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[4] Bartlett, M.S. (1970). Renewal Theory. Meten & Co. Ltd. Science paperbacks, ISBN 412 20570 X.
[5] Bris, R., Chatelet, E. & Yalaoui, F. (2003). New method to minimize the preventive maintenance cost of series-parallel systems. In Reliability Engineering & System Safety, ELSEVIER, Vol. 82, Issue 3, p.247-255.
[6] Bris, R. & Cepin, M. (2006). Stochastic ageing models under two types of failures. Proc. of the 31st ESReDA SEMINAR on Ageing, Smolenice Castle, Slovakia.
[7] Gertsbakh, I.B. Asymptotic methods in reliability: A review. Adv. Appl. Prob. 16, 147-175.
[8] Pyke, R. Markov renewal processes: Definitions and preliminary properties. Ann Math Statist 32, 1231-1242.
Appendix
Renewal Process
Renewal process serves for example to model mathematically a device behaviour which is maintained in such a way that it stays running as most effectively and longest as possible. May a file of components or the whole device with a time to a failure X (non-negative absolutely continuous random variable) with a dispersion given by a probability density f(t) exists and may a symbol t denotes for clearness a time. The first component is put into operation at time t = 0. Further, Xi is a period when the first component comes to the failure and at the same time it is substituted by a new identical component from a given file. It means that a renewal period (in this case a change period) is negligible, or equal to zero. This second component breaks down after the period X2 since it started to operate. At the time Xi + X2 the second component is renewed by the exchange for the third one and the process continues further in such a way. The r-th renewal will happen at the time S,=X1+X2+... +Xr.
If XL X2 are independent non-negative equally distributed random variables with a finite expected value and dispersion,
Sn=0, Sn = YJXj,n<eN,
i=1
then a random process is called a renewal
process in a renewal theory. Sometimes an order of stated random variables ¡X„j" „ „ is denoted in this way. In the case when a time distribution until the failure is exponential, we speak about Poisson process. A function F„(t) indicates a distribution function of a random variable S„.. There are a few other random variables connected with the renewal process, which describe its behaviour (at time). Let we
call Nt o number of renewals in the interval [0, t] for a firm t S 0, it means
Nt = max{«: Sn • f}
From this we also get that SNt < t < Sx, , Regarding the fact that the interval [0, t] contains n failures (as well as renewals) only if n'h failure happens at the latest at the time t
P{Nt>n}=P{Sn' t}=Fn(t)
and the probability that at the time t there are n renewals in the given renewal process can be described in the following way
= F,Ml-F„+l(t)] = F„(t)-F„+l(t)
Provided that XL X2 are independent non-negative equally distributed random variables and Pr(X¡ 0J 1, then a random variable TV, has finite moments of all the series (Stein s theorem).
And if Nt , t >.0 gives a number of renewals in the interval [0, t], then a function
H(t) = ENt, t> 0
is called a renewal function. As it is apparent it gives an expected number of renewals in the interval [0, t]. The expected number of renewals in the interval [ti, t2|. 0< 11 t2 can be quantified from H (t¡) - H (t2), because a number of renewals in this interval is NP -Ntl.
A renewal function can be also expressed from distributional functions F„(t) of random variables S„
oo
H{t)=^nP{Nt=n)
n=0
CO r CO
= S«[F„ (0-^(01 = 2^(0.
n=\ n 1
A renewal equation is important for the renewal function computation H(t). It provides a mutual unique relation between distributional function of a time to a renewal and a renewal function: if a distributional function of a time to the renewal F(t) is continuous, then a renewal function H(t) is convenient with an integral equation
H(t) = F(t) + \H{t - ii)F{n)du.
o
This equation can be easily derived from the previous equation with help of its integral transformation (e.g. Laplace).
An asymptotic behaviour of a renewal process is substantial. An asymptotic behaviour of a renewal process is discussed in an Elementary theorem about a renewal: if a time distribution to a renewal has a finite expected value ¡x, then
lim
H(t) 1
t->a> f
It is a Blackwell theorem, which testifies about a limited behaviour of an expected number of renewals at a finite interval (t, I At/: if a time to a renewal has a non-lattice distribution with a definite positive expected value ¡x, then V h • 0 is
lim\H(t + h)-H(t)] = — . 0 |0,
If a derivation of a renewal function exists (i.e. XL X2 are absolutely continuous random variables), then for the arbitrary time t > 0 a function h(t) that is defined by a relation
... H{t)-H{t + At) ,,„, h(t)= lim ——---- = H (t)
Af->0+ At
is a renewal density. Then with a help of a probability density f„(t) = F'„(t) we have
oo
M0 = E/„(0.
«=i
A renewal density most often appears in the following integral equation
t
h(t) = f(t) + \h(t - u)f(u)du,
o
so called a renewal equation for a renewal density. Here f(t) is a probability density of a absolutely continuous non-negative time to the renewal X. We can describe the equation approximately by words in such a way that for At—>0 renewal probability h(t)At in the interval (t, t + At ] is equal to a probability sum f(t) At that in the interval (i. t + At ] the first renewal happens and the sum of probabilities for V« e (0, t) that the renewal happens at the time t -u followed by a time to the failure of the length u.
