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Stochastic analysis of an information computer network system
Microelectron. Reliab., Vol. 33, No. 2, pp. 133-139, 1993. Printed in Great Britain.
0026-2714/9356.00+ .00 © 1993PergamonPress Ltd
STOCHASTIC ANALYSIS OF AN INFORMATION COMPUTER NETWORK SYSTEM VIRENDRA SINGH R A N A Government of India (Planning Commission), National Informatics Centre, District Guna (M.P.), India
(Received for publication 1 August 1991) Abstract--This paper investigates a mathematical model of an information computer network system. In this model, the system has two parts---a controller and a terminal. It is also linked with a satellite by a disk antenna. Each part has two failure modes; one is transient failure and second is latent failure. After latent failure, the terminal is recoverable, but the controller is not recoverable. The system may also fail due to a common-cause failure when it is in the normal state or in a partially normal state. Failure rates are constant and recovery rates are general. Using the regenerative point technique several measures of system effectiveness are obtained.
INTRODUCTION Many researchers, including [1] and [2], have studied different models related to network systems, using methods of reliability optimization and probability theorems. Now, information technology has an important role in decision making and policy planning. An information network is used for receiving and transmitting information, on a country-wide level. There are two main parts of this system; the controller and a terminal, by which we communicate information to anywhere in a country via the satellite. The terminal is the input and output medium of information and the controller is the processer of such information. In the present paper, this network model is studied using the following set of assumptions. 1. The controller and terminal of a system may work with reduced efficiency if either the controller or terminal is affected by transient failure. This state of the system is called the partial failure state. 2. The system is considered to be down completely if either the controller or both parts (controller and terminal) are down completely. 3. Recovery of the system is possible from the states: (a) when either the controller or the terminal is down partially; or (b) when the terminal is down completely. 4. Recovery of the system is not possible when the controller is down completely. 5. Recovery starts immediately after transient failure of the controller or terminal. Failure time distributions are assumed to be negative exponential, while all the recovery time
distributions are taken to be general. The following characteristics of interest to the system designers and reliability engineers, are obtained by using the regenerative point technique: (1) mean time to system failure; (2) pointwise and steady state availability of the system; (3) expected number of recoveries of the controller; (4) expected number of recoveries of the terminal; and (5) expected number of recoveries of the system.
DESCRIPTION OF THE MODEL The network system can be described as follows. (1) A system having two main components---a controller and a terminal, which are associated with a satellite. (2) The terminal may fail partially due to a transient failure, i.e. operational mishandling, or fail completely due to a latent failure, i.e. software or hardware problems. (3) The controller may fail partially due to a transient failure, i.e. electromagnetic interferences, and completely fail due to a latent failure, i.e. major hardware problems. (4) The system can fail completely due to a common cause, i.e. upset antenna direction or power fluctuations. (5) The recovery of the terminal is possible from both partial and complete failure states, while the recovery of the controller is only possible from a partial failure state. (6) The system is said to be down when either the controller, or both terminal and controller, are down completely. 133
134
V.S. RANA
(7) F a i l u r e rates are c o n s t a n t a n d recovery rates are general. (8) A f t e r recovery, the s y s t e m is as new.
Considering the above symbols, the system may be in one of the following states. Up states:
s0 = (No, No) NOTATION AND STATES OF THE SYSTEM % constant failure rate of the controller from a normal (N) mode to a partial normal (P) mode, and from the partial normal (P) mode to a down (D) mode, where i = 1, 2 /~ constant failure rate of terminal from N-mode to P-mode, and from P-mode to D-mode, where i=1,2 6 constant common-cause failure rate of the system, when both controller and terminal are completely down simultaneously G(-) c.d.f, o f recovery time of the controller from a P-mode to N-mode H~(.) c.d.f, ofrecovery time of the terminal from a P-mode to N-mode, and from D-mode to N-mode, where i=1,2 mij contribution to mean sojourn time in a state S~~ E and non-regenerative state, if one should occur, before transiting to a state Sj E E /z~ mean sojourn time in a state S ~ E or #t = £jm~j A~(t) p [system is up at time t i E 0 = &~E] RT(t) expected number of recoveries o f the controller from P-mode [ E 0 = S, R}(t) expected number of recoveries of the terminal from P-mode or from D-mode [ E0 = Sj. Other symbols used in this paper may be seen in Ref. [3]. No controller/terminal is in a normal operative mode Por controller/terminal is in a P-mode and undergoing a recovery process D controller is in a down mode and not recoverable Dr terminal is in a down mode and undergoing a recovery process D N system (controller and terminal) is in a down mode and not recoverable.
$6
S~
S l = (Po,, No) S3 = (No, Por) S, = (No, Dr)
s s = 0'o. Po0 $7 = (Por, Dr); and Down states: S2 = (D, No) $6 = (V, Pot)
ss
=
(DN).
Each state is thus a two-dimensional vector, with the first element denoting the state of the controller and the second denoting the state o f the terminal, A transition diagram showing the various states of the system is given by Fig. 1.
TRANSITION PROBABIIdTW~ AND MEAN S O J O U R N TIMES T h e t r a n s i t i o n p r o b a b i l i t y m a t r i x (t.p.m.) o f the e m b e d d e d M a r k o v c h a i n is:
P = ((P,j)) with n o n - z e r o elements:
Pol = ~l/(~l + ~l + 6), Pos = 6/(~] + p~ + 6),
/'03 = ~1/(~'1 +/~t + 6), P~o = C7(~ + ~2),
Pl2 = °~2[1 -- (~(~! + ~2)]/(~! + ~2), Pl5 = ~1[ 1 -- 0 ( ~ i + ~2)]/(~1 + ~2), P30 --/-I1 (f12 Jr ~1 ),
p ~ =/~2[1 - &(fl~ + ~)]/(/~ +
~,),
$2
¢)/: Io] --¢ Normal Par~ciat normal
] Fig. 1. Transition diagram.
