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Intermittently used redundant system
Microelectron. Reliab., Vol. 17, pp. 593 596 © Pergamon Press Ltd. 1978. Printed in Great Britain
0026-2714/78/0801 0593502.00/0
INTERMITTENTLY USED REDUNDANT SYSTEM P. K. KAPUR IMSOR, Technical University of Denmark+ and K. R. KAPOOR University of Delhi, India
(Received fi)r publication 13 January 1978) Abstract--This paper discusses the stochastic behaviour of a 2-unit cold-standby intermittently used system and derives the Laplace-Stieltjes Transform of (i) the distribution of first disappointment time; (ii) probability that the system is under disappointment at time t; (iii) the expected number of occurrences of disappointment during (0, t]. We employ Markov renewal processes technique to obtain these measures. In conclusion it has been shown how our intermittently used system changes to an intermittently available system.
INTRODUCTION A system in general can be classified according to its usage. One is where a system is constantly in use. Details of such systems can be found in [,1,4, 5, 8, 9]. Second is one which is intermittently used, e.g. telephone, and lastly a system which is intermittently available, e.g. computer. Gaver [3] first discussed a one-unit intermittently used system and introduced "Disappointment Time". Later Srinivasan [,,12] extended his results to a 2-unit redundant system with arbitrary failure and exponential repair time distributions to obtain the first passage time to disappointment and mean time to disappointment using a supplementary variable technique. Later [10] also discussed a one-unit intermittently used system to obtain other reliability measures, e.g. pointwise availability and expected no. of disappointments, etc., using renewal type equations of Markov renewal processes. Recently Kapur and Kapoor [-6] discussed an intermittently available 2-unit stand-by redundant system under same assumptions as those in [,12] using technique of Markov renewal processes as discussed in [9]. Later these results were generalized to an arbitrary repair time distribution improving upon the technique in [-9]. In this paper, we discuss the model due to Srinivasan [12] under more generalized assumptions of arbitrary repair time distributions using the technique in [7] to obtain (i) The distribution of first-disappointment-time ; (ii) probability that the system is under disappointment at time t; (iii) expected no. of disappointments.
(a) the system goes down during a usage period or (b) the system is needed while it is down. Some special cases are also discussed. In conclusion, it has also been shown, how an intermittently used system changes to an intermittently available system and vice versa under suitable modifications. More details can be found in [7]. Results presented in this paper are part of the research report [-7] by the authors.
ASSUMPTIONS
1. System has 2-identical units. 2. A standby unit never fails. 3. Once a unit is repaired, it is like-new and goes into standby state or operating state. 4. All the switchings operations are 100% reliable, instantaneous, and never damage anything. 5. The Cdf's of "failure time" and "repair time" are both arbitrary. 6. The Cdf's of occurrence time and holding time of a need are exponential. 7. The system is not needed at time 0. 8. All the random variables are mutually independent.
NOTATION
F(t) G(t) A(t) B(t) U(t) V(t)
Cdf of failure time of operating unit. Cdf of repair time of failed unit. (= 1 - e-'t) Cdf of occurrence time of need. (= 1 - e or) Cdf of holding time of need. Pr {Need occurs at time t[it occurred at time 0}. Pr {Need does not occur at time t lit did not occur at time 0}. _ Complement (e.g. F(t) =- 1 - F(t)). Stieltjes convolution
The disappointment of the system occurs in either case,
(e.g.A(t)*B(t)= f l d A ( u ) B ( t - u ) ) .
+ Present Address : Department of Operational Research, Faculty of Mathematics, University of Delhi, Delhi-110007, India.
[1 - A] ~- 1) 1 + A + A * A + A * A * A + " ' . 593
594
P.K. KAPUR and K. R. KAPOOR
Lower case
Laplace Stieltjes Transform (LST)
DERIVATION
U{t + At) = U(t)(1
ult)
[~
:
+/;
O F Uit) A N D
/]At) +
The time instants for States 0, 3, 4 and 5 are nonregeneration points, thus it is impossible to define Qii(t), i = O, 3. 4, 5. Therefore, we introduce new mass functions:
Q4ff(t):
V(t)
~7(t)c~At, At--+ 0
e '~'''],/[~
+
1~].
