Total time on test transforms ordering of semi-Markov system Текст научной статьи по специальности «Математика»

TOTAL TIME ON TEST TRANSFORMS ORDERING OF SEMI-MARKOV SYSTEM

V. M. Chacko

Department of Statistics St. Thomas' College, Thrissur, Kerala-680001

ABSTRACT

First passage time of semi-Markov performance process of a multistate system are considered. TTT (Total time on Test) transform ordering is discussed.

1. INTRODUCTION

First passage times of appropriate stochastic process have often been used to represent times to failure of devices or systems which are subject to shocks and wear, random repair time and random interruptions during their operations. The life distribution properties of these processes have therefore been widely investigated in reliability and maintenance literature. When the performance process of a multistate reliability system is Markov or semi-Markov process, we need to study the ageing properties of the first passage time distribution from up states to down states. Identification of failure rate model in statistical lifetime data analysis is major problem in the field of reliability and survival analysis. The total time on test (TTT) transform is used as a tool for identification of failure distribution model in binary system. When a system is modeled by Markov or semi-Markov process it is quiet interesting to get a procedure for the ordering of the failure distribution using its TTT which based on transition probability function.

Use of TTT transform for the identification of failure rate models (IFR/DFR/ Bathtub shaped/constant) in the binary system case is discussed by Barlow and Campo (1975). Later, Klefsjo (1982) presented some relationship between the TTT transform and other ageing properties (with their duals) of random variable. Barlow and Proschan (1996) discussed a wide application of IFR/DFR distributions in maintenance and replacement policies of a binary reliability system. Nair et al. (2008) studied the properties of TTT transforms of order n and examined their applications in reliability analysis.

But when we consider a complex system whose performance process is Markov or semi-Markov, we need the knowledge of ordering for applying suitable maintenance and repair/replacement policies. The ordering of distribution of performance process is Markov/semi-Markov will be helpful to the engineers and designers for applying suitable maintenance and repair or replacement policies.

In this paper, we consider a semi-Markov system whose first passage time distribution and the reliability function based on the transition probability function in the up states. We define the TTT using transition probability function. The conditions of ordering are discussed.

This paper is arranged as follows. Section 2 recall the existing results for identification of failure rate model of random variables based on TTT. In section 3, we discuss TTT transform ordering based on transition probability function of a semi-Markov process. In Section 4, we introduce some conditions for ordering properties of the semi-Markov system based on TTT built from transition probability function. Conclusions are given at the last section.

2. TTT TRANSFORM OF A LIFETIME RANDOM VARIABLE

Many parametric lifetime models such as Gamma, Weibull, and Truncated Normal distributions have monotone failure rate. The failure rate function X(t) will be continuous and twice differentiable for all t > 0 with the exception of the exponential distribution. Total time on test (TTT) transform is a fundamental tool in reliability investigation.

Given a random sample of size n from a nonnegative random variable with distribution F, let Xn1,Xn2,...,Xnr,...,Xnn be the order statistic corresponding to the sample. The total time on test to the r-th failure is

T(Xnr) = nXni + (n - 1)(Xn2 - Xni) + ... + (n - r + 1)(Xnr - Xn(r_„).

r

It is the sum of all observed lifetimes that can be expressed as T (Xnr) = ^ Xnk + (n - r) Xnr.

k=i

If F is exponential with mean 6 and we observe the first r ordered values, then it is well known that the maximum likelihood, minimum variance, unbiased estimator of 6 is,

0(r, n) =1T (Xnr). r

i 1 i rFT'(r / n)

Define H-1(r / n) = - T(Xm), then H;'(r /n) = fn [1 - Fn (u)]du, where Fn (u) is the n Jo

empirical distribution function defined as

0, u < Xm

Fn (u) = i i / n Xn ^ u < Xi+1,n and Fn 1 (u) = inf{X : Fn (x) > u} . 1 u > Xnn

The fact that Fn (x) ^ F (x) uniformly at continuity points of F, by Glivenko Cantelli

rF-\r / n) fF-1(t)

theorem, implies lim /I [1 - Fn (u)]du = I [1 - F(u)]du uniformly in t e [0,1].

00

i rF -1(t)

Define HF (t) = I [1 - F(u)]du, t e [0,1] to be the TTT of the distribution F.

0

Barlow and Campo (1975) proved the following result.

Theorem 2.1 There is a 1-1 correspondence between distributions F and their transform HFl.

Note that HF is a distribution with support on [0, /], where p is the mean of F, since

i rF -1(1)

HF (1) = f [1 - F(u)]du = u when F(0-) = 0.

0

H 1 (t)

Then (j)(t) = —is a continuous increasing function on [0; 1], which is 0 at t = 0 and 1 at

HF (1)

t = 1.

