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“Queuing theory was born in the early 1900s with the work of A. K. Erlang of the Copenhagen Telephone Company, who derived several important formulas for teletraffic engineering that today bear his name. The range of applications has grown to include not only telecommunications and computer science, but also manufacturing, air traffic control, military logistics, design of theme parks, and many other areas that involve service systems whose demands are random.” (Cooper, 2000)
Queuing Theory
Fun facts
What is Queuing theory?
Queuing theory is the study of lines – all kinds of lines. Examples include supermarket checkout lines, the virtual line of people waiting to purchase concert tickets online, a line of cars waiting to pay a toll at the tollbooth, or the group of people waiting for someone, anyone, to answer the phone at the local Domino’s pizza joint to order carry out (Stevenson, 2012).
In its broadest sense, queuing theory is the study of contention for the use of a shared, but limited, resource (Simley, 2013). It is comprised of models and formulas that describe the relationships between service requests, congestion, and delay (Beasley, n.d.)
“Queuing theory is used to study the phenomenon of waiting in lines. Some people use the information gathered from queuing theory in order to determine how to best serve customers and so prevent them from waiting in line longer than they have to. The theory allows researchers to analyze several things such as arriving in line, waiting in line, and the time it takes to service customers. This allows them to gather and derive information on a customer’s waiting time, the expected amount of customers that will be in a line, the probability of a customer encountering a line, as well as other data. This information is used in order to find ways to reduce lines and wait time.” (“What is Queuing”, 2013)
Visitors wait at the Eiffel Tower after an elevator broke down.
Photo by KenzobTribouillard/AFP/Getty Images.

Queuing theory, the mathematical study of waiting in lines, is a branch of operations research because the results often are used when making business decisions about the resources needed to provide service. At its most basic level, queuing theory involves arrivals at a facility (i.e., computer store, pharmacy, bank) and service requirements of that facility (i.e., technicians, pharmacists, tellers).
History behind Queuing Theory
“The first to develop a viable queuing theory was the French mathematician S.D. Poisson (1781-1840). Poisson created a distribution function to describe the probability of a prescribed outcome after repeated iterations of independent trials. Because Poisson used a statistical approach, the distributions he used could be applied to any situation where excessive demands are made on a limited resource.” (Simley, 2013)
MIT Professor Dick Larson is America's foremost scholar of queuing theory earning him the nickname “Dr. Queue.” According to Larson, queuing theory got its start about 100 years ago in Denmark, necessitated by a booming new technology: the telephone. The equations developed to calculate the probability of how many incoming calls, how many operators, and how many switches were required to handle daily call volume are the basis for the math behind queuing theory today. (Stevenson, 2012)
Queuing Management
Queuing management consists of three major components:

1. How customers arrive
2. How customers are serviced
3. The condition of the customer exiting the system

Arrivals are divided into two types:

1. Constant – exactly the same time period between successive arrivals (i.e., machine controlled).
2. Variable – random arrival distributions, which is a much more common form of arrival.
A good rule of thumb to remember the two distributions is that time between arrivals is exponentially distributed and the numbers of arrivals per unit of time is Poisson distributed.

Servicing or Queuing Systems
The servicing or queuing system consists of the line(s) and the available number of servers. Factors to consider include the line length, number of lines and the queue discipline. Queue discipline is the priority rule, or rules, for determining the order of service to customers in a waiting line. One of the most common used priority rules is “first come, first served” (FCFS). Others include a reservations first, treatment via triage (i.e., emergency rooms of hospitals), highest-profit customer first, largest orders first, “best” customers first and longest wait-time first. An important feature of the waiting structure is the time the customer spends with the server once the service has started. This is referred to as the service rate: the capacity of the server in numbers of units per time period (i.e., 15 orders per hour).