(1)
Similarly
8 . ) = ~[1 - e '~+~)']/'[~ + / q .
t2)
v(t) = [/~ + ~. e ('+~)']/[~ + / ~ ]
{3)
C d f o f M R P from state i into state j via state k" time instant for state i is a regeneration point whereas for state k it is a non-regeneralion point. Ql~'"(t): C d f o f M R P from state i into state j via states k and l; time instant for state i is a regeneration point whereas for states k and / these are non-regeneration points.
Q~l°)(t)=
and
e ,,+a,]/]~ + fl].
V(t) =/~[1
U(x) dFt.\),
(1 I)
V(x) dF(x),
3)
Ql°~{t) = ~ o Glx)V(x) dFlx),
14)
o
(4)
It may be noted that the occurrence of a need has the memoryless property because of a convenience of the analysis. To obtain the three measures of the system we define time instants at which the process of the system makes transition into : State - 1 : One unit goes on operation and the other goes on standby, while the system is not needed (i.e. initial state). State 0: The repair of the failed unit is completed (i.e. the unit goes on standby) and the other is operating, regardless of a need. State 1 : One unit goes on operation and the other goes on repair, while the system is needed. State 2 : One unit goes on operation and the other goes on repair, while the system is not needed. State 3 : System goes down while needed. State 4 : System goes down when it is not needed. State 5 : System is needed while it is down.
Q~m(t) =
Q~j(t): Cdf of M R P , i.e. the probability that after transiting into state i, the process next enters into state j, in an a m o u n t of time less than or equal to t.
,,(t)=f'C(x)dFC..),
(5)
= fl V(x) dE(.>:),
(6)
Ql3(t)=fis, x).WC~)dFCx),
~7)
Ql,(t)=flG(x)U(x)dF(x),
(8)
Q - 12(t)
j"G(x) o ~t
Q~'(t) = fi [fi" U(y) dF(y) l dG(x)
(16)
Q~3)~t) 22~ =
0.
[18}
~12°(4~(t~, =
fl{f 0
Q'~.~(t)=
U:(y) dF(y)
(19)
A(x) ,
1201
dG(x),
(23)
V v) dFlv)~ * A(x) dG(x).
(24)
t~x
• t
d
~, ,)
x
U(v} )
~21n~'*'s)~'~u~= ~ JD
Q2,lt) =
;i
P
dG{.\),
j "o G ( . , ) d l . l f o ~U { Y ) dF(v)} • ?t/
)= ~c~.sl(t 1
*A(x)
0
*
0
.
~ (I
By definition, the disappointment state, q), of the system is given by ~p = 3 u 5. In view of this we introduce following notations :
Qi¢(t) - Qi3(t) + Qz3(t)=flG(x)V(_.OdF(x ),
5)
Q~3)(t~ O,
It may be noted that the time instants for States 1 and 2 are regeneration points. Define
Q
f, G(x)
Q!~,(t)= Qn~3, (t) +
(9)
(4)
Qis (tL I~,s, (t), Qi~
i = 1,2. i = 1, ,."~
125) {26)
Now, we define G(x) V(x) dF(x).
(10)
Hijlt):
First-passage time Cdf from state i into state i :
Intermittently used redundant system the time instant for state i is a regeneration point.
H-lz(t)=Q
595
12(t)+Q-11(t)*Ha2(t),
Vile(t) = Ql~(t)- 9]~(t)
H 1,(t) = Q]°'(t) + Q]~)(t) + [Q]~(t) + Q]~(t)] * H2dt). = G(t)[f(i U(x)dF(x) + f l U(x)dF(x)* A(t)],
H2,(t) = Q~2°'(t)+ Q~2~)(t)+ [Q~z~(t)+ Q(z~(t)] * H2,(t).
n2~it)
Taking LST and solving, we obtain
]211(S) = q]l(S) -~- qlz(S)q21(S)/~]22(S),
(27)
h2 ds) = q2 l(S)/{I22(S),
(28)
= Q2~(t)
Q~(t)
=G(t) f l F ( x i d F ( x ) + j ~ : V ( x ) d F ( ' O * A ( t ) ].
and Taking LST of (32) we find p
where qh(s)=-
1,p(S)= [h
ll(S)h22(S)~lq~(S)
+ h i2(s)hll(s)~2~(s)]/hll(S)[~22(s).
