21 , , ,

1 1 1

ρ ρπ = π = π = π =

+ ρ + ρ + ρ + ρ + ρ + ρ

12

12

1 2

( ) 1 ( ) 1 ,s t rtsR t t e e

s r s− −

⎛ ⎞α= − π = −⎜ ⎟⎝ ⎠

,1,2 (2 1 1 4 )2

sβ

= ρ + + ρ∓

12 1 1 4 , .r s s −

= − = β + ρ ρ = αβ



The IM (generator) Q that corresponds to thistransitions graph has block�wise diagonal form,which blocks correspond to various states of theenvironment:

1,0 1 1,2

1 1,1 1 1,2

1 1,2 1,2

2,1 2,0 2

2,1 2 2,1 2

2,1 2 2,2

0 0 0

0 0

0 0 0,

0 0 0

0 0

0 0 0

Q

−γ α λ⎡ ⎤⎢ ⎥β − γ α λ⎢ ⎥

β − γ λ⎢ ⎥= ⎢ ⎥λ − γ α⎢ ⎥⎢ ⎥λ β − γ α⎢ ⎥

λ β − γ⎣ ⎦

where the variables γi, j equal the sum of all elements of the corresponding row of the matrix. The states {1, 2}and {2, 2} are the failure states of the system, respectively, when the system operates in the first and secondstate of environments.

Using the system of equations (3), one can calculate the non�stationary state probabilities of the reli�ability model operating in a random environment, while its stationary state probabilities, according to theresults of Section 4, can be found by the system of equations (10) solution. The formulas (14), allow tocalculate the system’s reliability function.

To find out how the system’s reliability and the rate of convergence to the stationary mode depend onthe random nature of the environment, we performed a series of computational experiments. To imple�ment them and represent the results in graphical form, a MATLAB module has been developed. Theresults of this experiments are given below.

To compare the reliability characteristics of the system operating in random and nonrandom (stable)environments, we need to match the respective failure and recovery parameters of the system’s elements.We set all parameters for the random environment and used the following averaged values for the stableenvironment:

(19)

In the figures below, we use the following designations for the graphs for different values of the param�eter k that gives the relation of failure intensity of the element operating in the first and second states ofthe random environment:

Designations used for curves for different values of the parameter k

Option 1. The homogeneous change of the environment λ1, 2 = λ2, 1; the failure and recovery intensitiesare comparable to the rate of change of the environment: α1 = 1, α2 = kα1, β1 = β2 = 1.

Table 1 gives the respective stationary probabilities of the system’s failure. Figures 2–5 give the graphs of the system availability and reliability functions in the stable and random

environments. The results show that the respective characteristics for the system operating in the stable

k = 0.1 k = 0.5 k = 1.0 k = 5.0

�� ............... –.–.–.–.–

2,1 1,2 2,1 1,21 2 1 2

1,2 2,1 1,2 2,1 1,2 2,1 1,2 2,1

, .λ λ λ λ

α = α + α β = β + βλ + λ λ + λ λ + λ λ + λ

Table 1. Stationary probabilities of the system’s failure for Option 1

k In the stable environment π2 = πfai

In the random environment

π12 π22 πfail = π12 + π22

0.1 0.1633 0.1067 0.0592 0.1659

0.5 0.2432 0.1352 0.1066 0.2418

1.0 0.3333 0.1667 0.1667 0.3334

5.0 0.6923 0.2607 0.3721 0.6328

β2

α1 α1

β1 β1

β2

α2 α2

λ1,2 λ2,1 λ1,2 λ2,1 λ1,2 λ2,1

1.0 1.1 1.2

2.0 2.1 2.2

Fig. 1. Transitions graph of the process X(t) = {I(t), J(t)}.

348



and random environments are fairly close, while they coincide for k = 1 as expected. As the failure inten�sity of elements—when the system operates in the random environment (state k = 2)—grows, the rate ofconvergence of non�stationary characteristics to stationary characteristics increases for both stable andrandom environments.

Option 2. The homogeneous change of the external environment λ1, 2 = λ2, 1 for “fast recovery” β1 = β2 =100; the failure intensities are comparable to the rate of change of the environment: α1 = 1, α2 = kα1.