Another important aspect of the servicing system is the line structure. There are four types: single-channel/single-phase; single-channel/multi-phase; multi-channel/single-phase; and multi-channel/multi-phase. The simplest type of waiting line structure is the single-channel, single-phase. Here, there is only one channel for arriving customers and one phase of the service system. An example is the drive-through window of a dry-cleaning store or bank.
There are two possible outcomes after a customer is served. The customer is either satisfied or not satisfied and requires re-service.
Waiting Line Models and Equations
Table 1 shows the four types of commonly used waiting line models, along with key properties and examples. Table 1: Four Types of Waiting Line Models

Types of Queuing Systems

Single Channel – Single Phase system: In which there is a single queue of customers waiting for service and only one phase of service is involved. An example is flu vaccination camp where a nurse practitioner is the server who does all the work (paper work and vaccination).

In this case there’s still a single queue
but the service involves multiple phases. For example a PCP’s office. Patients first arrive at the registration counter, get the registration done and then again wait
in a queue for being seen by the physician or for ancillary services. There is queue
formation or waiting time involved at each phase of the system.

Single Channel–Multiple Phase System:
In this type of queuing system,
customers form multiple queues, waiting for the service which involves only one
phase. Customers also have the liberty to switch from one line to the other. An
example is customers waiting at the pharmacy store.

Multiple Channel-Single Phase System:
Multiple Channel-Multiple Phase System:
This type of system has numerous
queues and a complex network of multiple phases of services involved. This type
of service is typically seen in a hospital setting, ER, multi-specialty outpatient
clinics, etc. For example in an hospital outpatient clinic, patient first forms the
queue for registration, then he/she is triaged for assessment, then for diagnostics,
review, treatment, intervention or prescription and finally exit from the system or triage to different provider.

Multiple Line Single Phase: Walgreens Drive-Thru Pharmacy
Team 4 Queuing Analysis. (n.d.). Retrieved July 27, 2013, from home.snc.edu
There are basically four types of queuing systems and different combinations of the same can be used for very complex networks.
Single Channel Single Phase: Trucks unloading shipments into a dock.
Team 4 Queuing Analysis. (n.d.). Retrieved July 27, 2013, from home.snc.edu
• Single Line Multiple Phase: Wendy’s Drive Thru -> Order + Pay/Pickup
Team 4 Queuing Analysis. (n.d.). Retrieved July 27, 2013, from home.snc.edu
• Multiple Line Multiple Phase: Hospital Outpatient Clinic, Multi-specialty
Team 4 Queuing Analysis. (n.d.). Retrieved July 27, 2013, from home.snc.edu
“In designing queuing systems we need to aim for a balance between service to customers (short queues implying many servers) and economic considerations (not too many servers).
In essence all queuing systems can be broken down into individual sub-systems consisting of entities queuing for some activity.

Designing a queuing system
Typically we can talk of this individual sub-system as dealing with customers queuing for service. To analyze this sub-system we need information relating to:
* how customers arrive e.g. singly or in groups (batch or bulk arrivals)
* how the arrivals are distributed in time (e.g. what is the probability distribution of time between successive arrivals (the interarrival time distribution))
* whether there is a finite population of customers or (effectively) an infinite number

The simplest arrival process is one where we have completely regular arrivals (i.e. the same constant time interval between successive arrivals). A Poisson stream of arrivals corresponds to arrivals at random. In a Poisson stream successive customers arrive after intervals which independently are exponentially distributed. The Poisson stream is important as it is a convenient mathematical model of many real life queuing systems and is described by a single parameter - the average arrival rate. Other important arrival processes are scheduled arrivals; batch arrivals; and time dependent arrival rates (i.e. the arrival rate varies according to the time of day).

