"Waiting Line Models" 
Index to Module Four Notes 
"No matter what line I get in, it becomes the slowest."Carol Harrington
Shipboard in the Bahamas,
February 6, 2000
4.1: Structure of Waiting Line
Systems
Perhaps the most significant difference
between systems that produce products and systems that produce
services is that product manufacturers can buffer their manufacturing
processes from customers through use of inventories. In fact, a major
category of inventory is called "buffer" or safety stock as we will
see in the next module of this course.
On the other hand, in pure service systems, where the customer
receives services directly from the service provider, the customer is
in the "boundary" of the provider. In this case, when demand for
service exceeds the capacity for service, one or more waiting lines
form or customers leave the system unserved. Medical doctors cannot
"inventory" medical care; lawyers and other professional counselors
cannot "inventory" consultations; and airports cannot "inventory"
aircraft parking positions at terminals.
My first experience with waiting line systems was in 1969 when I was
Commander of an Aerial Port Detachment at Ramstein Air Base Germany.
One of my first tasks was to design a parking plan for aircraft
involved in the evacuation of troops, equipment, bombs and other
explosives from a fighter training facility in Libya. Waiting line
models helped me determine how many parking positions we would need
(the waiting line) to avoid holding aircraft on the runway (also the
waiting line), until cargo handlers (service providers) could unload
and turnaround the aircraft. The models worked fairly well, although
I should have done more sensitivity analysis to handle increases in
the arrival rate as things got a little "hot" the last day of the
evacuation in Libya.
Enough nostalgia  on to the basic structure of waiting line systems.
Waiting line systems, also called queuing systems from the underlying
modeling basis of queuing theory, involve a population
source, an arrival process, a waiting
area, and a service area or
channel.
Waiting line systems also have costs, operating
characteristics, and management response
strategies.
Waiting Line System Costs and Management Strategies
There are basically two costs that must be balanced in waiting
line system  the cost of service and the cost of
waiting. Note that I am not considering another
possible cost component  the cost of a scheduling system.
Theoretically, a scheduling system is a management strategy designed
to avoid waiting lines (meaning you should never wait
in the doctor's office  yeah, right!) and is not covered in this
module. Scheduling systems are useful when the customer is known to
the system and the short and long run costs of waiting are relatively
high. We will study scheduling system applications in linear
programming later on in the course.
Rather, in this module, we will study the more interesting scenario
where customer arrivals to the system are random  that is, the
customer is not known before arriving to the system and the arrival
process is random as we will discuss later. In this case, to avoid
longer than desired waiting lines, the service provider may exercise
management strategies to increase the service rate by using faster
servers, more servers, automated service, or some combination of
strategies. Each of these strategies increase the cost of service,
but produce the benefit of reducing the cost of waiting.
When the cost of service and the cost of waiting are known and
measurable, the waiting line models in this set of module notes help
us determine the optimal, or close to optimal waiting system
configuration and rate of service. This can be tricky because the
cost of service has a positive relationship with the rate of service,
whereas the cost of waiting has a negative relationship with the rate
of service. That is, the faster the service rate (or the more
service), the higher the cost of service; whereas the faster the
service rate, the lower the cost of waiting. Of course, the opposite
is true: the slower the service rate, the lower the cost of service
but the higher the cost of waiting.
The cost relationship can be sketched out in a graph, with dollars on
the vertical axis and the rate of service on the horizontal axis, as
shown in Figure 4.4.1.
Figure 4.4.1
Dollars


