7«
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A Model
of
for the Configuration
Incoming
WATS Lines
by
Roger H. Blake
Stephen C. Graves
P. Clark Santos
WP #3134-90-MSA
March 1990
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
m r^^'^i)
^_ l,.?;'5®;^/f,^>:
•^^
A Model
for the Configuration
of Incoming
WATS Lines
by
Roger H. Blake
Stephen C. Graves
P. Clark Santos
WP #3134-90-MSA
March 1990
V--
A Model
for the Configuration of
Incoming
WATS
Lines
Roger H. Blake*
WearGuard
Director of Decision Support Systems
Longwater Drive
Norwell,
MA 02061
Stephen C. Graves
Massachusetts Institute of Technology
Sloan School of Management
Room E53-390
02139
Cambridge,
MA
P.
Clark Santos
currently
employed by
AT&T
Consumer Products
Woodhollow Road
5
Room
2J41
Parsippany, NJ 07054
January 1990
Abstract
WearGuard
is
a direct marketer
cloths, v^hich relies primarily
and
retailer of
on phone orders
for sales.
uniforms and work
For this purpose it
"800-number" lines, known as WATS lines, to
receive its incoming calls. These lines are of several types, where each type
serves a different portion of the country and has a different usage fee. In this
paper we determine how many of each type of WATS lines should be
employed. After defining the problem more completely, we develop a
queueing model to describe the system and a dynamic program to solve the
configuration problem to optimality. The model has been applied to the
problem by WearGuard since 1984. We present an example and examine the
maintains a series of
toll-free
We
sensitivity of the solution to variations in various parameters.
the
model by comparing
the results of this
model
to other
validate
approximate
models.
Key words: queueing
application, overflow, queues,
WATS
lines.
*As of 1990, Manager of Strategic Services, Andersen Consulting, Boston,
MA
BACKGROUND AND INTRODUCTION
1.
In this paper
we
describe an application of queueing theory to the
WATS
problem of determining the number of
We
operation.
for a
lines
phone-order
develop>ed the model in 1984 as part of the thesis project for
one of us (Santos
[5]),
where the application
WearGuard served
to
as the
primary motivation. Since then, WearGuard has periodically used the model
and
to evaluate
(re)
number and type of WATS lines in its
Typically, the model is used once or twice a year,
configure the
telemarketing operations.
by increases
triggered
due
Recently,
lines)
by
in call rates
and by changes
to the introduction of a
new
in the cost structure for calls.
type of service (unbanded T-1
long-distance carrier, the crux of the problem has been eliminated
its
for
WearGuard. Nevertheless, the general structure of the problem remains
for
many
telemarketing operations.
WATS
(a)
which
is
lines
.
AT&T
Communications
AT&T
the division within
is
responsible for long distance services. In addition to the typical long
distance service that most private customers use,
AT&T
Communications
(as
Wide Area Telephone
used by many industries. One type of WATS
well as the other major long distance carriers) offers
Services
(WATS) which
are
to the caller, so-called toll-free "800"
free
is
anywhere within
no
cost
numbers. There are seven types of
toll-
WATS
line
service allows a firm to receive calls from
a region at
numbers, which depend upon the portion of the country the
For WearGuard, whose telemarketing base
to serve.
the country
instance.
Jersey,
broken up into
is
Zone
and
all
of
I
six
zones plus the state of Massachusetts.
consists of Eastern Pennsylvania, Eastern
New
England except Massachusetts; Zone
New
II is
York,
York, Virginia, and West Virginia. Figure
can be used for
line
calls originating
from Zones
can handle
from Zone
calls
I
1
shows the
calls originating
I
and H,
etc.
through Zone
There
V
five zones.
from Zone
is
also a
I,
a
For
New
Delaware, the
Columbia, Ohio, Maryland, Western Pennsylvania, Western
District of
WATS
in Massachusetts,
is
Band
Band VI
A
II
New
Band
I
line for
line
which
plus Alaska and Hawaii, and an
additional service (Band IX) for calls coming only from the local state
(Massachusetts). For the purpose of this study, however, these
two bands
will
not be considered because the service of in-state calls represents a separate
problem and the number of
justify the
calls that
come from Alaska and Hawaii do not
expense of maintaining the service to these states for WearGuard.
1
^^^^Ve',vi?.\II^^J^^
SERVICE AREAS:
^^° °^SIDE WATS
MASSACHUSETTS
Nevertheless,
whereas the
it
would be very easy
intrastate lines.