Arrival process:

a description of the resources needed for service to begin
how long the service will take (the service time distribution)
the number of servers available
whether the servers are in series (each server has a separate queue) or in parallel (one queue for all servers)
whether preemption is allowed (a server can stop processing a customer to deal with another "emergency" customer)

Service Mechanism:
Assuming that the service times for customers are independent and do not depend upon the arrival process is common. Another common assumption about service times is that they are exponentially distributed.
*how, from the set of customers waiting for service, do we choose the one to be served next (e.g. FIFO (first-in first-out) - also known as FCFS (first-come first served); LIFO (last-in first-out); randomly) (this is often called the queue discipline)
* do we have:
 *balking (customers deciding not to join the queue if it is too long)
 *reneging (customers leave the queue if they have waited too long for service)
 *jockeying (customers switch between queues if they think they will get served faster by so doing)
 *a queue of finite capacity or (effectively) of infinite capacity
Changing the queue discipline (the rule by which we select the next customer to be served) can often reduce congestion. Often the queue discipline "choose the customer with the lowest service time" results in the smallest value for the time (on average) a customer spends queuing.”

Queue Characteristics:
The Importance of Queuing Theory
Why is queuing theory relevant in business today? Virtually any type of operational question will bump into queuing theory at some point. How should you handle the bottleneck of the slowest piece of equipment on a production line? How many elevators will be required to handle peak traffic within a skyscraper in Manhattan’s financial district? How can you maximize impulsive shopping while waiting in line at the store? Larson determined that not only is it critical to address the problem of the line, but also to address how people experience the wait.
Queuing theory may be extended to cover a wide variety of contention situations, such as how customer check-out lines form (and how they can be minimized), how many calls a telephone switch can handle, how many computer users can share a mainframe, and how many doors an office building should have. More generally, queuing theory is used in business settings primarily in operations management and research problems such as production scheduling, logistics/distribution, and computer network management. These are diverse applications, but their solutions all involve the same dynamics.
“There are several queuing disciplines that have been developed because of queuing theory—four of which are First In First Out (FIFO), Last In First Out (LIFO), Processor Sharing, and Priority. FIFO describes the practice of serving customers in the order they arrive in so that the person waiting longest is served first, while LIFO describes the practice of serving customers so that the person who comes in last leaves first, such as in the case of riding an elevator. Processor sharing serves customers at the same time to that the average waiting time for all customers is about the same. The Priority discipline serves the customer with the highest priority first. It is important to note that these disciplines can be applied to applications other than customer service.” (“What is Queuing”, 2013)
Queuing theory in business
“Examples of how queuing theory works is present in many aspects of everyday life. At bank tellers and credit unions, one may see one line and multiple tellers. This happens to help ensure that one slow transaction does not hold up the entire line. Some stores open more registers if there are more than three people waiting in a line. There are also other stores that have roaming clerks. These clerks ring up purchases and give customers a number so that the cashier can complete the transaction quickly, thus reducing waiting times for all.” (“What is Queuing”, 2013)
“Applications of queuing theory are used in many aspects of business, customer service, commerce, industry, healthcare, and engineering. Customer service applications can especially make use of the information gathered by queuing theory. This information can be used in order to make decisions on the kind of resources needed to provide service for customers. The data can be also be applied to call centers, network server queuing, telecommunications, and traffic flow. It can even be used to dictate what type of line customers will be standing in while waiting for different types of service.” (“What is Queuing”, 2013)
“Queuing theory in relation to restaurants involves numerous factors, including when large numbers of customers can typically be expected to arrive, the amount of time customers usually spend in the restaurant and the number of parties expected to linger at their table long after they finish eating. Fast-food restaurants commonly have their patrons waiting in a single queue to receive their orders and then sit down or leave with their meals. Other facilities must address psychological issues associated with queuing theory to ensure customer satisfaction. A goal of using queuing theory for restaurant analysis is to improve customer wait times. This might be accomplished by adding more servers at certain times of the day, on busier days or after special events taking place in the area. If it's not possible to speed up service, you also can work to improve the waiting experience”. (Moore, n.d.)
The Cost of Waiting in Line
