Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol27.Issue2.2000:

Operations Research Letters 27 (2000) 67–71
www.elsevier.com/locate/dsw

A Sokhotski–Plemelj problem related to a robot-safety
device system
E.J. Vanderperre ∗
Department of Quantitative Management, University of South Africa
Received 1 November 1999; received in revised form 1 January 2000

Abstract
We introduce a robot-safety device system attended by two dierent repairmen. The twin system is characterized by the natural feature of cold standby and by an admissible “risky” state. In order to analyse the random behaviour of the entire system
(robot, safety device, repair facility), we employ a stochastic process endowed with probability measures satisfying general
steady-state dierential equations. The solution procedure is based on the theory of sectionally holomorphic functions. An
application of the Sokhotski–Plemelj formulae determines the long-run availability of the robot-safety device. Finally, we conc 2000 Published by Elsevier Science B.V. All rights reserved.
sider the particular but important case of deterministic repair.
Keywords: Robot; Safety device; Invariant measure; Availability; Risk-criterion

1. Introduction
Innovations in the eld of microelectronics and
micromechanics have enhanced the involvement of
“smart” robots in all kind of advanced technical

systems [2].
Unfortunately, no robot is completely reliable.
Therefore, up-to-date robots are often connected with
a safety device [3–5]. Such a device prevents possible damage, caused by a robot failure, in the robot’s
neighbouring environment. However, the random behaviour of the entire system (robot, safety device,
repair facility) could jeopardize some prescribed
∗ Correspondence address: Ruzettelaan 183, Bus 158, B-8370
Blankenberge, Belgium.
E-mail address: evanderperre@hotmail.com (E.J. Vanderperre).

safety requirements. For instance, if we allow the
robot to operate during the repair time of the failed
safety device. Such a “risky” state is called admissible, if the associate event: “The robot is operative but
the safety device is under repair”, constitutes a rare
event. Therefore, an appropriate statistical analysis
of robot-safety device systems is quite indispensable
to support the system designer in problems of risk
acceptance and safety assessments.
In order to avoid undesirable delays in repairing
failed units, we introduce a robot-safety device attended by two dierent repairmen (henceforth called

a T-system). The T-system satises the usual conditions, i.e. independent identically distributed random
variables and perfect repair [6].
Each repairman has his own particular task. Repairman S is skilled in repairing the safety unit, whereas

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E.J. Vanderperre / Operations Research Letters 27 (2000) 67–71

repairman R is an expert in repairing robots. Both
repairmen are jointly busy if, and only if, both units
(robot + safety device) are down. In the other case, at
least one repairman is idle. In any case, the safety device always waits (in cold standby [1]) until the repair
of the robot has been completed.
In order to analyse the random behaviour of the
T-system, we introduce a stochastic process endowed with probability measures satisfying general
steady-state dierential equations. The solution procedure is based on the theory of sectionally holomorphic functions [7]. An application of the Sokhotski–

Plemelj formulae determines the long-run availability
of the robot-safety device.
Finally, we consider the particular but important
case of deterministic repair.

Consider a T-system satisfying the usual conditions.
The robot has a constant failure rate  ¿ 0 and a
general repair time distribution R(•); R(0) = 0 with
mean  and variance 2 .
The operative safety device has a constant failure
rate s ¿ 0 but a zero failure rate in standby (the
so-called cold standby state) and a general repair time
distribution Rs (•); Rs (0) = 0 with mean s and variance s2 . The corresponding repair times are denoted
by r and rs .
Characteristic functions (and their duals) are formulated in terms of a complex transform variable. For
instance,
Z ∞
i!r
ei!x dR(x); Im !¿0:
Ee =

0

Note that
Z ∞
e−i!x dR(x)
Ee−i!r =

The state space of the underlying Markov process is
given by
{A} ∪ {(B; x); x¿0} ∪ {(C; y); y¿0}
∪{(D; x; y); x¿0; y¿0}:
Next, we consider the T-system in stationary state
(the so-called ergodicP
state) with invariant measure
{pK ; K = A; B; C; D}, K pK = 1, where
t→∞