In 1984 the cost of a
Band
IX,
WATS
to
extend the model to include Band VI,
must be
line consisted of a
upon
$36.80 and an usage fee which depended
usage of the
1).
lines,
For example,
treated separately.
the band, the total
and the time of day during which
a user acquired a
if
Band
I
line
monthly access
calls are
and used
it
made
fee of
monthly
(see Table
for fourteen
hours
over the course of a month, he would be charged an access fee of $36.80 and
an usage
fee,
which depends on when the fourteen hours were incurred. The
usage charge would be $17.93 for each hour the phone was used during the
business day, $12.51 for each hour the phone was used during the evening,
and $8.54
for each
hour of night and weekend usage.
used the phone for
a total of sixteen hours,
If,
on the other hand, he
he would be charged $36.80 plus
$16.36 for each hour of usage during the day, $11.79 for each hour in the
evening, and $8.54 for each hour
the
same
type, the usage rate
at night.
When
there are multiple lines of
depends on the average monthly usage over the
set of lines.
Table
Band
1:
Hourly
WATS
Usage Rates (1984)
WearGuard
(b)
work
retailer of
WearGuard
.
privately-held
a
is
marketer and
direct
clothing and uniforms with total sales in excess of $130
million per year. In addition to a string of wholesale and retail outlets, the
firm publishes a catalog and accepts orders through
Department which
located at the corporate headquarters in Norwell,
is
These phone
Massachusetts.
open twenty-four hours per day, seven
lines are
for
roughly one half of the company's catalog
The company produces some
thirteen major catalog mailings each year.
days a week and are responsible
sales.
Telephone Sales
its
Since a majority of orders are placed within three weeks of a catalog's
mailing, the mailings are staggered so as to maintain a steady arrival rate of
and orders from day
calls
Although the
is
to
day throughout the
year.
from day
arrival rate of calls is fairly constant
calls
come
Between
hour over the course of a day.
a great fluctuation in calls per
midnight and 8:00 am,
to day, there
in at a
very low
During the business
rate.
day, the rate steadily increases as establishments across the country open for
business and then decreases through the evening hours as companies start to
The
close.
more
calls
coming from the East
however,
accommodate
a
the
modeling
in, the
call,
is free.
system would
can
of call arrivals.)
company makes use
For example,
signal.
no Band
If
a line
HI, IV, or
of
is
AT&T's
zone of
The
maintains
WATS
message and puts the
this time, the
WATS
hne
band
is
and lengths of
calls as
By using the
III
line, etc.
would
receive a
to serve the
on hold.
It
use ^nd charges
actually utilized, not the
abandoned and the length of time each
waiting for an operator to answer.
Band
Band IV
caller
in
from Zone
If all
In addition to assigning calls to lines, the
statistics of origins
calls that are
III line.
no available operators
cost of the call reflects the
call origin.
Band
to arrive
lines available, the caller
free but there are
should be noted that during
were
a call
try to assign the call to a
V
the system plays a recorded
accumulate.
if
try to assign the call to a
were busy, the system would
there were
busy
the
we develop
that
is
"hunt" system, which accepts calls and assigns them to the least
expensive line which
lines
in the
a constant distribution
techniques
non-homogeneous pattern
As do many telemarketers,
line
morning and from the West
company, so
origins by time of day from the
assumed;
If
in the
day with
(Unfortunately, at the time of the study data were not available on
evening.
call
distribution of call origins also varies throughout the
system also
well as the
caller
statistics
number
of
spends "on hold"
provided by the
system,
it is
possible to generate most of the data required by the
model we
developed here.
Problem
(c)
defir^ition
As mentioned
.
in the last section, the
hunt system
assigns each incoming call to the least expensive available line w^hich can
service a call
coming from
depend not only on
phone
number
of
phone
which the company
lines for
which can be used
WATS
the
a given origin.
to
Clearly the total telephone costs
lines,
contracts.
but also the configuration of
This paper develops a model
determine the optimal configuration of incoming
minimize telephone costs while maintaining an acceptable
lines to
level of service.
WearGuard, "acceptable
In the case of
defined as a level at which no more than
signal during any given period of the day,
X%
of
level of service"
receive a busy
all callers
where the
is
service percentage
X
is
a
managerial decision variable.
To model
Clearly, the
this
problem,
number
we
effectively ignore consideration of operators.
of operators
configuration of lines.
If
the
and
their
number
scheduling
of operators
more time on hold and more lines are needed.
staffing level makes it possible to deliver service
fewer
lines.
A
is
influence the optimal
reduced, callers spend
Similarly, an increase in
of the
same
complete optimization problem would therefore solve the
optimal staffing level as well as the optimal configuration of
we do
not address the staffing question here.
staffing
the
and scheduling
number
quality with
of lines;
Rather,
lines.
we assume
we assume
how many
that having decided
terms of expected time on hold,
lines to have,
is
measured
on the
total
We
a
few seconds), so
time to service a
that focuses
ours.
this subject.
Morrison
[4]
has done
on characterizing the queueing behavior, but he has
at Cornell University
Recently,
by Lampbell
we have come
[1],
which
is
across a study
very similar
Lampbell proposes the same queueing model as we do,
approximating the load on a given
line configuration.