Alternating Renewal Process
Provided that there are two kinds of components with various independent time to a failure X, Y ,
respectively adequate distributional functions F(t), G(t) (densities f(t), g(t)), at the time i 0 the component of the first type is activated and every time at the time of failure is substituted by the component of the opposite type, resulting process is named Alternating renewal process.
We can simulate a renewal process with a definite time to a renewal with such a model. At the time t = 0 the component begins to work to the moment of failure Xi. The final time to the renewal 7; follows. At the moment Xi + 7; the renewal ends and a new (or repaired) component is activated with a time to a failure X2. XiX2... rcsp. Y,. Y2... arc independent nonnegative random variables with a distr. function F(t) resp. G(t). The n'h failure happens at the moment
Sn=X1+Y1+... + Xn_1+Yn_1+Xn,
for nu renewal we have
Tn=Xx+Yx
■X
n-1
1 n-1
A random process {.V,. /',, S2. T2.....} is then an
alternating renewal process. A coefficient of availability K(t) (or also Aft) - availability) is a basic characteristic of a renewal process with a finite time to a renewal. It determines a probability that at the time t the component will work. It is consequently equal to a sum of probabilities that Xi > t, it means that the first component has a time to a failure greater than t, and that the renewal happens in the interval (u, ii + AnJ. Au —*■ 0. 0 < ii < t and a renewed component will have a time to a failure greater than t - u. Written by an integral equation:
K(t) = 1 - F(t) + \h{x)[ 1 - F(t - x)\ix
o
= R(t) + \h(x)R(t-x)dx,
o
h(x) is a renewal process density of a renewal {T„}°°„=o, F(t) is a distribution function of the time to a failure, resp. 1 -F(t) = R(t) is reliability function. In particular, an asymptotic coefficient of availability of the alternating renewal process is important practical reliability characteristics,
K = lim K(t).
! y*
It describes behaviour of the alternating renewal process in the situation when the system is stabilized in "a distant time moment t", i.e. a stationary case, when the influence of the beginning configuration subsides.
Budny Tymoteusz
Gdynia Maritime University, Faculty of Navigation, Gdynia, Poland
Two various approaches to VTS Zatoka radar system reliability analysis
Keywords
VTS system, system reliability, shipping safety Abstract
In the paper we propose two ways of reliability calculation of radar system in Vessel Traffic Services Zatoka. Reliability and availability of the system were calculated on the base of reliability of the system components. In the first approach there was assumed that system is series, in the second approach system is treated as a series-"m out of n". We obtain different results. Conclusion is that choosing proper method of approach to system reliability and availability analysis is decisive in appropriate evaluation of those properties.
1. Introduction
One of the most important properties of the devices and technical systems is their reliability. Reliability is extremely important when concerns systems, which assure people safety or/and natural environment protection. Vessel Traffic Services System - VTS Zatoka is that type of system. Its main task is to assure safe navigation for all ships that sails to ports of Gdynia and Gdansk. The most important part of the VTS Zatoka system are shore based maritime radars. Reliability of that system can be evaluated in different ways. In the paper there are proposed two possible approaches to calculate that reliability [1].
2. Systems' definitions
We assume that [2]
Ei, i = 1,2,...,n, n e N,
are two-state components of the system having reliability functions
R(t) = P(T > t), t e (-¥,¥), i = 1,2,...n, where T, i = 1,2,...,n,
are independent random variables representing the lifetimes of components Ei with distribution functions
F(t) = P(T < t), t e (-¥,¥), i = 1,2,...,n.
Definition 1. A two-state system is called series if its lifetime T is given by
T = min{T }.
1<i < n
E1 E2 - ... - E
Figure 1. The scheme of a series system
The above definition means that the series system is not failed if and only if all its components are not failed, and therefore its reliability function is given by
R(t) = nR.(t),t e (-¥,¥). (1)
i=1
Definition 2. A two-state series system is called non-homogeneous if it is composed of a, 1 < a < n, different types of components and the fraction of the ith type component in the system is equal to qi, where
a
qi > 0, X qt = 1. Moreover
i=1 i
R)(t) = 1 - F(i)(t), t e (-¥, ¥), i = 1,2,...,a, (2) is the reliability function of the ith type component.
The scheme of a non-homogeneous series system is given in Figure 2.
The scheme of a non-homogeneous "m out of n"
system is given in Figure 3, where
i1, i2,...,in e {1,2,...,n} and ij ^ ik for j ^ k.
