Case study of the complex systems efficiency

Aneta Antkowiak (Wrocław)

Case study of the complex systems efficiency

Abstract The subject of considerations shall be mathematical models describing the fluctuation of states of complex systems. The purpose of this work is to give a defined system (considering failures, the activation process and possibility of repair), its description along with the basic indications of the multi-level system analysis, and in particular the definition of steady-state probabilities and the performance analy- sis. The modeling tool here are continuous-time Markov chains with finite number of states.

2010 Mathematics Subject Classification: Primary: 90B25; Secondary: 62M25.

Key words and phrases: performance analysis; complex system; continuous-time Markov chains.

1. Preface. The subject of the analysis is the reliability and performance of a set of connected devices. Such issues were investigated by Stanisław Gładysz (v. [7]) in the 1960s as a case study for opencast mines in Turoszów and Bełchatów. A series of articles was then created on this subject, the results of which are summarized by papers [4], [11]. Despite the passage of time, the subject of those works is still valid in the applications of mathematics and in today’s work it is related to the design and maintenance of such engineering objects as mines. Therefore, you can see the need for modeling reliability and system performance, for example: Szybka et al. (2011), Witt (2011).

Chapter 2 presents the basic mathematical theory applied later in the case study analysis, elements of reliability theory, Markov processes and modeling of complex engineering systems, the problem of the efficiency of production systems (mines, mines), and the conservator’s problem in the technical sys- tem. In addition, the problem of using belt conveyors in mining has been described.

Chapter 2.2 is entirely devoted to the analysis of different cases for a double symmetric node on the example of a combination of conveyor belts.

The assumptions and model diagrams are described there. In order to out- line the function of this example technical system, 5 states model have been defined. Switches are the result of random events. For the modeled case, a graph (five states) was given to illustrate the intensity of failures and repairs.

By analysing such stochastic systems, probability limits of the states and

their corresponding efficiencies were determined, and for the case of the same

transition intensities, instantaneous probabilities were obtained by the solu- tion of Kolmogorov’s prospective equations using the Laplace transform. The examples show the convergence of instantaneous probabilities of staying in a given state to limit probabilities. For the nine-point graph, where in addition to failure and repair, both are still included in the work, the limit proba- bilities and the corresponding efficiency were calculated. The derived outputs for the case of the same intensities were compared in the last section.

The summary of the work and its conclusions can be found in the final chapter 5.

2. Models of complex systems. Complex technical systems are de- signed to perform specific tasks, so their description may depend on what functionality you want to get (v. [3], [15]). A system that can have a finite number of states is called the multi-state system. It can be, e.g., a device composed of many parts or a set of devices performing a given job. The im- plementation of each system may be different, but the mathematical model can be the same.

1 1 2 −1

p

₁₁

p

12

p

₂₁

p

₂₂

Figure 1: Graph of transition between states of the renewable one-element system with transition probabilities.

Example 2.1 One of the simpler examples of the system can be a device whose graph of states is shown in fig. 1. This is the discrete time model with random events. The state of work is marked "1" on the scheme. The device may crash with probability p

₁₂

, going to the "−1" state. If it is damaged, it can be repaired with probability p

₂₁

, and thus return to state 1. The device remains in state 1 with probability p

₁₁

and in state −1 with probability p

₂₂

. Each row of elements of the matrix adds up to 1 (we say then that the matrix in question is a row-stochastic one).

When time is continuous then the description of transition between states

is based on the transition intensities (the transition rate) (Fig. 2): λ – for

device failure, and µ – for its repair. The device remains in the operating

state for time h with the probability approximately equal to 1 − λh + o(h) and

analogously, remains in the failure state with probability close to 1−µh+o(h)

- rows in the transition matrix add up to 0.

1 1 2 −1

−λ

λ

µ

−µ

Figure 2: Graph of transition between states of a renewable one-element sys- tem with transition intensities.

In practice, there are many different situations in which a multi-state system can be considered. Numerous examples of case studies can be found in the positions [14], [16].

2.1. The performance of complex systems To analyze the perfor- mance of a complex system another characteristic is necessary which was defined in monograph [16] – performance rate ν

_j

(t) of j-th element at the moment t ≥ 0, which is a random variable that takes values from the set w

j

= {w

j1

, w

j2

, . . . , w

jkj

}, where w

_ji

is the performance level of j-th element in i-th state, i ∈ 1, 2, . . . , k

_j

. However, over the entire system operation time range [0, T ] ν

_j

(t) is a stochastic process.

In some applications, the performance levels of an element can be defined by vectors. Then, the performance of the element is defined as a vector of stochastic processes ν

_j

(t).

