This document provides an overview of queuing theory and the key components of a queuing system. It discusses the calling population or customers arriving for service, including characteristics like size, behavior, and arrival patterns. The queuing process and queue discipline are also examined. Finally, the document outlines various performance measures used to evaluate queuing systems, such as average wait time, number of customers, probability of waiting, and operating costs.
Department of Management- Queuing Theory
Queue is formed because:-
The service facility is limited & the arrivals are infinite.
The mismatch between service facility & arrivals
TERMINOLOGY
QUEUEING SYSTEM
ARRIVAL PROCESS
CATEGORIES OF CUSTOMERS
SERVICE PROCESS
REPRESENTATION OF QUEUEING SYSTEM
NOTATION
MCM,MCA,MSc, MMM, MPhil, PhD (Computer Applications)
Working as Associate Professor at Zeal Education Society, Pune for MCA Progrmme.
Having 18 Years teaching experience
Department of Management- Queuing Theory
Queue is formed because:-
The service facility is limited & the arrivals are infinite.
The mismatch between service facility & arrivals
TERMINOLOGY
QUEUEING SYSTEM
ARRIVAL PROCESS
CATEGORIES OF CUSTOMERS
SERVICE PROCESS
REPRESENTATION OF QUEUEING SYSTEM
NOTATION
MCM,MCA,MSc, MMM, MPhil, PhD (Computer Applications)
Working as Associate Professor at Zeal Education Society, Pune for MCA Progrmme.
Having 18 Years teaching experience
Hospital waiting times in Ireland - and why we need to do something about this and hold those that govern us accountable to tight targets and timelines for improvement
OPD is the mirror of the hospital, which reflects the functioning of the hospital being the first point of contact between the patient and the hospital staff.
Patients visit the OPD for various purposes, like consultation, day care treatment, investigation, referral, admission and post discharge follow up. Not only for treatment but also for preventing and promotive services like, health check up, Immunisation, Physio-therapy and so on.
Validate data
Questionnaire checking
Edit acceptable questionnaires
Code the questionnaires
Keypunch the data
Clean the data set
Statistically adjust the data
Store the data set for analysis
Analyse data
3 Things Every Sales Team Needs to Be Thinking About in 2017Drift
Thinking about your sales team's goals for 2017? Drift's VP of Sales shares 3 things you can do to improve conversion rates and drive more revenue.
Read the full story on the Drift blog here: http://blog.drift.com/sales-team-tips
Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...IOSR Journals
Waiting lines and service systems are important parts of the business world. In this article we describe several common queuing situations and present mathematical models for analyzing waiting lines following certain assumptions. Those assumptions are that (1) arrivals come from an infinite or very large population, (2) arrivals are Poisson distributed, (3) arrivals are treated on a FIFO basis and do not balk or renege, (4) service times follow the negative exponential distribution or are constant, and (5) the average service rate is faster than the average arrival rate. The model illustrated in this Bank for customers on a level with service is the multiple-channel queuing model with Poisson Arrival and Exponential Service Times (M/M/S). After a series of operating characteristics are computed, total expected costs are studied, total costs is the sum of the cost of providing service plus the cost of waiting time. Finally we find the total minimum expected cost.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A Survey on Patient Queue Management SystemIJAEMSJORNAL
In this paper, we study the various types of Queue and Queue Management System. Queue system can successfully reduce waiting time of the patients in the hospitals. We aim to implement a model that initializes alert notification via SMS to patients of a hospital. It will minimize the queue of patients in the waiting area of hospital and also patient can book an appointment from anywhere at a given time. Patients can book an appointment via android application. and accordingly patients and also it will provide navigation towards that nearest hospital. Hence, for developing such a model a survey has been done on various types of queues.
Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.