He
differ
somewhat from
the
to
for
also develops
optimizing and heuristic methods for selecting the number of
methods
be
call is small.
not addressed optimization issues.
performed
set to
that the effect of the scheduling of operators
have found very few papers on
some work
in
maintained throughout the day.
Furthermore, the service target for expected time on hold will be
(i.e.,
that the
of operators are secondary issues to the question of
the of)erators are scheduled so that an acceptable level of service,
very small
However,
lines.
dynamic program developed
These
in
this
The methods given by Lampbell permit
paper.
between the
different types of
WATS
the case faced by telemarketers using
One
for
incoming
work and
where the
lines,
reason for the difference in assumptions
concerned with lines for outgoing
is
WATS
calls,
is
is
specialized to
service areas are
that
Lampbell
is
whereas the WearGuard application
Nevertheless, there
calls.
relationship
which the service areas of
lines, in
need not overlap. Our dynamic program
different line types
nested.
more general
a
great similarity between our
is
that of Lampbell.
The remainder
of the paper consists of three sections.
develop a queueing model
programming techniques
In section 2
we
system and then use dynamic
to describe the
Then
to find the best solution.
in section 3
we
describe the data required by the model, give an example from WearGuard,
and check the
sensitivity of the solution to various
Finally, in section 4
we
validate the solution,
Larson's hypercube model,
WearGuard
choose staffing
at
2.
MODEL DEVELOPMENT
to
The model development
queueing model
lines.
We
entails
for finding the
embed
then
this
which determines the
with an approximation
to
and then with a simulator which has been
[2], [3]
used
first
parameters of the model.
levels.
two components.
usage levels for
a
First,
we propose
a
given configuration of
queueing model within a dynamic program,
least-cost
We
configuration of lines.
begin by
presenting the queueing model.
(a)
The queueing model
development of
degree of
a
model
tractability.
The
.
first
step in solving the
that describes the
In order to
simplifying assumptions.
The
first
do
so,
problem
is
the
system while maintaining some
is
it
of these
necessary to
to
is
make
several
break the day into "time
blocks" within which calls arrive as a Poisson process v^th a constant rate.
Different time blocks have different arrival rates.
The second assumption
state.
Within each time block,
that the
system
to the next, the
new
states that the
steady
is
calls arrive at a
is
generally in a steady
constant rate and
we assume
ergodic in each time block. In moving from one time block
system
state.
system
may
As long
experience transient behavior before achieving a
as there are relatively
few time blocks, each one of
these blocks will be sufficiently long that the system will spend most of the
time in steady
In building our model,
state.
time the system spends not in a steady state
The
third
therefore
band overflows
calls
Unes of a given band are busy, a
all
Morrison
to the next.
assume
that the
small and can be ignored.
assumption involves the overflow of
When
another.
is
we
[4]
from one band
to
call that arrives to that
has developed polynomial
expressions to describe overflow queues in the two band case, and his
work
could conceivably be extended to describe a five band system, but the resulting
equations would be very complex and optimization of the
would seem
impossible.
To preserve
an independent Poisson process.
calls as
actually the overflow process
overflow to Band
j
only
of calls from
Band
in a Poisson
manner,
sum
tractability,
to
j-1
it
when Band
j-1 is full.
as Poisson
j
Lampbell
[1]
made
Since
and since
Poisson
is
we model
the overflow
calls arrive
from Zone
follows that the arrival of calls to each band
same assumption
the
an approximation since
is
two independent Poisson processes and
of
of lines
the overflow of
a "censored" Poisson process: there
is
Band
This
we model
number
is
j
the
is
therefore itself Poisson.
approximate the overflow process
to
as a Poisson process.
Our fourth assumption
This assumption
identically distributed.
each phone line and hence,
fewer operators than
some period
and
One
final
we assumed
all
assumption
Using these
is
If,
one operator
for
however, there are
dependent upon the
and
arrival
would not be independent
when deciding the number of lines,
WearGuard would always staff enough
that
of the lines or at least to ensure that the time
is
created by the price breaks.
section.
there
Nevertheless,
very small portion of the
a
is
thus, the service times
identically distributed.
is
if
and
are independent
then callers will occasionally be put on hold for
and
operators either to cover
valid
is
never put on hold.
This time on hold
calls,
for service reasons
hold
calls are
lines,
of time.
duration of other
that service times
is
We
time spent being served.
total
that
it
on
is
valid to neglect the fringe effects
will discuss this
five assumptions,
we
assumption
later in the
build the queueing model of the
telephone system.