The probabilities for the given performance levels of j-th element at the time t can be represented as:

p

j

(t) = {p

j1

(t), p

j2

(t), . . . , p

_jkj

(t)}, (1) where:

p

ji

(t) = P(ν

j

(t) = w

ji

). (2) The expression (2) defines the probability function for the discrete random variable ν

_j

(t) for each moment t, and the pair (w

ji

, p

ji

(t)), i = 1, 2, . . . , k

j

completely define the probability distribution of performance of j-th element at the time t. Production systems usually work at different capacities, whose set level can determine a given state.

When a system is composed of n elements, its performance levels are

unambiguously determined by the performance levels of its elements, hence

at a given moment the states of the whole system are determined by the

states of its elements. Assuming that the system has K states, and w

_i

is the

performance level of the entire system in i-th state, one can find that the

performance of the entire system at t is determined by a random variable

that takes values from the set {w

₁

, . . . , w

_K

}

The general model of a complex system should include the performance of ν

_j

(t) of its j elements, where j = 1, . . . , n. Let us denote by L

ⁿ

= {w

₁₁

, . . . , w

_1k1

}×

{w

₂₁

, . . . , w

2k2

} × . . . × {w

_n1

, . . . , w

nkn

} the space of possible combinations of performance levels of all system components, and by M = {w

₁

, ldots, w

_K

} the space of possible values of performance levels of the entire system. Per- formance in the general model (integral) can then be written as:

ν(t) = f (ν

1

(t), . . . , ν

n

(t)), (3) where f (ν

₁

(t), . . . , ν

n

(t)) : L

ⁿ

→ M .

Performance can be understood differently here, e.g. as the amount of transported aggregate in the mine, the amount of energy generated in the power plant — all depending on the actual system that we describe.

One can use the expected instantaneous performance at time t to deter- mine the system performance:

ν

mean

(t) = E(ν(t)), (4)

and the steady-state expected performance:

ν

∞

=

K

X

k=1

p

_k

w

_k

. (5)

2.2. The problem of the conveyor belt assembly The problem of reliability of the DBD team (Digger(excavator)-Belt conveyor-Dumping ma- chine)

¹

in the mines was analyzed in the 1960s by Polish engineers and math- ematicians. He dealt with this, among others Stanisław Gładysz with the team, among others, analyzed various types of flow nodes resulting from the parallel connection of conveyors belt [10]. The topic is also current today.

Transport of aggregates in mines is an extremely important issue, as ac- cording to the analyzes of companies dealing in the production of mining machinery as much as 50% of costs are incurred for transport. In addition, in the era of ever-higher ecological standards, mines are forced to reduce harmful activities to the environment. It allows for the exchange of car transport on conveyor belts [21]. Their advantages are, among others:

• continuity of spoil disposal,

• ease of automation,

• the possibility of overcoming long distances and slopes,

• high performance,

• work safety,

1loading machine - crusher - belt conveying

• opportunity to work in different climatic conditions and in different areas.

However, attention is also paid to the high costs of their use, as well as the need for frequent repairs and maintenance related to the conveyor belt, which is characterized by low resistance to harmful environmental, chemical and mechanical factors (v. [17]). The conveyors deal with the spoil disposal not only in coal mines, but also in copper ore mines. An important issue is therefore the optimal selection of repair time to minimize their time and costs, but still ensure trouble-free operation and reduce downtime [19].

Conveyor belts are complex devices, hence one can consider their reliabil- ity in a holistic aspect, analyze them as a system of their connections – and monitor the number of failures. Then, the repair time can be easily managed (although it is not optimal) – when the empirical intensity of failure changes its character from a permanent function to an increasing function. It can be then assumed that maintenance and repair work should be carried out in the nearest future (v. [19]). Another way is a more accurate mathematical anal- ysis of failure times and repairs using Markov processes as recommended by the international standard [1].

3. Analysis of the system with a symmetrical node. The chapter considers the double symmetric node. Symmetry is understood here as follows:

the incoming stream (mining spoil) flows only through one of the branches, and when it is damaged, through another branch available in a cyclic order (assumed cold reserve). Here, we assume that there are two conveyor belts available, hence the model is relatively simple: the stream flows through one branch, when it fails, another branch is included in the work, if available.

When both branches are in a state of failure, there is a downtime until one of them is recovered.

The following assumptions were made for the analysis [10]:

1. Stability of processes over time.

2. Independence of individual branch parameters from load.

3. Independence of failure removal (there are as many teams as branches in the node, here 2).

4. Repair restores the conveyor to full efficiency.

Each of the conveyors can be in three states:

1 – the state of work; 0 – cold reserve status; −1 – failure status.

We also accept the following designations for i-th conveyor, where i = 1, 2:

λ

i

– failure intensity; β

i

– startup intensity; µ

i

– intensity of the failure.