Down
An information computer network system
135
for we(t) are obtained:
Pss ffi ai [1 --/'~l (f12 + a~ )]/(~ff2+ ¢x,),
we(t) == Qm (t) (~) al (t)-i-Qos(t) (~) as(t) -I-Qei(t)
p~o = : h (~l),
.~(t) = Qio(t) ® we(t) + Q,s(t) ® as(t) + Q,2(t)
as(t)= Q~o(t) ® ~o(t) + Q.(t) ® n4(t)
p~ f dHi(t)e-m+#,+~G(t), =
+ Q s s ( t ) (~) as(t)
a4(t) ffiQ4o(t) (~) ~o(t) + Q47(t) (~) n,(t)
f dG (t)e- (==+ #2+ s),Rt(t),
Pss
~s(t) = Qs,(t) ® n,(t) + Qss(t) ® .~(t)
Ps~ = f ~2e-(=2+ #2+ s),dt~_r(t)~i (t),
+ Qs7(t) ® av(t) + Qs~(t)+ Qs~(t)
n,(t) = Q,i(t) ® n~(t) + Q,.(t) ® a,(t) +
Q~(t). (I)
Psv = f f12e -¢=2+ #2+ ~)~dt~.r(t)R~(t),
By taking a L.S.T. of the above relations and simplifying for ~Zo(s),we have:
Pss = f 6 e -(=2+~+~)t dt~(t)Hl(t),
~o(s) = N~(s)/Di(s),
~,, =
J'dn~(t)~-'~'C.(,),
p,, =
f dG(t)e-=2'II2(t),
PTs
(2)
where
+ O.,s (~os ~s, Oi2 - O.ss ~o, O.i~)]
= f.2 e-a2t dt~(t)R2(t).
+ Oos(~.~,,
+ O.O.s,)(O,, + 0,, O.,.)
It can be easily verified that Poi +.17o3+Pos mPlo -t- P12 -t- PlS --P~o +P34 -I-P35 =P4o +P~, ~Psi +Pss +Ps6 +Ps~ +Pss ffi 1,
(~,, (Ts,)]
and
PTI +P74 +P78 ffi 1. To calculate the mean sojourn time ~ in state So, we discover that so long as the system is in a state So, there is no transition to Sl, $3 or Ss. Hence, if T denotes the sojourn time in state So, then ffi . I p ( T > t) dt ffi 1/(al +
-
i), = (1 - ~o,)[1 - {03, ~ss
(~+, ~,+) - 0,, ~,, }]
+ o - ~,,O,,)[O.os~(O,s~s,
- 1)
~l + 6), - ~,sOs,~,, {1 - Oos(O~ + ~,,O.o)}
~l ffi [1 - G(~2 +/h)]/(~2 + / ~ ) ,
-- ~o, 0s404o(1 -- 0.,s0m)
Us = [ 1 -- i~l (ai + fl2)]/(ai + iff2),
- O..O,,O..(O.o,O.,s + O.os~.)
~+ = B - O~(~,)l/~,,
#s = r e - " , +,, ++ dtG(t)Hi( t ) , ~,
=
(for brevity, the argument s is omitted), now the mean time to system failure, given that Eo = So, is
fe-'~'~(t)A~(t).
MEAN TIME TO SYSTEM FAILURE The time to system failure can be regarded as the first time to the failed state. To obtain this, we regard the down states $2, S~ and Ss as absorbing. By employing the arguments used for the regenerative process, the following recursive relations
M T S F = - d ~ o ( S ) ', ds ',-o
N, Dx'
(3)
where N~ ffi motao + mo3a I + t r ~ a 2 -t- mloa 3 + ml2aI + m~sas + msoa+ + m~a7 + mssa8 -~-m4oa9 + m47am + restart -t- mssat2 -I- ms6al3 + msTa~++ mssals + mTlal6 -t- mT+a~7+ m~sa,,
136
V.S. RANA
ao = (1 --P4~P74){Pls(P53P3o +P56 + P s s ) + P , o --
a12 ----(1 -- P47P74) [P35 { 1 -- Pos -- Pot (Plo + Pl2)}
P35P53 (Plo + Pl2)} + Pl5 {Ps3P34(P~ + P47P78 )
+ PolPlsP30] + PlsP34P47PTI(1 -- Po~)
+ Ps"/(P78 + P74P4o)} a I ----(l -- P47P74) {P3o ( 1 -- P lsPsl ) + P35PsI (PI0 + PI2) + P35 (P56 + P58)} d- ( l -- Pl5P51 ) (P47P7s
+ PolPl 5P34(P40 + P4~PTs )
at 3 = ( | -- P47P74)(PolPl5 + Po3P35) + Po3P34P47P71P15 al4 = (Pol Pl 5 + Po3P35) (P74P40 + PTs )
+ P40)P~ + P35P57 (PTs + P74P40)
+ PTI [Pl ~{ l -- Pos -- P03(P34P40 + P30)}
+ P35P57P71(Pl2 -[-Plo) -- PlsP57PTI (P34P40+P30)
+Po3P35(PIo+Pt2)],
+P34P4"IP'll {Pl0 +P12 + Pls(P56 + P5S)}
al6 = P03 {P34P47PTI(P56 + Pss ) + (Pro + Pl 2) (P~P47
a 2 = (l --p47P74)(1 --PlsP51 --P35P53)
+ P35P57) -- Pl sP57 (P30 + P34P40)}
-- P IS(P53P34P47 + P57)PTI a 3 ~- (1 -- P47P74) {Pol (1 -- P35P53) + Po3P35PsI}
+ (1 - Pos)Pl 5(P53P34P47 + P57 ) al7 -- (PolPI5 + Po3P35) {P57P40 - P47 (P56 + P58)}
+ P03 (P34P47 + P35P57)PT1
+ P47 {(1 - P35P53 (1 - Po8 - PolPlo)
a 4 ----(1 -- P47P74) {(Po3Psl -- Pol P53 }
+ (1 - PIsP51 ) (1 - Pos - Po3P30)
•4- ]703(P34P47 -F P35P57)PTI
+ PolP53 (PI2P35 - PI5P30)
a 5 = ( 1 -- P47P74) {Psi (1 -- Pos -- Po3P30) +:Pol (P53P30 +P56 +P58)} + (l -- Pos)(P57 +P53P34P47)PTI
- Po3P35PsI(Plo + P12)} als ----PolPl5 (P57 + P53P34P47)
+ Po3P34P47PTI(P56 -t- Pss ) -- P03 (P34P40 + P30 )P57P71 -- Po3P34(P47PTs + P4o)Psl
al5 = al3
+ P03 {P35P57 "Jr-P34P47 (1 -- PI5P51 )} and
