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Analysis of the machine repair problem: a diffusion process approach
Mathematics and Computers North-Holland
in Simulation
339
27 (1985) 339-364
ANALYSIS OF THE MACHINE REPAIR PROBLEM: A DIFFUSION PROCESS APPROACH HARYONO Department
and B.D. SIVAZLIAN
of Industrial and Systems Engineering,
An approximate analyzing
a non-Markovian
approximation N(t),
is that
is almost
becomes
large,
process
with
estimation used
[N(t),
of the
tend
parameters
becomes
of failed
the
used
machines When
in this
at time time
distributed
assumption
tractable,
for
argument
a normally
With
system
the
repair
based
For selected
systems,
repair
are compared
the
to become
basic
conditions.
is to obtain
factor
to show
number
is presented
t,
t
random
of normality
and the method
the
can be
problems.
study
stage
the
The
heavy-traffic
and variances.
methodology. of the
equation
problem.
stage,
under
O] will
repair
diffusion
repair
repair
means
of this
factor
on the
non-empty t >
machine
objective
simulation
accuracy
stage
the
formulas
of this
the
steady
on the diffusion
approximate with
for the
values
true
of the
values
state
approximation
utilization
obtained
through
approximation.
INTRODUCTION
In many problem.
industrial
The basic
processes,
machine
maintenance
of a group
combination
of objectives.
to idle machines
such
as meeting
management spare
may
machines
profitable the machine this
in the
always
of the diffusion
utilization
due
based machine
appropriate
to solve The
I.
method
The University of Florida, Gainesville, FL 32611, U.S.A
demand.
in the
long
repair
The objectives laborers,
With
the
be purchased run.
problem
practical
is to decide
so as to optimize
idle
to estimate
shoud
problem
of machines
and
want
an important
repair
may
objective
or rented
In the is related
be to maximize cost,
of extending
additional
literature
to make
(Barlow
to queueing
production, subject
and
processes,
should the
Proschan,
Q 1985, IMACS/Elsevier
Science Publishers
or some
to minimize
capacity,
losses
for example,
be employed
1965)
and the purpose
B.V. (North-Holland)
to the
to a set of constraints
production
relationship.
0378-4754/85/$3.30
repair
to assign
of performances
production
repairmen
in order
is the machine repairmen
some measure
or to minimize
how many
problem how many
or how many
system
more
it is shown
that
is to exploit
340
Hatyono, B. D. Sivazlian / Analysis of the machine repair problem Usually a queueing process is treated as a continuous time, discrete-state space Markov
process.
Based on this concept the mathematical model which represents the dynamic behavior
of this process consists of a set of differential-difference equations relating the rates of flow into and out of the system states of the processes.
This system of equations, in
principle, can be solved numerically for the fraction of time each system state is occupied.
Although some queueing systems can be treated as a Markov process, many important
queueing systems are of a non-Markovian type, and exact solutions sometimes are extremely difficult to obtain.
In fact, many interesting queueing phenomena have not been solved
exactly since most real systems are very large and have complex state spaces (For the GI/G/R system, for example, a model has been given by Kiefer and Wolfowitz (1955) whose solution requires solving a complex integral equation which reduces to Lindley's integral equation for the GI/G/l system).
The present study attempts to provide a solution to a non-Markovian
system using the diffusion model approach. The interest in the diffusion model arises because of its ability to handle complex queueing systems having general interarrival times (general time-to-failure distributions) and general interdeparture times (general repair time distributions).
Moreover, the model
is simpler to handle than the model from the discrete-state space approach. approximation
The diffusion
is based on the Chapman-Kolmogorov equation and reduces a complex discrete-
state space problem to a continuous-state space problem by regarding the number of customers (failed machines) as a continuous random variable rather than a discrete one.
The usual
approach attempts to find approximate solutions to the original system of equations. The difficulty in using the diffusion model approximation to describe a queueing process is that the arrival process and the departure process depend on each other.
Under
heavy-traffic conditions, the arrival process and the departure process are approximately independent renewal processes (Kleinrock, 1976).
Further, when time t becomes large, these
processes can be approximated by normal (Gaussian) processes with appropriate means and variances (using the renewal limit theorem argument).
The assumption of normality is the
basis of this approximation, and with this assumption, the estimation of the diffusion parameters becomes tractable. Statement of the Problem and Objectives of the Study The purpose of this study is to use the diffusion model to analyze a machine repair problem;
In the machine repair problem, the system consists of N=(M+S) identical machines
with M machines initially operating, S machines as spares and at most R repair facilities being simultaneously available.
For the diffusion approximation, it is assumed that the
time intervals between breakdowns are independent, identically distributed random variables having a general distribution.
Similarly, it is assumed that the repair times are
independent, identically distributed random variables having a general distribution.
Using
Hatyono, B. D. Sivazlian / Analysis of the machine repair problem the
diffusion
density
approximation
of x, the
In particular machihes
and the
selected values
number
the
Organization This
the
approximate
consists
process
to model
the mathematical
this
study.
It discusses
processes;
Section
specific
The
of the
the
which
gives
the
at steady
state
is constructed.
formulas
stage
in the
utilization
accuracy
for the mean steady
factor
of this
probability
number
state
of failed For
condition.
are compared
with
the
approximation.
The
model,
process
the
the
required
results These
problem.
for:
obtained
II.
in Section
parameters
boundary
results
application
and the definitions
1) constructing
the diffusion
the
on the
is reviewed
procedures,
the methodology
III presents
literature
of the
This used
the diffusion
that
characterize
section
extensively equation
also in for
queueing
conditions.
by applying
are compared
the diffusion
to the
approximation
simulation
results
to
using
configuration.
General This
propose
repair
sections.
2) obtaining
repair
METHODOLOGY
II.
values
a queueing
and 3) obtaining
the machine
system
approximation
of the
to show
of two
forth
processes;
in the
equation
of the Study
puts
queueing
factor
by simulation
a diffusion
machines
is to seek
utilization
study
diffusion
of failed
objective
systems, obtained
methodology,
341
AND
LITERATURE
Theory section
in this
briefly study.
parameter,
a continuous
this
the
study
REVIEW
reviews
some
of the theory
A diffusion
process
state
contained
diffusion
space
process
of the diffusion
is a Markov
process
in (-m, m),
is primarily
used
process
with
which
a continuous
and continuous
as an approximation
sample
we
time paths.
In
to a discrete
process. First state
suppose
that
continuous-time
transition
[X(t);
Markov
Prob[X(t)
( x\X(tO) t given
(x, t) represents in equation connects
process,
consider
the
process. following
By analogy conditional
with
a discrete-
continuous-state
probability
F(xo,tO;x,t)
at time
t ) 01 is a diffusion
2.1
that
= x01 is the it took
time
value
that
of the
the
x0 at time
will
t,tO
= x0],
of transition. dependent)
function
'1. Thus
< x(X(tO)
probability
on the
the direction (possibly-time
the distribution
an intermediate
= ProbCX(t)
process
Because
process
takes
The order
to.
satisfy
> 0
2.1
on a value of (x0, to)
of the Markov the
at time
property
Chapman-Kolmogorov t with
6 x
an earlier
and the
function
equation time
which
t0 through
342
Hatyono, B. D. Sivazlian / Analysis of the machine repair problem
F(xo,tO;x,t) where
=
I=F(Y,T;
x,tMyFbo.tO;w)
-m
T, t > T > t 0, is an arbitrary
Furthermore, associated
with
the
final
instant
and X(t)
be the transition
let f(x0, t0,* x, t) state
x, that
2.2
= x, X(r)
probability
= Y,
X(t0)
density
= x0.
function
(p.d.f.)
is
aF(xO,tO;x,t) f(xg,t0;x,t)
It can be shown
=
that
f(x0,
f(xO,tO;x,t)
where any
process'
transition
origin,
state
t, where that,
as well
this
At is a small,
if At approaches
f(x0,
condition
the
Chapman-Kolmogorov
equation
2.4
as to make
but a finite
further
First
more
to;
x, t) satisfies
[Akb,t)fbo,
I_
at
2.4 must
statements
regarding
Doing
for
the Markov
reflecting
change
holding
the diffusion
the
2.4,
increment.
besides
equation
include
explicit
In equation
condition
to determine
assumptions,
to properly
temporal
f(x0,
I_-
sufficient
its description.
in order
approximation.
0 then
as a compatibility
is not
t0; x, t):
equation
for example, under
this
to achieve
differential
behavior
satisfies
2.4 is to be considered
p.d.f.,
are necessary
assumption,
also
x, t)
j'= f(y,T;x,t)f(xO,tO;y,T)dy -m
However,
process.
as a partial the
=
t0,*
Equation
t > T > t0.
Markov
2.3
ax
be rewritten boundary the
at
steady
t into t + At and 'I into
so, Kleinrock
(1976)
showed
the equation
t o,
-x,t)l
2.5
k=1,2,...
2.6
where
Akb,t) is called
the
Equation we may in this equation
=
hi”+,&-
infinitesimal 2.5 holds
derive
various
study
is the
Irn (y-x)kf(x,t;y,t+At)dy, -m
moment
of the
for continuous-state
approximations second-order
2.5 to be non zero
process. space
for queueing approximation
and for which
Markov
processes
processes. in which
we assume
that
this
The approximation
we take Ak(x,
and from
the
first
equation
of interest
two terms
t) = 0, for k = 3, 4,
in ....
giving
af(xo,tO;x,t) at-
CAl(x,t)f(xo,tO;x,t)l
= - -+
;
-2;
ax
CA2(x,t)f(xo,t0;x,t)l
+ 2.7
343
Haryono, B. D. Sioazlian / Analysis of the machine repair problem Equation
2.7
diffusion are called If
is known
equation the
the
between
the
then
time
the
Fokker-Plank
equation)
and variance
is stationary,
present
therefore,
mean
the transition
f(x,t;y,t+At)
and
Kolmogorov
infinitesimal
process
translation,
as the one dimensional
(Forward
i.e.,
initial
of the
equation
Al(x,
density
time;
which
is referred
t) and A2(x,
t),
to as a
respectively,
process.
if its probability
probability
and the
where
laws
function
are invariant
depends
only
under
on the
a time
difference
thus
2.8
= f(x,U;y,At)
infinitesimal
moments
of the
process
Ak(X,
t)
do not depend
on the
time
variable. Upon
setting
= X(t
y-x
and
for
+ At) - X(t)
At sufficiently
A+)
= lim
small
there
results
ECrX(t+At)-X(t)~klX(t)=xl, , ,-.a At k=l
-__
A,-+,,
-
2.9
2
or
This interval
relation
says that
At, conditional
the moments
upon
X(t)
of the
process'
= x, are proportional
increment
over
to the time
a small
interval
time
itself.
In
particular,
so that
Al(x
= E[X(t+At)
A2(x)At
= E[IX(t+At)
the
conditional
u2cx (tat)
- X(t)(X(t)
- X(t)>21X(t)
variance
- X(t)(X(t)
= xl
of the
= x] increment
process
= x] = A2(x)At
is
- [AI(x)
Therefore
Al(x) =
lim At+0
A2(x)
;;“+,
=
E[X(t+At)
- X(t)IX(t) -----At
02[X(t+At) -A:(t)lx(t)
= xl 2.10
= Xl
344
Hatyono, B. D. Sivarlian / Analysis of the machine repair problem Let
failed basis N(t)
N(t)
the number
in the
system
of the diffusion
process
of the density
is in the
approximating
steady
state
density
2
where
Al(x) The
when
lim f(xo, t+m
will
take
queueing
The
reflective
the discrete
we can estimate of X(t).