In constructing this model,
operators available
lines as
number
time block
equal to the
effectively
Band
t
j
lines.
where Xu
Assume
is
assume
that the
number
of
We model each band of
waiting room, where n; equals the
number
M/G/n; queue with no
an
of
is
we
of lines.
that calls arrive at a rate yu
=
Zone
j
the arrival rate of calls from
^jt
+ C
and
j-l
^"
t
C,:^,
j
is
Band
the overflow rate from
we
Poisson and
we assume
Since
j-1.
ignore the transient behavior,
probability of having
n
Band
busy
lines
j
the overflow process
follows that Pjnt, the
it
any time during period
at
is
is
t
equal to
Pjnt =
(1)
1
(Y-t/^i)Vi!
i=0
where
l/fi is the
Given
average length of a
that this
is
true, calls will
the arrival rate of calls
busy,
i.e.
call (e.g.,
overflow from Zone
j
and
That
Erlang's loss formula.
< n <
at a rate
n;.
equal to
all n;
lines
is,
^jt='^jtPjn.t
where
is
Pj^.t
determine the
To
determined by
total
with n =
(1)
flow to Band
to
the proper rate.
also a
know
expected usage
fee
cost.
To compute
the total monthly usage of each
Yet, the total
random
monthly usage
monthly band usage
variable.
As
level.
is
a simplification,
by assuming the usage
good estimate
gives a
can use
calculate the expected cost of a given configuration,
we need
is
we
Hence,
n;.
(,u
to
we need
to
j+l, etc.
determine the expected monthly usage
rate
p. 361)
[6]
multiplied by the probability of having
(2)
fee,
Tijms
a
the expected usage
band
in order to charge
random
we
variable so the
calculate the expected
rate that corresponds to the
Such an approximation
is
very easy to calculate and
of the expected cost except for the case in
which E(HU)
is
very close to a price break and both Pr(HU < price break) and Pr(HU > price
break) are significantly greater than zero,
Even
in
such a case, the error
break which
is
the usage cost.
must be
is less
The expected number of
level.
10% and
is
never greater than 13% of
approximations in the model, an error which
to other
sigruficantly less than
being the monthly usage
than the savings associated with the price
generally equal to about
Due
HU
10% seems
tolerable.
lines that are
is
8
busy
for
any time block and band
(liPiit)
(3)
i=l
The monthly usage
for a time block is
number
of busy lines
month.
The
We
blocks.
to
1,
a
the
is
sum
determine the appropriate
To determine
is
the
for all time
WATS
of the cost of the
charge. Using this model,
phone system
number
generate
as a whole.
(2) for
j
=
of calls per hour that are blocked or lost
This overflow
5.
(i.e.,
receive busy
t.
The dynamic program The problem is
number of lines for each of five telephone bands
(b)
.
of the
we
the service level provided by a given line configuration,
signal) in time block
cost
time block over a
monthly usages
of
use the overflow rate for Band V, as given by
rate
in that
then use the total monthly usage, along with the rates from Table
good approximation
we
and the number of hours
monthly usage
total
simply the product of the expected
Such
system.
programming approach
determine the optimal
so as to minimize the total
itself
well
which an optimization problem
in
series of stages, each of
problem lends
a
to
which
is
to
a
dynamic
divided into a
is
solved sequentially to arrive at a final
solution.
We
1,2,.. .,i
define the system cost function fim(n) as the total cost for Bands
with a
m
total of
remaining m-n
lines
where Band
has n lines (n
i
lines are optimally assigned to
definition of iim^n),
n that minimizes
we
define i[^ as
Thus, i^^
fj^^^ (n).
the optimal assignment of
m
min
is
bands
(fim^r^)),
<.
l,2,....,i-l.
and njnQ
m) and
the
Given
this
as the value of
the total cost for bands
1,2,... .i
with
lines.
There are four components to the cost function fim(n): the access fee for
n Band
the
calls
i
lines, the
which arrive
associated with the
to
usage cost for Band
Band
first i-1
i
i,
the overflow cost for serving
but overflow to a higher band, and the cost
bands.
The access fee for n Band lines is the fixed monthly cost for having n
lines, which we denote by AFjCn). This cost is linear with respect to n and is
i
the
same
for all bands.
The usage
Band
cost for
on the nunnber of Band
the
Zone
on the
i
i
depends on the
The
lines.
i
call arrival rate to
and the overflow rate from Band
configuration for Bands l,2,...,i-l- We let
overflows rate from
Band
allocated to Bands
1,2,. ..,i.
in time block
i
t,
then use these arrival rates in
described previously.
We
(1) to
sum
of
w^hich depends
^^
denote the
lines are optimally
to
t
Band
i,
1,2,.. .i-1, is
determine the usage rates in each
i
lines,
and m-n lines allocated optimally to Bands
We need to include in fim(n) the cost for the
lines,
To estimate
but overflow to a higher band.
that calls
l,2,....,i-l.
calls
this
which arrive
overflow
cost,
corresponds
Typically,
prespecified.
is
charged
to the rate
note, though, that this
for
we assume
we assume
another approximation since
is
that the
usage fee
we
we
cannot
know how
We
need
lines allocated optimally to
overflow from Band
the caller.
one exception
to note
V
is
Hence, there
may
not
cost of a lost sale.
call
(i=5): if
indirect cost
is
is
a
busy signal
due
to a lost call.
back, in which case the cost for an "overflow"
with subsequent orders, so future sales
costs,
WearGuard
during any time period no more than
This constraint
An
for
not a direct charge associated with an overflow
Rather than try to estimate these
receive a busy signal.
with n
1,2,.. .i-1.