+ Pol (P57P74 + P53P34)P4o D 1 = (l -- POl ) [l -- {P35P53 (P47P74) -- P47P74 }] + POl (P57 + P53P34P47)PTs
a 6 = (l -- P4"IP74){PolPIsP53 + Po3 (1 -- PIsP51 )} - -
+ (l -- P47P74) [P03P30 (PlsPsl -- l) -- Pl5Psl -- PIoPo3P35PsI -- PoIPlsP53P30]
Po3PI5P57P'tl -- [PIsP53P34(P47PTl + P4oPoI)
aT = Po3PTI []747{Pl5 (P~ + PSS) + Pl2 }
-- PIsP57PTI { 1 - P03 (P30 + P34P40)}] + P47Plo -- Pl5P5TP4o] + PIsP53P47PTI(1 -- P0S) - Po3P34P40(l - Pl5P51 ) + (P40 +P47PTs){P03( 1 --PIsPsI) +PoIPIsP53} P57P74P4o(PolP15+ Po3P35)
- -
as = (1 --p47P74){pn(1 --Pos) -- (Pro "~-Pl2) × (P01P53 -- P03Ps1 ) + P03 (P56 + PSS)}
-- PloP03 (P34P47 + P35P57)P71"
+ P03P57 {(P7$ + P74P40) + PTt (Pro + Pl2)}
0-9 = PolPls(P57P74 + Ps3P34) + Po3[P35P57Pu +P34{1 --Pls(Psl +P57P71)}]
AVAILABILITY OF THE SYSTEM A c c o r d i n g to the definition o f At(t), we have the following recursive relations:
alo = (P~6 + Pss){Po3P34P71P15 -- (PolPls+ Po3P35)P74}
Ao(t) = Mo(t) + qol(t) © A l ( t ) + qo3(t) © A3(t)
"Jr"(1 --PlsP51 ) {Po3PMP78"[-P74 ( l --P08 --P03P30)}
A, (t ) = M l ( t ) + qlo(t) © Ao(t) + qls(t) © As(t)
+ (PI0 + Pl2 )P03 (P~4PTt -- P3sPsIP74)
A3(t) ----M3(t) + q30(t) (~
at"PIsP53P34{PTI (l --Po~) + P01PTs }
Ao(I) + q34(t) © A4(t)
+ q35(t) © As(t)
-- P35P53P74{ l -- Pos -- P01 (Pl0 + Pl2 )}
A4(t )
-- PoIP74(Plo + PI5P53P30)
As(t) -- M s ( t ) + qst (t) © A l ( t ) + q53 (t) (~ a 3 ( t )
all ~- (l --P4"rP74){Po3P35(Plo +PI2) + P l s ( 1 --P0s -- P03P30)} -- Po3P34(P47P7s + P40)PI5
=
M 4 ( t ) + q40(t ) (~ Ao(t ) + q47(t) (~) AT(t)
+ q n ( t ) © A7(t)
A , ( t ) = MT(t) + qT, (t) © A, (t) + qv4(t) © A4(t), (4)
An information computer network system where, as simple probabilistic considerations show: M o ( t ) = e -(~ +#~+6),,
and D2 = molbol + mo3bl + mlob2 + ml5b3 + m3ob4
M t (t) = e -(~2 + #~)~¢~(t),
M3(t) = e - t ~ + # 2 ) t R t ( t ) ,
137
M4(t) = e-~Fl2(t),
+ mub5 + m35b6 -b m4ob7 q- m47b8 + mslb9
M 5(t) = e -(~2 + #2+ 6)t//1 (t)G(t),
+ m53blo + m57bl! -}- m71 bl2 -}- m74b13
M 7(t) = e - :2,//2 ( t ) ¢ ( t ) .
b 0 =- (1 - P47P74){Pro (1 - P35P53) + P t5P53P30}
By taking the L.T. o f the a b o v e set of equations (4), we have: A ~ ( s ) = N2(s)/D2(s ),
+ Pt s (P53P34 + P57P74)P40 bt = (1 - P4~P74){P30(l - PtsP51 ) + PtoP35Pst }
(5) + { P34 (1 -- PIsPst) + P35P5~PT,}P4o
where + P lo(P34P47 + P35P57)P71 N2(S ) = (1 - q4~Q~,)[(Mo * * * + q*t M * ) ( 1 - q;'sq*3)
b2 = (1 -P47P74) {Pol (l - P35P53) + PoaP35P5t}
• * * * + M*) + qotqts(q53M3
-}-P03 (P34P47 -I- P35P57)P7t
+ qf~{M* + q35(qslM* + M * ) }
b3 - (1 - Po3P3o) {pst (1 - P4~P74) "}"P57PT! } -- qlsqst(Mo + q*3M*)] •
*
*
*
*
*
+ Pol (Ps3P34 +P57P74)P4o +P53P3oPot(1 -P47P74)
*
+ (qotqt5 + qo3q35)q57(q47M4 + M~*) •
*
*
- P34{P4oPo3(P57PT, + Ps, ) q- P47PTtP53}
*
qoaq ~4q4vqTt (1 -- q'~sMs) •
*
*
*
b4 = (1 - P47P74){PotPtsP53 + Po3(1 - Pt5P5t )}
*
+ q ~ ( M ~ + q47M~ )[qo~qtsq53 - Po3Pt5P57PTt
+ q * ~ ( 1 - q * s q * t )] + Mo* Qls(qs3q34q4v * * * * •
*
*
*
b5 = Pt5P53(P47PTI + P4oPot) + Po3 {(P4o + P47PTIPlo)
*
+ q*~) -- qo3[qts(M3 + q34M4 )
+P4oPIs(P51 + P57PTI)}
-- q35Mt ]q57q71 D2(s ) = (1 -- q't)[1 - {q35q53(q47q74 -- 1 ) - q*7q*4 }] •
*
*
*
*
*
-t- (1 -- q47q74)[qo3q30(qtsqsl -- 1) -- q'~sq~t
*
*
*
*
*
-- P34Pls(Pst+ P57P71 )}
*
-- qtsq53q34(q47qT~ -F q4oqot)
bs = (1 - P35P53)P74(1 -- PotPlo ) + P71(ProP03
- q t s q 5 7 q T t { l - qo3(q3o * * -4- q ' q * ) }
+ PtsP53)P34 - P30 (P03 + PotPtsP53)P74
-- qo3q34q 4o(1 - q*sq*~ ) •
*
•
*
*
*
*
*
- Psi {Pt5 (1 - Po3P30) + PtoPo3P35}P74
*
-- qsTq74q40(qot qt5 + qo3q35) *
*
*
*
+ P57(PtoPo3P:t + P74P40) b7 = PoIPls (P57P74 "}"P53P34) + P03{(Pu + P35P57P74)
-- qtoqo3q35qs~ -- q~l qtsq53q30] * * * •
b. = (1 - p.TpT.) {.5~ ( l - pot p lo) + psi p.op03 }
b9 = (1 -P47P74) {Pts( 1 - Po3P30) + PloPo3P35}
*
-- qtoqo3(qa4q47 + q35q57)q71"
T h e r e f o r e steady state availability of the system is: A 0(00) = N 2/O2,