The
number
= n,,].
stochastic
P(nU,
tU;
of
On the process
n, t) from
approximation
to;
by equation
in applying
a
of interest
lim P(no, to; n, t) = P(n) and for the t+m x, t) = f(x). Under steady conditions, equation
2.7
to represent
2.11
2.10.
this
the diffusion
means
approximation
to queueing
on the diffusion the
state
parameters
More
1973).
the diffusion Cox
and Miller
process;
space
Al(x)
(1965)
of the
processes
so that
the
are: process
process;
and A2(x)
(x)f(x)
boundary
or has a state the
space
number
system
when
the
that
a reflecting
characterize
the
in the
that
interval in the
variables).
By using
the
integrating
is empty)
beyond
boundary
that
as a reflecting
at the condition
the
solution
of the
of one or more
(those)
values;
is emptied,
barrier
(Cox and Miller,
barrier(s).
therefore boundary
it is represented
For
it is (Gaver
and
by letting
origin. for a reflecting
barrier
at the
is
guarantees
of customers
pass
used
barriers,
The existence
on non-negative
system
CA2bV(x III
+ +-+
condition
cannot
takes
the
normally
by one or more
condition(s).
the
show that
process
condition
process
process
(where
approach
for a diffusion
- A1
the
the
precisely,
process
boundary
is restricted
to boundary
that
for example,
to set the origin
Shedler,
is the
process
is subject
barriers
system,
natural
random
x, t),
the
when
values
barrier
a diffusion
equation
reflecting
because
replace
t,i.e., = nlN(t0)
Condition
When
diffusion
This
i.e.,
at time
process.
1965).
origin
to;
conditions
How to estimate
The Boundary
GI/G/R
we
a way that
boundary
on non-negative
2.
system
n, t) = P[N(t)
+--; --z-d2CA2b)fb)l
raised
How to place
1.
t0;
methodology,
f(xo,
are defined
problems
in the
P(n0,
in such
solution,
CA#.)f(x)l
and A2(x)
basic
X(t)
function,
becomes
0 = -
of customers
and define
approximation
by the diffusion
knowledge here
denote
machines
factor
x=0
=o
the diffusion
(0, m). system
2.12
process
takes
We are interested takes-on
non-negative
on non-negative
in this values
boundary
values condition
(non-negative
Hatyono, B. D. Sivarlian
= exp {- I
s(x)
x
and
performing
some
where
algebraic
= $$--
f(x)
k > 0 is the
2.11
subject
The Approximation The to model to
lost.
In particular
[X(t);
t ) 01 with
we use
a scheme
process.
The
under
assumption
estimation
III.
renewal
For
Detailed
choice issue
process the
solution
general
discussion
on this
limit
others
theorem
in this
[N(t);
the
infinitesimal
among
for the diffusion
of equation
boundary
t > 0] with
condition
mean
is used
of heavy-traffic
(1964)
(Ross,
becomes
the diffusion
for the
1983)
to obtain
case
are not
process.
In this
study
a stochastic
the diffusion
heavy-traffic
we want
process
for approximating
Under
and A2(x)
process
process
t ) 01 by the
and variance
condition.
parameters
[N(t);
AI(x)
In this
of the original
process
by Bailey
parameters
approximation.
characteristics
to replace
same
the
problem
stage)
(cold
when
operational
standby)
parameters
approximation,
the
tractable.
(operation
stage).
is freed.
immediately
repairmen
into
('first-in
first-out')
succession
of repair
from
the
sytem
with
when
general finite
where
queue
Further,
independent, distribution.
capacity
the output
and
from
the
one
being
system while
immediately joins
status in the
becomes
under
of breakdown
distributed that
in which the
until
this
input
forms
machines
to the
and the
variables
system
N=M+S
as good
a FIFO
times
random
a
it is put
is considered
is served
(repair it is
repair
line
is completed
of the machine
succession
source
into the
a waiting
machine
system
It is obvious
stage
facilities
system
identically
limited
R repair
of M
are backed-up
in the
of a failed
and the
the
a maximum
machines
it is out of the
new breakdown
repair
stage that
discipline. are
the
goes
of which
operating
to be a customer
whereas
machine
each
machines, These
has at most
is considered
A failed
an operation
times
system
to be repaired
we suppose
a separate
system
and the
are busy,
Moreover,
In addition,
are N identical at any time.
Each machine
waiting
repairman
there
machines
available.
If all
as new.
study,
or productive
it is down
facility.
queueing
under
can be operating
simultaneously
around
stochastic
of the diffusion
by S spare
drawn
is the most
APPLICATION
machines
back
This
2.12.
is a crucial
we want the
2.13
(1981).
as way that
suggested
that
dyl
of an approximate
the discrete
t ) 01 in such
be shown
and A2(x) ---
process
[X(t);
the
condition
for Al(x)
a queueing
it can
of integration.
and Taylor
determination
replace
manipulation,
boundary
in Karlin
dyl
Iofq2#
exp
constant
to the
is.available
~A&Y) --bum
0
345
/ Analysis of the machine repair problem
other
each
a cyclic
circulate stage.
This
346
Hatyono, B. D. Sioarlian / Analysis of the machine repair problem
system always reaches a steady state because no more than N machines exist in the system. By using Kendall's notation this system corresponds to the GI/G/R/N/N/FIFO system. To motivate the diffusion model employed in approximating the GI/G/R/N/N/FIFO system, we first consider an approximation for the M/M/R/N/N/FIFO system.
A scheme suggested by
Bailey (1964) and Heyman (1975) for approximating a stochastic process is used.
In the
M/M/R/N/N/FIFO system, breakdowns occur according to a Poisson procefs with rate X and repair times have the negative exponential distribution with mean lo .
Let A(t) and D(t),
respectively, denote the number of breakdowns and the number of service completions during the interval (O,t].
Then the queue size at time t, which corresponds to the number of
failed machines, is given by N(t) = A(t) - D(t ) It
3.1
is well-known that the stochastic process [N(t); t ) 0] forms a homogeneous birth-
death process as a mathematical model.
In this case, the number of failed machines in the
system corresponds to the state of the birth-death process (discrete state space).
This
system has (N+l) states, n=O, l,Z,...,N. Let us define the transition probability P(n,N-n;t) = Prob[N(t)&lN(t,)=n,]
3.2
(n,no=O,l,...,N)
that the process is in state n at time t starting from state no at time t0.
Furthermore,
let X,,and n,,be the birth rate (the breakdown rate) and the death rate (the repair rate) in state n, respectively.
The parameters 1,'s and n,,'scan take different forms depending on
the values taken by n with respect to S and R as described in what follows: Case A.
When S > R
In this case we have three possible situations for n. 1.
n < R < S or (N-n) > M.
For this condition the number of failed machines in the
system is less than the number of repairmen and also is less than the number of spares. Therefore the operation stage is still fully loaded but some of the repairmen are always idle.
Hence, we have X,=MX 'n
= nu
n = 0,1,2,...,M-1 n = 0,1,2,...,M-1
where A,,and u, represent respectively the mean breakdown rates of the operation stab: :.ad the mean repair rates of the repair stage.
Haryono, R 6 n < S or
2. system the
is greater
number
than
of spares. A,,=
3.
is more
repair
stage
Mh
than
(N-n)
< M.
For this
of spares.
fully
So, the
loaded.
Hence,
n=S+l,S+2,...,N-1 n=S+l,S+2,...,N
3-l.
these
The Mean
conditions
Breakdown
State
R-l
n = 0,1,2,...,
of repairmen
of failed
and also
is equal
in Table
Rates
System
1,'s
when
condition
the
operation
number
stage
of failed
is partially
and the Mean
Repair
Rates
---
n
'n
'n nu
n = R,Rtl,...,S
or
R
MA
R!J
n '= S,S+l,...,N
or
R
by the
*
.
following
(R-1)~
2u (M-l)1
RlJ
diagram
MA
(Figure
3-l.)
MA
RU
b
(M-2)~
4_@st2
.
3-l.
state-transition
MX
RlJ
Figure
(N-n)X
MA
MA
loaded
S > R.
MX
u
machines
~,,'s for
n < R c S
MA
in the
to or less
3-l.
or
is represented
machines
we have
= Ru
summarize
number
than
n = R,R+l,...,S
M/M/R/N/N/FIFO
This
number
the
347
n = R,R+l ,*a*, S
that
remains
to the
condition
repair problem
we have
A, = (N-n)1 57
Table
or equal
/ Analysis of the machine
For this
) M.
Hence,
R < S < n or
system
We can
(N-n)
=Rv
%
B. D. Sivarlian
State-Transition
RlJ
RlJ
Rate-Diagram
RP
for M/M/R/N/N/FIFO
System
when
S ) R.
in the
but the
348
Hatyono, B. D. Siuadian / Analysis of the machine repair problem By using the principle that the time derivative of the probability of a particular
state is equal to the difference between flow rate into and flow rate out of that state (Kleinrock, 1976) we establish the following system of differential-difference equations (forward Kolmogorov equations) which represents the dynamics of the process
dP(O,N;t) = - MxP(0 , N.t)tpP(l,N_l.t) t f 3 dt
--dP(n&AGt
n=O
= - (MXtnu)P(n,N-n;t)t(n+l)pP(n+l,N-n-l;t) t MXP(n-l,N-ntl;t)
3.3
ncR
,
dP(n;:-n;t-). = - (MXtRP)P(n,N-n;t)tRuP(n+l, N-n-l;t)
t MxP(n-l,N-n+l;t)
RtncS
,
dP(n N-n-t) _ --L-L- - ((N-n)XtRP)P(n,N-n;t)+RuP(n+l,N-n-1;t) dt + (N-n+l)xP(n-l,N-n+l;t) ,
R
If we assume that n machines begin to operate simultaneously at time t0, then the initial condition is given by Prob[N(to)=nlN(to)=no~ = 6,, n 0'
no,n=O,l,...,N
where 6 no,n
=
1
n=no
0
n#no
1
and the boundary condition Prob[N(t)=n(N(tO)=nO] = 0
if n < 0,
t>o
The idea of the diffusion approximation is to replace the system of equations 3.3 by a system of partial differential equations that are easier to solve.
This can be accomplished
by replacing the discrete random variable N(t) by the continuous random variable X(t) and P(n, N-n; t) by f(x, N-K; t).