Model Identification

Let G(x) = 1 -exp(-x/0), x,0 > 0be the exponential distribution with mean p. Then HG (t) = |G (t)e x/0dx = |G (t)0dG(x) = 0t and scaled TTT,

j(t) = = t, t e [0,1]. (3.1)

' HG!(1)

The scaled TTT of the Exponential distribution is a 450 line on [0,1]. The normalized total time on test is the boundary between the corresponding transforms of IFR and DFR distributions. TTT that permits to classify distributions according to their failure rate is that its slope evaluated at t = F(x) is the reciprocal of the failure rate at X

±H Vt)l = (1 ~t) | = 1 -F(x) = (32)

dtF (t)lt=F(x) f[F-1(t)]UF(x) f(x) A(xy (3 ) where X is the failure rate of F.

Semi-Markov system

We are concerned with a multistate system (MSS) having M + 1 states 0,1,...,M where '0' is the best state and 'M is the worst state, see Barlow and Wu (1978) for details of MSSs. At time zero the system begins at its best state and as time passes the system begins to deteriorate. It is assumed that the time spent by the system in each state is random with arbitrary sojourn time distribution. The system stays in some acceptable states for some time and then it moves to unacceptable (down) state. The first time at which the MSS enters the down state after spending a random amount of time in acceptable states is termed as the first passage time (failure time) to the down state of the MSS. We study the aging properties of the first passage time distribution of the MSS modeled by the semi-Markov process(7t, t > 0}. In the MSS with states {0,1,...,k,k+1, ...,M} where (0,1,...,kj is the acceptable states, the sojourn time between state 'i' to state j' is assumed to be distributed with arbitrary distribution Fij.

First Passage Time And Reliability Function

Let E = {0,1,...,M} be a set representing the state of the MSS and probability space with probability function P, on which we define a bivariate time homogeneous Markov chain (X,T) = {Xn,Tn,n e {0,1,2,...}}, Xn takes values of E and Tn on the half real line R+ = [0,rc),

with 0 < T < T <... < Tn <... Put Un = Tn Tn-1 for n >1. This Markov process is called a Markov renewal process (MRP) with transition function, the semi-Markov kernel, Q = [ Q^ ], where Qj (t) = P[Xn+1 = j,Un < 11 Xn = i],i, j e E, t > 0 and Qa (t) = 0,i e E, t > 0.

Now we consider the semi-Markov process (SMP), as defined in Pyke (1961). It is the generalization of Markov process with countable state space. SMP is a stochastic process which moves from one state to another of a countable number of states with successive states visiting form a Markov chain, and that the process stays in a given state a random length of time, the distribution of which may depend on this state as well as on the one to be visited in the next. Define Zt = XN ,Nt = sup{n,Tn = U1 + U2 +... + Un < t}, it is the semi-Markov process associated with the

MRP defined above. In terms of Z, the times TX,T2,... are successive times of transitions for Z, and X0, Xj,...are successive states visited. If Q has the form QtJ (t) = P[Xn+j = j | Xn = i][1 - e~X(I)t ],i, j e E, t > 0, for some function X(i), j e E then the process Zt is a Markov process. That is, in a Markov process, the distributions of the sojourn times are all exponential independent of the next state. The word semi-Markov comes from the somewhat limited Markov property which Z enjoys, namely, that the future of Z is independent of its past given the present state provided the "present" is the time of jump. Let Itj =indicator function of

{i = j}. Define the transition probability that system occupied state j e E at time t > 0, given that it is started at state i at time zero, as, i, j e E, t > 0

Pj (t) = P[ Zt = j \ Z 0 = i ] = P[ XNt = j | X 0 = i ] = h (t) Ii} + Q * P(t )(i, j), where h. (t) = 1-£ Qik (t), P(t) = [py. (t)]and 0 * P(t)(i, j) = ££0* (dc)P# (t-x)

k keE

To obtain the reliability function of the semi-Markov system described above, we must define a new process, Y with state space U uV, where U denotes set of all up states {0; 1; :::; k} and Vis the absorbing state in which all the states {k + 1, ...;M} of the system is united. Let TD denote the time of first entry to the down states of Z process.

That is, Yt = Zt (o) if t < TD (o) and Yt = V if t > TD (o).

Let 1 = (1,1,...,1)1, a unit row vector with appropriate dimension. The process Ytis a semi-Markov process with semi-Markov kernel

UP Down

Q1(t) Q2(t)

Up ,a1

We denote a = (a(0),...,a(k),a(k + 1),...,a(M)) where a(i) = P(Y0 = i). The reliability function is

R(t) = P[Vu e [0,t],Za eU] = P[Yt eU] = £P[Yt = j]

jeU

= ZZ P[Y = j,Y = i]

ieU jeU

= ZZ pv (t mo.

ieU jeU

(4.3)

0

0

3. TTT ORDERING OF SEMI-MARKOV SYSTEM

In order to identify the failure rate behavior of a semi-Markov system based on the transition probability function, we define the TTT based on transition probability function in up states as follows. Let F be the first passage time distribution of a semi-Markov system, define

where

But

Then

i F (t)

Hp1 (t ) = J0 Pj (u)du,Vi, j eU, t e [0,1]

(4.4)

F-\t) = inf{(1- R(t)) > t}

= inf ! x :

1-ZZ Pj ( xW)

v ieU jeU

> t !