Furthermore, due to the f)oor service the customer
less likely to try to call
is,
i
to this treatment of overflows.
from Band V, but there may be a very large
customer
Bands
not handled by a higher band, but
is
We
configure the higher bands.
denote our approximation to the overflow cost by OFi(m,n) for Band
and m-n
We
usage of eighty hours per line-month.
the overflow calls will be handled until
lines,
Band
to
which overflow are served by the next higher band and are charged
usage fee that
that
and
from which we estimate the usage cost as
denote by UCj(m,n) the usage cost for Band i with
Band
a
the
i,
Y-j(m-n) = X-j + C-.ij(m-n)
time block for the n
i,
Cit
is
i
Then, the arrival rate in time period
(4)
n
i-1,
m
given that
given that m-n lines are allocated optimally to Bands
We
Band
arrival rate
lines
Band
call arrival rate to
This approach
is
X%
may
is
The
the
may be
also be lost.
sets a service target of
X%;
of the arriving calls should
typical for telemarketing operations.
easy to incorporate into the evaluation of fim^J^) ^o^ Band
the overflow rate exceeds
X%, we
10
set fiiTi(n)
= +
»
V
The
first i-1
component of the
final
bands.
cost function
is
the cost associated with the
For the dynamic programming recursion,
we assume
have previously determined the best allocation of the m-n
we know
l,2,"-/i-l
and
that
given by
f^.i
m-n-
the expected
However,
Band
i,
cost used in finding
gi-1
m-n
fim^")'
m-n
that overflow
*o
avoid double counting.
Now we
i-1 to
Band
Thus,
1,2,..., i-1
lines, exclusive of the costs for calls
from Band
for this allocation,
i-1
the expected monthly cost for Bands
allocation of
over Bands
by the higher bands. For computing the
^^ need to subtract the estimate of the overflow
^-n
fj.]
we
this cost includes the estimate of the cost for
handling the overflow from Band
costs through
monthly cost
lines
that
we
denote by
with the optimal
from Zones
l,2,....,i-l
or higher.
i
< n < m:
can write the recursion for fi^^^^' for
(5)
^im^^) = AFj(n) + UCi(m,n) + OFi(m,n) + gi-i^m-n'
(6)
fin,
= min {fim(n)} = fim^^im)n
and
gin, = fim
(7)
-
OF^(^, n,^).
For Band V, an overflow results in a lost
overflow rate to be no more than
X%
In this case,
call.
we
constrain the
of the total calls that arrive to the
system during any time block.
Given these formulae, we find the optimal solution by solving the
equations for Band
i,
we
solve equation
rate of calls
value of
.01
is less
and working out
I
(5) for m=0,l,....,
m
V.
where
m
employed).
is
For each successive value of
m
than an arbitrarily small value
calls/hour
various values of
Band
to
is
such that the overflow
(in the
case of this study, a
The minimum value of
i^^^ over the
represents the optimal solution for the system.
be defined as the optimal
V
optimal number of Band
optimal number of Band
total
i
number
lines
of lines (fsm*
— %m
^°^ ^^^
would then be given by n^* =
lines is equal to
1
1
Let
^-
n5ni»-
m*
^^^
The
n* =n-j,,wherek = m*- n^
(8)
Using
this
we
method,
find the optimal
"'^i+i-
estimate the total expected monthly cost (=^51x1*^ ^^'^
number
of lines for each band.
This dynamic programming algorithm allows for a fairly efficient
method of solving
maximum
functions
Assume
the optimization problem.
conceivable value of m.
must be evaluated
as
M
that
the
is
For each value of m, a
maximum
of
ranges from
Therefore,
we
can
means
the
n
to
m.
m+1
M
Z
anticipate a total of
i
+ 1 evaluations for each band, which
i=0
0(KTM2) where K
number of time blocks.
algorithm will perform as a function of
bands
in the
efficient
We
system and
T
is
the
is
number
the
This
is far
of
more
than complete enumeration.
have implemented the algorithm
code appears
and usage
fees, the
FORTRAN, and
the complete
This program requires as input the
in Santos [5].
bands and time periods
in
in a day, the
maximum
average service time for a
number
call,
of
the access
allowable loss rate, the arrival rates of calls
from each zone during each time block, the number of hours of each time
block in a month, and the presumed usage rate for each band and time block.
The program reads
this
data from an input
file,
solves the problem using the
algorithm described in this section, and gives as a solution the optimal
number
of lines in each band, the overflow rate from each
time block, and the
and including
total cost of the
that band.