Therefore equations 3.3 become
349
Haryono, B.D. Sivazlian / Analysis of the machine repair problem
x=0
,
= Mhf(O,N;t)+pf(l,N-1;t)
----af(;;N;t)
af(x,N-x;t) = - (MXtxu)f(x,N-x;t)+(x+l)uf(x+l,N-x-1;t) at t MXf(x-l,N-x+l;t)
af(x N-x-t) __LL
x
,
_ - - (MxtRu)f(x,N-x;t)+Rnf(x+l,N-x-l;t)
at
+ MXf(x-l,N-x+l;t) af(x N-x-t) _-L-L
_ - _ {(N-x)ktRu)f(x,N-x;t)+Ruf(x+l,N-x-1;t)
at
t (N-x+l)xf(x-l,N-x+l;t)
If we now expand about the
x and
retain
following
which
the term
only
steady
terms
state
-$
{(MA-xp)f(x))
o=_
-$
{(MA-Rn)f(x)
are
- $
{((N-x)x-RuIf
in fact
We observe
of the of the
equations
O=-
o=
3.4
Rcx
,
(t+m)
d2
hand
side
order where
of equation
(second lim f(x, t+m
f(x))
{VA
order
3.4 in a Taylor approximation),
series
we obtain
N-x; t) = f(x)
Otx
,
f(x)] ,
3.5
R
dx2
(x), + -
of the diffusion
that
right second
d2 (Mx+xu) + -d--z- t-2
I +-
R
,
the equations
d2
Cm*;=
f(x)] ,
type. in 3.5
have
identical
forms
Al(x)
= (MI-xv)
and A2(x)
= (MXtxu)
,
O
Al(x)
= (MI-RP)
and A2(x)
= (MI+RP)
,
R
Al(x)
= {(N-x)X-RuI
and A2(x)
R < S < x
dx2
= {(N-x)~+RuI
,
with
R < S < X
equation
2.11
where
3.6
350
Haryono, We first
consider
a heuristic
interval
(t,t+At],
arrivals
(new breakdowns)
D(t)
the
= n > 0 this
Table
3-2.
State
B. D. Sivazlian
number
change
Expectation
method
of failed
minus
the
/ Analysis of the machine
machines
number
has expectation
and Variance
for obtaining
equation
in the
of repair
repair problem
system
for M/M/R/N/N/FIFO
given
System
During
the time
by the
number
and when
by Table
when
N(t)
of
= A(t)
-
3-2.
S > R.
Variance
Expectation
n
changes
completions
and variance
3.5.
I_n
(MA-nu)At+o(At)
(MX+nu)At+o(At)
R
(MbRp)At+o(
(MA+Rp)At+o(At)
At)
{(N-n)X+RulAt+o(At)
{(N-n)X-RuIAt+o(At)
R
Formally
we have
For n < R < S E fN(t+At)-f44tm_
lim At+0 lim
At+0
o2
(MA-w)
[NetAt)-N(t)(N(t)=n] ._ --
At
= (MX+np
For R c n ( S lim
E
lim
o2
At.+0 At+0
w+At)-N(t)lN(t)=nl
At
L!&tZLL:N~o-=ti
= (MX-mu ) = (MX+R,,)
For R c S < n lim
At+0
Therefore [X(t); equation a good
E LNLt+At)-NCt)INCt~
At
to approximate
t ) 0] with 3.6.
This
approximation
the
same
scheme
= ((N_n)A_Rp)
a stochastic
process
infinitesimal suggests
that
mean
[N(t);
an appropriate
for the GI/G/R/N/N/FIFO
t > 0] by a diffusion
and variance,
system.
we set Al(x)
choice
of Al(x)
process
and A2(x)
as in
and AZ(X) Will yield
Hatyono,
B. D. Sivarlian
For the GI/G/R/N/N/FIFO system, We attempt
complicated. the
succession
random (l-1,
0;
< -).
independently with
drawn
server
time
t is given
selection
form
a general
we assume
that
we have
a sequence
the
parameters
AI(x)
employed.
We first
of independent
distribution
with
succession
351
(mean,
and
times
assume
identically
variance)
of repair
and AZ(X)
given
form
from the
during
the
are
clearly
i-th
interval
=
!
operation
stage
Then
(0,t).
Ai
i=l
interests
the
process
Because buted
during
A(t)
assumption
on each
-
R z
the
and the number
number
of repair
of failed
completions
machines
in the
random
by the
system
at
3.7
Di(t)
i=l
R
(0,t)
other.
Ai
and the departure
In heavy
arrival
theorem
Based
respectively, for A(t)
i=1,2,...,M
we have
theorem
basis
limit
= MX
and
a2[A(t)]
- MX30$,
EM(t)1
s( MXt
and
a2[A(t)]
B MA 32tM
ECA(t)]
- (N-n)At
theorem
D(t)
the mean
theorem and the the
(Gaussian)
of the diffusion independent
E i=l
and
Di (t)
process
and it is this
limiting
normal
=
the departure
we can describe
independent
is the
ED(t)1
and 02[A(t)]
a renewal
i=1,2,..., R are
(renewal
however,
to approximate
on this as two
process
and vice-versa,
to apply
and D(t)
and Di(t),
traffic,
process
can be used
as t+m.
and D(t),
variables,
E i=l
it is reasonable
This
of normality
Ai(
=
of the
Therefore,
and Di(t).
processes
A(t)
independent
us.
to Ai
process
dependent
approximately
distributed by
a sequence
M arrival
that
by
N(t)=A(t)-D(t)
The
is more
distributed random variables drawn from a general distribution -1 let Ai and Di(t), denote the by (P , 0: < -). Furthermore,
given
of arrivals
i-th
the
of the
repair problem
identically
(mean,variance)
number
times
from
Next
and
determination
to justify
of breakdown
variables
/ Analysis of the machine
case
is
which
(Ross,
1983)
variance
of
stochastic processes.
The
approximation. identically
distri-
argument)
O
,
= (N-")x30$,
R
R
and
3.8 ECD(t)]
= npt. and
a2[D(t)]
= nu3$t,
Otn
E[D(t)]
= Rut
and 02[D(t)]
= Ru3d$,
Rtn
ECD(t)]
- Rut
2 and u [D(t)]
a Rp3a2t R'
R
352
Haryono, B. D. Sivarlian / Analysis of the machine repair problem
This can be summarized in the following table (Table 3-3) (where CM and CR are the squared coefficients of variation of the breakdown and repair times, respectively). Table 3-3.
The Mean and the Variance for GI/G/R/N/N/FIFO System if t Becomes Large
and S > R where
CM = X2$
and CR = u*oi. ----_-----
State
W(t)1
n
CD(t)1
~*CNW
o*CWl
-
n
MXt
nut
M XCMt
n pCRt
R
MAt
Rut
M $t
R vCRt
R
(N-n)xt
Rut
(N-n)XCMt
R uCRt
Formally A(t)
D t+_-->
N(Mxt,
A(t)
--$;->
N(MXt, MXCMt),
R
A(t)
---&;-->
N( (N-n)xt,
R
D(t)
---c:_->
D(t)
+&-->
N(Rut,
RuCRt),
R
D(t)
+&->
N(Rut,
RuCRt),
R
MXCMt),
Otn
(N-n)ACMt),
and
where
D I-----> t+=J
denotes
It is well-known processes,
say A(t)
distributed
process
combination expressed
O
N(npt, nuCRt),
convergence
that
if we have
and D(t) with
some
we are interested as the
number
in distribution
then
two
independent
any linear
mean
in is N(t)
= A(t)
machines
normally
combination
appropriate
of failed
if t+m.
of these
and variance. - D(t)
in the
distributed
is also
‘Of course,
> 0 which
system
two
at time
stochastic
represents t.
a normally
one linear the backlog
Therefore,
we have
Haryono, B. D. Sivazlian
353
/ Analysis of the machine repair problem
N(t) = A(t)-D(t) -+-
> N((Mx-nu)t, (MxCM+nuCR)t.) ,
0 < n < R < S
N(t) = A(t)-D(t) +&-
> N((Mx-Ru)t, (MxCM+RuCR)t) , R < n < S
N(t) = A(t)-D(t) -+-
> N(((N-n)X-Rv)t, ((N-n)xCf,i+RuCR)t) ,
3.9
R < S < n
or we can write for the steady state approximation results as:
;z
ECN;t)l -
12
ECNit)I
-
LE
%!!lu
= ((N_n)l_Rp)
(MA+,)
and
(MA-R,,)
and
;E
2k.f
iz
w
and 12
= (MXM+npCR)
,
=
, R < n
(MKM+RpCR)
0 < n < R < S
u*CN~t)3 = ((N-n)xCM+RvCR)
3.10
,R
This suggests for continuous approximation to the GI/G/R/N/N/FIFO systern the following diffusion parameters Am
= (MA-xv) and $(x)
Am
= (MA-Rv) and A*(x) = (MlCM+RuCR),
Al(x) = ((N-x)1-Ru)
= (Mx$+x?~cR)
O
,
and A*(x) = ((N-x)XCM+RvCR),
By using Al(x) and A*(x) in equation
3.11
RcxGS
R < S < x
3.5 we construct the steady state equations for
the probability density function f(x) of x (the number of failed machines in the system in the steady state)
o= -
-$
{(MA-xv)f(x)) + 2
do
( MXM+XUCR)
I---2-----
f(x)1
o=
d2 ( MKM+RuCR) - -$ ((MA-Ru)f(x)I + --2 +--~2f(x))
o=
- --$ {((N-x)&Ru)f(x))
+ 2
d2
((N-x)EM+kCR) 1--___-- 2
,
,
O
Rtx
f(x))
3.12
,
R
Haryono, B. D. Sivazlian
354
/ Analysis of the machine repair problem
The respective solutions of 3.12 are given by kl
Otx
exp ( I 0
f(x) = -(Mm
e
= kl (MACM+xpCR)
,
3.13
O
f(x) - klhl(x) ,
' 2(Mx-Ru) dy) , exp rol MxCM+RpCR
E(Mx-Rv)xl
= k2 exp ~~~~ +R;~ M
f(x) e k2h2(xL
R
RtxtS
R
,
3.14
R
dy,
f(x) = 2iR!JCR [--+ 211R_ (XM)2 "M =
Ocx
f(x) 5 k3h3 lx),
2x
11
5 e
k3{(N-X)~CM+RL$~
RcS(x
,
R
3.15
R
Thus, we have three probability density functions forO
3.16
for R < S < x kl, k2 and kg remain undetermined requiring that three conditions be given for their specification.
There are three constants to be determined since each member function of the com-
position solution in equation 3.16 contributes one undetermined constant of integration.
Haryono, B. D. Sivarlian
355
/ Analysis of the machine repair problem
The first condition comes from the obvious normalization criterion and takes the form kl
0
R / hl(x)dx + k2
R
S I h2(x)dx + k3
N I s
3.17
h3(x)dx = 1
To determine the other conditions, we observe that the probability density function in equation 3.16 is continuous (Halachamy and Franta, 1978) at x=R and x=S.
Therefore
klhl(R) = k2h2(R) 3.18 k2h2(S) = k3h3(S) By using the conditions in equations 3.17 and 3.18 we can determine the value of kl,k2 and k3 as follows klhl(R) = k2h2(R)
hl(R)kl + k2 = h2(~~
kl(M$,+bCR) k2 = ~__-----_-----_______-
exp
3.19
@_(MkVRl)
MXM+RpCR
k2 z kill, where ll is already known from the problem. klhl(R)h2(S) k3 =---_---_-_=
h2(fW3W
k 1
Similarly we have
h2(S)
1 1 h3W
k3 I klll12, where l2 is already known from the problem.