(5)
- Po3P34P40Pt5 blo = (1 -- P47P74) {/733(1 -- PotPto) + P30PotP15} + PIsP34 (P40Pot + P47P71 )
where, in terms of M*(0)=po,
M*(0)--U1,
M*(O)=#5,
M*(0)=#4,
M~(0)=#5,
M~'(0)=#7
N 2 ----(1 -- P47P74) [(P~ "4-Pot #1) (1 -- P35P53) + Pot Pt 5(P53 f13 "at"//5) "4-P03 {~3 -t- P35 (Psi UI + / t s ) } --PtsPs~ (Po "4-P03/'t3)] + (PotP~5 +Po3P35)P57(P47/A4 -t- U7)
+ P34 (P4 + P47/~7) [Pot Pt 5P53 "4-P03 (l -- P t5Ps~ )] + P'oPl 5(P53P34P47 -I- P57 )
-- P03 [Pt5 ~ 3 + P34/Z4) -- P35 fll ]P57P71 "4"Po3P34P47PTI (1 -- Pl5 US)
bn = Pt5 [P74P~oPot+ PT, { 1 - P03 + P~P~ }] d- P03P35 (P71 Plo + P?4P4o) bt2 = P15 {(P57 + P53P34P47) - Po3P57 (P30 q- P34P40) d- ProP03 (P34P47 d- P35P57)} b13 = (1 --P35P53)P47(I --PolPlo) +P40(potPt5 -t- P03P35 )P57 -- P30 (P03 q- P0! P lSP53)P47 -- Pst { P| 5( l -- P03P30) q- PloPo3P35 }/747' EXPECTED NUMBER
OF RECOVERIES
(1) F o r the controller: to find an expression for the expected n u m b e r of recoveries of the controller
138
V.S. RANA in (0, t], we have the following recursive relation in R~)(t): R~(t) = Qol(t) (~ R](t) + Q03(t) (~) R~(t)
R[(t) = Q3o(t) (~ [1 + R[(t)] + Q~(t) (~) R~(t) + Q35(t) (~) Rts(t) Rt4(t) = Q4o(t) (~ [1 + R[(t)] + Q4~(t) (g) R[(t)
R~(t) = Qio(t) (~ [1 + R~)(t)] + Q15(t) ~) R~(t)
Rts(t) = Qsl(t) ~) [1 + R[(t)] + Q53(t) (~) R[(t)
R~(t) = Q3o(t) (~ R~(t) + Q~(t) (~ R~(t)
+ Q57(t) ~) RtT(t)
+ Q35(t) (~) R~(t)
RtT(t) = Q71(t) (~ [1 + R~ (t)] + Q74(t) (~) Rt4(t). (9)
R~(t) = Q4o(t) ~) R[(t) + Q47(t) (~) R~(t) R~(t) = Qst(t) (~) R](t) + Q53(t) (~) [1 + R~(t)]
By taking the L.S.T. of relation (9) and simplifying for R~(s), we have:
+ Q57(t) (~) R~(t)
R~(t) = N4(s)/Dt(s),
R~(t) = Q74(t) (~) [1 + R](t)] + Q,I (t) (~) R](t). (6) By taking the L.S.T. of relations (6) and solving for R[(s), we have: R~)(s) = N3(s)/D 1(s),
(10)
where
N,(,) = (0o, - ~o3 0~)0,,{0,,07, + Qs,(1- 047 0u)} + ( 1 - {~47{~u)
× {0o3(03o + 03,0,,)
(7)
+ 0o, 0,,053030} + (0,707, + 0,o)
where
× 0,(003 + 00, 0,50~3)
N3(s) = (010 + {2,5Q53)
+ 003 {035 05, (0,, + 0,, 0~)
x {003 {~v*{~47{~,1 + ~o1( 1 -- 0470,4)}
+ 00d0.{0.0710,0 + (~ - 047 0~4)(0~3 + 0~1010)}
- 0~0~0,5(0,1 + 05707,)}. In the steady state, the expected number of recoveries of the terminal is given by: R~(oo) = N4/D,,
+ Q~ Q,7 Q74 (l - 0,5 051 )]
+ 057074(001 015 + 003035) and, Dt is obtained earlier in equation (2). In the steady state, the expected number of recoveries is given by:
(11)
where N4 = (Pol - Po3P3o)P15{P57P71+ P51(1 - P47Pu)} + (1 -- P47P74){P03(P30 + P35P51) + PolP15P53P3o} + (P47Pu + P4o)
R[(oo) =hmit R~(t)/ t~oo
x P34(P03 +PolP15P53) t = limit sR~(s) = N3/D 1, (8)
$~0
+ P03{P35P57(P71 + PUP40) -- P34P40P15(P51 + P57P71)}.
where N3 = (PIo + P15P53){P03P34P47P71 + pol (1 - p47pu)} + po3[p35{P57PnPlo
+ (l --P47Pu)(P53 +P51Plo)} + P~P47pu(1 -- PlsPsl)] + P57P74(PolP15 + Po3P35) and, DI is calculated in equation (3). (2) For the terminal: to obtain an expression for the expected number of recoveries of the terminal in (0, t], the recursive relations in Rto(t) are as follows: R~(t) = Qot(t) (~ Ri(t) + Qo3(t) (~ Rt3(t) R~(t) = Qlo(t) (~) R~(t) + Qls(t) (~) R~(t)
The expected number of recoveries of the system can be obtained by adding equation (7) and (10); also, the expected number of recoveries of the system in a steady state can be obtained by adding the relations (8) and (11). Acknowledgement--The author thanks Dr V. V. S. Rao, Director Government Computer Center, Bhopal, for giving him valuable suggestions regarding this work. REFERENCES 1. Hoang Pham and Shambhu J. Upadhyaya, Reliability
analysis of a class of fault tolerant system, IEEE Trans. Reliab. 38, 333-337 (1989). 2. Zoram Paulvik prcdreg Rokic, An approach of fault tolerant system reliability modelling, Microelectron. Reliab. 29, 243-348 (1989).