Developed to provide models for forecasting behaviors of systems subject to random demand
The first problems addressed concerned congestion of telephone traffic
Erlang observed that a telephone system can be modeled by Poisson customer arrivals and exponentially distributed service times
Molina, Pollaczek, Kolmogorov, Khintchine, Palm, Crommelin followed the track
Common phenomenon of everyday life
Line maybe People / Items
Examples
– Grocery shop, Bank, Petrol refilling units, Automobile Service station, Airplane, Train etc.
A Case Study of Employing A Single Server Nonpreemptive Priority Queuing Mode...IJERA Editor
This paper discusses a case study of employing a single server nonpreemptivepriorityqueuing model [1]at ATM
machine which originally operates on M/M/1 model. In this study we have taken two priority classes of people
in following order:-
.Priority class 1- woman
.Priority class 2- man
Sometimea long queue is formed at ATMmachine (single server)but the bank management don’t have enough
money to invest on installing new ATM machine.In this situation we want to apply single server nonpreemptive
priority queuing model.The security guard at the ATM will divide the customers in two category and arrange the
customers in the above said priority order Thuspriority class 1 people willreceive theatm service ahead of
priority class 2 people.This will reduce the waiting time of priority class 1 people. Of course by doing this the
waiting time of priority class 2will increase.
Managing Waiting Lines A big part of most people’.docxjoyjonna282
Managing Waiting Lines
A big part of most people’s day and consequently, life is spent waiting— waiting for the stop
light to turn green, waiting in a traffic jam, waiting at store checkouts, waiting for rides at
amusement parks. . . the list goes on and on. In a life time, it is estimated that a person
spends about five years waiting in lines.
If the queue is long, people get restless, anxious and even bored. Sometimes people will not
even join the queue if they see a long line (balking) or even walk away in disgust after
having spent some time in the queue (reneging).
Traditionally, a queue is pictured as a counter or a checkout with a server and a line of
customers in front and one person being served at a time. But queues can take other forms,
such as the following:
• Bus transportation or rides, where the customers receive the service in batches or in
bulk.
• Fire or ambulance service, when the service provider goes to the customer to provide
service.
• Customer service, when the customers are in a queue over the phone instead of
physically in a line.
• Medical clinic or hospital, where the patients may join a network of queues as they
move within the system to receive different types of service.
For businesses, especially service businesses, waiting lines impact on customer satisfaction
and the operating costs. If the number of servers is increased, the operating costs increase
but customer waits decrease and the customer satisfaction is impacted positively. The
reverse is equally true as well. Since services are more labor intensive in general compared
to manufacturing, balancing the number of servers and the customer waiting time can have
a significant impact on the health and the bottom line of the organization.
It is not difficult to quantify the cost of providing the service since the wage and other
expenses are known. If the customer is an internal employee, it is possible to quantify the
cost of keeping the customer waiting in terms of the productive wage that is lost in waiting.
What is really difficult to quantify is the cost of keeping the customers waiting when the
customers are external.
QSO 610 Module Nine 1
When the cost of keeping the customers waiting can be determined relatively accurately, as
in case of internal customers, the minimization of the total cost approach can be used. When
it is difficult to quantify the cost of keeping the customers waiting, as in case of external
customers, the service level approach is frequently used. In this approach, instead of
quantifying the cost of keeping the customers waiting, a business strives to achieve a
reasonable level of service, such as not letting the average customer wait time exceed five
minutes.
Wherever possible, the wait times should be reduced such as by taking appointments or
reservations. To the extent the wait is inevitable, customer’s reactions to waiting depends on
his or her expectations ...
Business Analytics
Business Intelligence
OLAP
OLTP
BI
BA
architecture of BI
techniques of BI and BA
star schema
online analytical processing
online transactional processing
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
3. Introduction
3By: Jignesh Kariya
A common situation that occurs in everyday life is that of waiting in a line
either at bus stops, petrol pumps, restaurants, ticket booths, bank,
hospital and so on.