Therefore, by using +.diion
kl can be determined, as well as k2 and kg, that is
k1 or
0
N S R / hl(x)dx + kill RI h2(x)dx + klll12 sI h3(x)dx = 1
3.17
356
Haryono, B. D. Sivazlian S
2 O’Q-RFl)x) dx
’ exp (MXC+RuC R M R
klll
/ Analysis of the machine repair problem
+
2xRuCR C----~-+-;gL 2" {(N-x)U+,+RuCRI tAc,)
klll12 S
11 M
2x T dx eM
=l
The probability of finding n failed machines, P(n), in the system in the steady state can be obtained by discretizing f(x) (Heyman and Sobel, 1982) as i(n) =
nt0.5 / f(x)dx n-O.5
Therefore the approximate formula, L, for the mean number of failed machines in the system can be obtained as L =
N E nP1 (n) n=O
3.21
Hence the approximate formula for the utilization of the repair stage is 3.22
U=L/R When R > S
Case B.
Similar to condition A there are three possibilities for n. 1. 'n
n < S < R or (N-n) ) M.
For this condition the mean breakdown rate
and the mean repair rate u,,are the same as with Case A.l. An = MX
n = 0,1,2,...,S
= nv
n = 0,1,2,...,S
Pn 2.
S < n < R or (N-n) < M.
Hence
For this condition the number of failed machines is more
than the number of spares.
So, the operation stage is partially loaded and some of the
repairmen are always idle.
Hence
X,,= (N-n)1 % 3.
n = Stl,
St2,
. . . . R-l
n = Stl,
St2,
. . . . R-l
= nn
S < R c n or (N-n) < M.
than that of spares. remains fully loaded.
For this condition the number of failed machines is more
So, the operation stage is partially loaded but the repair stage Hence
Haryono, B.D. Sivazlian hn = (N-n)1
n = R, R+l,
. . . . N-l
= RP
n = R, R+l,
.... N
'n
The
corresponding
.
.
state
transition
The
3-2.
steady
of failed
0 = z
state
diffusion
machines
d
d
Thus,
for
the
the
solution
Rate-Diagram
equations
in the
{(MA-xu)f(x)l
system)
d2 + dx2
)A-xl.df(x)l
[{(N-x )I-RpIf(x)l
We observe row.
becomes
that
in this
condition
d2 + --2-
by
be shown
3.23
[
System
density
to be in a way
when
function
similar
R > S.
of x (the number
to Case
A
Otx
(x)1,
I(N-x)XCM+xuCRI
2 I__
f(x)19
I(N-x)K~+RucRI [----- p-2
are the
for the condition
S < R G x the
is given
probability
MXM+x~CR I---f 2
d2
equations case
for M/M/R/N/N/FIFO
for the
may
+ --F-
RU
b
RU
State-Transition
d 0 = xx- [((N-x
0 = z
diagram
357
.
RU
Figure
/ Analysis of the machine repair problem
solution
same
f(x)],
as equations
0 G x < S < R the 3.15
remains
valid.
S
3.23
S < R < x
3.12
except
solution For the
3.13
in the remains
conditions
second valid
and
S < x < R
358
Haryono, B.D. Sivazlian / Analysis of the machine repair problem
f(x)
=
k4 (N-x)XC,,,+x"CR
R
2( x+u)Nx$
2NX
c
dx)
-11
= k4 {(N-x)XM+x~CR)
we have
f(x)
where
=
3.24
.S
three
probability
density
functions.
klhlb)
forO
k4h4(x)
for S ( x < R
k3h3(x)
for S < R < x
I
kl, k4 and
2(x+u)x $-pcR
e
f(x) = k4h4(x) Therefore
~
- -I=
kg remain
Similar
undetermined.
3.25
to Case
A, we have
the
following
conditions
kl
Is h+x)dx
+ k4
0
,
and
R / h4(x)dx
IN h3(x)dx
3.26
= 1
R
kqhq(S)
klhl(S)
=
k4h4(R)
= k3h3(R)
By using
t k3
S
condition
in equation
3.27
3.26
and
3.27 we can determine
the
values
of kl, k2 and
kg as follows
klhlW
hlW = k4h4(S) + k4 = fia7srkl
-11 k4 =
-$$ e
2Nl ($-VCR)
2(x+lJ)S QUCR
-11 e
3.28
Hatyono, B. D. Sivarlian k4 = kl14,
where
14 is already
known
hl(S)h4(Wkl k3 =
h4(S)h3(R)
Similarly
problem.
we have
h4W
c =
for the
__-_ h3(R)
= klk4
kl141(N-R)lCM
359
/ Analysis of the machine repair problem
+ RuCRl
Z(x+p)AN$,
2NX
-
(q&)*
-($-VCR)
2(X+n)R -_--
-I’
$,,-$ e
3.29
e
k3 z k11413, can
where
l3 is already
kl, as well
determine
known
as k4 and
S I k1
The
hl(x)dx
by using
equation
3.26
we
N + k11413
RI
h3(x)dx
= 1
S
probability
obtained
Therefore
problem.
k3; we obtain
R I h4(x)dx
f k114
0
for the
of finding
by discretizing
n failed f(x)
machines,
P(n),
in the
system
in the
steady
state
can be
as
nt0.5 i(n)
= I
f(x)
dx
n-O.5
Therefore is given
the
the
formula,
L, for the mean
number
of failed
machines
in the
system
by
L =
Hence
aporoximate
"c n;(n) n=O
approximate
U = L/R
3.30
formula
for the
utilization
of the
repair
stage
is
360
Haryono, B. D. Sivarlian / Analysis of the machine repair problem
Case .-
C.
S=O
This
problem
available.
stage
O
of the machine
are two
possibilities
condition
of failed
loaded.
the
machines
= nu
n =
O,l,Z,...,R-1
n > R.
is partially
For this
condition
of repairmen
loaded.
the
and thus
the
n=R
,...,M-I
= RP
n=R
,a**, M
corresponding
MA
3-3.
problem
where
no spares
state-transition
are
for n: of machines
is less
than
the
operating number
in the
operation
of repairmen.
number repair
of failed facility
machines is loaded
in the
diagram
(M-1)x
State-Transition Rate-Diagram When S=O and R < M.
is
(M-R+2)X
(M-R+l)X
for M/M/R/N/N/FIFO
System
system
but the
Hence
X,.,= (M-n)X 'n
number
repair
Thus,
Hence
O,l,Z,...,R-1
the number
Figure
number
are partially
case
and R < M
n =
2.
The
there
Problem)
An = (M-n)X %
than
case
For this
is M-n and the stages
Interference
is a special
In this
1.
both
(Machine
(M-R)X
is more
operation
stage
-Haryono, B. D. Sivazlian
361
/ Analysis of the machine repair problem
Similarly, it may be shown that for 0 < x < R
k4h4(x)
3.31
f(x) = I
for R < x < M
k3h3(x)
and k4 and k3 can be determined by the following conditions R k4
0
I
h4
(x)dx + k3
R
M I h3(x)dx = 1
3.32
and k4h4(R) =
3.33
k3h3(R)
By using equations 3.32 and 3.33 the value of k4 and k3 can be determined as follows
2(A+p)MXC c_-----_-+
_
( XM-UCR)L k3=--
k4f( M-R)XM+RuCRl ______________ 2RUCR C---t2vR (XM)2
2MA - 11 xcM-lJcR e
-____.___--._-.__-ll__
ACM
-11
{(M-R)$,+RuCR)
e
R h4(x)dx + k414
3. 34
@M
k3 E k414, where l4 is already known for the problem.
k4 o1
2(X_u)R EM-n_
R
Therefore
M I h3(x)dx = 1
The probability of finding n failed machines, P(n), in the system in the steady state condition can be obtained by discretizing f(x) as . P(n) =
nt0.5 / n-O.5
f(x)dx
Therefore, the approximate formula, L, for the mean number of failed machines in the system is
362
Haryono, B. D. Sivazlian / Analysis of the machine repair problem
L=
Hence
the
M Z n;(n) n=O
approximate
3.35
formula
for the
utilization
of the
repair
stage
is
U = L/R
Numerical
3.36
Examples
In Tables
3-4 and 3-5,
the M/M/R/N/N/FIFO
the
utilization
Table
3-4.
system factor
comparisons
and the
is obtained
of this
approximation
E2/E2/R/N/N/FIFO under
heavy
system
traffic
with
showed
a simulation
that
a close
condition.
Comparisons of Diffusion Approximation and Simulat ion Resu 1ts for M/M/R/N/N/FIFO System U where SIM = Simulation Results and DIF = Diffusion Results. M=6
R
4
6
vu
s=o
s=2
s=5
S=J
0.5
SIM DIF
0.505 0.587
0.614 0.687
0.707 0.801
0.712 0.796
0.6
SIM DIF
0.558 0.561
0.699 0.704
0.792 0.798
0.840 0.851
0.7
SIM DIF
0.614 0.619
0.771 0.779
0.887 0.894
0.897 0.911
0.5
SIM DIF
0.323 0.381
0.433 0.527
0.464 0.516
0.483 0.531
0.6
SIM DIF
0.373 0.378
0.478 0.484
0.545 0.551
0.563 0.568
0.7
SIM DIF
0.415 0.421
0.534 0.541
0.623 0.637
0.645 0.649
U = utilization
of the
R = number
of repair
S = number
of spares
repair
facilities
stage
model agreement
for for
Haryono, B. D. Sivazlian Table
3-5
363
/ Analysis of the machine repair problem
Comparisons of Diffusion Approximation and Simulation Results for E2/E2/R/N/N/FIFO System U where SIM = Simulation Results and DIF = Diffusion Results.
M=6
R
x/u
0.5
4
0.6 0.7
6
s=o
s=2
s=5
s=7
SIM
0.492
0.617
0.728
0.712
DIF
0.541
0.678
0.806
0.784
SIM
0.545
0.709
0.846
0.874
DIF
0.551
0.717
0.851
0.877
SIM
0.623
0.782
0.894
DIF
0.631
0.789
0.898
0.918 0.923
0.5
SIM DIF
0.338 0.393
0.428 0.471
0.495 0.537
0.482 0.526
0.6
SIM DIF
0.370 0.378
0.484 0.492
0.565 0.573
0.578 0.584
0.7
SIM DIF
0.497 0.412
0.566 0.520
0.668 0.679
0.684 0.669
u=
utilization
R=
number
of repair
s=
number
of spares
of the
repair
stage
facilities
REFERENCES
Cl1
Bailey, N.T.J.: The Elements of Stochastic Processes Sciences, J. Wiley and Sons, New York (1964).
[2]
Barlow, R.E.; New York,
[3]
Cox,
[4]
Gaver, D.P., and G.S. Shedler: "Processor Utilization in Multiprogramming Diffusion Approximation,". Oper. Res. 21:5691576 (1973).
[5]
Halachamy, Manag.
163
Heyman, D.P.: Bell Syst.
and F. Proschan: (1965).
D.R., and H.D. Miller: New York, (1965).
Mathematical
The
Theory
Theory
of Stochastic
G., and W.R. Franta: "A Diffusion Sci. 24:522-529 (1978). "A Diffusion Model Approximation Tech. J. 54:1637-1646 (1975).
with
Applications
of Reliability,
Processes,
Approximation
for the
to the
J. Wiley
Natural
and Sons,
J. Wiley and Sons,
to the
System
Multi-Server
GI/G/l Queue
in Heavy
via
Queue,"
Traffic,"
364
Haryono, B. D. Sivazlian / Analysis of the machine repair problem
[73
Heyman, D.P., and M.J. Sobel: Stochastic Models in Operations Research, Vol. 1, Academic Press, New York (1982).