An information computer network system 3. Rakesh Gupta and L. R. Goel, Profit analysis of a two-unit standby system with administrative delay in repair, Int. J. Systems. Sci. 20, 1703-1712 (1989). 4. Ali M. Rushdi, Threshold systems and their reliability, Microelectron. Reliab. 30, 299-312 (1990).
139
5. C. Mohan and K. Shanker, Reliability optimization of complex systems using random search technique, Microelectron. Reliab. 28, 513-518 (1988). 6. Vladimir Cherkassky and Miroslaw Malek, A measure of graceful degradation in parallel-computer systems, IEEE Trans. Reliab. 38, 76~1 (1989).
0026-2714/9356.00+ .00 © 1993PergamonPress Ltd
STOCHASTIC ANALYSIS OF AN INFORMATION COMPUTER NETWORK SYSTEM VIRENDRA SINGH R A N A Government of India (Planning Commission), National Informatics Centre, District Guna (M.P.), India
(Received for publication 1 August 1991) Abstract--This paper investigates a mathematical model of an information computer network system. In this model, the system has two parts---a controller and a terminal. It is also linked with a satellite by a disk antenna. Each part has two failure modes; one is transient failure and second is latent failure. After latent failure, the terminal is recoverable, but the controller is not recoverable. The system may also fail due to a common-cause failure when it is in the normal state or in a partially normal state. Failure rates are constant and recovery rates are general. Using the regenerative point technique several measures of system effectiveness are obtained.
INTRODUCTION Many researchers, including [1] and [2], have studied different models related to network systems, using methods of reliability optimization and probability theorems. Now, information technology has an important role in decision making and policy planning. An information network is used for receiving and transmitting information, on a country-wide level. There are two main parts of this system; the controller and a terminal, by which we communicate information to anywhere in a country via the satellite. The terminal is the input and output medium of information and the controller is the processer of such information. In the present paper, this network model is studied using the following set of assumptions. 1. The controller and terminal of a system may work with reduced efficiency if either the controller or terminal is affected by transient failure. This state of the system is called the partial failure state. 2. The system is considered to be down completely if either the controller or both parts (controller and terminal) are down completely. 3. Recovery of the system is possible from the states: (a) when either the controller or the terminal is down partially; or (b) when the terminal is down completely. 4. Recovery of the system is not possible when the controller is down completely. 5. Recovery starts immediately after transient failure of the controller or terminal. Failure time distributions are assumed to be negative exponential, while all the recovery time
distributions are taken to be general. The following characteristics of interest to the system designers and reliability engineers, are obtained by using the regenerative point technique: (1) mean time to system failure; (2) pointwise and steady state availability of the system; (3) expected number of recoveries of the controller; (4) expected number of recoveries of the terminal; and (5) expected number of recoveries of the system.
DESCRIPTION OF THE MODEL The network system can be described as follows. (1) A system having two main components---a controller and a terminal, which are associated with a satellite. (2) The terminal may fail partially due to a transient failure, i.e. operational mishandling, or fail completely due to a latent failure, i.e. software or hardware problems. (3) The controller may fail partially due to a transient failure, i.e. electromagnetic interferences, and completely fail due to a latent failure, i.e. major hardware problems. (4) The system can fail completely due to a common cause, i.e. upset antenna direction or power fluctuations. (5) The recovery of the terminal is possible from both partial and complete failure states, while the recovery of the controller is only possible from a partial failure state. (6) The system is said to be down when either the controller, or both terminal and controller, are down completely. 133
134
V.S. RANA
(7) F a i l u r e rates are c o n s t a n t a n d recovery rates are general. (8) A f t e r recovery, the s y s t e m is as new.
Considering the above symbols, the system may be in one of the following states. Up states:
s0 = (No, No) NOTATION AND STATES OF THE SYSTEM % constant failure rate of the controller from a normal (N) mode to a partial normal (P) mode, and from the partial normal (P) mode to a down (D) mode, where i = 1, 2 /~ constant failure rate of terminal from N-mode to P-mode, and from P-mode to D-mode, where i=1,2 6 constant common-cause failure rate of the system, when both controller and terminal are completely down simultaneously G(-) c.d.f, o f recovery time of the controller from a P-mode to N-mode H~(.) c.d.f, ofrecovery time of the terminal from a P-mode to N-mode, and from D-mode to N-mode, where i=1,2 mij contribution to mean sojourn time in a state S~~ E and non-regenerative state, if one should occur, before transiting to a state Sj E E /z~ mean sojourn time in a state S ~ E or #t = £jm~j A~(t) p [system is up at time t i E 0 = &~E] RT(t) expected number of recoveries o f the controller from P-mode [ E 0 = S, R}(t) expected number of recoveries of the terminal from P-mode or from D-mode [ E0 = Sj. Other symbols used in this paper may be seen in Ref. [3]. No controller/terminal is in a normal operative mode Por controller/terminal is in a P-mode and undergoing a recovery process D controller is in a down mode and not recoverable Dr terminal is in a down mode and undergoing a recovery process D N system (controller and terminal) is in a down mode and not recoverable.
$6
S~
S l = (Po,, No) S3 = (No, Por) S, = (No, Dr)
s s = 0'o. Po0 $7 = (Por, Dr); and Down states: S2 = (D, No) $6 = (V, Pot)
ss
=
(DN).
Each state is thus a two-dimensional vector, with the first element denoting the state of the controller and the second denoting the state o f the terminal, A transition diagram showing the various states of the system is given by Fig. 1.
TRANSITION PROBABIIdTW~ AND MEAN S O J O U R N TIMES T h e t r a n s i t i o n p r o b a b i l i t y m a t r i x (t.p.m.) o f the e m b e d d e d M a r k o v c h a i n is:
P = ((P,j)) with n o n - z e r o elements:
Pol = ~l/(~l + ~l + 6), Pos = 6/(~] + p~ + 6),
/'03 = ~1/(~'1 +/~t + 6), P~o = C7(~ + ~2),
Pl2 = °~2[1 -- (~(~! + ~2)]/(~! + ~2), Pl5 = ~1[ 1 -- 0 ( ~ i + ~2)]/(~1 + ~2), P30 --/-I1 (f12 Jr ~1 ),
p ~ =/~2[1 - &(fl~ + ~)]/(/~ +
~,),
$2
¢)/: Io] --¢ Normal Par~ciat normal
] Fig. 1. Transition diagram.
