Queuing theory is the mathematical study of waiting lines in systems like telephone networks and hospitals. It aims to predict queue lengths and waiting times. Agner Krarup Erlang created early models for telephone exchanges. Queues are described using Kendall's notation of arrival and service distributions and number of servers. Little's Law states the average number of customers in a stable system equals the arrival rate multiplied by the average time in the system. Queuing theory is used to optimize resource allocation and reduce waiting times in many applications.
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
This slide was prepared as a part of presentation in a course work of Master in IT, IIT, University of Dhaka. The main slide was prepared by Dr. Sam Labi of Purdue University and Dr. Fred of MIT.
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
This slide was prepared as a part of presentation in a course work of Master in IT, IIT, University of Dhaka. The main slide was prepared by Dr. Sam Labi of Purdue University and Dr. Fred of MIT.
Data Structure, Algorithm , Asymptotic notation, Pointers, Dynamic memory allocation, recursion, time and space complexity and trade-off, array and linked list with its operations.
Web engineering UNIT V as per RGPV syllabusNANDINI SHARMA
E- Commerce, E-commerce Business Models, The Internet and World Wide Web: E-commerce
Infrastructure, Building an E-commerce Web Site , Electronic Commerce environment and opportunities. Modes of Electronic Commerce, Approaches to safe Electronic Commerce, Electronic Cash and Electronic Payment Schemes ,Online Security and Payment Systems, Ecommerce Marketing Concepts, Advertising on the Internet: issues an Technologies, Ecommerce Marketing Concepts Electronic Publishing issues, approaches, legalities and technologies ,Privacy and Security Topics: Introduction, Web Security , Encryption schemes, Secure Web document, Digital Signatures and Firewalls, Cyber crime and laws, IT Act.
Web Engineering UNIT III as per RGPV SyllabusNANDINI SHARMA
Technologies for Web Applications: HTML and DHTML, HTML Basic Concepts, Static and dynamic HTML, Structure of HTML documents, HTML Elements, Linking in HTML, Anchor Attributes, Image Maps, Meta Information, Image Preliminaries, Layouts, Backgrounds, Colors and Text, Fonts, Tables, Frames and layers, Audio and Video Support with HTML Database integration, CSS, Positioning with Style sheets, Forms Control, Form. Elements. Introduction to CGI PERL, JAVA SCRIPT, PHP, ASP , Cookies Creating and Reading Cookies
Number System, Positional and non-positional number system, conversion number system from binary to another base and vice versa, decimal to another base and vice versa, convert another base than 10 to another base than 10, binary arithmetic operation such as binary addition, subtraction, multiplication, division
Basic Computer Engineering Unit II as per RGPV SyllabusNANDINI SHARMA
Algorithm, Flowchart, Categories of Programming Languages, OOPs vs POP, concepts of OOPs, Inheritance, C++ Programming, How to write C++ program as a beginner, Array, Structure, etc
Web engineering UNIT IV as per RGPV syllabusNANDINI SHARMA
Technologies for Web Applications: Introduction of XML, Validation of XML documents, DTD, Ways to use XML, XML for data files, HTML Vs XML, Embedding XML into HTML documents, Converting XML to HTML for Display, Displaying XML using CSS and XSL, Rewriting HTML as XML, Relationship between HTML, SGML and XML, web personalization , Semantic web,
Semantic Web Services, Ontology.
Data Link Layer, Error correction and detection like LRC, VRC, CRC, checksum and Hamming coding, Data link protocols, stop and wait ARQ, sliding window ARQ, Petrinet models, HDLC, etc
Fundamental of Computer, Types of computer, Bus, registers, Memory of computer, Instruction set, Operating system, functions of OS, Types of OS. study of Microsoft office word, power-point, excel etc
Computer Network notes (handwritten) UNIT 1NANDINI SHARMA
Introduction of computer network, layered architecture, topology, guided and unguided media, signals, multiplexing, OSI vs TCP/IP , IP address, TCP , UDP, DHCP, DNS, HTTP, etc.
Introduction, architecture of multimedia, multimedia input and output devices, ADSL, ATM, multimedia database, animation techniques, aliasing and anti-aliasing, morphing, video on demand
Talks about what is Queuing and its application, practical life usage, with a complex problem statement with its solution. Pre-emptive and non-preemptive queue models and its algorithm.
An Efficient Method to Compute the Rate Matrix for Multi-Server Retrial Queue...IJCNCJournal
This study presents the usage of retrial queues with cloud computing systems in which the operating unit (the server) and the storing unit (buffer) are independently considered. In fact, the tasks cannot occupy the server to the system. Instead, they are stored in the buffer and sent back to the server after a random time. Upon a service completion, the server does not always get to work while waiting for a new task or a task from the buffer. After the idle time, the server instantly starts searching for a task from the buffer. The analysis model proposed in this study refers to a retrial queue system searching for tasks from theorbit with limited size under a multi-server context, and the model is modelized into the 3-dimension Markov chain. The solution is based on building an algorithm under the analytical methodology of the quasi birthdeath (QBD) process that utilizes the Q-matrix to calculate the probability of states toward the proposed model.