Determining the state space, we should take into account the pairs of

possible states for both conveyors. Considering the fact, that at the same

time, it is possible to change the state for one conveyor belt (we assume that

the simultaneous change in the states of both conveyor belts is an event with a

zero probability) we obtain a state graph presented in Fig. 7. In this graph, we distinguish two states of (0, 0) to meet the assumption about the symmetry of the node (we want to point out that even if the damaged conveyor has been repaired earlier, the next conveyor should be started anyway).

1 1, 0

−1, 0 2

−1, 1 3

0, 1 4

0, −1 5

1, −1 6

−1, −1

7 λ

₁

β

₂

µ

₁

λ

2

λ

₂

β

₁

λ

1

µ

2

µ

₁

µ

₂

Figure 3: Graph of states introduced at the model applied in [10].

The analysis by Gładysz in [10] did not assume the possibility of repairing the broken conveyor belt at the moment when the second conveyor is started - there was no transition from the states (0, −1), (−1, 0) to the appropriate states of (0, 0) in Fig. 7 – as can be seen in Fig. 3 (another notation was used, but the transitions are exactly the same as those proposed in the paper).

This approach is at odds with the assumption that we have as many teams as there are branches. It is possible that the author decided to omit these cases, stating that the time to start-up is much shorter than that for repair, but this is not mentioned in the article. In addition, stating that the time to run the branch can be omitted in this case, is a suggestion for the reader that this time could be omitted also in other cases, receiving the five-state model – much simpler in the analysis and implementation (fig. 4).

The node has been taken from the article [10]. Gładysz [4, 11] did not

analyze larger connections than three conveyors at once, which is consistent

with today’s applications, because as one can read in [21], the maximum

number of conveyors directly connected to the mobile crusher controlled by

one operator is 3.

3.1. Five-state graph

3.1.1. General case As it was indicated above, in the situation when the time of switching the conveyor to work is negligibly small, the graph can be simplified to five states, which was presented in fig. 4.

1 1, 0

−1, 1 2

0, 1 3

1, −1 4

−1, −1 5 λ

₁

λ

₂

µ

1

λ

₂

λ

₁

µ

2

µ

₂

µ

₁

Figure 4: Derived five-state graph.

For such a graph, the following Kolmogorov forward equations can be derived:



 

 

 

 

 

 

 

 

 

 

 

  dp

1

dt = −λ

1

p

1

(t) + µ

2

p

4

(t) dp

2

dt = λ

1

p

1

(t) + (−λ

2

− µ

₁

)p

2

(t) + µ

2

p

5

(t) dp

₃

dt = µ

₁

p

₂

(t) + (−λ

₂

)p

₃

(t) dp

4

dt = λ

2

p

3

(t) + (−λ

1

− µ

₂

)p

4

(t) + µ

1

p

5

(t) dp

5

dt = λ

2

p

2

(t) + λ

1

p

4

(t) + (−µ

1

− µ

₂

)p

5

(t)

(6)

As one can see the equations for p

₁

(t) and p

3

(t) and for p

2

(t) and p

4

(t) are symmetrical, so symmetrical solutions are expected.

One can calculate the long-run probabilities because the graph is irre-

ducible (the Ergodic Theorem is here satisfied). In order to calculate them,

it is assumed that the derivatives are equal to 0, and one of the equations is

shifted into the equation of the sum of probabilities (26), because the deter-

minant of the matrix Λ is 0 (therefore there is a change of symbols and the

matrix with the replaced equation is marked as Λ

₁

). We get:











 0 0 0 0 1













=













−λ

₁

0 0 µ

2

0

λ

1

−λ

₂

− µ

₁

0 0 µ

2

0 µ

₁

−λ

₂

0 0

0 0 λ

2

−λ

₁

− µ

₂

µ

1

1 1 1 1 1













·











 p

1

p

2

p

₃

p

4

p

5











 .

In matrix notation:

b = Λ

^T₁

p, (7)

where T is the transposition of the matrix, and b is the appropriate column vector of zeros with 1 at the last position. Equivalently, one can write:

p = (Λ

^T₁

)

⁻¹

b. (8)

Using the above form, you can enter matrices into a mathematical package (here the Wolfram Mathematica [22]

²

) and calculate the long-run probabili- ties. The solution for such a system of equations is:











 p

₁

p

2

p

3

p

₄

p

5













= 1

m(λ

₁

, µ

₁

, λ

₂

, µ

₂

) ·













µ

₁

µ

₂

λ

²₂

+ µ

₁

µ

²₂

λ

₂

+ µ

²₁

µ

₂

λ

₂

λ

2

µ

2

λ

²₁

+ λ

2

µ

²₂

λ

1

+ λ

2

µ

1

µ

2

λ

1

µ

1

µ

2

λ

²₁

+ µ

1

µ

²₂

λ

1

+ µ

²₁

µ

2

λ

1

λ

₁

µ

₁

λ

²₂

+ λ

₁

µ

²₁

λ

₂

+ λ

₁

µ

₁

µ

₂

λ

₂

λ

²₁

λ

²₂

+ λ

1

µ

2

λ

²₂

+ λ

²₁

µ

1

λ

2











 , (9)