. + q;~'(s),~ qil~o~(,s)
(llI) Expected mmTber (?f occurrences of system's disappointments in (0, t], M _ 1¢,(t)
qi2 (S) + qi2 (s),J i = 1.2. Similarly, hi 2(s) = q]2(s)/gt] l(s),
(29)
h22(s) = q 2 2 ( s ) + q21(s)q]x(s)/(l] l(s)
(30)
We also have,
m-l~,(tl= H l l ( t ) * [ 1 - H l d t ) ] I ll*Qle(t) + H 12(t)*[1-H22(t)] I l~*Q2o(t) m . l~(s) ~- [h
1,(s)h22(s)ql~(s)
+ 12- 12(s)hl l(s)q2~o(s)]/hl 1(S}]122(S).
Hl~(t ) = Q1~(t) + Q]°'(t)* H,~(t) + [Q~°~(t) + Q]~It)] • 14~(t),
Example: When the system is in constant-usage
r,qlo)tt) Hz~(t) = Q2~(t) + Q~2°)(t)* Hl~(t) + I_,z22~
throughout the process then
(36)
+ Q~,~It)] • H~,(t).
Taking LST and solving, we obtain
Ibm(s) = [q~z(S)Ch¢(S) + q] 2(s)q2~(s)]/[gl]°'(s)gt22(s) - q~ 2(s)q(z°)(s)], (31) hz~(S) = [q]°'(s)q2¢(s ) + q~2°'(s)q~(s)]/[q]°~¢s)~"22(s)
- q~ 2(s)q~(s)].
(32)
Suppose that the process starts in state - 1 at time 0. Then using (5)-(32), we derive the following stochastic behaviour of the disappointment time. (I) Cd,l'~?lfirst disappointment rime H_ ~,~(t)
~2(t)*H2~(t)
H l¢(t)=Q-~(t)*H~o(t)+Q
h_l~o(s)=q
ll(S)hl~o(s)+q
A(t) = -B(t) =- 1
Vt
U(t) = 1 and
V(t) - O
As a result, the states 2, 4 and 5, of the model vanish and the states - 1, 0, 1 and 3 take the form as given below : State - 1 : One unit goes on operation and the other goes on standby. State 0: One unit is operative and the other goes on standby. State 1 : One unit goes on operation and the other goes on repair. State 3: System goes down.
Q- 11(o = F(t),
h 1,e(s)= [q ll(S){(]22(s)ql~o(s) -4- q;2(s)q2~(s)}
Q13(t) =
-? q ~2(s){{]~°)(s)q2~o(S)
+ q~2°'(s)q,~(s)',]/Egl]°'(S)Oz2(S) -- ql 2(s)q(2°)(S)].
(33)
(ll) Probability that system is' under disappointment at time t, P_ l~(t) Let Pij(t) denote the probability that the process is in state i at time t given that the process started in state i at time 0. Using renewal theoretic arguments, we have P_ ~,~(t) = H ~l(t) * [1 - Ha ~(t)] ( - ~)* gIa,~(t) + H ~2(t)*[1 - Hz2(t)]' '~*Hz~(t), where ~(tj=Q_as(r)+Q
~.I.R. 176 II
~2(t)*H2~(t),
Vt.
In view of these changes equations (5)-(24) take the form :
12(s)h2~o(s).
Using (29) and (30) it gives
H
(351
(34)
;o
(~(x) dF(x),
Q~°)(t)= f l G(x)dF(x), Q~ t(t) =
F(x) dG(x).
All the rest vanish. These coincide with the mass functions of Nakagawa and Osaki [10]. Hence all of their results follow. In conclusion, we discuss how our (main) model changes to an intermittently available system, studied by the authors [7]. We assume that the system serves two type of customers such that it satisfies assumptions 1 - 8 of [7] ; the need period of the system is taken to be serving period of type 1 customers, whereas the no
596
P . K . KAPUR and K. R. KAPOOR
d e m a n d period is taken to be servicing period of type 2 customers. As a result, following states o f the system follow from our main m o d e l : State - 1 : O n e unit goes o n o p e r a t i o n a n d the o t h e r goes on s t a n d b y serving type 2 customers. State 0 : O n e unit is in o p e r a t i o n a n d the other goes on s t a n d b y serving any type o f customers. State 1 : O n e unit goes on o p e r a t i o n and the other goes on repair serving type 1 customers. State 2 : O n e unit goes on o p e r a t i o n a n d the other goes o n repair serving type 2 customers. State 3: System goes d o w n while serving type 1 customers. State 4: System goes d o w n while serving type 2 customers. State 5: Interference b e t w e e n two classes o f customers by type 1 customers. State 6: Interference b e t w e e n two classes of cust o m e r s by type 2 customers. T h e e p o c h at which the system enters state 5 u 6 c o r r e s p o n d s to interference time. Now, let us s u p p o s e that AIt) B(t) Uit}
t = l - e ~') C d f of servicing time of type 2 customers. {-1 e ~,') C d f of servicing time of type 1 customers. Pr '~System serves type 1 c u s t o m e r s at time t I it was serving the same at time 0',.