[8]
Karlin, S., and H.M. Taylor: Press, New York (1981).
[9]
Kiefer, J., and J. Wolfowitz: "On the Theory of Queues with Many Servers," Trans. Amer. Math. Sot. 78:1-18 (1955).
[lo]
Kleinrock, L.:
:
Cl11 Cl23
Ross, S.M.:
A Second Course in Stochastic Processes, Academic
Queueing Systems, Vol. I, J. Wiley and Sons, New York (1975). Queueing Systems, Vol. II, J. Wi!ey and Sons, New York (1976).
Stochastic Processes, J. Wiley and Sons, New York (1983).
in Simulation
339
27 (1985) 339-364
ANALYSIS OF THE MACHINE REPAIR PROBLEM: A DIFFUSION PROCESS APPROACH HARYONO Department
and B.D. SIVAZLIAN
of Industrial and Systems Engineering,
An approximate analyzing
a non-Markovian
approximation N(t),
is that
is almost
becomes
large,
process
with
estimation used
[N(t),
of the
tend
parameters
becomes
of failed
the
used
machines When
in this
at time time
distributed
assumption
tractable,
for
argument
a normally
With
system
the
repair
based
For selected
systems,
repair
are compared
the
to become
basic
conditions.
is to obtain
factor
to show
number
is presented
t,
t
random
of normality
and the method
the
can be
problems.
study
stage
the
The
heavy-traffic
and variances.
methodology. of the
equation
problem.
stage,
under
O] will
repair
diffusion
repair
repair
means
of this
factor
on the
non-empty t >
machine
objective
simulation
accuracy
stage
the
formulas
of this
the
steady
on the diffusion
approximate with
for the
values
true
of the
values
state
approximation
utilization
obtained
through
approximation.
INTRODUCTION
In many problem.
industrial
The basic
processes,
machine
maintenance
of a group
combination
of objectives.
to idle machines
such
as meeting
management spare
may
machines
profitable the machine this
in the
always
of the diffusion
utilization
due
based machine
appropriate
to solve The
I.
method
The University of Florida, Gainesville, FL 32611, U.S.A
demand.
in the
long
repair
The objectives laborers,
With
the
be purchased run.
problem
practical
is to decide
so as to optimize
idle
to estimate
shoud
problem
of machines
and
want
an important
repair
may
objective
or rented
In the is related
be to maximize cost,
of extending
additional
literature
to make
(Barlow
to queueing
production, subject
and
processes,
should the
Proschan,
Q 1985, IMACS/Elsevier
Science Publishers
or some
to minimize
capacity,
losses
for example,
be employed
1965)
and the purpose
B.V. (North-Holland)
to the
to a set of constraints
production
relationship.
0378-4754/85/$3.30
repair
to assign
of performances
production
repairmen
in order
is the machine repairmen
some measure
or to minimize
how many
problem how many
or how many
system
more
it is shown
that
is to exploit
340
Hatyono, B. D. Sivazlian / Analysis of the machine repair problem Usually a queueing process is treated as a continuous time, discrete-state space Markov
process.
Based on this concept the mathematical model which represents the dynamic behavior
of this process consists of a set of differential-difference equations relating the rates of flow into and out of the system states of the processes.
This system of equations, in
principle, can be solved numerically for the fraction of time each system state is occupied.
Although some queueing systems can be treated as a Markov process, many important
queueing systems are of a non-Markovian type, and exact solutions sometimes are extremely difficult to obtain.
In fact, many interesting queueing phenomena have not been solved
exactly since most real systems are very large and have complex state spaces (For the GI/G/R system, for example, a model has been given by Kiefer and Wolfowitz (1955) whose solution requires solving a complex integral equation which reduces to Lindley's integral equation for the GI/G/l system).
The present study attempts to provide a solution to a non-Markovian
system using the diffusion model approach. The interest in the diffusion model arises because of its ability to handle complex queueing systems having general interarrival times (general time-to-failure distributions) and general interdeparture times (general repair time distributions).
Moreover, the model
is simpler to handle than the model from the discrete-state space approach. approximation
The diffusion
is based on the Chapman-Kolmogorov equation and reduces a complex discrete-
state space problem to a continuous-state space problem by regarding the number of customers (failed machines) as a continuous random variable rather than a discrete one.
The usual
approach attempts to find approximate solutions to the original system of equations. The difficulty in using the diffusion model approximation to describe a queueing process is that the arrival process and the departure process depend on each other.
Under
heavy-traffic conditions, the arrival process and the departure process are approximately independent renewal processes (Kleinrock, 1976).
Further, when time t becomes large, these
processes can be approximated by normal (Gaussian) processes with appropriate means and variances (using the renewal limit theorem argument).
The assumption of normality is the
basis of this approximation, and with this assumption, the estimation of the diffusion parameters becomes tractable. Statement of the Problem and Objectives of the Study The purpose of this study is to use the diffusion model to analyze a machine repair problem;
In the machine repair problem, the system consists of N=(M+S) identical machines
with M machines initially operating, S machines as spares and at most R repair facilities being simultaneously available.
For the diffusion approximation, it is assumed that the
time intervals between breakdowns are independent, identically distributed random variables having a general distribution.
Similarly, it is assumed that the repair times are
independent, identically distributed random variables having a general distribution.
Using
Hatyono, B. D. Sivazlian / Analysis of the machine repair problem the
diffusion
density
approximation
of x, the
In particular machihes
and the
selected values
number
the
Organization This
the
approximate
consists
process
to model
the mathematical
this
study.
It discusses
processes;
Section
specific
The
of the
the
which
gives
the
at steady
state
is constructed.
formulas
stage
in the
utilization
accuracy
for the mean steady
factor
of this
probability
number
state
of failed For
condition.
are compared
with
the
approximation.
The
model,
process
the
the
required
results These
problem.
for:
obtained
II.
in Section
parameters
boundary
results
application
and the definitions
1) constructing
the diffusion
the
on the
is reviewed
procedures,
the methodology
III presents
literature
of the
This used
the diffusion
that
characterize
section
extensively equation
also in for
queueing
conditions.
by applying
are compared
the diffusion
to the
approximation
simulation
results
to
using
configuration.
General This
propose
repair
sections.
2) obtaining
repair
METHODOLOGY
II.
values
a queueing
and 3) obtaining
the machine
system
approximation
of the
to show
of two
forth
processes;
in the
equation
of the Study
puts
queueing
factor
by simulation
a diffusion
machines
is to seek
utilization
study
diffusion
of failed
objective
systems, obtained
methodology,
341
AND
LITERATURE
Theory section
in this
briefly study.
parameter,
a continuous
this
the
study
REVIEW
reviews
some
of the theory
A diffusion
process
state
contained
diffusion
space
process
of the diffusion
is a Markov
process
in (-m, m),
is primarily
used
process
with
which
a continuous
and continuous
as an approximation
sample
we
time paths.
In
to a discrete
process. First state
suppose
that
continuous-time
transition
[X(t);
Markov
Prob[X(t)
( x\X(tO) t given
(x, t) represents in equation connects
process,
consider
the
process. following
By analogy conditional
with
a discrete-
continuous-state
probability
F(xo,tO;x,t)
at time
t ) 01 is a diffusion
2.1
that
= x01 is the it took
time
value
that
of the
the
x0 at time
will
t,tO
= x0],
of transition. dependent)
function
'1. Thus
< x(X(tO)
probability
on the
the direction (possibly-time
the distribution
an intermediate
= ProbCX(t)
process
Because
process
takes
The order
to.
satisfy
> 0
2.1
on a value of (x0, to)
of the Markov the
at time
property
Chapman-Kolmogorov t with
6 x
an earlier
and the
function
equation time
which
t0 through
342
Hatyono, B. D. Sivazlian / Analysis of the machine repair problem
F(xo,tO;x,t) where
=
I=F(Y,T;
x,tMyFbo.tO;w)
-m
T, t > T > t 0, is an arbitrary
Furthermore, associated
with
the
final
instant
and X(t)
be the transition
let f(x0, t0,* x, t) state
x, that
2.2
= x, X(r)
probability
= Y,
X(t0)
density
= x0.
function
(p.d.f.)
is
aF(xO,tO;x,t) f(xg,t0;x,t)
It can be shown
=
that
f(x0,
f(xO,tO;x,t)
where any
process'
transition
origin,
state
t, where that,
as well
this
At is a small,
if At approaches
f(x0,
condition
the
Chapman-Kolmogorov
equation
2.4
as to make
but a finite
further
First
more
to;
x, t) satisfies
[Akb,t)fbo,
I_
at
2.4 must
statements
regarding
Doing
for
the Markov
reflecting
change
holding
the diffusion
the
2.4,
increment.
besides
equation
include
explicit
In equation
condition
to determine
assumptions,
to properly
temporal
f(x0,
I_-
sufficient
its description.
in order
approximation.
0 then
as a compatibility
is not
t0; x, t):
equation
for example, under
this
to achieve
differential
behavior
satisfies
2.4 is to be considered
p.d.f.,
are necessary
assumption,
also
x, t)
j'= f(y,T;x,t)f(xO,tO;y,T)dy -m
However,
process.
as a partial the
=
t0,*
Equation
t > T > t0.
Markov
2.3
ax
be rewritten boundary the
at
steady
t into t + At and 'I into
so, Kleinrock
(1976)
showed
the equation
t o,
-x,t)l
2.5
k=1,2,...
2.6
where
Akb,t) is called
the
Equation we may in this equation
=
hi”+,&-
infinitesimal 2.5 holds
derive
various
study
is the
Irn (y-x)kf(x,t;y,t+At)dy, -m
moment
of the
for continuous-state
approximations second-order
2.5 to be non zero
process. space
for queueing approximation
and for which
Markov
processes
processes. in which
we assume
that
this
The approximation
we take Ak(x,
and from
the
first
equation
of interest
two terms
t) = 0, for k = 3, 4,
in ....
giving
af(xo,tO;x,t) at-
CAl(x,t)f(xo,tO;x,t)l
= - -+
;
-2;
ax
CA2(x,t)f(xo,t0;x,t)l
+ 2.7
343
Haryono, B. D. Sioazlian / Analysis of the machine repair problem Equation
2.7
diffusion are called If
is known
equation the
the
between
the
then
time
the
Fokker-Plank
equation)
and variance
is stationary,
present
therefore,
mean
the transition
f(x,t;y,t+At)
and
Kolmogorov
infinitesimal
process
translation,
as the one dimensional
(Forward
i.e.,
initial
of the
equation
Al(x,
density
time;
which
is referred
t) and A2(x,
t),
to as a
respectively,
process.
if its probability
probability
and the
where
laws
function
are invariant
depends
only
under
on the
a time
difference
thus
2.8
= f(x,U;y,At)
infinitesimal
moments
of the
process
Ak(X,
t)
do not depend
on the
time
variable. Upon
setting
= X(t
y-x
and
for
+ At) - X(t)
At sufficiently
A+)
= lim
small
there
results
ECrX(t+At)-X(t)~klX(t)=xl, , ,-.a At k=l
-__
A,-+,,
-
2.9
2
or
This interval
relation
says that
At, conditional
the moments
upon
X(t)
of the
process'
= x, are proportional
increment
over
to the time
a small
interval
time
itself.