where the denominator is:

m(λ

₁

, µ

₁

, λ

₂

, µ

₂

) = λ

²₂

λ

²₁

+ λ

²₂

λ

₁

µ

₁

+ λ

²₂

λ

₁

µ

₂

+ λ

²₂

µ

₁

µ

₂

+

+ λ

2

λ

²₁

µ

1

+ λ

2

λ

²₁

µ

2

+ λ

2

λ

1

µ

²₁

+ 2λ

2

λ

1

µ

1

µ

2

+ λ

2

λ

1

µ

²₂

+ + λ

₂

µ

²₁

µ

₂

+ λ

₂

µ

₁

µ

²₂

+ λ

²₁

µ

₁

µ

₂

+ λ

₁

µ

²₁

µ

₂

+ λ

₁

µ

₁

µ

²₂

.

As expected, symmetrical solutions were received for p

₁

(t) and p

₃

(t) and for p

2

(t) and p

4

(t).

The set of performance levels for the five-state system is three-piece w

_j

= {w

₁

, w

₂

, 0} with the performance level of w

₁

for the first conveyor belt, w

₂

for the second and with zero in the failure state of both conveyors :

P(ν = w

1

) = p

1

+ p

4

, P(ν = w

2

) = p

2

+ p

3

,

P(ν = 0) = p

₅

. The steady-state expected performance:

ν

∞

= w

₁

λ

₂

µ

₁

`

₁₂

(`

₂₁

+ µ

₂

) + w

₂

λ

₁

µ

₂

`

₂₁

(`

₁₂

+ µ

₁

)

λ

2

`

12

(λ

1

+ µ

1

)(`

21

+ µ

2

) + λ

1

µ

1

µ

2

(`

12

+ µ

1

) , (10)

2Inverse[A].B, where A is a properly defined transposed transition matrix with a sub- stituted column of ones, and B is the appropriate column vector

where `

_ij

= λ

_i

+ µ

_j

, for i 6= j. `

₂₁

and `

₁₂

can be interpreted as the intensity of leaving the states 2 and 4 respectively.

3.1.2. Case with equal intensities of transition Assuming that equal failure intensities for both conveyors (λ

₁

= λ

₂

= λ and µ

₁

= µ

₂

= µ), we get the simplest the model discussed in this article. Such a case can be considered if, for example, we have equipment with the same parameters, bought at the same time, similarly exploited, etc. After all, calculating the exact probability of staying in a given state at time t is not a trivial task. To derive them, you can use the Wolfram Mathematica package, the Laplace transform, and the inverse transform implemented there. Thanks to this, it was possible to derive formulas in a relatively clear form.

In order to determine the solution, let us assume that the first state is the initial state. Using the properties of the Laplace transform, we get:

s ·











 ˆ p

₁

(s)

ˆ p

2

(s)

ˆ p

3

(s)

ˆ p

₄

(s)

ˆ p

5

(s)













−











 1 0 0 0 0













=













−λ 0 0 µ 0

λ −λ − µ 0 0 µ

0 µ −λ 0 0

0 0 λ −λ − µ µ

0 λ 0 λ −2µ













·











 ˆ p

₁

(s) ˆ p

2

(s) ˆ p

3

(s) ˆ p

₄

(s) ˆ p

5

(s)













In matrix notation:

sˆ p(s) − p(0) = Λ

^T

p(s) ˆ (11)

and equivalently:

(sI − Λ

^T

)ˆ p(s) = p(0), (12)

p(s) = (sI − Λ ˆ

^T

)

⁻¹

p(0),

³

(13)

where I is a unit matrix, and T is the transposition of the matrix.

3In Mathematica [22], the command: InverseLaplaceTransform[Inverse[-A].B, s, t], where A is a properly defined matrix of intensity of transitions with subtracted s on the diagonal, and B is a column vector of initial probabilities. Here, in addition notation is simplified.