Queues (waiting Lines) are also found in workshops where the machines
wait to be repaired ; at a tool crib(cheat) where the mechanics wait to
receive tools; in a warehouse where items wait to be used . Incoming
calls wait to mature in the telephone exchange, trucks wait to be
unloaded, airplanes wait either to take off or land and so on.
Queuing theory can be applied to a variety of situations where it is not
possible to accurately predict the arrival rate (or time) of customers and
service rate (or time) of service facility or facilities.
Queuing theory can be used to determine the level of services that
balances the following two conflicting costs :
4. Introduction
4By: Jignesh Kariya
1. Cost of offering the service
2. Cost incurred due to delay in offering service.
The first cost is associated with the service facilities and their operation,
and the second represents the cost of customers waiting for service.
Obviously an increase in the existing service facilities would reduce the
customer’s waiting time and decreasing the level of service would result
in long queue. This means in the level of service increases the cost of
operating service facilities but the decreases the cost of customers
waiting for service.
Since customer waiting cost for service is difficult to estimate, it is usually
measured in terms of loss of sales or goodwill when the customer is a
human being and has no sympathy with the service system. But if the
customer is machine waiting for repair then cost of waiting is measured
in terms of cost production.
6. The Structure Of A Queuing System
6By: Jignesh Kariya
The major components of any waiting line(queuing) system are :
1. Calling population (or input score)
2. Queuing Process
3. Queue discipline
4. Service Process (or Mechanism)
7. The Structure Of A Queuing System
7By: Jignesh Kariya
Potential customers who arrive to the queuing system is referred as ‘Calling
Population’ also known as ‘customer (input) source’.
The manner in which customers arrive at the service facility, individually, or in
batches, at scheduled or unscheduled time is called the arrival process. The
customer's entry into the queuing system depends upon the queue conditions.
Customers, from a queue, are selected for service according to certain rules
known as queue discipline.
A service facility may be without server (self service), or may include one or
more servers operating either in a series (as a team) or in parallel (multiple
service channels). The rate (constant or random) at which service is rendered is
known as the service process. After the service is rendered, the customer leaves
the system.
If the server is idle at the time of the customer's arrival, then the customer is
served immediately, otherwise the customer is asked to join a queue or wailing
line, which may have single, multiple or even priority lines.
8. The Structure Of A Queuing System
8By: Jignesh Kariya
Calling population Characteristics
The arrivals or inputs to the system are characterized by:
• Size of calling population
• behavior of the arrivals
• Pattern of arrivals at the system
The calling population need not be homogeneous and may consist of
several subpopulations. For example, patients arriving at the OPD of a
hospital are usually of three categories: walk-in patients, patients with
appointments and emergency patients. Each patient class places different
demands on service facility, but the waiting expectations of each category
differ significantly.
Size of calling population
The size of calling population, whether it is homogeneous or consists of
several subpopulations, is considered to be either finite (limited) or infinite
(unlimited).
9. The Structure Of A Queuing System
9By: Jignesh Kariya
Calling population Characteristics cont..
If customer's arrival depends on the number of customers already in the
system (in service plus in queue), the calling population is called limited or
finite.
An example of a finite calling population is a factory only has four
machines, which often require repair/service and two of them (say) are in
working condition. Then at any point in time, there are only two machines
that could possibly require service.
Alternately, if new customer's arrival is independent of the number of
customers already in the system, the calling population is called unlimited
or infinite.
Examples of infinite population include customers arriving at a bank or
super market, students arriving to get admission at a university, cars
arriving at a highway petrol pump, etc.
10. The Structure Of A Queuing System
10By: Jignesh Kariya
Calling population Characteristics cont..
• behavior of the arrivals
If a customer, on arriving at the service system waits in the queue until
served and docs not switch between waiting lines. He is called a patient
customer.
Ex: Machines arrived at the maintenance shop are examples of patient
customers.
Whereas the customer, who waits for a certain time in the queue and
leaves the service system without getting service due to certain reasons is
called an impatient customer.