/
Cost of Service


/


/

\
/

\
/

\
/

+ 
/
\
_
_
_
_
_
Cost of Waiting

/

/
_
_
_
_
_
_
_
_
_
_
_
Service Rate: Faster >
We will examine the cost of service and cost of
waiting components as we examine the main waiting line models later.
It should be cautioned, however, that it is often difficult to
measure the cost of waiting for customers that are external to the
service provider's company or organization. For example, if customers
in a car repair waiting line system are defined as mechanics waiting
for tools from a tool crib to repair a car, it is fairly easy to
measure the cost of mechanic waiting since the repair company employs
the mechanic. The cost of waiting may simply be the time spent idle
in the line times the employee's salary for that time unit.
However, if the owner of the car being repaired is defined as the
customer of the waiting line system, the cost of waiting may be more
difficult to measure. Retired "snow birds" may have discretionary
time to spend in a waiting room and their cost of waiting may be
relatively low. On the other hand, for someone who is employed and
depends on their car to get to work, the cost of waiting may be
relatively high, depending on their income level.
If the cost of waiting also incorporates the cost of losing a
customer because long waiting time drives the customer away,
measurement becomes even more difficult. As an alternative to
managing the waiting system by measuring, analyzing and minimizing
its combined total costs, the service provider may try to manage the
service system by setting threshold parameters for system operating
characteristics, and then use faster servers, more servers,
automation of the service activity or some combination of strategies
to achieve those parameters.
For example, a consumer products store may decide to open another
cash register checkout station when the number of customers in line
at the first register goes over six; or a bank may decide to open
another teller position when the waiting time in line exceeds five
minutes. Other waiting line system operating characteristics are
discussed next.
Waiting Line System Operating Characteristics
Operational characteristics of waiting lines include:
1. the probability that no customers (or units) are in the system,
2. the average number of customers in the lines,
3. the average number of customers in the system (customers in line plus those being served,
4. the average time a customer spends in the waiting line,
5. the average time a customer spends in the system (waiting time plus time in the service facility,
6. the probability that an arriving customer has to wait for service,
7. the probability of n customers in the system, where n could be any real integer such as 1 customer, 2, 3, ...
We will examine the operating characteristics of each of the main
waiting line models presented in this module. Before that, we will
examine the main features of the structure within a waiting line
system: the population, arrival process, waiting line configuration,
service area and its configuration, and exit.
The Population
The population that generates customers to waiting line systems may
be infinite or finite. In most cases, populations can be considered
infinite, even though they are really finite. For example, if we were
to study the characteristics of waiting lines forming at the Fort
Myers side of the Cape Coral Bridge at morning rush hour, we know
that the population generating the arrivals is the finite population
of Cape Coral, around 100,000. However, since there is no actual
limit placed on the customers arriving at the toll booths, we
assume the population is infinite. All but one of the models
we will study make this assumption. The finite model is the
appropriate model to use when the population is relatively small,
such as 20 total computers in a network office that feed a computer
repair person server.
Arrival Process
Arrivals to the waiting line system from the population source may be
on an individual or batch basis. We will assume arrivals are on
an individual basis. The difference is best illustrated by
the arrival of a car to a parking lot at a restaurant. One driver
leaving the car to enter the restaurant would represent the arrival
of one unit or customer to the waiting line system. If a bus pulls
in, there could be a batch arrival of 30 customers. Did you ever
notice that the bus stalls are behind the Cracker
Barrel Restaurants on the interstate highways  just so you can't see
all those batch arrivals before you pull off!
It is also assumed that the arrivals are nonscheduled,
and the arrival of one unit is independent of, or does
not impact, the arrival of other units. Whenever these assumptions
are made, arrivals are assumed to follow the Poisson Probability
Distribution, a member of the family of discrete probability
distributions. The Poisson Probability Distribution is completely
described by its mean, which is given the Greek symbol lambda. In a
waiting line system, the mean we are referring to is the mean
arrival rate. For example, we may say that the mean arrival
rate is 4 calls per hour to a catalog company's telephone bank.
Another way of representing the mean arrival rate is to take its
inverse, which gives us the mean time between arrivals.
So, if I invert the mean rate of 4 calls per hour, I get ¼
hours. The mean time between arrivals is 1/4^{th} of an hour,
or one arrival every 15 minutes.
Mean Time Between Arrivals = 1 / Mean Arrival Rate
The probability distribution that is used to
describe this time between arrivals in a waiting line system is the
Exponential Distribution. The Exponential Distribution is used to
model the probabilities of continuous variables such as time, in the
case of waiting line systems. The Greek Symbol Mu, is used to
describe the mean of the Exponential Distribution.
