On determination of some characteristics of semi-Markov process for different distributions of transient probabilities

Mateusz Zajac; Tymoteusz Budny

ON DETERMINATION OF SOME CHARACTERISTICS OF SEMI-MARKOV PROCESS FOR DIFFERENT DISTRIBUTIONS OF TRANSIENT PROBABILITIES

Mateusz Zajac

Wroclaw University of Technology, Wroclaw, Poland e-mail: [email protected]

Tymoteusz Budny

Gdynia Maritime University, Gdynia, Poland e-mail: [email protected]

ABSTRACT

There is a model of transport system presented in the paper. The possible semi - Markov process definitions are included. The system is defined by semi -Markov processes, while functions distributions are assumed. There are attempts to assess factors for other than exponential functions distributions. The paper consist discussion on Weibull and Gamma distribution in semi - Markov calculations. It appears that some forms of distribution functions makes computations extremely difficult.

1 INTRODUCTION

The reliability model of intermodal transport was presented during ESREL'06 conference (Zajac 2006b). The model is described by semi - Markov processes. During the presentation assumed, that, probabilities of transition between states were exponential. Complex technical systems are usually assumed, that probabilities of transition between states or sojourn times' probabilities are exponential. Lack of information, too little number of samples or inaccurate assessment of data may cause that such assumption is abused. In some cases, when exponential distribution is assumed, there is also possibility to assess factors according to different distributions (Weibull, Gamma, etc.). Probabilities of transition between states are one of the fundamental reliability characteristic. The paper includes example of determination of above mention characteristic for one of the phases of combined transportation systems reliability model.

2 TRANSHIPMENT PHASE CHARACTERISTIC

There are three methods to define semi - Markov processes (Grabski 2002, Grabski&Jazwinski 2003):

- by pair (p, Q(t)),

when: p - vector of initial distribution, Q(t) - matrix of distribution functions of transition times between states;

- by threes (p, P, F(t)),

where: p - vector of initial distribution, P - matrix of transition probabilities, F(t) - matrix of distribution functions of sojourn times in state i-th, when j-th state is next;

- by threes (p, e(t), G(t)),

where: p - vector of initial distribution, e(t) - matrix of probabilities of transition between i-th and j-th states, when sojourn time in state i-th is x, G(t) - matrix of sojourn times distribution functions. For transshipment phase semi - Markov process is defined by (p, P, F(t)). Phase of transshipment includes following states:

1. standby,

2. dislocation works,

3. transshipment,

4. preventive maintenance,

5. repair (after failure).

Activities which are involved into each of above states are described in papers (Zajac 2006a, Zajac 2007). The graph of state is presented in Figure 1.

Figure 1. Graph of states in transshipment phase 3 CONDITIONS DETERMINATION FOR TRANSSHIPMENT PHASE RELIABILITY

Transshipment phase elements can stay in reliability states from the set S (0,1), where:

0 - unserviceability state,

1 - serviceability state.

Operation states takes values from the set T (1,2,3,4,5). Cartesian product of both states creates following pairs: (0,1), (1,1), (0,2), (1,2), (0,3), (1,3), (0,4), (1,4), (0,5), (1,5). The model allows for existence of following pairs, only: S1p - (1,1), S2p - (1,2), S3p - (1,3), S4p - (0,4), S5p - (0,5). Means of transport are in first operation state (standby) during time described by random variable Zp1. The distribution function of random variable is

F(pl(t) = P{Zpl < t}, t > 0.

The time of the second state (dislocation) is described by Zp2. The distribution function of random variable takes form

Fp2(t) = P\ZP2 < t}, t > 0

The time of third state (transshipment) is described by Zp3, where distribution function of random variable is given by formula:

FCp3(t) = P[Cp3 < t}, t > 0

If the time of realization of preventive maintenance is known (and lasts yp), than the distribution function of sojourn time in the fourth state (preventive maintenance) is

FrP (t ) = Ph < 4 t >

Some of activities can be interrupted by failures. It was assumed, that time of work without failure in states 2-nd and 3-rd is described by npi, i = 2,3. The distribution function is given by formula:

Fm(t) = P{npi< t}, t > 0, i = 2,3.

If there is known time, when the system is broken down, and that time is given by xp, then the distribution function of state 5-th (repair) is

FxP (t) = P{XP * 4 t > 0.

States 4-th and 5-th are states of unserviceability, however only state 5-th requires repair after failure. We assume that random variables £pi, r)Pi and xP are independent.

3.1 Kernel determination and the definition of semi - Markov process in transshipment phase

The phase of transshipment can be described by semi - Markov process [X(t)\ t >0} with the finite set of states Sp = {1, 2, 3, 4, 5}. The kernel of the process is described by matrix

Qp(t)=

0 Qp12 Qpi3 Qp14 0

Qp 21 0 0 0 Qp 25

QP3i 0 0 0 Q p35

Qp 41 0 0 0 0

Qp51 0 0 0 0

(1)

Transshipments from 1-st state to 2-nd, 3-rd and 4-th can be described by

Qp12 (t) = Pl2FZp1 (tX

Qp13 (t) = P13 F(P1 (t), Qp14(t) = Pl4F(pi (t). Transshipments from 2-nd state to 1-st and 5-th:

Qp 21 (t) = P 21 FCp2 (t),

Qp25 (t) = P25FCp2 (t). Transshipments from 3-rd state to 1-st and 5-th:

Qp31 (t) = P31 Fzp3 (t), Qp3s(t) = P35Fzp3 (t).