AN EFFICIENT M COMPUTE THE RATE MATRIX FOR MULTI-SERVER RETRIAL QUEUES WITH C...IJCNCJournal
This study presents the usage of retrial queues with cloud computing systems in which the operating unit
(the server) and the storing unit (buffer) are independently considered. In fact, the tasks cannot occupy the
server to the system. Instead, they are stored in the buffer and sent back to the server after a random time.
Upon a service completion, the server does not always get to work while waiting for a new task or a task
from the buffer. After the idle time, the server instantly starts searching for a task from the buffer. The
analysis model proposed in this study refers to a retrial queue system searching for tasks from theorbit
with limited size under a multi-server context, and the model is modelized into the 3-dimension Markov
chain. The solution is based on building an algorithm under the analytical methodology of the quasi birthdeath (QBD) process that utilizes the Q-matrix to calculate the probability of states toward the proposed
model.
A MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENTIAEME Publication
In this paper, we investigated a queuing model of fuzzy environment-based a multiple channel queuing model (M/M/C) ( /FCFS) and study its performance under realistic conditions. It applies a nonagonal fuzzy number to analyse the relevant performance of a multiple channel queuing model (M/M/C) ( /FCFS). Based on the sub interval average ranking method for nonagonal fuzzy number, we convert fuzzy number to crisp one. Numerical results reveal that the efficiency of this method. Intuitively, the fuzzy environment adapts well to a multiple channel queuing models (M/M/C) ( /FCFS) are very well.
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.
This work constructs the membership functions of the system characteristics of a batch-arrival queuing system with multiple servers, in which the batch-arrival rate and customer service rate are all fuzzy numbers. The -cut approach is used to transform a fuzzy queue into a family of conventional crisp queues in this context. By means of the membership functions of the system characteristics, a set of parametric nonlinear programs is developed to describe the family of crisp batch-arrival queues with multiple servers. A numerical example is solved successfully to illustrate the validity of the proposed approach. Because the system characteristics are expressed and governed by the membership functions, the fuzzy batch-arrival queues with multiple servers are represented more accurately and the analytic results are more useful for system designers and practitioners.
Queuing theory: What is a Queuing system???
Waiting for service is part of our daily life….
Example:
we wait to eat in restaurants….
We queue up in grocery stores…
Jobs wait to be processed on machine…
Vehicles queue up at traffic signal….
Planes circle in a stack before given permission to land at an airport….
Unfortunately, we can not eliminate waiting time without incurring expenses…
But, we can hope to reduce the queue time to a tolerable levels… so that we can avoid adverse impact….
Why study???? What analytics can be drawn??? Analytics means ---- measures of performance such as
1. Average queue length
2. Average waiting time in the queue
3. Average facility utilization….
Computer Network Notes (Handwritten) UNIT 2NANDINI SHARMA
Data link layer: flow control, error control, line discipline, stop and wait, sliding window protocol, stop and wait arq, sliding window arq, BSC, HDLC, bit stuffing, elemenary data link protocol etc
Introduction to distributed systems
Architecture for Distributed System, Goals of Distributed system, Hardware and Software
concepts, Distributed Computing Model, Advantages & Disadvantage distributed system, Issues
in designing Distributed System,
Cloud computing notes unit I as per RGPV syllabusNANDINI SHARMA
Cloud Computing
Historical development ,Vision of Cloud Computing, Characteristics of cloud
computing as per NIST , Cloud computing reference model ,Cloud computing environments,
Cloud services requirements, Cloud and dynamic infrastructure, Cloud Adoption and rudiments
.Overview of cloud applications: ECG Analysis in the cloud, Protein structure prediction, Gene
Expression Data Analysis ,Satellite Image Processing ,CRM and ERP ,Social networking .
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Water Industry Process Automation and Control Monthly - May 2024.pdf
Queueing theory
1. Truba College of Science & Technology, Bhopal Queuing
Model
Compiled By : Ms. Nandini Sharma Page 1
Queuing theory is the mathematical study of waiting lines, or queues. In queuing theory a
model is constructed so that queue lengths and waiting times can be predicted. Queuing theory is
generally considered a branch of operations research because the results are often used when
making business decisions about the resources needed to provide a service.
Queuing theory has its origins in research by Agner Krarup Erlang when he created models to
describe the Copenhagen telephone exchange. The ideas have since seen applications
including telecommunication, traffic engineering, computing and the design of factories, shops,
offices and hospitals.