In
particular,
so that
Al(x
= E[X(t+At)
A2(x)At
= E[IX(t+At)
the
conditional
u2cx (tat)
- X(t)(X(t)
- X(t)>21X(t)
variance
- X(t)(X(t)
= xl
of the
= x] increment
process
= x] = A2(x)At
is
- [AI(x)
Therefore
Al(x) =
lim At+0
A2(x)
;;“+,
=
E[X(t+At)
- X(t)IX(t) -----At
02[X(t+At) -A:(t)lx(t)
= xl 2.10
= Xl
344
Hatyono, B. D. Sivarlian / Analysis of the machine repair problem Let
failed basis N(t)
N(t)
the number
in the
system
of the diffusion
process
of the density
is in the
approximating
steady
state
density
2
where
Al(x) The
when
lim f(xo, t+m
will
take
queueing
The
reflective
the discrete
we can estimate of X(t).
The
number
= n,,].
stochastic
P(nU,
tU;
of
On the process
n, t) from
approximation
to;
by equation
in applying
a
of interest
lim P(no, to; n, t) = P(n) and for the t+m x, t) = f(x). Under steady conditions, equation
2.7
to represent
2.11
2.10.
this
the diffusion
means
approximation
to queueing
on the diffusion the
state
parameters
More
1973).
the diffusion Cox
and Miller
process;
space
Al(x)
(1965)
of the
processes
so that
the
are: process
process;
and A2(x)
(x)f(x)
boundary
or has a state the
space
number
system
when
the
that
a reflecting
characterize
the
in the
that
interval in the
variables).
By using
the
integrating
is empty)
beyond
boundary
that
as a reflecting
at the condition
the
solution
of the
of one or more
(those)
values;
is emptied,
barrier
(Cox and Miller,
barrier(s).
therefore boundary
it is represented
For
it is (Gaver
and
by letting
origin. for a reflecting
barrier
at the
is
guarantees
of customers
pass
used
barriers,
The existence
on non-negative
system
CA2bV(x III
+ +-+
condition
cannot
takes
the
normally
by one or more
condition(s).
the
show that
process
condition
process
process
(where
approach
for a diffusion
- A1
the
the
precisely,
process
boundary
is restricted
to boundary
that
for example,
to set the origin
Shedler,
is the
process
is subject
barriers
system,
natural
random
x, t),
the
when
values
barrier
a diffusion
equation
reflecting
because
replace
t,i.e., = nlN(t0)
Condition
When
diffusion
This
i.e.,
at time
process.
1965).
origin
to;
conditions
How to estimate
The Boundary
GI/G/R
we
a way that
boundary
on non-negative
2.
system
n, t) = P[N(t)
+--; --z-d2CA2b)fb)l
raised
How to place
1.
t0;
methodology,
f(xo,
are defined
problems
in the
P(n0,
in such
solution,
CA#.)f(x)l
and A2(x)
basic
X(t)
function,
becomes
0 = -
of customers
and define
approximation
by the diffusion
knowledge here
denote
machines
factor
x=0
=o
the diffusion
(0, m). system
2.12
process
takes
We are interested takes-on
non-negative
on non-negative
in this values
boundary
values condition
(non-negative
Hatyono, B. D. Sivarlian
= exp {- I
s(x)
x
and
performing
some
where
algebraic
= $$--
f(x)
k > 0 is the
2.11
subject
The Approximation The to model to
lost.
In particular
[X(t);
t ) 01 with
we use
a scheme
process.
The
under
assumption
estimation
III.
renewal
For
Detailed
choice issue
process the
solution
general
discussion
on this
limit
others
theorem
in this
[N(t);
the
infinitesimal
among
for the diffusion
of equation
boundary
t > 0] with
condition
mean
is used
of heavy-traffic
(1964)
(Ross,
becomes
the diffusion
for the
1983)
to obtain
case
are not
process.
In this
study
a stochastic
the diffusion
heavy-traffic
we want
process
for approximating
Under
and A2(x)
process
process
t ) 01 by the
and variance
condition.
parameters
[N(t);
AI(x)
In this
of the original
process
by Bailey
parameters
approximation.
characteristics
to replace
same
the
problem
stage)
(cold
when
operational
standby)
parameters
approximation,
the
tractable.
(operation
stage).
is freed.
immediately
repairmen
into
('first-in
first-out')
succession
of repair
from
the
sytem
with
when
general finite
where
queue
Further,
independent, distribution.
capacity
the output
and
from
the
one
being
system while
immediately joins
status in the
becomes
under
of breakdown
distributed that
in which the
until
this
input
forms
machines
to the
and the
variables
system
N=M+S
as good
a FIFO
times
random
a
it is put
is considered
is served
(repair it is
repair
line
is completed
of the machine
succession
source
into the
a waiting
machine
system
It is obvious
stage
facilities
system
identically
limited
R repair
of M
are backed-up
in the
of a failed
and the
the
a maximum
machines
it is out of the
new breakdown
repair
stage that
discipline. are
the
goes
of which
operating
to be a customer
whereas
machine
each
machines, These
has at most
is considered
A failed
an operation
times
system
to be repaired
we suppose
a separate
system
and the
are busy,
Moreover,
In addition,
are N identical at any time.
Each machine
waiting
repairman
there
machines
available.
If all
as new.
study,
or productive
it is down
facility.
queueing
under
can be operating
simultaneously
around
stochastic
of the diffusion
by S spare
drawn
is the most
APPLICATION
machines
back
This
2.12.
is a crucial
we want the
2.13
(1981).
as way that
suggested
that
dyl
of an approximate
the discrete
t ) 01 in such
be shown
and A2(x) ---
process
[X(t);
the
condition
for Al(x)
a queueing
it can
of integration.
and Taylor
determination
replace
manipulation,
boundary
in Karlin
dyl
Iofq2#
exp
constant
to the
is.available
~A&Y) --bum
0
345
/ Analysis of the machine repair problem
other
each
a cyclic
circulate stage.
This
346
Hatyono, B. D. Sioarlian / Analysis of the machine repair problem
system always reaches a steady state because no more than N machines exist in the system. By using Kendall's notation this system corresponds to the GI/G/R/N/N/FIFO system. To motivate the diffusion model employed in approximating the GI/G/R/N/N/FIFO system, we first consider an approximation for the M/M/R/N/N/FIFO system.
A scheme suggested by
Bailey (1964) and Heyman (1975) for approximating a stochastic process is used.
In the
M/M/R/N/N/FIFO system, breakdowns occur according to a Poisson procefs with rate X and repair times have the negative exponential distribution with mean lo .
Let A(t) and D(t),
respectively, denote the number of breakdowns and the number of service completions during the interval (O,t].
Then the queue size at time t, which corresponds to the number of
failed machines, is given by N(t) = A(t) - D(t ) It
3.1
is well-known that the stochastic process [N(t); t ) 0] forms a homogeneous birth-
death process as a mathematical model.
In this case, the number of failed machines in the
system corresponds to the state of the birth-death process (discrete state space).
This
system has (N+l) states, n=O, l,Z,...,N. Let us define the transition probability P(n,N-n;t) = Prob[N(t)&lN(t,)=n,]
3.2
(n,no=O,l,...,N)
that the process is in state n at time t starting from state no at time t0.
Furthermore,
let X,,and n,,be the birth rate (the breakdown rate) and the death rate (the repair rate) in state n, respectively.
The parameters 1,'s and n,,'scan take different forms depending on
the values taken by n with respect to S and R as described in what follows: Case A.
When S > R
In this case we have three possible situations for n. 1.
n < R < S or (N-n) > M.
For this condition the number of failed machines in the
system is less than the number of repairmen and also is less than the number of spares. Therefore the operation stage is still fully loaded but some of the repairmen are always idle.
Hence, we have X,=MX 'n
= nu
n = 0,1,2,...,M-1 n = 0,1,2,...,M-1
where A,,and u, represent respectively the mean breakdown rates of the operation stab: :.ad the mean repair rates of the repair stage.
Haryono, R 6 n < S or
2. system the
is greater
number
than
of spares. A,,=
3.
is more
repair
stage
Mh
than
(N-n)
< M.
For this
of spares.
fully
So, the
loaded.
Hence,
n=S+l,S+2,...,N-1 n=S+l,S+2,...,N
3-l.
these
The Mean
conditions
Breakdown
State
R-l
n = 0,1,2,...,
of repairmen
of failed
and also
is equal
in Table
Rates
System
1,'s
when
condition
the
operation
number
stage
of failed
is partially
and the Mean
Repair
Rates
---
n
'n
'n nu
n = R,Rtl,...,S
or
R
MA
R!J
n '= S,S+l,...,N
or
R
by the
*
.
following
(R-1)~
2u (M-l)1
RlJ
diagram
MA
(Figure
3-l.)
MA
RU
b
(M-2)~
4_@st2
.
3-l.
state-transition
MX
RlJ
Figure
(N-n)X
MA
MA
loaded
S > R.
MX
u
machines
~,,'s for
n < R c S
MA
in the
to or less
3-l.
or
is represented
machines
we have
= Ru
summarize
number
than
n = R,R+l,...,S
M/M/R/N/N/FIFO
This
number
the
347
n = R,R+l ,*a*, S
that
remains
to the
condition
repair problem
we have
A, = (N-n)1 57
Table
or equal
/ Analysis of the machine
For this
) M.
Hence,
R < S < n or
system
We can
(N-n)
=Rv
%
B. D. Sivarlian
State-Transition
RlJ
RlJ
Rate-Diagram
RP
for M/M/R/N/N/FIFO
System
when
S ) R.
in the
but the
348
Hatyono, B. D. Siuadian / Analysis of the machine repair problem By using the principle that the time derivative of the probability of a particular
state is equal to the difference between flow rate into and flow rate out of that state (Kleinrock, 1976) we establish the following system of differential-difference equations (forward Kolmogorov equations) which represents the dynamics of the process
dP(O,N;t) = - MxP(0 , N.t)tpP(l,N_l.t) t f 3 dt
--dP(n&AGt
n=O
= - (MXtnu)P(n,N-n;t)t(n+l)pP(n+l,N-n-l;t) t MXP(n-l,N-ntl;t)
3.3
ncR
,
dP(n;:-n;t-). = - (MXtRP)P(n,N-n;t)tRuP(n+l, N-n-l;t)
t MxP(n-l,N-n+l;t)
RtncS
,
dP(n N-n-t) _ --L-L- - ((N-n)XtRP)P(n,N-n;t)+RuP(n+l,N-n-1;t) dt + (N-n+l)xP(n-l,N-n+l;t) ,
R
If we assume that n machines begin to operate simultaneously at time t0, then the initial condition is given by Prob[N(to)=nlN(to)=no~ = 6,, n 0'
no,n=O,l,...,N
where 6 no,n
=
1
n=no
0
n#no
1
and the boundary condition Prob[N(t)=n(N(tO)=nO] = 0
if n < 0,
t>o
The idea of the diffusion approximation is to replace the system of equations 3.3 by a system of partial differential equations that are easier to solve.
