chelst et al 1981 a coal unloader a finite queueing system with breakdowns

8/9/2019 Chelst Et Al 1981 a Coal Unloader a Finite Queueing System With Breakdowns

1/15

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.

Accessed: 04/01/2015 14:57

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at.http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

INFORMSis collaborating with JSTOR to digitize, preserve and extend access to Interfaces.

http://www.jstor.org

This content downloaded from 192.245.60.72 on Sun, 4 Jan 2015 14:57:41 PMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/action/showPublisher?publisherCode=informshttp://www.jstor.org/stable/25060139?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/25060139?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=informs


2/15

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

This content downloaded from 192.245.60.72 on Sun, 4 Jan 2015 14:57:41 PMAll use subject toJSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp


3/15

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



4/15

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



5/15

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



6/15

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



7/15

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*"")


8/15

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



9/15

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

?

10

chelst et al 1981 a coal unloader a finite queueing system with breakdowns

Documents