If you have the mean time between arrivals, you can find the mean
arrival rate by the similar procedure  taking the inverse. For
example, what if we knew that the average time between arrivals to a
bank teller was 5 minutes. The mean arrival rate would be computed as
follows:
Mean Arrival Time = 5 minutes = 5/60^{th} hours
Mean Arrival Rate = 60/5 = 12 customers per hour
Customers arriving from a population next join
the waiting line in the waiting line system.
Waiting Line Configuration
Waiting lines may be infinite or truncated. For all of the models
we will examine except one, we will assume infinite line
length. One of the models we will examine is designed to
model situations where no waiting is allowed  the ultimate of
truncated systems. My telephone allows no waiting  if I am talking
to someone, the next caller gets a busy signal. However, airline
reservation systems allow callers to a busy reservation agent to wait
in a queue.
The waiting line system may use a single line/single server
channel configuration which means one line forms in front of
a single
service channel. The word service channel
is used rather than server to avoid confusion. A single service
channel may have many servers, but room for only one customer. That
is called a single service channel. Grocery stores have multiple
single line/single channel configurations. Banks, on the other hand,
employ a single line/multiple channel or teller
configuration.
Customers discipline in the waiting line configuration
may vary from patient, to balk (view the
line, then leave), renege (join the line, then leave),
jockey (join the line, then move to another line when
you think it is moving faster  that's me!), or collude (give your
groceries to another customer  scum of the earth to students of
quantitative methods)! Waiting line models assume that the
customer is patient (my wife).
Departures from the waiting line to the server are assumed to
be first in, first out or first come, first served rather
than last in first out, or like my deli  service in random
order!
Service Channel
The main feature of a service channel in a waiting line system is
the service time, also assumed to follow the Exponential
Distribution when the time to perform service for one
customer is independent from the time to perform service for others.
If the average time to serve one customer is 10 minutes, then the
mean service rate in hours may be found by converting 10 minutes to
hours (10/60^{th} of an hour), then taking the
inverse:
Mean Service Rate = 60/10 = 6 customers per hour
Note that rates are always stated as per hour,
… per minute, … or per whatever time unit; where as service
times and times between arrivals are stated simply as hours, minutes,
or whatever time unit.
Once service is completed, it is assumed that customers exit
the system and return to the population. Of course, they may
exit one system and feed another. If the second system
is independent
of the first, then there may be two separate and distinct single
line/single server systems.
That finishes our coverage of the basis structure of the waiting line
system. Waiting line models have been designed as quantitative
methods to analyze the operating characteristics and costs of waiting
line systems. The models are categorized by the probability
distributions that describe the arrival rate and service time
processes, the number of channels, and whether the population is
infinite or finite. The first model we will examine is one which
follows the structure of a single service/single channel system.
4.2 Single Server/Single
Channel System with Poisson Arrivals and Exponential Service
Times
The title of this section is a long way of describing a very common
waiting line system: a single line forming in front of a single
server. The quantitative methods described in this section are used
to compute the operating characteristics and costs of this
system.
Pause and Reflect
The assumptions needed for using quantitative methods to analyze operating characteristics of this single server/single channel system include an infinite population, Poisson arrival rates, infinite line length, patient customer discipline, FIFO departure from the waiting line, Exponential Service times and customer departure to the population.
Let's examine the following example problem. A
document clerk earns $15/hour, processes an average of 5 documents
per hour throughout the workday, and receives documents at the rate
of 4 per hour. There is a cost of waiting for the documents (by an
office paralegal) of $25.00 per hour. For this problem, the mean or
average service rate is mu = 5 documents per hour, and the
mean or average arrival rate is lambda = 4 documents per
hour.
The law firm is interested in knowing the operating characteristics
of this waiting line system, such as average time a document spends
in the waiting line, number of documents in the waiting line, the
utilization rate of the document clerk, the probability of 3
documents in the system, and the cost of the system. The following
formulas are used to model the operating characteristics of the
single line/single channel waiting line system with Poisson Arrivals
and Exponential Service time.
1. The probability of an idle system (no documents in the system, P_{0}):P_{0 }= 1  (lambda/mu) = 1  (4/5) = 0.20There is a 20% chance that the system will be idle at any one point in time.
Note: look at the formula carefully to see that we divide lambda, the mean arrival rate, by mu, the mean service rate. Recall that the arrival rate must be less than the service rate or the waiting line explodes. This can be shown mathematically in the formula. If lambda were greater than mu, the probability of an idle system would be a negative number greater than one which is infeasible.2. The average number of documents in the waiting line (L_{q}):