The problem in virtually every queuing situation is a trade-off decision. The manager must weigh the added cost of providing more rapid service (i.e., more checkout counters, more production staff) against the inherent cost of waiting. For example, if employees are spending their time manually entering data, a business manager or process improvement expert could compare the cost of investing in bar-code scanners against the benefits of increased productivity. Likewise, if customers are walking away disgusted because of insufficient customer support personnel, the business could compare the cost of hiring more staff to the value of increased revenues and maintaining customer loyalty.
The relationship between service capacity and queuing cost can be expressed graphically (Figure 1). Initially, the cost of waiting in line is at a maximum when the organization is at minimal service capacity. As service capacity increases, there is a reduction in the number of customers in the line and in their wait times, which decreases queuing cost. The optimal total cost is found at the intersection between the service capacity and waiting line curves.
Figure 1: Service Capacity vs. Cost
Source: Richard B. Chase and Nicholas J. Aquilano, Production and Operations Management, 1973, page 131.
Think about all the ways businesses strive to entertain you and prevent boredom while you are waiting in line. Everything from the TV in the dentist’s office, hold music while on the phone, mirrors next to elevators, and the absolute master of queuing theory – Walt Disney, who can make a two hour wait for an 8 minute ride seem like fun! Did you know that Disney pads the wait times displayed on popular rides so that you always get to the front of the line in less time than you expected?
Expectations are everything. There have been incidents of Queue Rage when someone has cut in front of another person in a long line, sometimes even resulting in knives or guns being drawn! We make exceptions to the American fairness rule of first come first serve when it comes to priority of service at the emergency room or with the addition of an express lane in the grocery store. The serpentine line was invented to solve people cutting in lines at banks, theaters and fast food restaurants because the “line” is marked off with ropes or cones to direct the travel path.
Ever had to wait in a line in a foreign country? In China, it’s considered socially acceptable to push your way to the front of a line. This is because what appears to an American to be a line is really a one-dimensional mob and any sign of weakness will be pounced upon and shoved in front of. (Chen, 2010) Conversely, in England, queuing is an art form and taken very seriously. A man waiting by himself for the bus will form an orderly queue of one.
Why can’t someone apply queuing theory formulas to the ladies public restroom? There are never enough stalls to handle the volume of women – no matter what the venue.
Rite Aid Case Study
In March 2010, Rite Aid Pharmacy introduced a bold new marketing promotion strategy: Rite Aid will fill your prescriptions in 15 minutes, guaranteed (customers are given a $5 Rite Aid Gift Card for prescriptions not filled within 15 minutes). To understand how Rite Aid could make such a guarantee, I visited a few stores in the Atlanta metro area, where I live. As a good Six Sigma practitioner, I carefully watched the process of how prescriptions were filled, talked with the employees and took notes. Rite Aid uses a single-channel/multi-phase queuing model with one technician and one pharmacist. Prescription filling is not a serial process. While the technician is handling customers (drop off and pick up), the pharmacist fills prescriptions. Table 3 reflects the high-level, three-step process and key activities.
From my cursory analysis using queuing model equations, I was able to see how Rite Aid could make such a bold guarantee. If Rite Aid wanted to reduce the waiting time to, say, 10 minutes, they could either reduce the arrival rate (i.e., get customers “out-of-line” by encouraging prescriptions through telephone or email; embrace the use of automatic prescription routing from the doctor via e-commerce applications) or improve the service rate through technology (i.e., robotic pharmacy dispensing systems
“In manufacturing, queuing is a necessary element of flexible systems in which factors of production may be continually adjusted to handle periodic increases in demand for manufacturing capacity. Excess capacity in periods of low demand may be converted into other forms of working capital, rather than be forced to spent those periods as idle, nonproductive assets.
The concept of flexible manufacturing systems is very interesting. Consider that today a company such as Boeing endures long periods of low demand for its commercial aircraft (a result of cycles in the air transportation industry), but must quickly tool up for expanded production when demand rises. The company must alternately open and mothball millions of dollars worth of manufacturing capacity (and hire and lay off thousands of workers) through every demand cycle.
During periods of low demand, floor space, machinery and inventories remain tied up. If, on the other hand, demand flows could be better managed, Boeing could convert these assets to more productive applications. The primary focus of queuing theory on flexible manufacturing remains centered on machine reliability and depreciation and processing and cycle times.” (Simley, 2013)
The nature of the queue is one of cost shifting and burden averaging. A provider of some service whose resources are limited may serve only a small number of people at a time. Any number of people beyond that are obliged to wait their turn.