Table 2 and Figs. 6–9 give results similar to the previous ones. The results also show that the charac�teristics of the system’s operation in the stable and random environments are fairly close; however, “fastrecovery” leads to fast convergence of non�stationary state probabilities to stationary ones. As before, asthe variable k grows, the rate of convergence of non�stationary characteristics to stationary ones increasesfor both stable and random environments. This deserves special attention.

Option 3. Slow homogeneous change of the environment λ1, 2 = λ2, 1 = 0.01; the failure and recoveryintensities are comparable: α1 = 1, α2 = kα1, β1 = β2 = 1.

Table 3 and Figs. 10–13 give the calculation results for the systems operating in the stable and randomenvironments.

1.00.90.80.7

0.2

5.04.53.50 4.00.5 1.0 1.5 2.0 2.5 3.0

0.60.50.40.3

0.1

t

Fig. 2. System availability function (1 – πfail(t)) in thestable environment.

1.00.90.80.7

0.2

5.04.53.50 4.00.5 1.0 1.5 2.0 2.5 3.0

0.60.50.40.3

0.1

t

Fig. 3. System availability function (1 – πfail(t)) in therandom environment.

1.00.90.80.7

0.2

10970 81 2 3 4 5 6

0.60.50.40.3

0.1

t

Fig. 4. The system’s reliability function R(t) = 1 – πfail(t)in the stable environment.

1.00.90.80.7

0.2

10970 81 2 3 4 5 6

0.60.50.40.3

0.1

t

Fig. 5. The system’s reliability function R(t) = 1 – πfail(t)in the random environment.


k In the stable environment π2 = πfail (10–5)In the random environment

π12 (10–5) π22 (10–5) πfail = π12 + π22 (10–5)

0.1 3.0084 4.8588 0.10192 4.96072

0.5 5.5828 4.8898 1.2919 6.1817

1.0 9.9000 4.9500 4.9500 9.9000

5.0 87.302 6.2481 120 126.2481



0.99980.99970.9996

0.9991

0.10 0.080.02 0.04 0.06

0.99950.99940.99930.9992

0.9990

t

0.99991.0000



k In the stable environment π2 = πfail

In the random environment

π12 π22 πfail = π12 + π22

0.1 0.1633 0.1647 0.0063 0.17100.5 0.2432 0.1657 0.0726 0.23831.0 0.3333 0.1667 0.1667 0.33345.0 0.6923 0.1687 0.4025 0.5712

0.9992

0.99863.00 2.00.5 1.0 1.5

0.99900.9988

t

0.9994

1.00000.99980.9996

2.5


1.00.90.80.7

0.2

10000 800200 400 600

0.60.50.40.3

0.1

t


1.00.90.80.7

0.2

10000 800200 400 600

0.60.50.40.3

0.1


1.00.90.80.7

0.2

5.04.53.50 4.00.5 1.0 1.5 2.0 2.5 3.0

0.60.50.40.3

0.1

t


1.00.90.80.7

0.2

1000 8020 40 60

0.60.50.40.3

0.1

t10 30 50 70 90


t

In this case, the system’s behaviors in the stable and random environments also turn out to be fairlyclose. However, for slow homogeneous change of the environment, the nature of convergence of non�sta�tionary characteristics to stationary ones and the behavior of the reliability function in the random envi�ronment significantly differ from the respective characteristics for systems operating in the stable environ�ment.

350



1.00.90.80.7

0.2

10970 81 2 3 4 5 6

0.60.50.40.3

0.1

t


1.00.90.80.7

0.2

10970 81 2 3 4 5 6

0.60.50.40.3

0.1

t


0.99980.99970.9996

0.9991

0.100 0.080.02 0.04 0.06

0.99950.99940.99930.9992

0.9990

t

0.99991.0000


0.99980.99970.9996

0.9991

0.100 0.080.02 0.04 0.06

0.99950.99940.99930.9992

0.9990

t

0.99991.0000


1.00.90.80.7

0.2

10000 800200 400 600

0.60.50.40.3

0.1

t


1.00.90.80.7

0.2

10000 800200 400 600

0.60.50.40.3

0.1

t



k In the stable environment π2 = πfail (10–5)In the random environment

π12 (10–5) π22 (10–5) πfail = π12 + π22 (10–5)