V(t)
Pr ISystem serves type 2 c u s t o m e r s at time t! it was serving the same at time 0~.
We further introduce ('it} Dit)
( = 1 - e ~'~ called-into-use time C d f of type 2 customers. {= 1 - e ~,') called-into-use time C d f o f type 1 customers.
Now. following the p r o c e d u r e of our main model, one can easily evaluate h ~6(s), p ,6(s) and ,1 ~,(s}. For details a n d usefulness of the technique refer [ 7].
REFERENCES 1. R. E. Barlow and L. C. Hunter. Reliability analysis of one-unit system, Opns Res. 9, 201 208 (1961}. 2. R. E. Barlow and F. Prochan, Mathematical Theo,3 ,?/ Reliability, Wiley, New York (1965). 3. D. P. Gaver, Jr, Opns Res. 12, 534 542 (1964). 4. D. P. Gaver, Jr, IEEE Trans. Reliah. R-12, 30 (1963). 5. D. P Gaver. Jr, IEEE Trans. Reliab. R-13, 14 [1964}. 6. P. K. Kapur and K. R. Kapom'. IMSOR Rep. pp. I 29 (March 1977l, and Errata. 7. P. K. Kaput and K. R. Kapoor, IMSOR Rep. pp. I 3{) (August 1977). X. B. V. Gnedenko, Yu. K. Belyayev and A. D. Solovyc~. Mathematical Methods ,~! Reliability Them'v. Academic Press, New York (1969). 9. T. Nakagawa and S. Osaki, I N k O R 12, 68 71)11974). 10. T. Nakagawaand S. Osaki, RAIRO 2, 101 112(1975}. 11. D. R. Cox, Renewal Theory, Metheun, London {19621. 12. V. S. Sriniwtsan, Opns Res. 14, 1024 1036 (19661.
0026-2714/78/0801 0593502.00/0
INTERMITTENTLY USED REDUNDANT SYSTEM P. K. KAPUR IMSOR, Technical University of Denmark+ and K. R. KAPOOR University of Delhi, India
(Received fi)r publication 13 January 1978) Abstract--This paper discusses the stochastic behaviour of a 2-unit cold-standby intermittently used system and derives the Laplace-Stieltjes Transform of (i) the distribution of first disappointment time; (ii) probability that the system is under disappointment at time t; (iii) the expected number of occurrences of disappointment during (0, t]. We employ Markov renewal processes technique to obtain these measures. In conclusion it has been shown how our intermittently used system changes to an intermittently available system.
INTRODUCTION A system in general can be classified according to its usage. One is where a system is constantly in use. Details of such systems can be found in [,1,4, 5, 8, 9]. Second is one which is intermittently used, e.g. telephone, and lastly a system which is intermittently available, e.g. computer. Gaver [3] first discussed a one-unit intermittently used system and introduced "Disappointment Time". Later Srinivasan [,,12] extended his results to a 2-unit redundant system with arbitrary failure and exponential repair time distributions to obtain the first passage time to disappointment and mean time to disappointment using a supplementary variable technique. Later [10] also discussed a one-unit intermittently used system to obtain other reliability measures, e.g. pointwise availability and expected no. of disappointments, etc., using renewal type equations of Markov renewal processes. Recently Kapur and Kapoor [-6] discussed an intermittently available 2-unit stand-by redundant system under same assumptions as those in [,12] using technique of Markov renewal processes as discussed in [9]. Later these results were generalized to an arbitrary repair time distribution improving upon the technique in [-9]. In this paper, we discuss the model due to Srinivasan [12] under more generalized assumptions of arbitrary repair time distributions using the technique in [7] to obtain (i) The distribution of first-disappointment-time ; (ii) probability that the system is under disappointment at time t; (iii) expected no. of disappointments.
(a) the system goes down during a usage period or (b) the system is needed while it is down. Some special cases are also discussed. In conclusion, it has also been shown, how an intermittently used system changes to an intermittently available system and vice versa under suitable modifications. More details can be found in [7]. Results presented in this paper are part of the research report [-7] by the authors.