The reliability function of the non-homogeneous two-state "m out of n" system is given either by
Figure 2. The scheme of a non-homogeneous series system
It is easy to show that the reliability function of the non-homogeneous two-state series system is given by
R'nm) (t)
= 1 - E n(qi"i )[R(0(t)p [F(i)(t)]«'"-r' , (7)
0<r <qfn i=1 r1+r2 +---+ra <m-1
t G (-¥, ¥),
R'k„l„ (t) = n R W , t G (-¥, ¥).
(3)
A two-state system is called an "m out of n" system if its lifetime T is given by
T = T(n-m+l), m = 1,2,...,n,
where T(n-m+1) is the mth maximal order statistic in the sequence of component lifetimes Tj,T2,...,Tn.
or by
Rf) (t)
E n
0<r <qin i=1
r1+r2 +...+ra <m
t G (-¥, ¥),
where m = n - m.
(qrn)[F(1 )(t)]ri[R(i)(t)]qin-ri , (8)
i=1
The above definition means that the two-state "m out of n" system is not failed if and only if at least m out of its n components are not failed. The two-state "m out of n" system becomes a parallel system if m = 1, whereas it becomes a series system if m = n. The reliability function of the two-state "m out of n" system is given either by
R nm)(t) = 1 -
t G (-¥, ¥),
or by
1
E
n [R (t)]ri [F (t)]1-ri ,(4)
1. r2. 1 + r2 +..
,= 0 i 7 < m-1
E1 Ei2 q1
q 2
E,
E, qa
n
R nm )(t) =
1
E
n [F (t )]ri [ R. (t)]
1- ri
(5)
1.r2 ..... rn = L i=1 + r2 +... + rn <m
t g (-¥. ¥). m = n - m.
Definition 3. A two-state "m out of n" system is called non-homogeneous if it is composed of a, 1 < a < n, different types of components and the fraction of the ith type component in the system is equal to qi, where
a
qi > 0, E qt = 1. Moreover
i=1
R(i)(t) = 1 - F(i)(t), t e (-¥,¥), i = 1,2,...,a, (6)
Figure 3. The scheme of a non-homogeneous "m out of n" system
Definition 4. A multi-state system is called series- "m out of kn " if its lifetime T is given by
T = T
(kn-m+1) -
m = 1.2..... kn.
where T
(kn - m+1)
is m-th maximal statistics in the
random variables set
T = min (Ty}. i =1.2.....kn.
1< j<ii
The above definition means that series-" m out of kn"
system is composed of kn series subsystems and it is not failed if and only if at least m out of its kn series subsystems are not failed.
The reliability of the series-"m out of kn" system is given either by
R(m) (t) = 1 -
^kn ,11,12,..., hy> 1
1 kn h
I n[ n R (Of [1 -n R (t)]1
(9)
r1, — rkn =0 i=1 j=1 r + r2 +... + rkn <m-1
t e (-¥, ¥), or by
(t)=
j=1
R (m )
kn ,ll,l2,..., lkn
1 kn ,
I n[1 -nRj(t)f[nRj(t)]1-r', (10)
r1,r2,..., rkn =0 i=1 j=1 j=1
t e (-¥, ¥),
where m = k - m .
2. System of VTS Zatoka radars
Radars' system is the basic subsystem of whole VTS system and also part of identification and watching system at the Gulf of Gdansk region. The purpose of
Figure 4. Positions of VTS Zatoka shore based radars (dots)
that system is assuring real time information about ships' traffic in that region [3]. VTS Zatoka system works involving five shores based radars, which are put in following places:
• Lighthouse Hel, (radar height 42,5 m a.s.l.);
• Port of Gdynia Harbourmaster Office building (HMO), (32,5 m a.s.l.);
• Northern Port of Gdansk Harbourmaster Office building, (66 m a.s.l.);
• Western Hills?, (17,5 m a.s.l.);
• Lighthouse Krynica Morska, (26 m a.s.l.). Radars work permanently and their range cover whole responsibility area assigned to VTS Zatoka. Two, or even three, radars cover most part of Gulf of Gdansk simultaneously. That situation has great matter in case of failure of single radar.
For the VTS Zatoka systems' radars, apart from standard equipment, additional Radar Data Processor (RDP) has been installed. RDP changes radar data from analogue to digital form. This digital information is next transferred to VTS centre by wireless line or light cable, which connect two Harbourmaster's offices of Ports in Gdynia and Gdansk. Signals from radars, after preliminary treatment, are transferred to the VTS Centre. Then after final processing signals are sending out and visualized (with use of computer program ARAMIS) at VTS Centre itself, Harbourmaster's offices of Gdynia and Gdansk ports and at Harbourmaster's office of Krynica Morska port.
Sc heme of the radar' subsystem and data transmission is showed on Figure 5.
- net cable HMO - Harbormaster Office building
Figure 5. Scheme of VTS Zatoka radars' system
3. VTS Zatoka radar system reliability
In order to analyse the considered system reliability we will firstly calculate reliability parameters of single radar.
3.1. Single radar reliability
VTS Zatoka system has been designed and constructed by Holland Institute of Traffic Technology (HITT). It
+ r2 +... + rk <m