The use of an inverse transform gives the following solution:

p

₁

(t) = µ

²

λ

²

+ 2λµ + 2µ

²

+ e

^{−12t(2λ+µ)}



√ µ sinh

¹₂

√ µt √

µ − 4λ
2 √

µ − 4λ + (14)

+ cosh

¹₂

√ µt √

µ − 4λ
2



+ λe

^{−12t(2λ+3µ)}



√ µ(3λ + 2µ) sinh

¹₂

√ µt √

4λ + µ
2 √

4λ + µ (λ

²

+ 2λµ + 2µ

²

) + (λ + 2µ) √

4λ + µ cosh

¹₂

√ µt √

4λ + µ
2 √

4λ + µ (λ

²

+ 2λµ + 2µ

²

)



p

2

(t) = λµ

λ

²

+ 2λµ + 2µ

²

+ λe

^{−12t(2λ+µ)}

sinh

¹₂

√ µt √

µ − 4λ

√ µ √

µ − 4λ + (15)

+ λe

^{−λt−3µt2}

 (λ − µ)(λ + µ) sinh

¹₂

√ µt √

4λ + µ

√ µ √

4λ + µ (λ

²

+ 2λµ + 2µ

²

) +

− µ cosh

¹₂

√ µt √

4λ + µ
λ

²

+ 2λµ + 2µ

²



p

3

(t) = µ

²

λ

²

+ 2λµ + 2µ

²

− e

^{−12t(2λ+µ)}



√ µ sinh

¹₂

√ µt √

µ − 4λ
2 √

µ − 4λ + (16)

+ cosh

¹₂

√ µt √

µ − 4λ
2

 + + λe

^{−12t(2λ+3µ)}



√ µ(3λ + 2µ) sinh

¹₂

√ µt √

4λ + µ
2 √

4λ + µ (λ

²

+ 2λµ + 2µ

²

) + + (λ + 2µ) cosh

¹₂

√

µt √

4λ + µ
2 (λ

²

+ 2λµ + 2µ

²

)



p

₄

(t) = λ

²

µ

²

λ

²

+ 2λµ + 2µ

²

− λ

²

√

µe

^{−12t(2λ+µ)}

sinh

¹₂

√ µt √

µ − 4λ

√ µ − 4λ + (17)

+ λ

²

e

^{−λt−3µt2}



√ µ(λ − µ)(λ + µ) sinh

¹₂

√ µt √

4λ + µ

√ 4λ + µ (λ

²

+ 2λµ + 2µ

²

) +

− µ

²

cosh

¹₂

√ µt √

4λ + µ
λ

²

+ 2λµ + 2µ

²



p

₅

(t) = − λ

²



e

^−12t

(

^{√µ√4λ+µ+2λ+3µ}

) − 1 

λ

²

+ 2λµ + 2µ

²

+ (18)

−

λ

²

e

^−12t

(

^{√µ√4λ+µ+2λ+3µ}

) 2λ + 3µ + √ µ √

4λ + µ
 e

^õt

√4λ+µ

− 1  2 √

µ √

4λ + µ (λ

²

+ 2λµ + 2µ

²

)

The solution is quite complicated, but after checking, the sum of the above

probabilities is 1.

Some denominators may be equal to 0. However, these uncomfortable expressions containing √

µ − 4λ are shortened when sinh develops into the Taylor series.

Substituting λ

₁

= λ

₂

= λ i µ

₁

= µ

₂

= µ to the previously obtained formula (9) we get the steady-state probabilities:











 p

1

p

₂

p

3

p

4

p

₅













= 1

λ

²

+ 2λµ + 2µ

²











 µ

²

λµ µ

²

λµ λ

²













. (19)

Using the Sevastjanov Model (Fig. 5) and the formula (29) one can de- termine steady-state probabilities for a model in which we have the same transition intensity, one machine working and the other in cold reserve, as well as two conservators, where the subscript means the number of nonfunc- tional devices:





 p

^(S)₀

p

^(S)₁

p

^(S)₂





 = 1

λ

²

+ 2λµ + 2µ

²



 2µ

²

2λµ

λ

²



 . (20)

From the formulas (20) and (19) one can see that:

0 1, 0 1 −1, 1 2 −1, −1

λ

µ

λ

µ

Figure 5: The graph of transitions between states for the Sevastianov system with two devices (the order in the pair does not matter).

p

^(S)₀

= p

1

+ p

3

, p

^(S)₁

= p

₂

+ p

₄

, p

^(S)₂

= p

₅

.

Which means that in the discussed example of the symmetric node there are

more cases due to the additional assumption of symmetry, but the sum of the

probabilities of staying in states where there is the same number of inefficient

devices is identical to those set out from the Sevastjanov Model. This may

indicate the correctness of calculations.

The steady-state expected performance for the considered case is given by the following formula:

ν

∞

= (w

1

+ w

2

)µ(λ + µ)

λ

²

+ 2λµ + 2µ

²

. (21)

Figure 6: Derived instantaneous probabilities and their convergence to steady- state probabilities.

Example 3.1 Let’s take λ

1

= λ

₂

= 1 [

_month¹

] and µ

₁

= µ

₂

= 3 [

_month¹

] . Then the steady-state probabilities from the formula (19) are equal to:

p

1

= p

3

= 9 25 , p

₂

= p

₄

= 3

25 , p

₅

= 1

25 .

Using the derived formulae (14)-(18) on the probability of staying in a given state at the time t, one can plot probabilities depending on time, as shown on Fig. 6.