Finally, we introduce the measures

0

ei!x d(1 − R((−x)−));

{Nt }, if Nt = A (i.e. if the event {Nt = A} occurs).
{Nt ; Xt }, if Nt = B, where Xt denotes the remaining
repair time of the safety device in progressive repair
at time t.
{Nt ; Yt }, if Nt = C, where Yt denotes the remaining repair time of the robot in progressive repair at
time t.
{(Nt ; Xt ; Yt )}, if Nt = D.

pK := P{N = K} := lim P{Nt = K|N0 = A}:

0

=

{Nt = A}: “Both units are operating in parallel at
time t”.
{Nt = B}: “The robot is operative but the safety

device is under repair at time t”. State B is the
so-called risky state.
{Nt = C}: “The safety device is in cold standby and
the robot is under repair at time t”.
{Nt = D}: “Both units are simultaneously down at
time t”.
A Markov characterization of the process {Nt } is
piecewise and conditionally dened by:

2. Formulation

Z

tions have bounded densities (in the Radon–Nikodym
sense) dened on [0; ∞).
In order to analyse the random behaviour of
the T-system, we introduce a stochastic process {Nt ; t¿0} with arbitrary discrete state space
{A; B; C; D} ⊂ [0; ∞), characterized by the following
events:

Im !60:

−∞

The corresponding Fourier–Stieltjes transforms are
called dual transforms.
Without loss of generality (see our forthcoming remarks), we may assume that both repair time distribu-

’B (x) d x := P{N = B; X ∈ d x}
:= lim P{Nt = B; Xt ∈ d x|N0 = A};
t→∞

’C (y) dy := P{N = C; Y ∈ dy}
:= lim P{Nt = C; Yt ∈ dy|N0 = A};
t→∞

E.J. Vanderperre / Operations Research Letters 27 (2000) 67–71

’D (x; y) d x dy := P{N = D; X ∈ d x; Y ∈ dy}
:= lim P{Nt = D; Xt ∈ d x; Yt ∈ dy|N0 = A}:

t→∞

Notations. The indicator of an event E is denoted by
(
1 if E occurs;
5(E) :=
0 otherwise:
Note that, for instance,
E{ei!X eiY 5(N = D)}
Z ∞Z ∞
ei!x eiy ’D (x; y) d x dy;
=
Im

0

4. Solution procedure
Note that our equations are well adapted to an
integral transformation. The integrability of the
’-functions and their corresponding derivatives implies that each ’-function vanishes at innity irrespective of the asymptotic behaviour of the underlying repair time densities! Applying a routine Fourier

transform technique to the equations and invoking
the boundary condition (s + )pA = ’B (0) + ’C (0),
reveals that
i(! + )E{ei!X eiY 5(N = D)}

0

0

so that
Z
pD =

69

!¿0; Im
∞

Z

+((1 − Eeir ) + i!)E{ei!X 5(N = B)}

¿0

+ iE{eiY 5(N = C)} + pA (1 − Eeir )

∞

pD (x; y) d x dy:

+s pA (1 − Eei!rs ) = 0:

0

The real line and the complex plane are denoted by
R and C with obvious superscript notations C + , C − ,
C + ∪ R, C − ∪ R. For instance, C + := {! ∈ C :
Im ! ¿ 0}.
The robot and the safety device are only jointly
available (operative) in state A.

Therefore, the long-run availability of the robot-safety
device, denoted by A, is given by pA .
Finally, we propose the following risk-criterion:
State B is admissible if pB satises the relation
pB ¡  ≪ 1 for some  ¿ 0, called the security level.