A
typical
system (excluding overflow cost) up
problem requires
CPU
time to be solved on a
3.
EXAMPLE OF MODEL APPUCATION
In this section
with
when
we
PRIME
band during each
less
to
than one minute of
850 computer.
present and discuss the
irutial
building and testing the model in 1984.
we worked
example we first
example
For this
that
describe the input data required by the model, and then present the optimal
solution from the
dynamic program.
alternate solution,
namely the
We
compare
line configuration
12
this solution
with an
which WearGuard had
at
the time of the study.
we examine
Finally,
the sensitivity of the solution to
various changes in the input data.
any operation
for
and usage
fees
we
For the given model,
Data.
(a)
can solve the configuration problem for
which the necessary input data are
available.
The
access
by the telephone company; we provide an
rates are set
and Table
illustration of these in Section 1.1
Other data, namely arrival
1.
time blocks, source distribution, and service time information are
rates,
specific to the firm being studied
data sources.
The
and must be extracted from
their available
input parameters are the estimated overflow penalty
final
and the maximum percentage of
We
calls lost.
initially set
5%
as an
acceptable rate of lost calls during the peak period. Thus, any solution with a
maximum
or peak loss rate greater than
The estimated overflow penalty
next higher band at the
is set
maximum
5%
is
considered to be not feasible.
equal to the cost of serving a
price break
(i.e.,
call at
the
more than eighty hours
per month).
The parameters
that remain are specific to the
was mentioned previously,
maintains relevant
are maintained
statistics
company
the telephone system used
which are tabulated every day.
in question.
As
by WearGuard
These
statistics
on Daily Summary Sheets which contain such data as the
average length of a
call
and the average time spent on hold, and Daily
Reports which contain the arrival rate of
throughout the day.
calls for
Profile
each fifteen minute interval
Using these summary sheets over the course of several
weeks, along with other data which has been collected
previously at the company,
we determined
in studies
performed
the input parameters.
We
describe next the values for arrival rates, distribution of call origins, and
mean
service times.
The
of the
arrival rate data define
how many
calls are
received over the course
day and when during the day they are received. As an
graph of these
calls for
arrival rates appears in Figure 2,
which gives the
each fifteen minute interval over a day.
variation in the arrival rate pattern from
day
to
There
is
illustration, a
arrival rate of
remarkably
little
day during the week, and the
distribution has proven to remain stable over the past five years (although
daily call
volumes have
increased).
We
considered the cost of operating the
telephone system over the weekend negligible and therefore ignored
instead
assumed
that an average
month
each with the same arrival rate of
calls.
13
it.
We
consisted of close to twenty-two days,
Calls
per
Quarter Hour
C
iS
00 -
3
<D
2.
<
0)
IS
o
33
fi>
<s
o
o
o
5"
14
From
we assumed
we
data
this
divided the day into time blocks within each of which
a constant arrival rate of calls.
We needed
a sufficient
number
of
blocks to describe the changing rates throughout the day while having few
enough blocks
to
pm, and
5:00
period
is
number
a
keep the problem
of calls will be
blocks,
12:00 noon,
10:15
calls
usage fees change.
We
lost, it
was necessary
when
the greatest
to isolate that period into a
day
(7
am
to 5
pm)
and the other consisting
am and
12:00-5:00 pm).
of the rest of the business
During the evening hours, the
day
two time blocks
7:00
am
is
of equal length,
8:00
pm
seemed prudent
arrival rate of
to split the
pm
one running from 5:00
until 11:00
low and generally
It
pm. The
arrival rate
fairly constant.
For
to
7:00-
(i.e.,
pm
decreases monotonically from 40.9 calls per fifteen minutes at 5:00
pm.
into
am
one consisting of the peak period and running from 10:15
and the other from
and
Since the absolute peak
therefore broke the business
3 calls per fifteen minutes at 11:00
into
Natural breaks occur at 7:00 am,
major source of expense and represents the time
unique time block.
two
pm when
11:00
tractable.
evening
until 8:00
between 11:00
this reason,
to
pm
pm
only one
time block was used for that interval. The arrival rates for an entire day were
thus compacted into five time blocks, the data for which appear in Table
The data
for this
sample period indicated an
day, distributed as displayed in Figure
WearGuard
five
calls
arrival rate of 2283 calls per
However,
at
desired to configure the phone system to
hundred
volumes
2.
calls
per day, which represented future
in 1989 are
per day,
we
around 5500 per day.) Thus,
the time of the study,
accommodate
thirty-
volumes.
(Call
call
for an arrival rate of 3500
adjusted each of the observed arrival rates.
rates also appear in Table 2 in the
column
15
titled
2.
These
arrival
"Adjusted Arrival Rate."