EX : a customer who has just arrived at a grocery store and finds that the
salesmen arc busy in serving the customers already in the system, will
either wait for service till his patience is exhausted or estimates that his
waiting time may be excessive and so leaves immediately to seek service
elsewhere.
11. The Structure Of A Queuing System
11By: Jignesh Kariya
Calling population Characteristics cont..
The behavior of the arrivals at any queuing system is categorized as :
• Balking Customers do not join the queue either by seeing the
number of customer already in service system or by estimating the
excessive waiting time for the desired service.
• Reneging Customers, after joining the queue, wait for sometime in
the queue but leave before being served on account of certain
reasons.
• Jockeying Customers move from one queue to another hoping to
receive service more quickly(a common scene at a railway booking
window).
12. The Structure Of A Queuing System
12By: Jignesh Kariya
Calling population Characteristics cont..
• Pattern of arrivals at the system
Customers may arrive in batches or individually. These customers may
arrive at a service facility either on scheduled time or on unscheduled time.
The arrival process of customers to the service system is classified into two
categories : Static and Dynamic
In static arrival pattern, the control depends on the nature of arrival rate
(random or constant): In random (or unscheduled) arrivals the times are
random variable and therefore we need to understand the average and
frequency distribution of the times.
In both the cases, the arrival process can be described either by the
average arrival time or by the average inter-arrival time(average time
between two consecutives arrival).
13. The Structure Of A Queuing System
13By: Jignesh Kariya
Calling population Characteristics cont..
The number of unscheduled arrivals to a service facility, in some fixed
period of time, can be studied by a statistical probability distribution such a
Poisson Distribution.
The dynamic arrival process is controlled by both the service facility and the
customers. The service facility adjusts its capacity to match changes in the
service intensity.
The arrival time distribution can be approximated by one of the following
probability distribution.
Poisson Distribution
Exponential Distribution
Erlang Distribution
14. The Structure Of A Queuing System
14By: Jignesh Kariya
Calling population Characteristics cont..
The poisson distribution, a discrete probability distribution, describes the
arrival rate variability i.e. number of random arrivals at a service facility in a
fixed period of time.
Another probability distribution that describes the average time between
arrivals (inter-arrival time) when arrival rate is poisson is called Exponential
Probability distribution.
Let n customers arrive in a time 0 to t. If λ is the expected number of
arrivals in a given time interval 0 to t will be λ *t
P(x=n) = ((λt)n e- λt) / n! For n=0,1,2..
P(x=0) = ((λt)0 e- λt) / 0! = e- λt
15. The Structure Of A Queuing System
15By: Jignesh Kariya
2. Queuing Process
The queuing process refers to the number of queues and their respective
lengths. The number of queues single, multiple or priority queues depend
upon the layout of a service system.
The length (or size) of a queue depends upon operational situations such as
physical space, legal restrictions, and attitude of the customers.
In certain cases, a service system is unable to accommodate more than the
required number of customers at a time. No further customers are allowed
to enter until more space is made available to accommodate new
customers. Such type of situations are referred to as finite (or limited)
source queue.
Examples of finite source queues are cinema halls, restaurants, etc.
16. The Structure Of A Queuing System
16By: Jignesh Kariya
Queuing Process cont..
On the other hand, if a service system is able to accommodate any number
of customers at a time, then it is referred to as infinite (or unlimited) source
queue. For example, in a sales department where the customer orders are
received, there is no restriction on the number of orders that can come in,
a queue of any size can be formed.
In many other situations, when arriving customers find long queue(s) in
front of a service facility, they often do not enter the service system even
though additional waiting space is available. The queue length in such cases
depends upon the attitude of the customers.
For example, when a motorist finds that there are many vehicles waiting at
the petrol station, in most of the cases, he does not stop at this station and
seeks service elsewhere.
17. The Structure Of A Queuing System
17By: Jignesh Kariya
Queuing Process cont..