L_{q} = lambda^{2}/ [mu(mu  lambda) ] = 4^{2} / [5 (54)] = 3.2 documents3. The average number of units in the system (L):
L = L_{q} + (lambda / mu) = 3.2 + (4/5) = 4 documentsFrom the formula, you can see that this includes the documents in line as well as in the service channel.
4. The average time a document spends in the waiting line:
W_{q} = L_{q} / lambda = 3.2 / 4 = 0.8 hours or 48 minutes.Note:
The initial time units of this and other timerelated operating characteristics are the same as the input units. If input arrival and service times and rates are in hours, then output times will be in hours. I converted 0.8 hours to minutes for ease in interpretation.5. The average time a document spends in the system:
W = W_{q} + (1/mu) = 0.8 + (1/5) = 0.8 + 0.2 = 1 hour
As with the number of units in the system parameter, this time includes the time a document spends in the line, plus the time in service.6. The probability that an arriving unit has to wait for service:
P_{w} = lambda / mu = 4/5 = 0.80This operating characteristic is also known as the utilization factor for the service channel. A critical requirement for single line/single channel service systems is that the utilization factor be less than one. Otherwise, the waiting line would explode. In fact, queuing systems are not very efficient anytime the utilization factor exceeds 75 or 80% due to the interaction of the two probability functions. We will illustrate this later.
7. The probability of n documents in the system. Let's say we are interested in knowing the probability of n = 3 documents in the system:
P_{n} = ( lambda/mu)^{n} * P_{0} = (4/5)^{3} * 0.2 = 0.1024Since we are working with discrete probabilities, to find the probability of three or less documents in the system:
P_{n} < 3 = P_{0} + P_{1} + P_{2} + P_{3}= 0.20 + (4/5)^{1}* 0.2 + (4/5)^{2} * 0.2 + 0.1024
= 0.20 + 0.16+ 0.128 + 0.1024 = 0.5904…and we follow the law of probability that says all probabilities of the distribution must sum to one, so the probability that there will be more than 3 documents in the system is:
P_{n} > 3 = 1  0.5904 = 0.4096
Costs of the Waiting Line System
The total cost of this waiting line system is the sum of the cost of
waiting and cost of service.
Total Cost = Cost of Waiting + Cost of Service= ( c_{w} L ) + (c_{s} k ) where
k =number of channels
c_{w} = cost of waiting = $25.00 per hour for 1 paralegal
c_{s} = cost of service = $15.00 per hour for clerkTotal Cost = ($25 * 4) + ($15 * 1) = $115
Sensitivity Analysis
Recall that I said with purely random arrivals waiting line
systems are most stable with utilization rates less than or equal to
75%. That may seem unusual to you, that we only want to work our
employees at 75% utilization. The problem is that the arrival rate
and service time parameters are averages. The variation is always
what kills us.
For example, the average arrival rate is 4 documents per hour, and
the service rate is 5 documents per hour. At any point in time, the
system may experience an arrival rate of 5 documents per hour and
would be at the verge of explosion unless at the next
point in time, the arrival rate slows down to 3 documents per hour.
And, this considers that the service rate remains constant, which it
doesn't in this model. The service rate could slow down to 4
documents per hour with an arrival rate of 4, and we have the same
problem. Of course, we could also have a speed up in the arrival rate
and a slow down in the service rate to result in chaos.
To buffer against explosion, which occurs when the waiting grows and
grows and we never catch up, waiting line systems with purely random
arrivals are generally kept to a maximum utilization rate of 75%.
Note that the above example illustrated a system with a utilization
rate of 80%. What if the arrival rate increases to 4.5 documents per
hour, which represents a 12.5% increase. The new utilization rate
is:
P_{w} = lambda / mu = 4.5 / 5 = .90, a 12.5% increase as well.
The line length now increases to:
L_{q} = lambda^{2}/ [mu(mu  lambda) ] = 4.5^{2} / [5 (54.5)] = 8.1documents, a 150% increase!
That's why we don't operate at utilization
rates above 75%. I really find this interesting from my operational
experience. People who say they are working their people at 100%
utilization are not working in random service systems  they can't be
or the system would be pure chaos all the time.
The formulas for the operating characteristics and costs of this
model are relatively simple. As we begin to change the assumptions
and develop more complicated models, it is important to understand
relationships and concepts, and then rely on the software to do the
number crunching, in my opinion.
Using The Management Scientist Software Package
We will be using The Management Scientist Waiting Line
Module to do the actual operating characteristic and cost analyses.
To illustrate the package for the first example, click Windows
Start/Programs/The Management Scientist/The Management Scientist
Icon/Continue/Select Module 9 Waiting Lines/OK/File/New and you
are ready to load this example problem.
The next dialog screen asks you to select a model. Highlight
"Poisson Arrivals/Exponential Service", and click OK.
Next comes the input dialog screen. Enter 1 for Number of
Channels, 4 for Mean Arrival Rate, 5 for Mean Service
Rate, click Economic Analysis. Then enter 25 for the
Cost per Time Period for Units in System (that is the
waiting cost per person per paralegal per hour), and 15 for Cost
per Time Period for a Channel (that is the service cost for the
clerk to operate the service channel). Then select Solve, and
you should get the following solution:
Printout 4.2.1
WAITING LINES
*************
NUMBER OF CHANNELS = 1
POISSON ARRIVALS WITH MEAN RATE = 4
EXPONENTIAL SERVICE TIMES WITH MEAN RATE = 5
COST FOR UNITS IN THE SYSTEM = $25 PER TIME PERIOD
COST FOR A CHANNEL = $15 PER TIME PERIOD
OPERATING CHARACTERISTICS