Transshipment from 4-th state to 1-st:

Qp41 (t) = P(Yp < t) = F/p (t).

Transshipment from 5-th state to 1-st:

Qp51 (t) = P(Xp < t) = FXp (t).

The vector p = [p1,p2, p3, p4, p5] is initial distribution of the process. In this case vector takes values p = [1, 0, 0, 0, 0].

The matrix of transient probabilities is given by

0 Pl2 P13 P14 0

P 21 0 0 0 P25

P31 0 0 0 P35

P41 0 0 0 0

P51 0 0 0 0

(2)

3.2 The transient probabilities

Transient probabilities are one of the most important characteristics of semi - Markov processes. They are defined as conditional probabilities

Pj (t) = P{X(t) = j | X(0) = 4 i, j e S (3)

Above probabilities obey Feller's equations (Grabski 2002, Grabski&Jazwinski 2003)

Pj (t ) = ôj [1 - Gt (t )] + Zj Pk] (t - x)dQik ( x) (4)

Solution of that set of equations can be found by applying the Laplace - Stieltjes transformation. After that transformation the set takes form

p j (s) = Sj [1 - ~ (s)] + I (s)pk] (s), i, j e S. (5)

kGS

In matrix notation this set of equation has form

p(s) = [I - g(s)] + P(s) p(s), (6)

hence

p(s) = [I - q(s)]-1[I - g(s)] . (7)

Determination of transient probabilities requires finding of the reverse Laplace - Stieltjes transformation of the elements of matrix p(s).

4 DATA AND ASSUMPTIONS FOR CALCULATIONS

Data were collected in 2006 in one of the Polish containers terminals. The data includes information about numbers of transient between states during 50 succeeded days. Selected data are presented in Table 1.

Table 1. Selected data about time of states [h]

state 1 stand by state 2 dislocation state 3 transshipment state 4 preventive maintenance state 5 repair

average 3.610 3.766 3.805 0.538 0.28

variance 0.387 0.022 0.020 0.046 0.287

min. value 1.862 3.482 3.482 0.3 0

max. value 4.411 4.042 4.020 1.1 2

dispersion 2.548 0.560 0.538 0.8 2

Collected data didn't allow for verifying probabilities distribution. The information gave possibility to estimate necessary parameters to assess factors for exponential, Weibull and Gamma distribution functions. Factors are presented in Table 2.

Table 2. Distribution parameters for different distribution function

state 1 state 2 state 3 state 4 state 5

Parameter of exponential distribution

0.28 0.27 0.26 1.86 0.86

Parameters of gamma distribution

a 9.260 152.059 181.992 13.968 4.120

X 33.208 531.455 665.314 9.129 4.878

Parameters of Weibull distribution

a 1.020 1.01 1.014 0.985 1.021

X 0.269 0.26 0.266 1.538 0.833

At first calculation has been done with assumption, that transient probabilities are exponential. The distribution function of sojourn times and their Laplace - Stieltjes transformation respectively, take form

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

Fwl(t) = 1 - e -0 28t , fwl*(t) = 0,28

^ + 0,28

Fw2(t) = 1 - e , fw2*(t)

0,27

FW3(t) = 1 - e -0^26t, fW3*(t) = Fw4(t) = 1 -e"1,i6t, fw4 *(t) =

^ + 0,27 0,26

^ + 0,26 1,86

FwS(f) = 1 - e -0,86t, fw5*(t) =

^ +1,86 0,86

^ + 0,86

Then, kernel of the process is given by matrix

Qp(t) =

0,81(1 - e ~0'28t) 0,16(1 - e ~0'28t) 0,04(1 - e ~0'28t)

0,98(1 - e -0 27t) 0 0 0 0,02(1 - e ~°,27t)

0,99(1 - e "0 26t) 0 0 0 0,01(1 - e ~°,26t)

(1 - e -U6t) 0 0 0 0

(1 - e-a86t) 0 0 0 0

(8)

Matrices q(s) and g(s) have been determined according to equations (5) - (7). In considered example we obtain

~(s) =

0,98-

1 0,80 0,27

0,99

s + 0,27 0,26

s + 0,26 1,86

s +1,86 0,86

s + 0,86

0,28 s + 0,28

1

0 0 0

0,16-

0,28 s + 0,28

0 1 0 0

0,04

0,28 s + 0,28

0 0 1 0

0,02-

0,01

0,27

s + 0,27 0,26

s + 0,26 0

1

(9)

and

0

g (s) =

0,28 s + 0,28

0 0 0 0

0,27 s + 0,27

0 0 0

0,26 s + 0,26 0

0

0 0

0

1,86 s +1,86

0 0 0 0

0,86

s + 0,86

(10)

According to (7), matrix p(s) is a result of multiplying of two matrices. Elements from first column of obtained matrix p(s) are shown on Figure 2.

L. macierz - Notatnik

Plik Edycia Format Widok Pomoc

column 1

2 3 4

0.1122915 + 1.038732s + 3.1114s + 3.25s + s

2 3 4