= inf ! x :

H F (t ) = J0F 1(t ) ZZ Pj (u )du,Vi, j eU, t e [0,1]

f > ZZPj(xMO * 1-1!

vieU jeU

ieU jeU

=ZZa(i)f0 (t)Pj(x)dx

ieU jeU

=ZZ^(i)HP1j (t )

ieU jeU

F-Vn H-ut) S£a(i)i (i)Pn(x)dx

Ht(D = EE«(i)j0 () Pn (x)dx = EE«(i)Hp; (1) Hf^ = -, t e [0,1].

leujeu HF (1) ££«(/) J P H (X)dX

ieU jeU 0

Let F and G be the distributions of first passage time of two semi-Markov processes with transition probability functions piJ (u) and qnj (u) Vi, j eU, t e [0,1]

Theorem 3.1. IF F _1(t) < G _1(t ),Vi, j eU, t e[0,1]and pn (u) > qn (u)Vi, j eU, t e [0,1] then

F -1(t) G ~\t)

H-j (t) = J Pj (u)du < Jqj (u)du =H- (t).

0

When interarrival time of semi-Markov p-system is less that of q-system, we have TTT of F is less than that of G.

Let HF\t) = EE«(i)Hp!(t) <EE P(i)Hjj(t) = HQ (t), t e [0,1]. When this holds we say

ieU jeU ieU jeU

that the first passage time of semi-Markov p-system is less that that of q-system in TTT order. But this is possible only when F - (t) < G - (t ),Vi, j eU, t e [0,1] and p1} (u) > qt] (u) Vi, j eU, t e [0,1].

As the rate of transition from state i to state j is greater in p-sysetm than in q-system then the failure of system occur rapidly. So that TTT of the p-system will be smaller than that of q-system.

Also F j1(t) < G_1(t)indicate that chance of occurrence of p-system early failure is greater than that of the q-system.

In a semi-Markov system it is very difficult to compute mean or variance or expected value of any convex function of first passage time random variable. In such situation TTT is found to be good ordering tool.

4. TTT ORDERING OF CONTINUOUS TIME MARKOV PROCESSES

Consider a Markov process in continuous time and discrete state space {1,2,...,M}, Doob (1953), p.241. The system starts in state '1' at time zero and as it enters 'M', it remains there. Consider the intensity matrix, H = [hjj ] with entries

hi} = 0, i e {1,2,...,M-1}, j * i +1, h,+1 = h, hu = 0.

The Kolmogorov's system of differential equation becomes, for p^ (t-u) = P[Yt = j | Yu = i], 0 < u < t and we take u = 0,

p)k (t) = -hpik (t) + hpMk (tX i < M, pMk (t) = 0 with initial conditions, pik (0) = 5ik, the indicator of {i=k}. Then,

pm (t) = 0, k * M,pmm (t) = 1

and it is easily verified that the solution is

Pk (t) =

0, k <i

(ht Ve-ht . , ^

-, i < k < M

T(k) '

e-t [eht -1- ht -...- (ht ^ ' 1 ], k = M.

T(M-i)

Here the process is of monotone paths. Now consider Vi, j e {0,1,...,M -1}

hi (i) = r(ht = *k-L r tk-e-htdt=r(k - i+1).

PiJ J0 r(k) r(k )J0 r(k )h

Therefore

Hp1 (t)_ T(k )h rF-\t) (hu )k-ie-hu

fF (t) W e du I JO rm

HP1 (1) r(k - i +1)->0 r(k)

hk-M ^F 1 (t) k-i -hu

-f "uk-e-hudu.

r(k - i + 1)Jo

It is the scaled TTT transform for q-system with r and k. Similarly, let

H-1(1) r(k - i+1)J° r(k)

Hqtj (t) _ r(k)r ) (ru)

r(k - i +1) fo

rk-M

r(k - i +1)

du

-f u e~rudu.

i +1) Jo

be the scaled TTT transform for q-system with r and k.

t ■ H P (t) Hq1(t) FV) . . , G-1(t) t ■

Since uk-,e-ru is unimodel curve, —Pj— < —^— if f uk-,e-hu < f uk-,e-"du, t e [0,1],h < r.

Hp; (t) Hqj (1) J0 J0 ' L'J'

This is possible only when F_1(t) < G_1(t), t e[0,1]. But F(x) < G(x) imply

F_1(t) < G_1(t), t e[0,1]. Thus Zp <st Zq would imply Zp <TTT Zq order of semi-Markov systems with restrictions.

5. CONCLUSION

The TTT transform ordering of first passage time distribution of a semi-Markov system is discussed. This ordering is applicable for Multi-state systems whose performance process is modeled using semi-Markov process.

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