Assuming everyone's time is worth something, those who must wait for service are expending a valuable possession—their time. By waiting in line, the service provider is ensured that none of his resources will stand idle. In effect, the waiting customer is forced to pay in time for the privilege of being served, shifting costs from the service provider to the customer.
In a post office, where there is usually one line but several clerks, the next person to be served is the one who has stood longest in line. The burden of the wait is shared by all those in line—the larger the line, the longer the average wait.
The burden is less equally shared in a grocery store, where each clerk has a line. If one line happens to have five simple orders, and another has five patrons with large orders, coupons, and fruit to weigh, the simple line will probably move much faster. Those lucky enough to be in that line will be served before those in the complex line. Thus, grocery shoppers will not share the burden of the wait as equally as in the post office.
Project 6 Team 1
Danielle Butterfield
Joe Castorena
Amy M Miller
Christopher Antony Reid
Liya Kolina Vizcarra
The question for the service provider is simple: how to provide good service. Indeed, the highest level of service would be achieved by providing resources equal to the number of patrons; a cashier for every shopper. However, this would be extremely costly and logistically impractical; dozens of cashiers would stand idle between orders.

To minimize costs, the manager may provide only one cashier, forcing everyone into a long, slow-moving line. Customers tiring of the wait would be likely to abandon their groceries and begin shopping at a new store. The question arises: what is an acceptable level of service at an acceptable cost to the provider.
These examples seem simple, but the questions they raise extend far beyond the average grocery store. Queuing theory is the basis for traffic management—the maintenance of smooth traffic flow, keeping congestion and bottlenecks to a minimum. Once the nature of the traffic flow is understood, solutions may be offered to ease the demands on a system, thereby increasing its efficiency and lowering the costs of operating it.” (Simley, 2013)
Moore, S. (n.d.). An Example of the Queuing Theory for Restaurants | Chron.com. Small Business - Chron.com. Retrieved July 28, 2013, from http://smallbusiness.chron.com/example-queuing-theory-restaurants-35246.html
Simley, J. (2013). QUEUING THEORY. Retrieved July 25, 2013, from http://www.referenceforbusiness.com/encyclopedia/Pro-Res/Queuing-Theory.html
Chen, R. (2010, December 9). What Appears Superficially to be a Line is Actually Just a One-Dimensional Mob. Retrieved July 22, 2013, from The Old New Thing: http://blogs.msdn.com/b/oldnewthing/archive/2010/12/09/10102459.aspx
Stevenson, S. (2012, June 1). What You Hate Most About Waiting in Line. Retrieved July 22, 2013, from Slate - Queuing Theory: http://www.slate.com/articles/business/operations/2012/06/queueing_theory_what_people_hate_most_about_waiting_in_line_.2.html
Beasley, J. (n.d.). Queueing theory.Brunel. Retrieved July 28, 2013, from http://people.brunel.ac.uk/~mastjjb/jeb/or/queue.html
Singh, V. (2006). Use of Queuing Models in Health Care. Retrieved July 28, 2013, from http://works.bepress.com/cgi/viewcontent.cgi?article=1003&context=vikas_singh&sei-redir=1&referer=http%3A%2F%2Fsearch.yahoo.com%2Fsearch%3B_ylt%3DA0oG7lV0P_VR30sA4BxXNyoA%3Fp%3Duses%2Bof%2Bqueuing%2Btheory%26ei%3DUTF-8%26fr%3Dyfp-t-900-s%26pstart%3D1%26b%3D21#search=%22uses%20queuing%20theory%22

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