ASSUMPTIONS
1. System has 2-identical units. 2. A standby unit never fails. 3. Once a unit is repaired, it is like-new and goes into standby state or operating state. 4. All the switchings operations are 100% reliable, instantaneous, and never damage anything. 5. The Cdf's of "failure time" and "repair time" are both arbitrary. 6. The Cdf's of occurrence time and holding time of a need are exponential. 7. The system is not needed at time 0. 8. All the random variables are mutually independent.
NOTATION
F(t) G(t) A(t) B(t) U(t) V(t)
Cdf of failure time of operating unit. Cdf of repair time of failed unit. (= 1 - e-'t) Cdf of occurrence time of need. (= 1 - e or) Cdf of holding time of need. Pr {Need occurs at time t[it occurred at time 0}. Pr {Need does not occur at time t lit did not occur at time 0}. _ Complement (e.g. F(t) =- 1 - F(t)). Stieltjes convolution
The disappointment of the system occurs in either case,
(e.g.A(t)*B(t)= f l d A ( u ) B ( t - u ) ) .
+ Present Address : Department of Operational Research, Faculty of Mathematics, University of Delhi, Delhi-110007, India.
[1 - A] ~- 1) 1 + A + A * A + A * A * A + " ' . 593
594
P.K. KAPUR and K. R. KAPOOR
Lower case
Laplace Stieltjes Transform (LST)
DERIVATION
U{t + At) = U(t)(1
ult)
[~
:
+/;
O F Uit) A N D
/]At) +
The time instants for States 0, 3, 4 and 5 are nonregeneration points, thus it is impossible to define Qii(t), i = O, 3. 4, 5. Therefore, we introduce new mass functions:
Q4ff(t):
V(t)
~7(t)c~At, At--+ 0
e '~'''],/[~
+
1~].
(1)
Similarly
8 . ) = ~[1 - e '~+~)']/'[~ + / q .
t2)
v(t) = [/~ + ~. e ('+~)']/[~ + / ~ ]
{3)
C d f o f M R P from state i into state j via state k" time instant for state i is a regeneration point whereas for state k it is a non-regeneralion point. Ql~'"(t): C d f o f M R P from state i into state j via states k and l; time instant for state i is a regeneration point whereas for states k and / these are non-regeneration points.
Q~l°)(t)=
and
e ,,+a,]/]~ + fl].
V(t) =/~[1
U(x) dFt.\),
(1 I)
V(x) dF(x),
3)
Ql°~{t) = ~ o Glx)V(x) dFlx),
14)
o
(4)
It may be noted that the occurrence of a need has the memoryless property because of a convenience of the analysis. To obtain the three measures of the system we define time instants at which the process of the system makes transition into : State - 1 : One unit goes on operation and the other goes on standby, while the system is not needed (i.e. initial state). State 0: The repair of the failed unit is completed (i.e. the unit goes on standby) and the other is operating, regardless of a need. State 1 : One unit goes on operation and the other goes on repair, while the system is needed. State 2 : One unit goes on operation and the other goes on repair, while the system is not needed. State 3 : System goes down while needed. State 4 : System goes down when it is not needed. State 5 : System is needed while it is down.
Q~m(t) =
Q~j(t): Cdf of M R P , i.e. the probability that after transiting into state i, the process next enters into state j, in an a m o u n t of time less than or equal to t.
,,(t)=f'C(x)dFC..),
(5)
= fl V(x) dE(.>:),
(6)
Ql3(t)=fis, x).WC~)dFCx),
~7)
Ql,(t)=flG(x)U(x)dF(x),
(8)
Q - 12(t)
j"G(x) o ~t
Q~'(t) = fi [fi" U(y) dF(y) l dG(x)
(16)
Q~3)~t) 22~ =
0.
[18}
~12°(4~(t~, =
fl{f 0
Q'~.~(t)=
U:(y) dF(y)
(19)
A(x) ,
1201
dG(x),
(23)
V v) dFlv)~ * A(x) dG(x).
(24)
t~x
• t
d
~, ,)
x
U(v} )
~21n~'*'s)~'~u~= ~ JD
Q2,lt) =
;i
P
dG{.\),
j "o G ( . , ) d l . l f o ~U { Y ) dF(v)} • ?t/
)= ~c~.sl(t 1
*A(x)
0
*
0
.