Down
An information computer network system
135
for we(t) are obtained:
Pss ffi ai [1 --/'~l (f12 + a~ )]/(~ff2+ ¢x,),
we(t) == Qm (t) (~) al (t)-i-Qos(t) (~) as(t) -I-Qei(t)
p~o = : h (~l),
.~(t) = Qio(t) ® we(t) + Q,s(t) ® as(t) + Q,2(t)
as(t)= Q~o(t) ® ~o(t) + Q.(t) ® n4(t)
p~ f dHi(t)e-m+#,+~G(t), =
+ Q s s ( t ) (~) as(t)
a4(t) ffiQ4o(t) (~) ~o(t) + Q47(t) (~) n,(t)
f dG (t)e- (==+ #2+ s),Rt(t),
Pss
~s(t) = Qs,(t) ® n,(t) + Qss(t) ® .~(t)
Ps~ = f ~2e-(=2+ #2+ s),dt~_r(t)~i (t),
+ Qs7(t) ® av(t) + Qs~(t)+ Qs~(t)
n,(t) = Q,i(t) ® n~(t) + Q,.(t) ® a,(t) +
Q~(t). (I)
Psv = f f12e -¢=2+ #2+ ~)~dt~.r(t)R~(t),
By taking a L.S.T. of the above relations and simplifying for ~Zo(s),we have:
Pss = f 6 e -(=2+~+~)t dt~(t)Hl(t),
~o(s) = N~(s)/Di(s),
~,, =
J'dn~(t)~-'~'C.(,),
p,, =
f dG(t)e-=2'II2(t),
PTs
(2)
where
+ O.,s (~os ~s, Oi2 - O.ss ~o, O.i~)]
= f.2 e-a2t dt~(t)R2(t).
+ Oos(~.~,,
+ O.O.s,)(O,, + 0,, O.,.)
It can be easily verified that Poi +.17o3+Pos mPlo -t- P12 -t- PlS --P~o +P34 -I-P35 =P4o +P~, ~Psi +Pss +Ps6 +Ps~ +Pss ffi 1,
(~,, (Ts,)]
and
PTI +P74 +P78 ffi 1. To calculate the mean sojourn time ~ in state So, we discover that so long as the system is in a state So, there is no transition to Sl, $3 or Ss. Hence, if T denotes the sojourn time in state So, then ffi . I p ( T > t) dt ffi 1/(al +
-
i), = (1 - ~o,)[1 - {03, ~ss
(~+, ~,+) - 0,, ~,, }]
+ o - ~,,O,,)[O.os~(O,s~s,
- 1)
~l + 6), - ~,sOs,~,, {1 - Oos(O~ + ~,,O.o)}
~l ffi [1 - G(~2 +/h)]/(~2 + / ~ ) ,
-- ~o, 0s404o(1 -- 0.,s0m)
Us = [ 1 -- i~l (ai + fl2)]/(ai + iff2),
- O..O,,O..(O.o,O.,s + O.os~.)
~+ = B - O~(~,)l/~,,
#s = r e - " , +,, ++ dtG(t)Hi( t ) , ~,
=
(for brevity, the argument s is omitted), now the mean time to system failure, given that Eo = So, is
fe-'~'~(t)A~(t).
MEAN TIME TO SYSTEM FAILURE The time to system failure can be regarded as the first time to the failed state. To obtain this, we regard the down states $2, S~ and Ss as absorbing. By employing the arguments used for the regenerative process, the following recursive relations
M T S F = - d ~ o ( S ) ', ds ',-o
N, Dx'
(3)
where N~ ffi motao + mo3a I + t r ~ a 2 -t- mloa 3 + ml2aI + m~sas + msoa+ + m~a7 + mssa8 -~-m4oa9 + m47am + restart -t- mssat2 -I- ms6al3 + msTa~++ mssals + mTlal6 -t- mT+a~7+ m~sa,,
136
V.S. RANA
ao = (1 --P4~P74){Pls(P53P3o +P56 + P s s ) + P , o --
a12 ----(1 -- P47P74) [P35 { 1 -- Pos -- Pot (Plo + Pl2)}
P35P53 (Plo + Pl2)} + Pl5 {Ps3P34(P~ + P47P78 )
+ PolPlsP30] + PlsP34P47PTI(1 -- Po~)
+ Ps"/(P78 + P74P4o)} a I ----(l -- P47P74) {P3o ( 1 -- P lsPsl ) + P35PsI (PI0 + PI2) + P35 (P56 + P58)} d- ( l -- Pl5P51 ) (P47P7s
+ PolPl 5P34(P40 + P4~PTs )
at 3 = ( | -- P47P74)(PolPl5 + Po3P35) + Po3P34P47P71P15 al4 = (Pol Pl 5 + Po3P35) (P74P40 + PTs )
+ P40)P~ + P35P57 (PTs + P74P40)
+ PTI [Pl ~{ l -- Pos -- P03(P34P40 + P30)}
+ P35P57P71(Pl2 -[-Plo) -- PlsP57PTI (P34P40+P30)
+Po3P35(PIo+Pt2)],
+P34P4"IP'll {Pl0 +P12 + Pls(P56 + P5S)}
al6 = P03 {P34P47PTI(P56 + Pss ) + (Pro + Pl 2) (P~P47
a 2 = (l --p47P74)(1 --PlsP51 --P35P53)
+ P35P57) -- Pl sP57 (P30 + P34P40)}
-- P IS(P53P34P47 + P57)PTI a 3 ~- (1 -- P47P74) {Pol (1 -- P35P53) + Po3P35PsI}
+ (1 - Pos)Pl 5(P53P34P47 + P57 ) al7 -- (PolPI5 + Po3P35) {P57P40 - P47 (P56 + P58)}
+ P03 (P34P47 + P35P57)PT1
+ P47 {(1 - P35P53 (1 - Po8 - PolPlo)
a 4 ----(1 -- P47P74) {(Po3Psl -- Pol P53 }
+ (1 - PIsP51 ) (1 - Pos - Po3P30)
•4- ]703(P34P47 -F P35P57)PTI
+ PolP53 (PI2P35 - PI5P30)
a 5 = ( 1 -- P47P74) {Psi (1 -- Pos -- Po3P30) +:Pol (P53P30 +P56 +P58)} + (l -- Pos)(P57 +P53P34P47)PTI
- Po3P35PsI(Plo + P12)} als ----PolPl5 (P57 + P53P34P47)