THE PROBABILITY OF NO UNITS IN THE SYSTEM 0.2000
THE AVERAGE NUMBER OF UNITS IN THE WAITING LINE 3.2000
THE AVERAGE NUMBER OF UNITS IN THE SYSTEM 4.0000
THE AVERAGE TIME A UNIT SPENDS IN THE WAITING LINE 0.8000
THE AVERAGE TIME A UNIT SPENDS IN THE SYSTEM 1.0000
THE PROBABILITY THAT AN ARRIVING UNIT HAS TO WAIT 0.8000
THE TOTAL COST PER TIME PERIOD $115.00
Number of Units in the System Probability
 
0 0.2000
1 0.1600
2 0.1280
3 0.1024
4 0.0819
5 0.0655
6 0.0524
7 0.0419
8 0.0336
9 0.0268
10 0.0215
11 0.0172
12 0.0137
13 0.0110
14 0.0088
15 0.0070
16 0.0056
17 0.0045
18 0.0036
19 0.0029
20 0.0023
21 OR MORE 0.0092
A couple of comments are in order. The
operating system characteristics are reported similar to those
calculated earlier in the module notes. However, the only cost given
in the program output is the total cost. If you wanted to get the
components, you would have to work the formulas given earlier in the
Module 4.2. Notes. Also note the probabilities are the probability of
exactly n units in the system. So, the probability of exactly 1 unit
in the system is 0.16. To get the probability of 1 or less units in
the system, you would have to add the probability of 1 unit in the
system to the probability of 0 units in the system. Printing follows
the same format used in the other three modules we have used in the
course.
Management Strategies
With The Management Scientist Software, management can quickly
analyze the impact of continuous improvement strategies. For example,
suppose management could train the document clerk to be more
efficient and process a document in an average of 10 minutes, versus
the current service time of 12 minutes. This change should not impact
the arrival rate of 4 documents per hour, at least in the short run.
Let's also keep the cost per document waiting at $25.00 per hour, and
the cost of service at $15.00 per hour. (I know, you bright MBA's
say, "wait a minute, training isn't free!" Auk  I'll address that
later).
Pause and Reflect
In this example, the service rate was: the document clerk can process 5 documents per hour. Remember to get the service time, take the inverse of the service rate. The inverse of 5 is 1/5^{th} hour, and 1/5^{th} hour is 12 minutes. So, what is the service rate for a service time of 10 minutes? First, convert 10 minutes to hours to keep the time units the same as the arrival rate. That gives 10/60^{th} hours. Now take the inverse of 10/60 which is 60/10 or 6 documents per hour.
Now we are ready to go to The Management
Scientist. Note again that all we need as input parameters are
the arrival rate, the service rate, the number of servers, and the
costs if we are going to perform an economic analysis. Printout 4.2.2
provides the operating characteristic and cost results.
Printout 4.2.2
WAITING LINES
*************
NUMBER OF CHANNELS = 1
POISSON ARRIVALS WITH MEAN RATE = 4
EXPONENTIAL SERVICE TIMES WITH MEAN RATE = 6
COST FOR UNITS IN THE SYSTEM = $25 PER TIME PERIOD
COST FOR A CHANNEL = $15 PER TIME PERIOD
OPERATING CHARACTERISTICS