Observe that Eq. (1) holds for any pair (!; ) ∈
C ×C : Im !¿0; Im ¿0. Therefore, substituting ! = t,  = −t(t ∈ R) into Eq. (1), yields the
functional equation
+

(t) −

(s + )pA = ’B (0) + ’C (0);


d
d
’B (x) = ’D (x; 0) + s pA Rs (x);
−
dx
dx
d
d
− ’C (y) = ’D (0; y) + pA R(y);
dy
dy


@
d
@
’D (x; y) = ’B (x) R(y):
− −
@x @y
dy

−

(t) = ’(t);

(2)

where
+

(t) := pA−1 E{eitX 5(N = B)};

−

(t) :=

3. Dierential equations
In order to determine the ’-functions, we rst construct a system of steady-state dierential equations
based on a time independent version of Hokstad’s
supplementary variable technique (see e.g. Ref. [8, p.
526]). For x ¿ 0; y ¿ 0, we obtain

(1)

1
p−1 E{e−itY 5(N = C)}
1 + ’− (t) A
−

’− (t)
;
1 + ’− (t)

’− (t) :=

1 − Ee−itr
;
it

’+ (t) :=

Eeitrs − 1
;
its

’(t) :=

’− (0) := 1;
’+ (0) := 1;

s s ’+ (t)
:
1 + ’− (t)

Eq. (2) constitutes a boundary value problem on
the real line which can be solved by the theory of
sectionally holomorphic functions. An application of
Rouche’s (see the appendix) theorem reveals that
the function 1 + ’− (!), Im !60, has no zeros in C − ∪ R. Consequently, the function − (!),

70

E.J. Vanderperre / Operations Research Letters 27 (2000) 67–71

Im !60, is analytic in C − , bounded and continuous on C − ∪ R, whereas + (!), Im !¿0, is analytic in C + , bounded and continuous on C + ∪ R and
−

lim

|!|→∞
6arg !62

(!) =

lim

|!|→∞
06arg !6

+

(!) = 0:

Moreover, ’ is (uniformly) Lipschitz continuous
on R. (Indeed, |’′ (t)| is bounded on R. Therefore,
our assertion follows from the mean value theorem).
Finally, ’ is Holder continuous at innity, i.e. |’(t)|=
O(|t|−1 ), if |t| → ∞.
Consequently, the Cauchy-type integral
Z
d
1
’()
;
2i
−!
(see the appendix) exists for all ! ∈ C (real or complex) and denes a sectionally holomorphic function
vanishing at innity. A straightforward application of
the Cauchy formulae for the regions C + and C − , entails that
Z
d
1
i!X
; ! ∈ C +;
’()
E{e 5(N = B)} = pA
2i
−!
E{e−i!Y 5(N = C)}

Z
1
d
−
’()
=pA (1 + ’ (!))
2i
−!

+’− (!) ; ! ∈ C − :
In particular, pB = pA + (0), where
Z
d
1
+
:
(0) = lim
’()
−!
!→0 2i
!∈C +

Note that by the Sokhotski–Plemelj formulae,
Z
1
d
1
+
(0) = ’(0) +
’() :
2
2i

Moreover,
pD = lim

lim E{ei!X eiY 5(N = D)}:

→0 !→0
∈C + !∈C +

Applying the limit procedure to Eq. (1) and invoking the condition pA + pB + pC + pD = 1, yields the

additional relations
1
pA + pB =
;
1 + 

pC + pD =


;
1 + 

pD + pB = s s pA :
Substituting pB = pA + (0) into the rst relation
yields pA = [(1 + )(1 + + (0))]−1 . Observe that
we have completely determined the invariant measure
simply and solely depending upon + (0). We state
the following
Theorem.
1
;
(1 + )(1 + + (0))
where
Z
1
d
1
+
(0) = ’(0) +
’() :
2
2i


A=

Remarks. Note that the kernel ’(t); t ∈ R, preserves
all the relevant properties to ensure the existence of
the Cauchy integral for arbitrary repair time distributions with nite mean and variance. First of all, the
order relation |’(t)| = O(|t|−1 ), |t| → ∞, also holds
for arbitrary characteristic functions. Moreover, the
H-continuity of ’ on R does not depend on the canonical structure (decomposition) of R or Rs . For instance,
the Holder inequality
|Eeit2 r − Eeit1 r |6|t2 − t1 |;

