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A Coal Unloader: A Finite Queueing System with Breakdowns
Author(s): Kenneth Chelst, Andrea Zundell Tilles and J. S. PipisSource: Interfaces, Vol. 11, No. 5 (Oct., 1981), pp. 12-25Published by: INFORMSStable URL: http://www.jstor.org/stable/25060139.
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8/9/2019 Chelst Et Al 1981 a Coal Unloader a Finite Queueing System With Breakdowns
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INTERFACES
Copyright ?
1981,
The
institute
of
Management
Sciences
Vol.
11,
No.
5,
October
1981
0092-2102/81/1105/0012$01.25
A
COAL
UNLOADER: A
FINITE
QUEUEING
SYSTEMWITH BREAKDOWNS*
Kenneth
Chelst
Department
of
Industrial
Engineering
and
Operations
Research,
Wayne
State
University,
Detroit,
Michigan
48202
Andrea Zundell
Tilles
Control
Data
Corporation,
Rock
ville,
Maryland
20850
and
J.
S.
Pipis
Detroit
Edison,
2000
Second
Avenue,
Detroit,
Michigan
48226
Abstract.
The
Detroit
Edison
Company
owns
and
operates
the
coal-fired Monroe
Power
Plant.
Coal
is
generally
brought
by
train
to
the
plant
from
mines in
nearby
states
and
is
unloaded
by
a
single
unloader
system.
There
were
difficulties
in
meeting
the
plant's
coal
needs with the
existing
rail
transport
system.
Management
observed
frequent
queues
of
trains
at the
unloader
system
which
they
attributed
to
breakdowns
of the
unloader
system.
A
queueing
model
was
developed
to
explore
the
impact
on
the
system
of unloader break
downs and the
potential
benefits associated with
adding
a second unloader
system.
The
model
was
also
used
to
study
the
relationship
between
the number of
trains,
coal
throughput,
and
queueing delays.
The Monroe Power Plant is
a
3,000-megawatt
facility
that
requires
approxi
mately
6.5
million
tons
of
coal
annually.
In
addition
to
being
one
of
the
world's
largest
coal-fired
plants,
it is
also
one
of
the first of
its
size
to
utilize
a
unit
train
to
supply
its
fuel.
At
present
there
are
between
four
and
eight
trains
allocated
to
moving
coal
from the mines
to
the
power
plant,
which
has
one
unloader
system
to
dump
the
coal. Originally, coal was to be brought into the plant entirely by rail. However, as
the
plant
increased its
generating capacity,
the
existing
rail
system
became
insuffi
cient. The
more
recent
plan
has
been
to
combine
rail
and vessel
delivery
of
coal
to
satisfy
the
plant's requirements.
Several
factors
are
believed
to
have
contributed
to
the
rail shortfall.
The
major
contributor
is
believed
to
be
the
designed single-car
unloader. Trains
were
frequently
queued
at
the
unloader
system,
which
management
attributed
to
unloader
break
downs. These
delays
were
costly
for
a
number
of
reasons:
Trains
waiting
to
be
unloaded
are
not
transporting
coal,
which results
in
a
decreased
throughput.
More
expensive backup
vessels must then be used to
satisfy
coal
requirements
at
an
added
cost
of
$30,000
to
$60,000
per
equiva
lent trainload.
If the unloader is
broken
for
long
periods
of
time,
the number of
queued
trains
may
exceed
local
holding-track
capacity.
These
trains
will
then
be
held
at
Toledo and
a
demurrage
cost
is
then incurred.
QUEUES?APPLICATIONS;
TRANSPORTATION
EQUIPMENT
*This
paper
was
refereed,
as
requested
by
the first
author.
12
INTERFACES
October
1981
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It
is
thought
that
trains
waiting
in
the winter
months
contribute
significantly
to
the
problem
of frozen
coal
and its
associated
added
costs.
To alleviate these
problems,
Detroit
Edison
is
considering
the addition
of
a
second multimillion dollar unloader
system.
Although
a
simulation
model of
the
entire
rail
coal
movement
system
had
already
been
built for other
purposes,
it
was
decided
to
build
an
analytic
model
which
focused
on
the
unloader
in
order
to
isolate
its
effect.