According to the predictions, probabilities from a certain moment are constant, they converge on the steady-state probabilities depending on the number of damaged conveyors and do not depend on the initial state.

The function p

₁

(t) is decreasing from the value p(0) = 1, because the first

state (1, 0) was considered the initial state. Its biggest drop can be seen in the

[0, 1] section, due to the selected failure intensity. The initial fastest growth and concavity of the function chart p

₂

(t) is due to the fact that the second state (−1, 1) is the only one to go directly from the first state. The graph also shows that probability functions converge almost with the same velocity (only p

₅

converges slower). This is of course due to the choice of parameters.

Additionally, you can see that in fact for t → ∞ derivatives of all probability functions are going to 0.

3.2. Nine-state graph 3.2.1. General case

1 1, 0

−1, 0 2

−1, 1 3

4 0, 0 0, 0 8

0, 1 5

0, −1 6

1, −1 7

−1, −1

9 λ

1

β

2

µ

₁

µ

1

λ

₂

β

2

β

₁

λ

2

β

1

µ

₂

λ

₁

µ

₂

µ

1

µ

2

Figure 7: Derived graph of transitions between states.

For an assumed nine-state graph of transitions between states (fig. 7) a

system of differential equations can be derived using the formula (28). By

proceeding in the same way as in the case of a five-state graph, we get:



























 0 0 0 0 0 0 0 0 1





























=





























−λ₁ 0 0 0 0 0 µ₂ β₁ 0

λ₁ −β₂− µ₁ 0 0 0 0 0 0 µ₂

0 β2 −λ2− µ1 0 0 0 0 0 0

0 µ1 0 −β2 0 0 0 0 0

0 0 µ1 β2 −λ2 0 0 0 0

0 0 0 0 λ2 −β1− µ2 0 0 µ1

0 0 0 0 0 β1 −λ1− µ2 0 0

0 0 0 0 0 µ2 0 −β1 0

1 1 1 1 1 1 1 1 1























































 p₁ p₂ p3

p4

p5

p6

p7

p8

p9



























 .

Limit probabilities were determined using an analogous solution as in the case of a five-state graph, the notation of which is much simpler than the probabilities at t. The last equation has been replaced by the equation of unity of probabilities:1 = p

₁

(t) + . . . + p

9

(t).

The obtained steady-state probabilities:



























 p1

p2

p3

p4

p5

p6

p7

p8

p9





























=





























α1α2λ2µ1µ2((λ2+ µ1)(µ1+ µ2) + α2λ2(λ2+ µ1+ µ2))(α1λ1+ λ1+ µ2) α1α2λ1λ2µ2(λ2+ µ1)((λ1+ µ2)(µ1+ µ2) + λ1α1(λ1+ µ1+ µ2)) α1α²₂λ1λ²₂µ2((λ1+ µ2)(µ1+ µ2) + λ1α1(λ1+ µ1+ µ2))

α1λ1µ1µ2(λ2+ µ1)((λ1+ µ2)(µ1+ µ2) + λ1α1(λ1+ µ1+ µ2))

α1α2λ1µ1µ2((λ1+ µ2)(µ1+ µ2) + λ1α1(λ1+ µ1+ µ2))(α2λ2+ λ2+ µ1) α1α2λ1λ2µ1(λ1+ µ2)((λ2+ µ1)(µ1+ µ2) + λ2α2(λ2+ µ1+ µ2)) α²₁α2λ²₁λ2µ1((λ2+ µ1)(µ1+ µ2) + λ2α2(λ2+ µ1+ µ2))

α2λ2µ1µ2(λ1+ µ2)((λ2+ µ1)(µ1+ µ2) + λ2α2(λ2+ µ1+ µ2)) α1α2λ1λ2 α2λ²₂(λ1+ µ2)(µ2+ α1λ1) + α1λ²₁λ2µ1(α2+ 1) + α1λ²₁µ²₁





























·

· 1

m(λ1, α1, µ1, λ2, α2, µ2) , (22)

where α

_i

=

^β_λⁱ

i

, i = 1, 2 were used to simplify the entry, and the denominator m(λ

1

, α

1

, µ

1

, λ

2

, α

2

, µ

2

) is equal to:

m(λ1, α1, µ1, λ2, α2, µ2) =

= λ³₂α²₂(λ1+ µ2)(α1(λ1+ µ1) + µ1)(λ1α1+ µ2)+

+ λ²₂α2(α2+ 1)(λ1+ µ2)(µ1+ µ2)(α1(λ1+ µ1) + µ1)(λ1α1+ µ2)+

+ λ₂µ₁[λ³₁α²₁ µ₁α₂+ µ₂(α₂+ 1)²
+ λ²₁α₁(α₁+ 1)(µ₁+ µ₂) µ₁α₂+ µ₂(α₂+ 1)²
+ + λ1µ2(µ1+ µ2) µ1(α1+ 1)²α2+ µ2α1(α2+ 1)²
+ µ1µ²₂(α1+ 1)α2(µ1+ µ2)]+