In some finite source queuing systems, the maximum permissible queue is
of zero length, i.e. no queue is allowed to form. For example, when a
parking space (service facility) cannot accommodate additional incoming
vehicles (customers), the motorists are diverted elsewhere.
Multiple queues at a service facility can also be finite or infinite. But this
has certain advantages such as
• Division of manpower is possible.
• Customer has the option of joining any queue and can also switch to the
end of any other queue.
• Balking behavior of the customers can be controlled.
18. The Structure Of A Queuing System
18By: Jignesh Kariya
3. Queue Discipline
Queue discipline refers to the selections of customers from a queue for
service.
The queue discipline is the order or manner in which customers from the
queue are selected for service.
There are number of ways in which customers in the queue are served.
Some of these are :
a. Static Queue Discipline :
These are based on the individual customer’s status in the queue. Few of
such discipline are :
FCFS : If the customers are served in the order of their arrival, then this
is known as the ‘First Come First Serve’ service discipline.
LCFS : ‘Last Come First Serve’
19. The Structure Of A Queuing System
19By: Jignesh Kariya
Queue Discipline Cont..
b. Dynamic Queue Discipline :
These are based on the individual customer’s attributes in the queue. Few
of such discipline are :
Service In Random Order (SIRO) : under this rule customers are selected
for service at random, irrespective of their arrivals in the service system.
Priority Service : Under this rule customers are grouped in priority
classes on the basis of some attributes such as service time or urgency.
The FCFS rule is used within each class to provide service. The payment
of telephone or electricity bills by cheque or cash are examples of this
discipline.
Pre-emptive priority (Emergency) : Under this rule, an important
customer is allowed to enter into the service immediately after entering
into the system, even if a customer with a lower priority is already in
service.
20. The Structure Of A Queuing System
20By: Jignesh Kariya
Queue Discipline Cont..
That is lower priority customer’s service is interrupted to start the service
for such a customer. This interrupted service is resumed after the priority
customer is served.
Non- pre-emptive priority : In this case an important customer is
allowed to go ahead in the queue but the service his started
immediately on completion of the current service.
4. Service Process (or Mechanism)
The Service process is concerned with the manner in which customers are
serviced and leave the system.
It is characterized by:
The arrangement of service facilities
The distribution of service time
Server’s behavior
Management Policies
21. The Structure Of A Queuing System
21By: Jignesh Kariya
Service Process Cont..
The arrangement of service facilities
The capacity of the service facility is measured in terms of customers who
can be served simultaneously and/or effectively.
The service facilities (or servers) commonly known as Service channels may
be in series, in parallel or mixed.
Single Queue, Single Service Facility
22. The Structure Of A Queuing System
22By: Jignesh Kariya
Service Process Cont..
Single Queue, Multiple Service
facilities in Parallel
23. The Structure Of A Queuing System
23By: Jignesh Kariya
Service Process Cont..
Multiple Queues, Multiple Service
facilities in Parallel
24. The Structure Of A Queuing System
24By: Jignesh Kariya
Service Process cont..
The distribution of service times
Service time is the elapsed time from the beginning to the end of a
customer’s service.
The time taken by the server from the commencement of service to the
completion of service for a customer is known as the “service time”.
A random service time may be described in two ways :
a. Average Service Rate : µ * t
a. Average Length of Service Time : E(T) = 1 / µ
25. Performance Measure of a Queuing System
25By: Jignesh Kariya
The performance measure of any queuing system, which are of a general
interest, for the evaluation of the performance of an existing queuing
system, and for designing a new system in terms of level of service a
customer receives as well as proper utilization of the service facilities are
listed as follows:
1. Time-related questions for the customers
(average (or expected) time spent by a customer in the queue and system)
2. Quantitative questions related to the number of customers
average (or expected) number of customers in the queue and system)
3. Questions Involving value of time both for customers and servers
Value of time both for customers and servers
4. Cost- related questions
Average cost required to operate the queuing system
26. Performance Measure of a Queuing System
26By: Jignesh Kariya
1. Time-related questions for the customers
(a) Wq : What is the average (or expected) time an arriving customer has to
wait in a queue (denoted by (Wq) before being served.