This can be accomplished
by replacing the discrete random variable N(t) by the continuous random variable X(t) and P(n, N-n; t) by f(x, N-K; t).
Therefore equations 3.3 become
349
Haryono, B.D. Sivazlian / Analysis of the machine repair problem
x=0
,
= Mhf(O,N;t)+pf(l,N-1;t)
----af(;;N;t)
af(x,N-x;t) = - (MXtxu)f(x,N-x;t)+(x+l)uf(x+l,N-x-1;t) at t MXf(x-l,N-x+l;t)
af(x N-x-t) __LL
x
,
_ - - (MxtRu)f(x,N-x;t)+Rnf(x+l,N-x-l;t)
at
+ MXf(x-l,N-x+l;t) af(x N-x-t) _-L-L
_ - _ {(N-x)ktRu)f(x,N-x;t)+Ruf(x+l,N-x-1;t)
at
t (N-x+l)xf(x-l,N-x+l;t)
If we now expand about the
x and
retain
following
which
the term
only
steady
terms
state
-$
{(MA-xp)f(x))
o=_
-$
{(MA-Rn)f(x)
are
- $
{((N-x)x-RuIf
in fact
We observe
of the of the
equations
O=-
o=
3.4
Rcx
,
(t+m)
d2
hand
side
order where
of equation
(second lim f(x, t+m
f(x))
{VA
order
3.4 in a Taylor approximation),
series
we obtain
N-x; t) = f(x)
Otx
,
f(x)] ,
3.5
R
dx2
(x), + -
of the diffusion
that
right second
d2 (Mx+xu) + -d--z- t-2
I +-
R
,
the equations
d2
Cm*;=
f(x)] ,
type. in 3.5
have
identical
forms
Al(x)
= (MI-xv)
and A2(x)
= (MXtxu)
,
O
Al(x)
= (MI-RP)
and A2(x)
= (MI+RP)
,
R
Al(x)
= {(N-x)X-RuI
and A2(x)
R < S < x
dx2
= {(N-x)~+RuI
,
with
R < S < X
equation
2.11
where
3.6
350
Haryono, We first
consider
a heuristic
interval
(t,t+At],
arrivals
(new breakdowns)
D(t)
the
= n > 0 this
Table
3-2.
State
B. D. Sivazlian
number
change
Expectation
method
of failed
minus
the
/ Analysis of the machine
machines
number
has expectation
and Variance
for obtaining
equation
in the
of repair
repair problem
system
for M/M/R/N/N/FIFO
given
System
During
the time
by the
number
and when
by Table
when
N(t)
of
= A(t)
-
3-2.
S > R.
Variance
Expectation
n
changes
completions
and variance
3.5.
I_n
(MA-nu)At+o(At)
(MX+nu)At+o(At)
R
(MbRp)At+o(
(MA+Rp)At+o(At)
At)
{(N-n)X+RulAt+o(At)
{(N-n)X-RuIAt+o(At)
R
Formally
we have
For n < R < S E fN(t+At)-f44tm_
lim At+0 lim
At+0
o2
(MA-w)
[NetAt)-N(t)(N(t)=n] ._ --
At
= (MX+np
For R c n ( S lim
E
lim
o2
At.+0 At+0
w+At)-N(t)lN(t)=nl
At
L!&tZLL:N~o-=ti
= (MX-mu ) = (MX+R,,)
For R c S < n lim
At+0
Therefore [X(t); equation a good
E LNLt+At)-NCt)INCt~
At
to approximate
t ) 0] with 3.6.
This
approximation
the
same
scheme
= ((N_n)A_Rp)
a stochastic
process
infinitesimal suggests
that
mean
[N(t);
an appropriate
for the GI/G/R/N/N/FIFO
t > 0] by a diffusion
and variance,
system.
we set Al(x)
choice
of Al(x)
process
and A2(x)
as in
and AZ(X) Will yield
Hatyono,
B. D. Sivarlian
For the GI/G/R/N/N/FIFO system, We attempt
complicated. the
succession
random (l-1,
0;
< -).
independently with
drawn
server
time
t is given
selection
form
a general
we assume
that
we have
a sequence
the
parameters
AI(x)
employed.
We first
of independent
distribution
with
succession
351
(mean,
and
times
assume
identically
variance)
of repair
and AZ(X)
given
form
from the
during
the
are
clearly
i-th
interval
=
!
operation
stage
Then
(0,t).
Ai
i=l
interests
the
process
Because buted
during
A(t)
assumption
on each
-
R z
the
and the number
number
of repair
of failed
completions
machines
in the
random
by the
system
at
3.7
Di(t)
i=l
R
(0,t)
other.
Ai
and the departure
In heavy
arrival
theorem
Based
respectively, for A(t)
i=1,2,...,M
we have
theorem
basis
limit
= MX
and
a2[A(t)]
- MX30$,
EM(t)1
s( MXt
and
a2[A(t)]
B MA 32tM
ECA(t)]
- (N-n)At
theorem
D(t)
the mean
theorem and the the
(Gaussian)
of the diffusion independent
E i=l
and
Di (t)
process
and it is this
limiting
normal
=
the departure
we can describe
independent
is the
ED(t)1
and 02[A(t)]
a renewal
i=1,2,..., R are
(renewal
however,
to approximate
on this as two
process
and vice-versa,
to apply
and D(t)
and Di(t),
traffic,
process
can be used
as t+m.
and D(t),
variables,
E i=l
it is reasonable
This
of normality
Ai(
=
of the
Therefore,
and Di(t).
processes
A(t)
independent
us.
to Ai
process
dependent
approximately
distributed by
a sequence
M arrival
that
by
N(t)=A(t)-D(t)
The
is more
distributed random variables drawn from a general distribution -1 let Ai and Di(t), denote the by (P , 0: < -). Furthermore,
given
of arrivals
i-th
the
of the
repair problem
identically
(mean,variance)
number
times
from
Next
and
determination
to justify
of breakdown
variables
/ Analysis of the machine
case
is
which
(Ross,
1983)
variance
of
stochastic processes.
The
approximation. identically
distri-
argument)
O
,
= (N-")x30$,
R
R
and
3.8 ECD(t)]
= npt. and
a2[D(t)]
= nu3$t,
Otn
E[D(t)]
= Rut
and 02[D(t)]
= Ru3d$,
Rtn
ECD(t)]
- Rut
2 and u [D(t)]
a Rp3a2t R'
R
352
Haryono, B. D. Sivarlian / Analysis of the machine repair problem
This can be summarized in the following table (Table 3-3) (where CM and CR are the squared coefficients of variation of the breakdown and repair times, respectively). Table 3-3.
The Mean and the Variance for GI/G/R/N/N/FIFO System if t Becomes Large
and S > R where
CM = X2$
and CR = u*oi. ----_-----
State
W(t)1
n
CD(t)1
~*CNW
o*CWl
-
n
MXt
nut
M XCMt
n pCRt
R
MAt
Rut
M $t
R vCRt
R
(N-n)xt
Rut
(N-n)XCMt
R uCRt
Formally A(t)
D t+_-->
N(Mxt,
A(t)
--$;->
N(MXt, MXCMt),
R
A(t)
---&;-->
N( (N-n)xt,
R
D(t)
---c:_->
D(t)
+&-->
N(Rut,
RuCRt),
R
D(t)
+&->
N(Rut,
RuCRt),
R
MXCMt),
Otn
(N-n)ACMt),
and
where
D I-----> t+=J
denotes
It is well-known processes,
say A(t)
distributed
process
combination expressed
O
N(npt, nuCRt),
convergence
that
if we have
and D(t) with
some
we are interested as the
number
in distribution
then
two
independent
any linear
mean
in is N(t)
= A(t)
machines
normally
combination
appropriate
of failed
if t+m.
of these
and variance. - D(t)
in the
distributed
is also
‘Of course,
> 0 which
system
two
at time
stochastic
represents t.
a normally
one linear the backlog
Therefore,
we have
Haryono, B. D. Sivazlian
353
/ Analysis of the machine repair problem
N(t) = A(t)-D(t) -+-
> N((Mx-nu)t, (MxCM+nuCR)t.) ,
0 < n < R < S
N(t) = A(t)-D(t) +&-
> N((Mx-Ru)t, (MxCM+RuCR)t) , R < n < S
N(t) = A(t)-D(t) -+-
> N(((N-n)X-Rv)t, ((N-n)xCf,i+RuCR)t) ,
3.9
R < S < n
or we can write for the steady state approximation results as:
;z
ECN;t)l -
12
ECNit)I
-
LE
%!!lu
= ((N_n)l_Rp)
(MA+,)
and
(MA-R,,)
and
;E
2k.f
iz
w
and 12
= (MXM+npCR)
,
=
, R < n
(MKM+RpCR)
0 < n < R < S
u*CN~t)3 = ((N-n)xCM+RvCR)
3.10
,R
This suggests for continuous approximation to the GI/G/R/N/N/FIFO systern the following diffusion parameters Am
= (MA-xv) and $(x)
Am
= (MA-Rv) and A*(x) = (MlCM+RuCR),
Al(x) = ((N-x)1-Ru)
= (Mx$+x?~cR)
O
,
and A*(x) = ((N-x)XCM+RvCR),
By using Al(x) and A*(x) in equation
3.11
RcxGS
R < S < x
3.5 we construct the steady state equations for
the probability density function f(x) of x (the number of failed machines in the system in the steady state)
o= -
-$
{(MA-xv)f(x)) + 2
do
( MXM+XUCR)
I---2-----
f(x)1
o=
d2 ( MKM+RuCR) - -$ ((MA-Ru)f(x)I + --2 +--~2f(x))
o=
- --$ {((N-x)&Ru)f(x))
+ 2
d2
((N-x)EM+kCR) 1--___-- 2
,
,
O
Rtx
f(x))
3.12
,
R
Haryono, B. D. Sivazlian
354
/ Analysis of the machine repair problem
The respective solutions of 3.12 are given by kl
Otx
exp ( I 0
f(x) = -(Mm
e
= kl (MACM+xpCR)
,
3.13
O
f(x) - klhl(x) ,
' 2(Mx-Ru) dy) , exp rol MxCM+RpCR
E(Mx-Rv)xl
= k2 exp ~~~~ +R;~ M
f(x) e k2h2(xL
R
RtxtS
R
,
3.14
R
dy,
f(x) = 2iR!JCR [--+ 211R_ (XM)2 "M =
Ocx
f(x) 5 k3h3 lx),
2x
11
5 e
k3{(N-X)~CM+RL$~
RcS(x
,
R
3.15
R
Thus, we have three probability density functions forO
3.16
for R < S < x kl, k2 and kg remain undetermined requiring that three conditions be given for their specification.
There are three constants to be determined since each member function of the com-
position solution in equation 3.16 contributes one undetermined constant of integration.
Haryono, B. D. Sivarlian
355
/ Analysis of the machine repair problem
The first condition comes from the obvious normalization criterion and takes the form kl
0
R / hl(x)dx + k2
R
S I h2(x)dx + k3
N I s
3.17
h3(x)dx = 1
To determine the other conditions, we observe that the probability density function in equation 3.16 is continuous (Halachamy and Franta, 1978) at x=R and x=S.