+ Po3P34P47PTI(P56 -t- Pss ) -- P03 (P34P40 + P30 )P57P71 -- Po3P34(P47PTs + P4o)Psl
al5 = al3
+ P03 {P35P57 "Jr-P34P47 (1 -- PI5P51 )} and
+ Pol (P57P74 + P53P34)P4o D 1 = (l -- POl ) [l -- {P35P53 (P47P74) -- P47P74 }] + POl (P57 + P53P34P47)PTs
a 6 = (l -- P4"IP74){PolPIsP53 + Po3 (1 -- PIsP51 )} - -
+ (l -- P47P74) [P03P30 (PlsPsl -- l) -- Pl5Psl -- PIoPo3P35PsI -- PoIPlsP53P30]
Po3PI5P57P'tl -- [PIsP53P34(P47PTl + P4oPoI)
aT = Po3PTI []747{Pl5 (P~ + PSS) + Pl2 }
-- PIsP57PTI { 1 - P03 (P30 + P34P40)}] + P47Plo -- Pl5P5TP4o] + PIsP53P47PTI(1 -- P0S) - Po3P34P40(l - Pl5P51 ) + (P40 +P47PTs){P03( 1 --PIsPsI) +PoIPIsP53} P57P74P4o(PolP15+ Po3P35)
- -
as = (1 --p47P74){pn(1 --Pos) -- (Pro "~-Pl2) × (P01P53 -- P03Ps1 ) + P03 (P56 + PSS)}
-- PloP03 (P34P47 + P35P57)P71"
+ P03P57 {(P7$ + P74P40) + PTt (Pro + Pl2)}
0-9 = PolPls(P57P74 + Ps3P34) + Po3[P35P57Pu +P34{1 --Pls(Psl +P57P71)}]
AVAILABILITY OF THE SYSTEM A c c o r d i n g to the definition o f At(t), we have the following recursive relations:
alo = (P~6 + Pss){Po3P34P71P15 -- (PolPls+ Po3P35)P74}
Ao(t) = Mo(t) + qol(t) © A l ( t ) + qo3(t) © A3(t)
"Jr"(1 --PlsP51 ) {Po3PMP78"[-P74 ( l --P08 --P03P30)}
A, (t ) = M l ( t ) + qlo(t) © Ao(t) + qls(t) © As(t)
+ (PI0 + Pl2 )P03 (P~4PTt -- P3sPsIP74)
A3(t) ----M3(t) + q30(t) (~
at"PIsP53P34{PTI (l --Po~) + P01PTs }
Ao(I) + q34(t) © A4(t)
+ q35(t) © As(t)
-- P35P53P74{ l -- Pos -- P01 (Pl0 + Pl2 )}
A4(t )
-- PoIP74(Plo + PI5P53P30)
As(t) -- M s ( t ) + qst (t) © A l ( t ) + q53 (t) (~ a 3 ( t )
all ~- (l --P4"rP74){Po3P35(Plo +PI2) + P l s ( 1 --P0s -- P03P30)} -- Po3P34(P47P7s + P40)PI5
=
M 4 ( t ) + q40(t ) (~ Ao(t ) + q47(t) (~) AT(t)
+ q n ( t ) © A7(t)
A , ( t ) = MT(t) + qT, (t) © A, (t) + qv4(t) © A4(t), (4)
An information computer network system where, as simple probabilistic considerations show: M o ( t ) = e -(~ +#~+6),,
and D2 = molbol + mo3bl + mlob2 + ml5b3 + m3ob4
M t (t) = e -(~2 + #~)~¢~(t),
M3(t) = e - t ~ + # 2 ) t R t ( t ) ,
137
M4(t) = e-~Fl2(t),
+ mub5 + m35b6 -b m4ob7 q- m47b8 + mslb9
M 5(t) = e -(~2 + #2+ 6)t//1 (t)G(t),
+ m53blo + m57bl! -}- m71 bl2 -}- m74b13
M 7(t) = e - :2,//2 ( t ) ¢ ( t ) .
b 0 =- (1 - P47P74){Pro (1 - P35P53) + P t5P53P30}
By taking the L.T. o f the a b o v e set of equations (4), we have: A ~ ( s ) = N2(s)/D2(s ),
+ Pt s (P53P34 + P57P74)P40 bt = (1 - P4~P74){P30(l - PtsP51 ) + PtoP35Pst }
(5) + { P34 (1 -- PIsPst) + P35P5~PT,}P4o
where + P lo(P34P47 + P35P57)P71 N2(S ) = (1 - q4~Q~,)[(Mo * * * + q*t M * ) ( 1 - q;'sq*3)
b2 = (1 -P47P74) {Pol (l - P35P53) + PoaP35P5t}
• * * * + M*) + qotqts(q53M3
-}-P03 (P34P47 -I- P35P57)P7t
+ qf~{M* + q35(qslM* + M * ) }
b3 - (1 - Po3P3o) {pst (1 - P4~P74) "}"P57PT! } -- qlsqst(Mo + q*3M*)] •
*
*
*
*
*
+ Pol (Ps3P34 +P57P74)P4o +P53P3oPot(1 -P47P74)
*
+ (qotqt5 + qo3q35)q57(q47M4 + M~*) •
*
*
- P34{P4oPo3(P57PT, + Ps, ) q- P47PTtP53}
*
qoaq ~4q4vqTt (1 -- q'~sMs) •
*
*
*
b4 = (1 - P47P74){PotPtsP53 + Po3(1 - Pt5P5t )}
*
+ q ~ ( M ~ + q47M~ )[qo~qtsq53 - Po3Pt5P57PTt
+ q * ~ ( 1 - q * s q * t )] + Mo* Qls(qs3q34q4v * * * * •
*
*
*
b5 = Pt5P53(P47PTI + P4oPot) + Po3 {(P4o + P47PTIPlo)
*
+ q*~) -- qo3[qts(M3 + q34M4 )
+P4oPIs(P51 + P57PTI)}
-- q35Mt ]q57q71 D2(s ) = (1 -- q't)[1 - {q35q53(q47q74 -- 1 ) - q*7q*4 }] •
*
*
*
*
*
-t- (1 -- q47q74)[qo3q30(qtsqsl -- 1) -- q'~sq~t
*
*
*
*
*
-- P34Pls(Pst+ P57P71 )}
*
-- qtsq53q34(q47qT~ -F q4oqot)
bs = (1 - P35P53)P74(1 -- PotPlo ) + P71(ProP03
- q t s q 5 7 q T t { l - qo3(q3o * * -4- q ' q * ) }
+ PtsP53)P34 - P30 (P03 + PotPtsP53)P74
-- qo3q34q 4o(1 - q*sq*~ ) •
*
•
*