(t1 ; t2 ∈ R);

always holds for any r with mean .
The requirement of nite variances is extremely
mild since the current probability distributions employed to model repair times [1] even have moments
of all orders!
Consequently, our initial assumptions concerning
the existence of repair time densities are totally super
uous to ensure the existence of an invariant measure.
As an example, we consider the case of deterministic repair, i.e., for t0 ¿ 0 and  ¡ 1, let
(
1 if x¿t0 ;
R(x) =
0 if x ¡ t0 ;
(
1 if x¿t0 ;
Rs (x) =
0 if x ¡ t0 :
Clearly, Eeitr = eitt0 ,  = t0 ; 2 = 0 and Ee−itrs = e−itt0 ,
s = t0 ; s2 = 0.

E.J. Vanderperre / Operations Research Letters 27 (2000) 67–71

71

where
L− (!) :=

Fig. 1. Region D with boundary D.

From the identity
Z
1
dt
’(t)
2i
t−!

−t0
− ei!t0 )

 is (e
! − i
=


−t0
s t0 e
we obtain

A=

(1 +

+

lim

(1 + ’− (!)) = 1:

if Im ! ¿ 0; ! 6= i;

References

if ! = i;

[1] A. Birolini, Quality and Reliability of Technical Systems,
Springer, Berlin, 1994.
[2] B.A. Brandin, The real-time supervisory control of an
experimental manufacturing cell, IEEE Trans. Robotics
Automation 12 (1996) 1–13.
[3] B.S. Dhillon, A.R.M. Fashandi, Robotic systems probability
analysis, Microelectron. Reliab. 37 (1997) 187–209.
[4] B.S. Dhillon, N. Yang, Availability analysis of a robot with
safety system, Microelectron. Reliab. 36 (1996) 169–177.
[5] B.S. Dhillon, N. Yang, Formulae for analyzing a
redundant robot conguration with a built-in safety system,
Microelectron. Reliab. 37 (1997) 557–563.
[6] I.B. Gertsbakh, Statistical Reliability Theory, Marcel Dekker,
New York, 1989.
[7] N.I. Muskhelishvili, Singular Integral Equations, Noordho,
Leiden, 1997.
[8] E.J. Vanderperre, S.S. Makhanov, S. Suchatvejapoom,
Long-run availability of a repairable parallel system,
Microelectron. Reliab. 37 (1997) 525–527.

1
:
− e−t0 ))(1 + t0 )

Appendix
Denition. Let f(t), t ∈ R be a bounded and continuous function. f is called -integrable, if
Z
dt
f(t)
; u ∈ R;
lim
t−u
T →∞ LT; 
↓0

exists, where LT;  := (−T; u − ] ∪ [u + ; T ). The corresponding integral, denoted by
Z
1
dt
f(t)
;
2i
t−u
is called a Cauchy principal value in double sense.

Lemma. The function 1+’− (!); Im !60; has no
zeros in C − ∪ R.
Proof. Clearly, 1 + ’− (0) = 1 +  ¿ 0. Hence,
! = 0 is not a zero. Consider region D with boundary
D as depicted in Fig. 1.
For ! 6= 0 we have
1 + ’− (!) =

Note that L− (t) is the characteristic function of a
non-lattice random variable. Consequently, L− (t) 6=
1 on (−T; −] ∪ [; T ). Therefore, 1 + ’− (t) has
no zeros in R. Furthermore, |L− (!)| ¡ 1, ∀! ∈ D.
Moreover, L− (!) is analytic in D and continuous on
D ∪ D. Applying Rouche’s theorem to the functions 1
and −L− (!) reveals that 1 + ’− (!) has no zeros
in C − ∪ R. In addition, note that
|!|→∞
6arg !62

(0) = −1 s (1 − e−t0 ). Hence,

−1 s (1


Ee−i!r ; Im !60:
 + i!

 + i!
(1 − L− (!));
i!