THE PROBABILITY OF NO UNITS IN THE SYSTEM 0.3333
THE AVERAGE NUMBER OF UNITS IN THE WAITING LINE 1.3333
THE AVERAGE NUMBER OF UNITS IN THE SYSTEM 2.0000
THE AVERAGE TIME A UNIT SPENDS IN THE WAITING LINE 0.3333
THE AVERAGE TIME A UNIT SPENDS IN THE SYSTEM 0.5000
THE PROBABILITY THAT AN ARRIVING UNIT HAS TO WAIT 0.6667
THE TOTAL COST PER TIME PERIOD $65.00
Number of Units in the System Probability
 
0 0.3333
1 0.2222
2 0.1481
3 0.0988
Results: a 20% increase in the service rate
(going from 5 to 6 documents per hour) results in a 58% reduction in
document waiting time in line (0.80 hours reduced to 0.333 hours).
The reduction in waiting time shows up in the system hourly cost of
$65 versus $115 before the training. Now, we can address the cost of
training as how much we are willing to spend in order to enjoy the
$50 per hour savings.
By the way, this analysis took five minutes, including getting a cup
of coffee! Seriously, the time consuming part is gathering the data.
Every waiting line analysis I have done means getting out a clip
board and a watch, and timing how many arrivals come into the system
in an hour (or a day, or a month  whatever the appropriate time
unit). Repeat this to get at least 30 observations of the arrival
rate. This could take some time if there are arrival rate differences
throughout the day. For example, a bank teller operation experiences
a different arrival rate between 4 and 5 p.m. on a Friday payday than
that which is experienced between 9 and 10 a.m. on a slow Tuesday. If
the bank wants to study the arrival rate on Friday paydays, they need
a lot of Fridays to gather enough observations to get the average
arrival rate. After you get the arrival rate, then you have to use
your watch and clipboard to measure the service time of the teller,
the document clerk, the aircraft unloading team  whatever your
server is. at 4 p.m. on a Friday payday, do this for maybe 30
observations (30 hours).
Back to management strategies. Suppose instead of training,
management decides to hire another document clerk. If we make the
assumption that documents can randomly arrive in a line in front of
each document clerk (actually would probably go into a file basket)
and the clerks allow the documents to be patient (one clerk doesn't
steel "easy to process" documents from the other's in basket, for
example), then we can use the models of this section to compute the
operating characteristics.
The only adjustment we need to make is to assume that the arrival
rate is cut in half to 2 documents per hour for each clerk. This
assumption makes sense if we assume that adding the clerk does not
impact the arrival rate of documents to the system in the short run.
So, our arrival rate is 2 per hour, the service rate is 5 per hour
(for each clerk), and the costs remain at $25 for waiting and $15 for
service.
Here are the results:
Printout 4.2.3
WAITING LINES
*************
NUMBER OF CHANNELS = 1
POISSON ARRIVALS WITH MEAN RATE = 2
EXPONENTIAL SERVICE TIMES WITH MEAN RATE = 5
COST FOR UNITS IN THE SYSTEM = $25 PER TIME PERIOD
COST FOR A CHANNEL = $15 PER TIME PERIOD
OPERATING CHARACTERISTICS