Table
Time
2:
Time Block Data
Table
3:
Zone
Distribution of Call Origins
Zone V and most
serves primarily calls from
and V. Such a strategy
possible)
results in
and requires more
lines,
The optimal solution has
16.3 calls per hour;
day.
however,
a
peak
loss rate of 16.3 calls per
5%
(lost-call) rate
peak rate holds
for
lost calls in the other
is
constraint,
only tv^o hours of each
time periods of the day.
all calls
4:
are being
However, the
hour was considered high enough
which we examine
Table
from Band V of
well within the constraint of 5%.
This overall loss rate
this
but seems to save money.
of the business day, roughly 3.7% of
lost.
lowering
come from Zones IV
lost calls (as close to the constraint as is
peak overflows
this
Table 6 shows the rate of
Over the remainder
more
lost calls
to
warrant
in the next section.
Results for Base Case
Peak Overflow
Total Cost
Band
Rate
Table
6:
%
Time Block
1
Loss Rates of the Base Case Solution
of Daily
Arrivals
Lost Calls/
Hour
%
of Calls
o
00
O
r<5
o
o
in
in
CO
o
o
in
o
in
en
00
fTi
C/5
>-
<
z
<
o
in
o
o
in
00
CO
in
en
00
o
o
C/5
Z
W
CO
<
o
o
in
CD
o
o
in
ro
s
in
en
00
en
en
00
en
en
o
After
are increased 3.5%.
some
was
a lower overflow rate
it
was decided
5%
preferable to the
subsequent sensitivity analysis,
We
discussion,
we assumed
that a solution with
rate originally
assumed. For
a service constraint of 1%.
next considered the effect from variations in the arrival rate.
Additional runs were performed with arrival rates of 3850 (10% greater than
the base rate) and 3150 (10% less) and 5500 calls per day.
runs also appear in Table
lines are
added
When
7.
to the system,
The
results of these
the arrival rate increases
by 10%, seven
two each on Bands
I
and U and one on Band
Two
so as to maintain a very low level of overflow to the outer bands.
are also
loss rate
added
to
Band V and two
are shifted from IV to
below the constrained value of 1%.
V
III,
lines
so as to keep the
Such a system would have an
expected cost of $73,800 per month which represents an increase of 13.5% over
the base case with a
cost
maximum
overflow rate of 5%; the increase in expected
only 10% over the base case with a
is
When
occurs.
the arrival rate
One
line is
removed from Band
is
reduced
removed from each
III,
1% maximum overflow
3150
to
of the
but three lines are
calls
first
per day, the opposite
two bands, two
moved from Band IV
The resulting 11-11-12-4-8 configuration has an expected
month which
is
about 7%
less
to
lines are
Band
than the cost of the base system with a
The service time
for this
V.
cost of $60,400 per
maximum overflow rate; the expected cosL is 10% less than
a 1% maximum overflow rate.
Lastly, we considered the sensitivity of the solution
service time.
rate.
5%
the base case with
to
changes in the
problem includes both the time
required to serve the customer and the time the caller spends on hold waiting
to
be served.
we assumed
In developing the model,
constitutes a very
minor portion of the
total service time.
days for which data were tabulated, the average
hold before being served.
that the time
Over the
caller spent four
when
the service time
Table
7,
we
Nevertheless, a change in operator scheduling
similar effect as a
10%
made
was increased
see that a
10%
fifteen
seconds on
could have an effect on the service time due to this delay time.
reason additional runs were
on hold
to
For this
observe the sensitivity of the solution
or decreased by 10%.
From
the results in
increase or decrease in service time has a very
shift in arrival rate.
The
resulting strategy
or the same, and the monthly costs vary by less than 1/2%.
21
is
very close
MODEL ACCURACY
4.
we examine
In this section
This will be done
the accuracy of the model.
by comparing the estimated cost from the model to that from another
approximate model and from a simulator that is used at WearGuard for the
purpose of scheduling operators.
Approximation of the hypercube model
(a)
analytic model, developed by Larson
spatially distributed
by setting up a
queues
in
[3]
[2],
The hypercube model
.
an
purpose of analyzing
for the
emergency vehicle systems. The model works
priority schedule for each
zone of a
city indicating the
which emergency vehicles should be dispatched given
order in
that the higher priority
vehicles have been previously dispatched to other emergencies.
is
is
This system
completely analogous to the telephone system that has been described in
this
paper where the different telephone bands correspond to zones of a
and the telephone
lines are similar to
model makes several assumptions.
emergency
city
The hypercube
vehicles.
Those that remain relevant
in
the
conversion from emergency vehicles to telephone lines are the following:
(1)
Independent Poisson
(2)
N
(3)
Single server dispatch to any
(4)
Fixed preference dispatching.
