Module Three Notes
"Waiting Line Models"

 

Index to Module Three Notes

3.1: Structure of Waiting Lines

3.2: Single Channel/Single Server

3.3: Other Waiting Line Models

 

"No matter what line I get in, it becomes the slowest."

Carol Harrington
Shipboard in the Bahamas,
February 6, 2000

 


 3.1: 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 un-served. 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 increases the cost of service, but produces 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 3.1.1.


Figure 3.1.1

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 Ό hours. The mean time between arrivals is 1/4th 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 = 5/60th 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 (10/60th 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.


3.2 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, P0):

P0 = 1 - (lambda/mu) = 1 - (4/5) = 0.20

There 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 (Lq):

Lq = lambda2/ [mu(mu - lambda) ] = 42 / [5 (5-4)] = 3.2 documents

3. The average number of units in the system (L):

L = Lq + (lambda / mu) = 3.2 + (4/5) = 4 documents

From 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:

Wq = Lq / lambda = 3.2 / 4 = 0.8 hours or 48 minutes.

Note:
The initial time units of this and other time-related 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 = Wq + (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:

Pw = lambda / mu = 4/5 = 0.80

This 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:

Pn = ( lambda/mu)n * P0 = (4/5)3 * 0.2 = 0.1024

Since we are working with discrete probabilities, to find the probability of three or less documents in the system:

Pn < 3 = P0 + P1 + P2 + P3

= 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:

Pn > 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

= ( cw L ) + (cs k ) where
k =number of channels
cw = cost of waiting = $25.00 per hour for 1 paralegal
cs = cost of service = $15.00 per hour for clerk

Total 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:

Pw = lambda / mu = 4.5 / 5 = .90, a 12.5% increase as well.

The line length now increases to:

Lq = lambda2/ [mu(mu - lambda) ] = 4.52 / [5 (5-4.5)] = 8.1

documents, 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 3.2.1

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 1/5th hour, and 1/5th 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 10/60th 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 3.2.2 provides the operating characteristic and cost results.


Printout 3.2.2

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 3.2.3

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.


3.3: 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 3.3.1

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 – sometimes referred to as the machine repair problem.   At Ramstein Air Base we had a detachment of 8 C-130 cargo aircraft. The aircraft maintenance operation was really a waiting line system with the average arrival rate being 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. The textbook offers additional reading for this special case and illustrates the slight modification to the formulas introduced in the general waiting line with infinite population – see pages 625 through 629, section 14.9.  This model is also included in The Management Scientist.

You should be ready to tackle the assignment for Module 3.

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