THE PROBABILITY OF NO UNITS IN THE SYSTEM 0.6000
THE AVERAGE NUMBER OF UNITS IN THE WAITING LINE 0.2667
THE AVERAGE NUMBER OF UNITS IN THE SYSTEM 0.6667
THE AVERAGE TIME A UNIT SPENDS IN THE WAITING LINE 0.1333
THE AVERAGE TIME A UNIT SPENDS IN THE SYSTEM 0.3333
THE PROBABILITY THAT AN ARRIVING UNIT HAS TO WAIT 0.4000
THE TOTAL COST PER TIME PERIOD $31.67
Number of Units in the System Probability
 
0 0.6000
1 0.2400
2 0.0960
3 0.0384
4 0.0154
5 0.0061
6 OR MORE 0.0041
We have to be a little careful since these
results are for the single server/single channel system, meaning for
one document clerk. We can assume that the average time a document
spends in line and in the system are representative for both servers
if both document clerks operate at the same service rate (if this
assumption isn't t true, then we simply run two different analysis,
one for each of the two different service rates). However, we would
have to double the average number of units waiting in line and in the
system since there are two systems in our new configuration. The cost
of this system looks favorable, but remember, we have to double it
since we have two systems. That gives a new total cost of $63.34 
just a little better than the training solution.
But wait a minute! This isn't the best way to add a server, although
this is what grocery stores do  when a new server is added (that is,
another cash register opens), a new line is forced to form in front
of that server.
However, rather than adding a line in front of each server in a
waiting line system, it is more efficient to keep a single line that
feeds the multiple servers as is done in banks and at Disney. Single
line/multiple service channel systems allow the line to discipline
the slowest server; and keep customers patient and
happy. Every time I go to Dunkin Donuts or the
Publix Deli, I try to talk the customers around me to form a single
line saying, "it will minimize your average waiting time in the queue
compared to the chaos of haphazard clustering around the counter 
trust me." Most of the time they just look at me with an expression
that says, "you need a hobby or something."
We are going to look at the single line/multiple server channel
system next.
4.3: Other Waiting Line
Models
Single Line/Multiple Channel System with Poisson Arrivals and
Exponential Service Time
My favorite waiting line model is this one  I wish more service
activities would adopt it. Everything is the same as with the Single
Line/Single Channel System except there are multiple service
channels.
So, for our example, the arrival rate is 4 documents per hour, the
service rate is 5 documents per hour, the cost of waiting is $25 per
hour, and the cost of service is $15 per hour per server channel.
Let's go to The Management Scientist to get the operating
characteristics and costs for a single line/two server channel
system.
Printout 4.3.1
WAITING LINES
*************
NUMBER OF CHANNELS = 2
POISSON ARRIVALS WITH MEAN RATE = 4
EXPONENTIAL SERVICE TIMES WITH MEAN RATE = 5 PER CHANNEL
COST FOR UNITS IN THE SYSTEM = $25 PER TIME PERIOD
COST FOR A CHANNEL = $15 PER TIME PERIOD
OPERATING CHARACTERISTICS