Therefore
klhl(R) = k2h2(R) 3.18 k2h2(S) = k3h3(S) By using the conditions in equations 3.17 and 3.18 we can determine the value of kl,k2 and k3 as follows klhl(R) = k2h2(R)
hl(R)kl + k2 = h2(~~
kl(M$,+bCR) k2 = ~__-----_-----_______-
exp
3.19
@_(MkVRl)
MXM+RpCR
k2 z kill, where ll is already known from the problem. klhl(R)h2(S) k3 =---_---_-_=
h2(fW3W
k 1
Similarly we have
h2(S)
1 1 h3W
k3 I klll12, where l2 is already known from the problem.
Therefore, by using +.diion
kl can be determined, as well as k2 and kg, that is
k1 or
0
N S R / hl(x)dx + kill RI h2(x)dx + klll12 sI h3(x)dx = 1
3.17
356
Haryono, B. D. Sivazlian S
2 O’Q-RFl)x) dx
’ exp (MXC+RuC R M R
klll
/ Analysis of the machine repair problem
+
2xRuCR C----~-+-;gL 2" {(N-x)U+,+RuCRI tAc,)
klll12 S
11 M
2x T dx eM
=l
The probability of finding n failed machines, P(n), in the system in the steady state can be obtained by discretizing f(x) (Heyman and Sobel, 1982) as i(n) =
nt0.5 / f(x)dx n-O.5
Therefore the approximate formula, L, for the mean number of failed machines in the system can be obtained as L =
N E nP1 (n) n=O
3.21
Hence the approximate formula for the utilization of the repair stage is 3.22
U=L/R When R > S
Case B.
Similar to condition A there are three possibilities for n. 1. 'n
n < S < R or (N-n) ) M.
For this condition the mean breakdown rate
and the mean repair rate u,,are the same as with Case A.l. An = MX
n = 0,1,2,...,S
= nv
n = 0,1,2,...,S
Pn 2.
S < n < R or (N-n) < M.
Hence
For this condition the number of failed machines is more
than the number of spares.
So, the operation stage is partially loaded and some of the
repairmen are always idle.
Hence
X,,= (N-n)1 % 3.
n = Stl,
St2,
. . . . R-l
n = Stl,
St2,
. . . . R-l
= nn
S < R c n or (N-n) < M.
than that of spares. remains fully loaded.
For this condition the number of failed machines is more
So, the operation stage is partially loaded but the repair stage Hence
Haryono, B.D. Sivazlian hn = (N-n)1
n = R, R+l,
. . . . N-l
= RP
n = R, R+l,
.... N
'n
The
corresponding
.
.
state
transition
The
3-2.
steady
of failed
0 = z
state
diffusion
machines
d
d
Thus,
for
the
the
solution
Rate-Diagram
equations
in the
{(MA-xu)f(x)l
system)
d2 + dx2
)A-xl.df(x)l
[{(N-x )I-RpIf(x)l
We observe row.
becomes
that
in this
condition
d2 + --2-
by
be shown
3.23
[
System
density
to be in a way
when
function
similar
R > S.
of x (the number
to Case
A
Otx
(x)1,
I(N-x)XCM+xuCRI
2 I__
f(x)19
I(N-x)K~+RucRI [----- p-2
are the
for the condition
S < R G x the
is given
probability
MXM+x~CR I---f 2
d2
equations case
for M/M/R/N/N/FIFO
for the
may
+ --F-
RU
b
RU
State-Transition
d 0 = xx- [((N-x
0 = z
diagram
357
.
RU
Figure
/ Analysis of the machine repair problem
solution
same
f(x)],
as equations
0 G x < S < R the 3.15
remains
valid.
S
3.23
S < R < x
3.12
except
solution For the
3.13
in the remains
conditions
second valid
and
S < x < R
358
Haryono, B.D. Sivazlian / Analysis of the machine repair problem
f(x)
=
k4 (N-x)XC,,,+x"CR
R
2( x+u)Nx$
2NX
c
dx)
-11
= k4 {(N-x)XM+x~CR)
we have
f(x)
where
=
3.24
.S
three
probability
density
functions.
klhlb)
forO
k4h4(x)
for S ( x < R
k3h3(x)
for S < R < x
I
kl, k4 and
2(x+u)x $-pcR
e
f(x) = k4h4(x) Therefore
~
- -I=
kg remain
Similar
undetermined.
3.25
to Case
A, we have
the
following
conditions
kl
Is h+x)dx
+ k4
0
,
and
R / h4(x)dx
IN h3(x)dx
3.26
= 1
R
kqhq(S)
klhl(S)
=
k4h4(R)
= k3h3(R)
By using
t k3
S
condition
in equation
3.27
3.26
and
3.27 we can determine
the
values
of kl, k2 and
kg as follows
klhlW
hlW = k4h4(S) + k4 = fia7srkl
-11 k4 =
-$$ e
2Nl ($-VCR)
2(x+lJ)S QUCR
-11 e
3.28
Hatyono, B. D. Sivarlian k4 = kl14,
where
14 is already
known
hl(S)h4(Wkl k3 =
h4(S)h3(R)
Similarly
problem.
we have
h4W
c =
for the
__-_ h3(R)
= klk4
kl141(N-R)lCM
359
/ Analysis of the machine repair problem
+ RuCRl
Z(x+p)AN$,
2NX
-
(q&)*
-($-VCR)
2(X+n)R -_--
-I’
$,,-$ e
3.29
e
k3 z k11413, can
where
l3 is already
kl, as well
determine
known
as k4 and
S I k1
The
hl(x)dx
by using
equation
3.26
we
N + k11413
RI
h3(x)dx
= 1
S
probability
obtained
Therefore
problem.
k3; we obtain
R I h4(x)dx
f k114
0
for the
of finding
by discretizing
n failed f(x)
machines,
P(n),
in the
system
in the
steady
state
can be
as
nt0.5 i(n)
= I
f(x)
dx
n-O.5
Therefore is given
the
the
formula,
L, for the mean
number
of failed
machines
in the
system
by
L =
Hence
aporoximate
"c n;(n) n=O
approximate
U = L/R
3.30
formula
for the
utilization
of the
repair
stage
is
360
Haryono, B. D. Sivarlian / Analysis of the machine repair problem
Case .-
C.
S=O
This
problem
available.
stage
O
of the machine
are two
possibilities
condition
of failed
loaded.
the
machines
= nu
n =
O,l,Z,...,R-1
n > R.
is partially
For this
condition
of repairmen
loaded.
the
and thus
the
n=R
,...,M-I
= RP
n=R
,a**, M
corresponding
MA
3-3.
problem
where
no spares
state-transition
are
for n: of machines
is less
than
the
operating number
in the
operation
of repairmen.
number repair
of failed facility
machines is loaded
in the
diagram
(M-1)x
State-Transition Rate-Diagram When S=O and R < M.
is
(M-R+2)X
(M-R+l)X
for M/M/R/N/N/FIFO
System
system
but the
Hence
X,.,= (M-n)X 'n
number
repair
Thus,
Hence
O,l,Z,...,R-1
the number
Figure
number
are partially
case
and R < M
n =
2.
The
there
Problem)
An = (M-n)X %
than
case
For this
is M-n and the stages
Interference
is a special
In this
1.
both
(Machine
(M-R)X
is more
operation
stage
-Haryono, B. D. Sivazlian
361
/ Analysis of the machine repair problem
Similarly, it may be shown that for 0 < x < R
k4h4(x)
3.31
f(x) = I
for R < x < M
k3h3(x)
and k4 and k3 can be determined by the following conditions R k4
0
I
h4
(x)dx + k3
R
M I h3(x)dx = 1
3.32
and k4h4(R) =
3.33
k3h3(R)
By using equations 3.32 and 3.33 the value of k4 and k3 can be determined as follows
2(A+p)MXC c_-----_-+
_
( XM-UCR)L k3=--
k4f( M-R)XM+RuCRl ______________ 2RUCR C---t2vR (XM)2
2MA - 11 xcM-lJcR e
-____.___--._-.__-ll__
ACM
-11
{(M-R)$,+RuCR)
e
R h4(x)dx + k414
3. 34
@M
k3 E k414, where l4 is already known for the problem.
k4 o1
2(X_u)R EM-n_
R
Therefore
M I h3(x)dx = 1
The probability of finding n failed machines, P(n), in the system in the steady state condition can be obtained by discretizing f(x) as . P(n) =
nt0.5 / n-O.5
f(x)dx
Therefore, the approximate formula, L, for the mean number of failed machines in the system is
362
Haryono, B. D. Sivazlian / Analysis of the machine repair problem
L=
Hence
the
M Z n;(n) n=O
approximate
3.35
formula
for the
utilization
of the
repair
stage
is
U = L/R
Numerical
3.36
Examples
In Tables
3-4 and 3-5,
the M/M/R/N/N/FIFO
the
utilization
Table
3-4.
system factor
comparisons
and the
is obtained
of this
approximation
E2/E2/R/N/N/FIFO under
heavy
system
traffic
with
showed
a simulation
that
a close
condition.
Comparisons of Diffusion Approximation and Simulat ion Resu 1ts for M/M/R/N/N/FIFO System U where SIM = Simulation Results and DIF = Diffusion Results. M=6
R
4
6
vu
s=o
s=2
s=5
S=J
0.5
SIM DIF
0.505 0.587
0.614 0.687
0.707 0.801
0.712 0.796
0.6
SIM DIF
0.558 0.561
0.699 0.704
0.792 0.798
0.840 0.851
0.7
SIM DIF
0.614 0.619
0.771 0.779
0.887 0.894
0.897 0.911
0.5
SIM DIF
0.323 0.381
0.433 0.527
0.464 0.516
0.483 0.531
0.6
SIM DIF
0.373 0.378
0.478 0.484
0.545 0.551
0.563 0.568
0.7
SIM DIF
0.415 0.421
0.534 0.541
0.623 0.637
0.645 0.649
U = utilization
of the
R = number
of repair
S = number
of spares
repair
facilities
stage
model agreement
for for
Haryono, B. D. Sivazlian Table
3-5
363
/ Analysis of the machine repair problem
Comparisons of Diffusion Approximation and Simulation Results for E2/E2/R/N/N/FIFO System U where SIM = Simulation Results and DIF = Diffusion Results.
M=6
R
x/u
0.5
4
0.6 0.7
6
s=o
s=2
s=5
s=7
SIM
0.492
0.617
0.728
0.712
DIF
0.541
0.678
0.806
0.784
SIM
0.545
0.709
0.846
0.874
DIF
0.551
0.717
0.851
0.877
SIM
0.623
0.782
0.894
DIF
0.631
0.789
0.898
0.918 0.923
0.5
SIM DIF
0.338 0.393
0.428 0.471
0.495 0.537
0.482 0.526
0.6
SIM DIF
0.370 0.378
0.484 0.492
0.565 0.573
0.578 0.584
0.7
SIM DIF
0.497 0.412
0.566 0.520
0.668 0.679
0.684 0.669
u=
utilization
R=
number
of repair
s=
number
of spares
of the
repair
stage
facilities
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Haryono, B. D. Sivazlian / Analysis of the machine repair problem
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