The
specific
parameters
the model
was
to
explore
were:
the
impact
of
a
second
unloader
system
on
the
coal
throughput;
L,
the
average
number
of
trains
in
queue;
and
W,
the
average
time
spent
in the
unloader
system.
In
addition,
the
impact
of
other
changes
were
studied,
including
increasing
the
number of
trains,
reducing
the
frequency
of
unloader
breakdowns,
reducing
the
repair
time,
and
changing
the
cycle
time between mine
and
power
plant (puchasing
coal
from different mine
fields).
A
schematic diagram of the unloader system is displayed in Figure 1.
FIGURE
1.
THE
COAL SUPPLY SYSTEM.
Literature
Review
The
nature
of
the unloader
system
suggested
a
queueing
model
subject
to
service
interruptions.
The
earliest
work
in
this
area
[White
and
Christie, 1958]
viewed
the
service
interruption
as
a
high
priority
class of
customer
which
preempts
the service
of
the
primary
customer,
which
in
this
case
is
a
loaded
train.
Our
problem
involved
a
finite
source
of
customers
(i.e.,
a
limited number
of
trains),
and the
primary
statistic
of
concern
is
the
average
arrival
rate
of trains
to
the
unloader
system,
which
is
equivalent
to
the coal
throughput.
This
statistic is
not
addressed
in
the
priority
queueing
literature.
Because
of
the
limited
number
of
trains
involved,
we
could
analyze
this
problem
using only
basic
queueing
theory
at
the level
of
Hillier
and
Lieberman's
text
on
Operations
Research
[1979].
INTERFACES ctober 1981
13
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MODEL
AND
ASSUMPTIONS
In
beginning
the
model
development,
we
adopted
an
approach
set
forth
by
Morris
[1967]
in
his
article "The
Art
of
Modeling."
To
make
the
problem
tractable
we
assumed
an
exponential
probability density
function
for four
time
components
of
the
system's operation:
Unloading
time
Repair
time
Cycle
time
Failure time
the time
to
unload
a
train
the time
to
repair
a
broken
unloader
the
time it
takes
for
an
unloaded train
to
go
to
the coal fields and
return with
a
trainload of coal
the
time
between breakdowns when the
un
loader
is
operating continuously.
Though the exponential assumption was motivated by tractability, 1977 data on 15
breakdowns showed
that
the
exponential
distribution
was a
reasonable
approximation
for the
repair
time
(Figure
2)
and
failure time.
Cycle
time
we
knew could
not
follow
an
exponential
distribution,
since
there
was an
obvious
minimum time
for
an
un
loaded
train
to
go
to
the
coal
fields
and
return
with
a
trainload
of coal.
FIGURE
2.
REPAIR
TIME:
A
COMPARISON
OF
EMPIRICAL
DATA
AND
THE
EXPONENTIALDENSITY.
>
i?
CO
as
LU
C3
\
\
\*^
EXP(-T/2.8)
2.8
1977
DATA
N^l
o
3
15
6
REPAIRIME N
DAYS
14
INTERFACES
October
1981
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Additional
assumptions
about
system
operation
follow:
Coal
availability
does
not
affect
the train
cycle
rate.
All
trains
are
the
same
size.
A
train
can
wait
at
the
facility
while
an
unloader
is
being
repaired
(this
assumption only affects the demurrage cost).
When
two
unloaders
are
broken,
two
crews
work
independently
on
them.
When
one
unloader is
broken,
only
one crew
repairs
it.
A
train that
is
in
a
facility
when
one
unloader breaks down
is rerouted
to
the
other
unloader
facility.
(The average
repair
time of
the unloader
is substan
tially
greater
than the time
necessary
to
reroute
the
train.)
The
queueing
model
we
built,
as
mentioned
earlier,
was
a
modified
version of
a
standard
single
and
multiple
server
finite
source
queueing
model which
allowed for
server
breakdowns. The
notation
we
use
in
describing
our
model is
as
follows:
Xj
is the
arrival
rate
of
an
individual
train
when
not
in
queue.