(b) Ws: What is the average (or expected) time an arriving customer
spends in the system (denoted by Ws), including waiting and service. This
data can be used to make economic comparison of alternative queuing
systems.
2. Quantitative questions related to the number of customers
(a) Lq : The expected number of customers who are in the queue (queue
length) for service. This is denoted by Lq
(b)Ls : The expected number of customers who are in the system either
waiting in the queue or being serviced (denoted by Ls,). This data can be
used for finding the mean customer time spent in the system.
27. Performance Measure of a Queuing System
27By: Jignesh Kariya
3. Questions Involving value of time both for customers and servers
(a) Pw : What is the probability that an arriving customer has to wait before
being served (denoted by Pw)? This is also called blocking probability.
(b) p = λ / μ : What is the probability that a server is busy at any particular
point in time (denoted by p )? This is the proportion of the time that a
server actually spends with the customer, i.e. the fraction of the time a
server is busy.
p = λ / μ
(c)Pn : What is the probability of n customers being in the queuing system
when it is in steady state condition? This is denoted by Pn, n = 0, 1....
(d) Pd : What is the probability of service denial when an arriving customer
cannot enter the system because the queue is full? This is denoted by Pd
28. Performance Measure of a Queuing System
28By: Jignesh Kariya
4. Cost-related questions
(a) What is the average cost needed to operate the system per unit of time?
(b) How many servers (service centers) are needed to achieve cost
effectiveness?
29. Performance Measure of a Queuing System
29By: Jignesh Kariya
Transient State and Steady States
At the beginning of service operations, a queuing system is influenced by
the initial conditions, such as number of customers in the system for
service and percentage of time servers are busy serving customers.
This period of transition is termed as transient-state. However, after
sufficient time has passed. the system becomes independent of
the initial conditions and and enters a steady state condition.
In the development of queuing theory models it is assumed that the system
has entered a steady-state.
31. Performance Measure of a Queuing System
31By: Jignesh Kariya
Relationship among Performance Measure
32. Performance Measure of a Queuing System
32By: Jignesh Kariya
Relationship among Performance Measure cont..
33. Classification of Queuing Models
33By: Jignesh Kariya
Different models in queuing theory are classified by using special
notations described initially by D.G.Kendall in 1953 in the form (a/b/c).
Later A.M. lee in 1966 added the symbols d and c to the kendallb
notation.
36. Performance Measure of a Queuing System
36By: Jignesh Kariya
Example – 2
Customers arrive at a box office window, being manned by a single individual,
according to a Poisson input process with a mean rate of 30 per hour. The time
required to serve a customer has an exponential distribution with a mean of 90
seconds. Find the average waiting time of a customer. Also, determine the average
number of customers in the system and the average queue length.
37. Performance Measure of a Queuing System
37By: Jignesh Kariya
Example – 3 A self service store employs one cashier at its counter. Nine
customers arrive on an average every five minutes but the cashier can serve 10
customers in 5 minutes. Assuming Poisson distribution for arrival rate and
exponential distribution for service time, find
i. Average number of customers in the system.
ii. Average number of customers in the queue or average queue length.
iii. Average time a customer spends in the system.
iv. Average time a customer waits before being served.
39. Performance Measure of a Queuing System
39By: Jignesh Kariya
Example – 5
Customers arrive at a clinic at the rate of 8 per hour (Poisson arrival) and the doctor
can serve at the rate of 9 per hour (exponential).
(1) What is the probability that a customer does not join the queue and walks into
the doctor’s room?
(2) What is the probability that there is no queue?
(3) What is the probability that there are 10 customers in the system?
(4) What is the expected number in the system?
(5) What is the expected waiting time in the queue?