~ (I
By definition, the disappointment state, q), of the system is given by ~p = 3 u 5. In view of this we introduce following notations :
Qi¢(t) - Qi3(t) + Qz3(t)=flG(x)V(_.OdF(x ),
5)
Q~3)(t~ O,
It may be noted that the time instants for States 1 and 2 are regeneration points. Define
Q
f, G(x)
Q!~,(t)= Qn~3, (t) +
(9)
(4)
Qis (tL I~,s, (t), Qi~
i = 1,2. i = 1, ,."~
125) {26)
Now, we define G(x) V(x) dF(x).
(10)
Hijlt):
First-passage time Cdf from state i into state i :
Intermittently used redundant system the time instant for state i is a regeneration point.
H-lz(t)=Q
595
12(t)+Q-11(t)*Ha2(t),
Vile(t) = Ql~(t)- 9]~(t)
H 1,(t) = Q]°'(t) + Q]~)(t) + [Q]~(t) + Q]~(t)] * H2dt). = G(t)[f(i U(x)dF(x) + f l U(x)dF(x)* A(t)],
H2,(t) = Q~2°'(t)+ Q~2~)(t)+ [Q~z~(t)+ Q(z~(t)] * H2,(t).
n2~it)
Taking LST and solving, we obtain
]211(S) = q]l(S) -~- qlz(S)q21(S)/~]22(S),
(27)
h2 ds) = q2 l(S)/{I22(S),
(28)
= Q2~(t)
Q~(t)
=G(t) f l F ( x i d F ( x ) + j ~ : V ( x ) d F ( ' O * A ( t ) ].
and Taking LST of (32) we find p
where qh(s)=-
1,p(S)= [h
ll(S)h22(S)~lq~(S)
+ h i2(s)hll(s)~2~(s)]/hll(S)[~22(s).
. + q;~'(s),~ qil~o~(,s)
(llI) Expected mmTber (?f occurrences of system's disappointments in (0, t], M _ 1¢,(t)
qi2 (S) + qi2 (s),J i = 1.2. Similarly, hi 2(s) = q]2(s)/gt] l(s),
(29)
h22(s) = q 2 2 ( s ) + q21(s)q]x(s)/(l] l(s)
(30)
We also have,
m-l~,(tl= H l l ( t ) * [ 1 - H l d t ) ] I ll*Qle(t) + H 12(t)*[1-H22(t)] I l~*Q2o(t) m . l~(s) ~- [h
1,(s)h22(s)ql~(s)
+ 12- 12(s)hl l(s)q2~o(s)]/hl 1(S}]122(S).
Hl~(t ) = Q1~(t) + Q]°'(t)* H,~(t) + [Q~°~(t) + Q]~It)] • 14~(t),
Example: When the system is in constant-usage
r,qlo)tt) Hz~(t) = Q2~(t) + Q~2°)(t)* Hl~(t) + I_,z22~
throughout the process then
(36)
+ Q~,~It)] • H~,(t).
Taking LST and solving, we obtain
Ibm(s) = [q~z(S)Ch¢(S) + q] 2(s)q2~(s)]/[gl]°'(s)gt22(s) - q~ 2(s)q(z°)(s)], (31) hz~(S) = [q]°'(s)q2¢(s ) + q~2°'(s)q~(s)]/[q]°~¢s)~"22(s)
- q~ 2(s)q~(s)].
(32)
Suppose that the process starts in state - 1 at time 0. Then using (5)-(32), we derive the following stochastic behaviour of the disappointment time. (I) Cd,l'~?lfirst disappointment rime H_ ~,~(t)
~2(t)*H2~(t)
H l¢(t)=Q-~(t)*H~o(t)+Q
h_l~o(s)=q
ll(S)hl~o(s)+q
A(t) = -B(t) =- 1
Vt
U(t) = 1 and
V(t) - O
As a result, the states 2, 4 and 5, of the model vanish and the states - 1, 0, 1 and 3 take the form as given below : State - 1 : One unit goes on operation and the other goes on standby. State 0: One unit is operative and the other goes on standby. State 1 : One unit goes on operation and the other goes on repair. State 3: System goes down.
Q- 11(o = F(t),
h 1,e(s)= [q ll(S){(]22(s)ql~o(s) -4- q;2(s)q2~(s)}
Q13(t) =
-? q ~2(s){{]~°)(s)q2~o(S)
+ q~2°'(s)q,~(s)',]/Egl]°'(S)Oz2(S) -- ql 2(s)q(2°)(S)].