(Equivalently,
1/X2
is the
mean
cycle
time
of
the
train.)
pi1
is
the
rate
at
which
a
train is unloaded.
k2
is
the
rate
at
which
unloader
breakdowns
occur
when
an
unloader
system
is
operating.
fjL2
is
the
rate
at
which
an
unloader is
repaired
(1/ju,2
is
the
mean
repair
time).
K
is the
total
number
of trains in
the
system.
In
order
to
describe the
system's
state,
we
need
to
specify
both
i,
the number
of
broken unloaders
andy,
the number
of
trains
queued
at
Monroe.
The
probability
of
being
in
a
particular
state
(i,j)
is written
as
Pitj.
In
Figures
3
and
4
we
present
state
transition
rate
diagrams
for
the
one-unloader
and
two-unloader
systems,
respectively.
If the
top
row
of
each
figure
were
isolated,
we
would
simply
have
a
standard
finite
source
queueing
model
[Hillier
and Lieber
man,
1979].
When,
for
example,
we are
in
state
(0,2),
two
trains
are
in
queue
and
new
trains
arrive
at
a
rate
of
(K?
2)X.2.
Trains
are
unloaded
at
a
rate
of
/?j
for the
single
unloader
system
(Figure
3);
transitions
occur
from
the
top
row
to
the second
row
whenever
the
unloader breaks
down,
which
occurs
at
a
rate
of
X2.
Within
the
second
row,
transitions
are
made
to
a
higher
state with the
arrival of
a
new
train
at
a
rate
of
(K?j)\j.
No
transitions
to
a
lower
state
can
occur,
since
no
train is unloaded
when the
single
unloader
is
broken.
FIGURE
3.
TRANSITION
RATE DIAGRAM
OF
SINGLE-UNLOADER
QUEUEING
SYSTEM.
INTERFACES
ctober
1981
15
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FIGURE
4.
TRANSITION
RATE
DIAGRAM
OF
TWO-UNLOADER
QUEUEING
SYSTEM.
Our discussion of
the transition
rates
within
row
1
and between
rows
1 and
2
applies
similarly
to
Figure
4
of
the two-unloader
system
with
two
minor
modifications. Whenever there are two or more trains
being
unloaded,
service is
completed
(i.e.,
downward
transitions)
at
a
rate
of
2pl9
and breakdowns
occur
at
a
rate
of
2X2
Row
2
ismodified
to
allow for
completion
of
unloading
a
train
at
a
rate
of
P
because
coal
can
still be
unloaded
when
only
one
unloader
is
broken.
In
addition,
we
need
to
add
state
(1,0),
since it is
now
possible
to
have
one
unloader
broken
and
no
trains
waiting
to
be
unloaded,
which
was
not
possible
in the
previous
model.
Lastly,
we
must
now
add
another
row
of
states to
allow
for
two
broken unloaders.
Breakdown
transitions
from
row
2
to
3
occur
at
a
rate
of
\2
>
while
repair
transitions
from
row
3
to
2
occur
at
a
rate
of
2p2-
Transitions within
row
3 result
from
new
train
arrivals. In
steady
state these models can be
represented
by
systems
of difference
equations
which
equate
the
rate
into
and
out
of each
state
(available
upon
request
from
the
authors).
For the
one-unloader
model
involving
K
trains,
the model results
in 2^+1
simultaneous
equations
with 2K+
1
unknowns.
Any
one
of these
equations
is redun
dant and
must
be
replaced
by
an
equation
that
sets
the
sum
of all
of
the
state
probabilities equal
to
one.
The two-unloader model
involves 3K+
2
equations
and
an
equal
number
of
unknowns. These
equations
were
rewritten
so
that
the
right-hand
side
contained all
zeroes
except
for
the last
equation,
whose
right-hand
side
was one.
A standard
program
to invert amatrix and which is available on
any
computer
could
have been
used
to
solve
these
equations.
Because
of
the
special
structure
of
the
right-hand
side,
the
entire
solution
of these
equations
(AX
=
b)
is contained
in
only
one
column
of
the inverse
of
the
A
matrix.
We
therefore
wrote
our own
simple
program
to
solve this
problem,
which
allowed
us
to
perform
basic
sensitivity
analysis
with little
added
cost.