+ λ1µ²₁µ2α1(α2+ 1)(λ1α1(λ1+ µ1+ µ2) + (λ1+ µ2)(µ1+ µ2)). (23)

The set of performance levels for a nine-state system can be specified

as w

_j

= {w

1

, w

2

, 0}. The system will operate at a capacity of w

1

when the

first conveyor operates, w

₂

when the second is working, and with zero during

downtime and failure of both belt conveyors. One can therefore write:

P(ν = w

1

) = p

1

+ p

7

, P(ν = w

2

) = p

3

+ p

5

,

P(ν = 0) = p

₂

+ p

₄

+ p

₆

+ p

₈

+ p

₉

.

When the steady-state expected performance is calculated from the formula (5), some components will be simplified because of zero performance. Ulti- mately, we’ll get:

ν

∞

= 1

m(λ

₁

, α

₁

, µ

₁

, λ

₂

, α

₂

, µ

₂

) · α

₁

α

2

·

· [w

₁

λ

₂

µ

₁

(λ

₁

+ µ

₂

)(λ

₁

α

₁

+ µ

₂

)(λ

₂

α

₂

(λ

₂

+ µ

₁

+ µ

₂

) + (λ

₂

+ µ

₁

)(µ

₁

+ µ

₂

)) + w

₂

λ

₁

µ

₂

(λ

₂

+ µ

₁

)(λ

₂

α

₂

+ µ

₁

)(λ

₁

α

₁

(λ

₁

+ µ

₁

+ µ

₂

) + (λ

₁

+ µ

₂

)(µ

₁

+ µ

₂

))],

where denominator m(λ

₁

, α

1

, µ

1

, λ

2

, α

2

, µ

2

) is the same as in the equation (23).

3.2.2. Case with equal intensities of transition For the case where λ

₁

= λ

₂

= λ, µ

₁

= µ

₂

= µ, β

₁

= β

₂

= β the equations are symmetric and the solution is simplified. For such a system, we have the following solution:



























 p

₁

p

₂

p

3

p

₄

p

₅

p

6

p

₇

p

₈

p

9





























= 1

m(β, λ, µ) ·





























βµ

²

(β + λ + µ) βλµ(λ + µ) β

²

λµ λµ

²

(λ + µ) βµ

²

(β + λ + µ) βλµ(λ + µ) β

²

λµ λµ

²

(λ + µ) β

²

λ

²



























 ,

where denominator:

m(β, λ, µ) = 2(β + λ)µ

³

+ 2(β + λ)

²

µ

²

+ 2βλ(β + λ)µ + β

²

λ

²

. Performance determined at long-run probabilities (we assume that despite the same intensity of transitions, conveyors work with different performance levels — the first of w

₁

the second of w

₂

):

ν

∞

= (w

₁

+ w

₂

)βµ(β + µ)(λ + µ)

β

²

λ

²

+ 2µ

³

(β + λ) + 2µ

²

(β + λ)

²

+ 2βλµ(β + λ) . (24)

4. Comparison of models Let’s return to the equation (24) and (21)

– let us denote the steady state expected performance for a nine-state graph

with the same transition intensities for both conveyors as ν

_∞^(β)

. If we assume that the average time of switching the conveyor to work is zero then β → ∞:

β→∞

lim ν

_∞^(β)

= lim

β→∞

(w

1

+ w

2

)βµ(β + µ)(λ + µ)

β

²

λ

²

+ 2µ

³

(β + λ) + 2µ

²

(β + λ)

²

+ 2βλµ(β + λ) =

= lim

β→∞

(w

1

+ w

2

)µ(1 +

_β^µ2

)(λ + µ)

λ

²

+ 2

^µ_β³2

(β + λ) + 2

^µ_β²2

(β + λ)

²

+ 2λµ(1 +

_β^λ2

) =

= (w

1

+ w

2

)µ(λ + µ)

λ

²

+ 2λµ + 2µ

²

= ν

∞

.

We received the same, which means that the five-state case is a borderline case of a nine-state graph.

Next, using the results previously obtained (24) and (21), we can designate an asymptotic relationship between ν

∞^(β)

and ν

∞

:

ν

_∞^(β)

≈ ν

∞

 1 + µ

β − 2µ(λ + µ)

²

β(λ

²

+ 2λµ + 2µ

²

)



=

= ν

∞



1 − µλ(2µ + λ) β(λ

²

+ 2λµ + 2µ

²

)



=

= ν

∞

 1 − c

β

 ,

where c =

_(λ2^µλ(2µ+λ)+2λµ+2µ²)

. This means that the performance calculated for a 9- state system is smaller than for 5-state by approximately

_β^c

> 0 – the 5-state model pad the performance, hence the model 9-state is more accurate.