0.1 3.0084 2.3813 1.2902 3.6715

0.5 5.5828 3.3056 2.4806 5.7862

1.0 9.9000 4.9500 4.9500 9.9000

5.0 87.302 36.916 62.462 99.378

Option 4. Fast homogeneous change of the external environment λ1, 2 = λ2, 1 = 100 for “fast recovery”β1 = β2 = 100; the failure and recovery intensities are comparable: α1 = 1, α2 = kα1.

Table 4 and Figs. 14–17 give results similar to the previous ones.



The results also show that the characteristics of the system’s operation in the stable and random envi�ronments are fairly close; however, “fast recovery” leads to fast convergence of non�stationary state prob�abilities to stationary ones. With increasing of the elements failure intensities, when the system operatesin the second state of the environment, the rate of convergence of non�stationary characteristics to thestationary ones also increases for both stable and random environments.

7. CONCLUSIONS

The general Markov model for reliability of systems operating in a random Markov environment hasbeen proposed. Relations for calculating of stationary and non�stationary reliability characteristics of suchsystems has been given. Numerical study and comparison of reliability characteristics for a cold redun�dancy system operating in the stable and random two�state environments has been performed. The resultsof numerical study represented in the form of tables and graphs showed both common features and differ�ences in how systems operate in random and stable environments.

REFERENCES

1. Eisen, M. and Tainiter, M., Stochastic variations in queuing processes, Operat. Res., 1963, vol. 11, no. 6,pp. 922–927.

2. Yehiali, U. and Naor, P., Queuing problems with heterogeneous arrivals and service, Operat. Res., 1971, vol.19,no. 3, pp. 722–734.

3. Yehiali, U., A Queuing�type birth�and�death process defined as a continuous�time Markov chain, Operat. Res.,1973, vol. 21, no. 2, pp. 604–629.

4. Newts, M., A Queue subject to extraneous phase changes, Adv. Appl. Prob., 1971, vol. 3, pp. 78–119.

5. Purdue, P., The M/M/1 queue in a Markovian environment, Operat. Res., 1974, vol. 22, no. 3, pp. 562–569.

6. O’Cinneide, C.A. and Purdue, P., The M/M/∞ queue in a random environment, J. Appl. Probab., 1986, vol. 23,no. 1, pp. 175–184.

7. Kim, C.S., Klimenok, V., Mushko, C., and Dudin, A., The BMAP/PH/N retrial queuing system operating inMarkovian random environment, Comp. Oper. Res., 2010. no. 37, pp. 1228–1237.

8. Kim, C.S., Dudin, A., Klimenok, V., and Khramova, V., Erlang loss queuing system with batch arrivals operat�ing in a random environment, Comp. Oper. Res., 2009, no. 36, pp. 674–697.

9. Rykov, V. and Efrosinin, D., Queuing model of the non�reliable hybrid data transmission channel with hetero�geneous links, Proc. 7th Int. Conf. “Mathematical Methods in Reliability (MMR�2011),” Beijing, China, 2011,pp. 272–279.

10. Vishnevsky, V., Kozyrev, D., and Rykov, V., On the reliability of hybrid system information transmission evalu�ation. Queues: flows, systems, networks (Modern probabilistic methods for analysis of telecommunication net�works), Proc. BWWQT�2013, Dudin, A., Klimenok, V., Tsarenkov, G., and Dudin, S., Eds., Minsk, 2013,pp. 192–202.

11. Andronov, A.M., Markov�modulated birth�death processes, Autom. Cont. Comp. Sci., 2011, vol. 45, no. 3,pp. 123–133.

12. Rykov, V.V., Technical System Safety and Technogenic Risk. Summary of Lectures, Moscow: Ross. Gos. Univ. Oiland Gas, 2001.

Translated by M. Talacheva

on reliability of binary systems in a random environment

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