16
INTERFACES
October
1981
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Once all of
the
Pu
are
calculated,
it
is
a
straightforward
task
to convert
these
into
more
meaningful
performance
measures.
The
following
statistics
were
calcu
lated for
both
unloader
systems;
we
present
the
appropriate
formulas
only
for the
simpler
one-unloader
model.
The
average
number
of trains
at
Monroe,
L,
is
given
by
K l
L=
1
j
I
Pu.
j
=
\ i=0
The fraction of
time the
unloader
is
idle
is
P0,o>
The
fraction
of
time
the
unloader
is
broken
is
K
I
Pu
The fraction of time the unloader is busy is
K
I
Poj
7
=
1
The
probability
of
a
queue
of
trains
is
1
-
P0,o
-
(Po,i
+
Put)
The
key
statistic
is
the
average
coal
throughput,
which
equals
the
average
arrival
rate:
r
*_1
i
*1
=
L
2
i*"")
-
8/9/2019 Chelst Et Al 1981 a Coal Unloader a Finite Queueing System With Breakdowns
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trains in the
system.
Three
plots
were
generated:
(1)
single
unloader;
(2)
two
un
loaders,
operating
one
at
a
time;
(3)
two
unloaders
operating
simultaneously.
The
queueing
models for
curves
1
and
3
were
described
earlier,
while the model
for
curve
2
requires only
a
slight
modification
of
these models.
From
Figure
5
we see
that
five
trains and
a
single
unloader
can
throughput
322
trainloads.
The
two
unloaders
can
throughput
355 trainloads
if
only
one
operates
at
a
time,
and 370
trainloads
if
both
operate
simultaneously.
FIGURE 5.
RELATIONSHIP
BETWEEN THE
NUMBER
OF TRAINS
AND
THE
ANNUAL
COAL
THROUGHPUT:
ONE- VS
TWO-UNLOADER
SYSTEMS.
600r
TWO
UNLOADERS-BOTH
MAY
OPERATESIMULTANEOUSLY
TWO
UNLOADERS-ONE
OPERATING T
A
TIME
ONE
UNLOADER
TRAINS
IN
SYSTEM
We
can
also
look
horizontally
at
the
curves
to
determine
how
many
additional
trains
are
required
for the
single-unloader
system
to
maintain
a
specified
level of
throughput.
To
maintain
an
annual 400 trainload
throughput,
the
single
unloader
requires
an
additional
two-thirds
of
a
train
or
approximately
one
additional
train
18
INTERFACES
October 1981
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operating
two-thirds
of
the
time
(these
two
things
are
not
equivalent)
when
compared
to
the
two-unloader
system.
As
the
throughput
required
increases
to
475
or
520
trainloads,
the
single
unloader
requires
one
and
one
and
a
half
additional
trains,
respectively.
Clearly,
the
difference
between the
two
systems
grows
as
the demand
increases
and
diminishing
return
from
each
train
sets
in
more
quickly
for
the
single
unloader system. The
eighth
train
only
adds about 50 trainloads to the total
throughput
for
the
single
unloader,
while
it
adds 60 trainloads
to
the total
throughput
for the dual
system.
Remember,
each train is
potentially
capable
of
carrying
74.6
trainloads
per
year.
These
differences become
especially
significant
if,
for
example,
after
a
coal
strike,
abnormally high
amounts
of coal
are
needed
in
a
relatively
short
time.
The
single
unloader
may
need
as
many
as
five
or
six additional
trains
to
bring
in
the
same
amount
of
coal.
The
average
wait
to
begin
unloading,
Wq,
is
even
more
dramatically
affected
by
the addition
of
a
second
unloader.
With
four
trains
in
operation,
adding
a
second
unloader reduces the wait from 6.7 hours to 0.1 hours
(see
Figure
6).
With
eight
trains,
the wait
is reduced
from
20.5
hours
to
1.6
hours.
These differences
may
be
critical when conditions
cause
coal
to
freeze.
FIGURE 6.
THE RELATIONSHIPBETWEEN
THE
NUMBER OF
TRAINS
AND
THE
AVERAGE
DELAY IN
QUEUE:
ONE-
VS
TWO-UNLOADER
SYSTEMS.
25 r
20
3
O
X 15
I
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10