The above asymptotic is satisfied if λ  β andµ  β, which can be accepted for a given application when the assumption that the average start- up time is definitely shorter than the average repair time and average time to the point of failure seems to be reasonable.

In addition, we see that

_β^c

constant approximates the relative error of the nine-state model to the five-state model:

 

 ν

∞^(β)

− ν

_∞

   ν

∞

≈    ν

∞

 1 −

_β^c



− ν

_∞

   ν

∞

= c

β = µλ(2µ + λ)

β(λ

²

+ 2λµ + 2µ

²

) . (25) The c constant can be limited by µ. Knowing thatµ  β, one can notice that the above relative error

_β^c

<

^µ_β

is not big, but still a nine-state model remains more accurate.

5. Conclusions The subject of modeling in the article was a double sym-

metrical node for the connection of belt conveyors proposed by Gładysz(1965).

For the model, we assumed: the independence of the conveyor belt from the load; as many specialists as there are branches; repair to the state of complete efficiency; immediate transitions from the state to state; constant intensity of transitions between states – stationarity (the period of normal operation) and process homogeneity; independence of the next transition from the entire process history and time spent in individual states, only from the last state.

Thanks to the above assumptions, it was possible to apply a lot of useful results in the theory of Markov processes and derive graphs of transitions between the states for two general cases – when only failures and repairs are taken into account, and when the time of switching a given branch into operation is also important.

The derived nine-state graph differed from that proposed by Gładysz in [10] because of two additional states – in two cases it did not assume the possibility of repairing the broken conveyor belt earlier than starting from the reserve, which was in conflict with the assumption of the number of specialists.

Hence, the calculations were based on a graph, which was expected to meet all assumptions. A system of Kolmogorov’s forward equations was derived, and then the probabilities of staying in each of the specified states and the corresponding performances were calculated.

A five-state graph was also derived for the case when the switching time is negligibly small. It was checked that this is a borderline case of the nine-state graph, when β → ∞.

For the case with equal intensities, the probabilities of being in a given state were calculated at the moment t using the Laplace transform. Then, the examples show the convergence of instantaneous probabilities to long-run probabilities. The results obtained for this case were also compared with the Sevastjanov model. The sum of the long-run probabilities for the same number of devices out-of-order turned out to be an appropriate long-run probability in the literature model.

Then, the derived performances for cases of equal intensities for a given branch were compared. An asymptotic dependence of the performance of the nine-state system on the five-state system was obtained. The performance for λ  β and µ  β differ by the received constant

_β^c

, which can be limited by

µ

β

. The same constant approximates the relative error of the nine-state model relative to the five-state model. This is not a big error, but it indicates greater accuracy of the model with more states.

The above results show that it is necessary to choose the right model and state space, because we can expose ourselves to a large inaccuracy of calculations when we omit significant states (e.g. here for switching branches into work). It is also important to check whether the actual system that we want to model satisfies the assumptions of the model that we want to choose.

For example, if we have variables in the intensity of transitions between states,

and we want to get a more accurate model and consider it as not constant, but

variable in time failure intensity λ(t), one should use a different distribution (Weibull distribution is often used) and semi-Markov processes that are a generalization of Markov processes (v. Grabski(2015), Lisnianski et al.(2010)).

The assumption with constant transition intensities may be satisfactory for the model of the system during the time of normal operation of the equip- ment, when the failures are accidental and do not result from either break-in or aging of the parts.

The system described for the conveyor belt node can also be used in the case of a system of other devices, if only the same assumptions and the established graph can be used for its description. Then, you can successfully use the derived formulae to approximate performance for another real system.

On the other hand, if all assumptions agree, except for those that do not interfere with the use of Markov chains (e.g. the occurrence of a symmetric node or the number of connected devices), an appropriate graph of possible states can be derived successfully and similar calculations can be made for the matrix in a mathematical package, e.g. Wolfram Mathematica or MATLAB.

The operating scheme will remain the same as in the article.

It should be emphasized that in economic applications, the accuracy of calculations may be important, hence the derived models on performance for all models were not approximated and allow for the substitution of values obtained from failure data and repairs that should be collected on a regular basis in the company.

Determining the system performance and probabilities of staying in a given state may be useful for estimating the expected profits, as well as the investment and operating costs. With the various choices of maintenance op- tions – different costs and the corresponding failure and repair intensities – you can conduct an analysis and check which is more cost-effective and plan optimally repairs and maintenance.

The analysis can be helpful already at the design stage – knowing the intensity of the repair and costs – one can make better decisions regarding the selection of equipment or the choice of the services of the company repairing the equipment.

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