(33)
(ll) Probability that system is' under disappointment at time t, P_ l~(t) Let Pij(t) denote the probability that the process is in state i at time t given that the process started in state i at time 0. Using renewal theoretic arguments, we have P_ ~,~(t) = H ~l(t) * [1 - Ha ~(t)] ( - ~)* gIa,~(t) + H ~2(t)*[1 - Hz2(t)]' '~*Hz~(t), where ~(tj=Q_as(r)+Q
~.I.R. 176 II
~2(t)*H2~(t),
Vt.
In view of these changes equations (5)-(24) take the form :
12(s)h2~o(s).
Using (29) and (30) it gives
H
(351
(34)
;o
(~(x) dF(x),
Q~°)(t)= f l G(x)dF(x), Q~ t(t) =
F(x) dG(x).
All the rest vanish. These coincide with the mass functions of Nakagawa and Osaki [10]. Hence all of their results follow. In conclusion, we discuss how our (main) model changes to an intermittently available system, studied by the authors [7]. We assume that the system serves two type of customers such that it satisfies assumptions 1 - 8 of [7] ; the need period of the system is taken to be serving period of type 1 customers, whereas the no
596
P . K . KAPUR and K. R. KAPOOR
d e m a n d period is taken to be servicing period of type 2 customers. As a result, following states o f the system follow from our main m o d e l : State - 1 : O n e unit goes o n o p e r a t i o n a n d the o t h e r goes on s t a n d b y serving type 2 customers. State 0 : O n e unit is in o p e r a t i o n a n d the other goes on s t a n d b y serving any type o f customers. State 1 : O n e unit goes on o p e r a t i o n and the other goes on repair serving type 1 customers. State 2 : O n e unit goes on o p e r a t i o n a n d the other goes o n repair serving type 2 customers. State 3: System goes d o w n while serving type 1 customers. State 4: System goes d o w n while serving type 2 customers. State 5: Interference b e t w e e n two classes o f customers by type 1 customers. State 6: Interference b e t w e e n two classes of cust o m e r s by type 2 customers. T h e e p o c h at which the system enters state 5 u 6 c o r r e s p o n d s to interference time. Now, let us s u p p o s e that AIt) B(t) Uit}
t = l - e ~') C d f of servicing time of type 2 customers. {-1 e ~,') C d f of servicing time of type 1 customers. Pr '~System serves type 1 c u s t o m e r s at time t I it was serving the same at time 0',.
V(t)
Pr ISystem serves type 2 c u s t o m e r s at time t! it was serving the same at time 0~.
We further introduce ('it} Dit)
( = 1 - e ~'~ called-into-use time C d f of type 2 customers. {= 1 - e ~,') called-into-use time C d f o f type 1 customers.
Now. following the p r o c e d u r e of our main model, one can easily evaluate h ~6(s), p ,6(s) and ,1 ~,(s}. For details a n d usefulness of the technique refer [ 7].
REFERENCES 1. R. E. Barlow and L. C. Hunter. Reliability analysis of one-unit system, Opns Res. 9, 201 208 (1961}. 2. R. E. Barlow and F. Prochan, Mathematical Theo,3 ,?/ Reliability, Wiley, New York (1965). 3. D. P. Gaver, Jr, Opns Res. 12, 534 542 (1964). 4. D. P. Gaver, Jr, IEEE Trans. Reliah. R-12, 30 (1963). 5. D. P Gaver. Jr, IEEE Trans. Reliab. R-13, 14 [1964}. 6. P. K. Kapur and K. R. Kapom'. IMSOR Rep. pp. I 29 (March 1977l, and Errata. 7. P. K. Kaput and K. R. Kapoor, IMSOR Rep. pp. I 3{) (August 1977). X. B. V. Gnedenko, Yu. K. Belyayev and A. D. Solovyc~. Mathematical Methods ,~! Reliability Them'v. Academic Press, New York (1969). 9. T. Nakagawa and S. Osaki, I N k O R 12, 68 71)11974). 10. T. Nakagawaand S. Osaki, RAIRO 2, 101 112(1975}. 11. D. R. Cox, Renewal Theory, Metheun, London {19621. 12. V. S. Sriniwtsan, Opns Res. 14, 1024 1036 (19661.