THE PROBABILITY OF NO UNITS IN THE SYSTEM 0.4286
THE AVERAGE NUMBER OF UNITS IN THE WAITING LINE 0.1524
THE AVERAGE NUMBER OF UNITS IN THE SYSTEM 0.9524
THE AVERAGE TIME A UNIT SPENDS IN THE WAITING LINE 0.0381
THE AVERAGE TIME A UNIT SPENDS IN THE SYSTEM 0.2381
THE PROBABILITY THAT AN ARRIVING UNIT HAS TO WAIT 0.2286
THE TOTAL COST PER TIME PERIOD $53.81
Number of Units in the System Probability
 
0 0.4286
1 0.3429
2 0.1371
3 0.0549
4 0.0219
5 0.0088
6 OR MORE 0.0059
As anticipated, this configuration gives the best results of all
alternatives tried so far. Sorter waiting times and fewer documents
in the line and system result in lower costs. Can we improve on the
single line/two service channel configuration by adding a third
server? Just go back to The Management Scientist, input 3 as
the number of service channels, and rerun the solution. The results
show the waiting time decreasing but the total cost increasing to
$65. This means, 2 servers with a single waiting line is the optimal
solution to this problem. While the waiting time did decrease, the
decrease from 2 to 3 servers is less than from 1 to 2 servers; but
the cost of service goes up at a flat rate of $15 per hour. The net
effect is that the total cost curve is beginning to rise when we go
to 3 servers.
I hope you are getting as excited about waiting line models as I am!
We've just scratched the surface but the remaining models just
respond to changes in the assumptions. Let me summarize other popular
waiting line models.
Poisson Arrivals, Deterministic Service Time, One Channel
The Management Scientist Waiting Line Module includes the
model used to analyze single line/single service channel systems that
assume deterministic service times. This is the case of automated
service, when there is no variation in the time to serve
customers.
Poisson Arrivals, Arbitrary Service Time, One Channel
The Management Scientist Waiting Line Module includes the
model used to analyze single line/single service channel systems that
assume service times are not random and independent. Recall that when
service times are random and independent, we can model them with the
Exponential Distribution. But what if your service times for your
project follow the Beta Distribution like in Pert/CPM. Recall that
activity times are represented by three time estimates: optimistic,
most likely and pessimistic. Recall also that we can compute the mean
and standard deviation for this distribution with no trouble at all.
To work this model in The Management Scientist, just select
Poisson Arrivals/Arbitrary Service (1 Channel) in the Model
selection screen, and the next screen asks you for the mean and
standard deviation of the service time.
Poisson Arrivals/Arbitrary Service (No Waiting)
This waiting line system is like a telephone system that does not
allow customers to wait if the telephone server is busy. The
Management Scientist has this option but don't expect a lot of
operating characteristic output. That is obviously because with no
waiting, there is no average wait time, average number of units in
the line, etc. Multiple channels are allowed.
Poisson Arrivals/Exponential Service (Finite Population)
This waiting line system is designed for scenarios in which there
is a finite customer base, where finite is defined as less than 30
customers. At Ramstein Air Base we had a detachment of 8 C130 cargo
aircraft. The aircraft maintenance operation was really a waiting
line system with the average arrival rate being the the number of
aircraft that would go into maintenance per day. The service time was
the average time to repair or inspect/maintain the aircraft. But
since there were only 8 "customers", we would use this model to
analyze the operating characteristics of the system. This model is
also included in The Management Scientist.
You should be ready to tackle the assignment for Module 4, "Airline
Reservations," in the text, pp. 646647. The case answers via email
and The Management Scientist computer output files are due
March 10, 2001. If you want a "free" review of your draft
responses/output, please forward as a draft by Tuesday, March 6,
2001.