*
*
*
*
- Psi {Pt5 (1 - Po3P30) + PtoPo3P35}P74
*
-- qsTq74q40(qot qt5 + qo3q35) *
*
*
*
+ P57(PtoPo3P:t + P74P40) b7 = PoIPls (P57P74 "}"P53P34) + P03{(Pu + P35P57P74)
-- qtoqo3q35qs~ -- q~l qtsq53q30] * * * •
b. = (1 - p.TpT.) {.5~ ( l - pot p lo) + psi p.op03 }
b9 = (1 -P47P74) {Pts( 1 - Po3P30) + PloPo3P35}
*
-- qtoqo3(qa4q47 + q35q57)q71"
T h e r e f o r e steady state availability of the system is: A 0(00) = N 2/O2,
(5)
- Po3P34P40Pt5 blo = (1 -- P47P74) {/733(1 -- PotPto) + P30PotP15} + PIsP34 (P40Pot + P47P71 )
where, in terms of M*(0)=po,
M*(0)--U1,
M*(O)=#5,
M*(0)=#4,
M~(0)=#5,
M~'(0)=#7
N 2 ----(1 -- P47P74) [(P~ "4-Pot #1) (1 -- P35P53) + Pot Pt 5(P53 f13 "at"//5) "4-P03 {~3 -t- P35 (Psi UI + / t s ) } --PtsPs~ (Po "4-P03/'t3)] + (PotP~5 +Po3P35)P57(P47/A4 -t- U7)
+ P34 (P4 + P47/~7) [Pot Pt 5P53 "4-P03 (l -- P t5Ps~ )] + P'oPl 5(P53P34P47 -I- P57 )
-- P03 [Pt5 ~ 3 + P34/Z4) -- P35 fll ]P57P71 "4"Po3P34P47PTI (1 -- Pl5 US)
bn = Pt5 [P74P~oPot+ PT, { 1 - P03 + P~P~ }] d- P03P35 (P71 Plo + P?4P4o) bt2 = P15 {(P57 + P53P34P47) - Po3P57 (P30 q- P34P40) d- ProP03 (P34P47 d- P35P57)} b13 = (1 --P35P53)P47(I --PolPlo) +P40(potPt5 -t- P03P35 )P57 -- P30 (P03 q- P0! P lSP53)P47 -- Pst { P| 5( l -- P03P30) q- PloPo3P35 }/747' EXPECTED NUMBER
OF RECOVERIES
(1) F o r the controller: to find an expression for the expected n u m b e r of recoveries of the controller
138
V.S. RANA in (0, t], we have the following recursive relation in R~)(t): R~(t) = Qol(t) (~ R](t) + Q03(t) (~) R~(t)
R[(t) = Q3o(t) (~ [1 + R[(t)] + Q~(t) (~) R~(t) + Q35(t) (~) Rts(t) Rt4(t) = Q4o(t) (~ [1 + R[(t)] + Q4~(t) (g) R[(t)
R~(t) = Qio(t) (~ [1 + R~)(t)] + Q15(t) ~) R~(t)
Rts(t) = Qsl(t) ~) [1 + R[(t)] + Q53(t) (~) R[(t)
R~(t) = Q3o(t) (~ R~(t) + Q~(t) (~ R~(t)
+ Q57(t) ~) RtT(t)
+ Q35(t) (~) R~(t)
RtT(t) = Q71(t) (~ [1 + R~ (t)] + Q74(t) (~) Rt4(t). (9)
R~(t) = Q4o(t) ~) R[(t) + Q47(t) (~) R~(t) R~(t) = Qst(t) (~) R](t) + Q53(t) (~) [1 + R~(t)]
By taking the L.S.T. of relation (9) and simplifying for R~(s), we have:
+ Q57(t) (~) R~(t)
R~(t) = N4(s)/Dt(s),
R~(t) = Q74(t) (~) [1 + R](t)] + Q,I (t) (~) R](t). (6) By taking the L.S.T. of relations (6) and solving for R[(s), we have: R~)(s) = N3(s)/D 1(s),
(10)
where
N,(,) = (0o, - ~o3 0~)0,,{0,,07, + Qs,(1- 047 0u)} + ( 1 - {~47{~u)
× {0o3(03o + 03,0,,)
(7)
+ 0o, 0,,053030} + (0,707, + 0,o)
where
× 0,(003 + 00, 0,50~3)
N3(s) = (010 + {2,5Q53)
+ 003 {035 05, (0,, + 0,, 0~)
x {003 {~v*{~47{~,1 + ~o1( 1 -- 0470,4)}
+ 00d0.{0.0710,0 + (~ - 047 0~4)(0~3 + 0~1010)}
- 0~0~0,5(0,1 + 05707,)}. In the steady state, the expected number of recoveries of the terminal is given by: R~(oo) = N4/D,,
+ Q~ Q,7 Q74 (l - 0,5 051 )]
+ 057074(001 015 + 003035) and, Dt is obtained earlier in equation (2). In the steady state, the expected number of recoveries is given by:
(11)
where N4 = (Pol - Po3P3o)P15{P57P71+ P51(1 - P47Pu)} + (1 -- P47P74){P03(P30 + P35P51) + PolP15P53P3o} + (P47Pu + P4o)
R[(oo) =hmit R~(t)/ t~oo
x P34(P03 +PolP15P53) t = limit sR~(s) = N3/D 1, (8)
$~0
+ P03{P35P57(P71 + PUP40) -- P34P40P15(P51 + P57P71)}.
where N3 = (PIo + P15P53){P03P34P47P71 + pol (1 - p47pu)} + po3[p35{P57PnPlo
+ (l --P47Pu)(P53 +P51Plo)} + P~P47pu(1 -- PlsPsl)] + P57P74(PolP15 + Po3P35) and, DI is calculated in equation (3). (2) For the terminal: to obtain an expression for the expected number of recoveries of the terminal in (0, t], the recursive relations in Rto(t) are as follows: R~(t) = Qot(t) (~ Ri(t) + Qo3(t) (~ Rt3(t) R~(t) = Qlo(t) (~) R~(t) + Qls(t) (~) R~(t)
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