(5)
Exponential or near exponential service times.
servers, each of
Of these assumptions,
system
is
j+l,....,5.
call
call
and
I,
a call
call;
is
fact a call
from Zone
in the region.
is lost if all
servers are busy.
contradictory to the telephone
j
can only be served by Bands
the
after the
Band VI
Band V
lines
in the priority lists for calls arriving
lines but before the
would be placed
at the
can only be served by an infeasible line
all
any zone
j,
This problem can be avoided by creating a buffer of "Band VI" phone
n through V
Zone
travel to
This claims that each of the telephone lines could
which would be placed
lines
A
while in
which can
the only one that
the second one.
handle any
arrivals.
the buffer lines are busy as well.
the probability of an overflow to a
If
Band
assumption of exponential service times
model, whereas
it
is
needed
for the
22
I
Band
I
lines.
the
Band V
enough buffer
is
For
calls
from
very end of the priority
if all
line
from Zones
becomes
lines are
list.
busy
lines are included,
insignificant.
The
not needed by our queueing
hypercube model.
However, the
actual
service times are probably close
enough
to
being exponential to
make
this
assumption reasonable.
The hypercube model describes
the system exactly, but requires that
simultaneous equations to be solved.
intractable for large values of
N,
e.g.,
For
N
this reason, the
2^
problem becomes
greater them fifteen servers.
Larson
has developed an approximation to the model which requires that only
N
equations be solved simultaneously and which generally solves the problem
to within
one or two percent of the exact
N
considered has a value of
solution procedure.
results.
equal to about 50,
Since the problem being
we
use
this
approximate
Table 8 compares the results of the hypercube evaluation
for the 12-12-14-7-5 configuration to that for
our approximate queueing
model. The model uses an iterative
Table
the
8:
Comparison of User Cost Estimates Using
Hypercube Model and the Approximate Queueing Model
Estimated User Cost
Hypercube
Band
Approximate
approximation
each of the
However, since the estimated
is.
costs of the
two models
for
four bands vary by less than 2.5% and the expected user cost of
first
the entire system as calculated using the hypercube
model
is
only 1.5% greater
than the expected cost from the approximate queueing model, the two models
do not contradict each other and
in fact there is
a very high level of
consistency.
(b)
model
Simulation
method
In addition
.
to the
hypercube model, another
of testing the accuracy of the approximate queueing
A
simulation.
time-driven simulator
model
through
is
was developed by WearGuard
for the
purpose of scheduling operators. Instead of dividing the day into large time
blocks during which the arrival rate of calls remains constant, the simulator
has a different arrival rate for each fifteen-minute time block over the day.
A
second difference between
is
model and the one developed
this
in this
paper
the fact that the simulator considers the service time to be a normally
distributed
random
to 3.83 minutes.
number
variable (truncated at zero) with
The simulator
mean and
variance equal
takes as input the configuration of lines, the
of operators over the course of the day, and the duration of time to
simulate.
By choosing the number of operators
telephone lines for a one-month simulation,
an estimate of the monthly
to
we
be equal to the number of
obtain from the simulator
cost of a given configuration.
One advantage
of
checking our solutions with the simulator was that the simulator gave a very
accurate calculation of the costs.
We
used the simulator
and the current system.
We
to estimate the cost of
both the optimal ^lution
also used the simulator to determine
whether or
not an improved solution could be found by adding or removing one line
from each of the bands.
The
results of this
procedure are
in
Table
9.
Once
again the estimated cost using this simulator comes very close to the estimate
of the queueing model; they differ
by
less
than 2%.
Since the result of the
simulator, the hypercube model, and the approximate queueing
within a range of
developed in
this
3%
of each other,
we concluded
paper gives an accurate
24
result.
that the
model
all fell
queueing model
Table
9:
Costs of Various Configurations as Estimated
Using the WearGuard Simulator
Configuration
Bands a-n-ni-iv-v)
References
1.
Lampbell, David M., "On the Selection of
Numbers
of Servers for the
N
Server-Type Problem," Technical Report No. 349, School of Operations Research
and
2.
Industrial Engineering, Cornell University, Ithaca
Research.
3.
C, "A Hypercube Queuing Model
Larson, Richard
Redistricting in
1 (1)
Urban Emergency
1974,
pp
Services,"
Bell
5.
Morrison,
J.
A., "Analysis of
Svstem Technical Tournal 59
Santos, P. Clark,
WATS
Lines,"
(5)
Tijms,
(8)
"An Optimization Model
&
Urban Emergency
w^ith
P.
for the
Configuration of Incoming
Sloan School of Management,
MA,
1984.
and Analysis:
A
Computational
Sons, Chichester, Great Britain, 1986.
BUG
Queuing," The
October 1980, pp 1430-1434.
Stochastic Modelling
Approach, John Wiley
of
Some Overflow Problems
unpublished M.S. Thesis, A.
Henk C,
and
September-Octoberl975, pp 845-868.
Massachusetts Institute of Technology, Cambridge
6.
for Facility Location
Computers and Operations
C, "Approximating Performance
Larson, Richard
1977.
67-95.
Service Systems," Operations Research, 23
4.
NY, August
090
26
Date Due
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