Single queueing nodes
Single queueing nodes are usually described using Kendall's notation in the
form A/S/C where A describes the time between arrivals to the queue, S the size of jobs and C the
number of servers at the node. Many theorems in queue theory can be proved by reducing queues
to mathematical systems known as Markov chains, first described by Andrey Markov in his 1906
paper.
Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange,
published the first paper on what would now be called queuing theory in 1909. He modeled the
number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1
queue in 1917 and M/D/k queueing model in 1920. In Kendall's notation.
M stands for Markov or memoryless and means arrivals occur according to a Poisson process.D
stands for deterministic and means jobs arriving at the queue require a fixed amount of
servicek describes the number of servers at the queueing node (k = 1, 2,...). If there are more jobs
at the node than there are servers then jobs will queue and wait for service.The M/M/1 queue is a
simple model where a single server serves jobs that arrive according to a Poisson process and
have exponentially distributed service requirements. In an M/G/1 queue the G stands for general
and indicates an arbitrary probability distribution. The M/G/1 model was solved by Felix
Pollaczek in 1930, a solution later recast in probabilistic terms by Aleksandr Khinchin and now
known as the Pollaczek–Khinchine formula. After World War II queueing theory became an area
of research interest to mathematicians. Work on queueing theory used in modern packet
switchingnetworks was performed in the early 1960s by Leonard Kleinrock. It was in this period
that John Little gave a proof of the formula which now bears his name: Little's law. In 1961 John
Kingman gave a formula for the mean waiting time in a G/G/1 queue: Kingman's formula.
The matrix geometric method and matrix analytic methods have allowed queues with phase-type
distributed interarrival and service time distributions to be considered. Problems such as
performance metrics for the M/G/k queue remain an open problem.
Service disciplines
Various scheduling policies can be used at queuing nodes:
2. Truba College of Science & Technology, Bhopal Queuing
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First in first out
This principle states that customers are served one at a time and that the customer that has been
waiting the longest is served first.
Last in first out
This principle also serves customers one at a time, however the customer with the shortest
waiting time will be served first. Also known as a stack.
Processor sharing
Service capacity is shared equally between customers.
Priority
Customers with high priority are served first. Priority queues can be of two types, non-
preemptive (where a job in service cannot be interrupted) and preemptive (where a job in service
can be interrupted by a higher priority job). No work is lost in either model.
Shortest job first
The next job to be served is the one with the smallest size
Preemptive shortest job first
The next job to be served is the one with the original smallest size[18]
Shortest remaining processing time
The next job to serve is the one with the smallest remaining processing requirement.[19]
Service facility
Single server:customers line up and there is only one server
Parallel servers:customers line up and there are several servers
Tandem queue:there are many counters and customers can
decide going where to queue
Customer’s behavior of waiting
Balking:customers deciding not to join the queue if it is too long
Jockeying:customers switch between queues if they think they
will get served faster by so doing
Reneging:customers leave the queue if they have waited too long
for service
Queueing networks
Networks of queues are systems in which a number of queues are connected by customer
routing. When a customer is serviced at one node it can join another node and queue for service,
or leave the network. For a network of m the state of the system can be described by an m–
dimensional vector (x1,x2,...,xm) where xirepresents the number of customers at each node. The
first significant results in this area were Jackson networks, for which an efficient product-form
3. Truba College of Science & Technology, Bhopal Queuing
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Compiled By : Ms. Nandini Sharma Page 3
stationary distribution exists and the mean value analysiswhich allows average metrics such as
throughput and sojourn times to be computed. If the total number of customers in the network
remains constant the network is called a closed network and has also been shown to have a
product–form stationary distribution in the Gordon–Newell theorem. This result was extended to
the BCMP network[25] where a network with very general service time, regimes and customer
routing is shown to also exhibit a product-form stationary distribution. Networks of customers
have also been investigated, Kelly networks where customers of different classes experience
different priority levels at different service nodes. Another type of network are G-networks first
proposed by Erol Gelenbe in 1993: these networks do not assume exponential time distributions
like the classic Jackson Network.
Example of M/M/1
Birth and Death process
A/B/C
A:distribution of arrival time
B:distribution of service time
C:the number of parallel servers
A system of inter-arrival time and service time showed exponential distribution, we
denoted M.
λ:the average arrival rate
µ:the average service rate of a single serviceP : the probability of n customers in system
n :the number of people in system
Let E represent the number of times
of entering state n, and L represent
the number of times of leaving state
n. We have .
When the system arrives at steady
state, which means t, we have ,
therefore arrival rate=removed rate.
Balance equation
situation 0:
4. Truba College of Science & Technology, Bhopal Queuing
Model
Compiled By : Ms. Nandini Sharma Page 4
situation 1:
situation n:
By balance
equation,
By mathematical induction,
Because
we get
In queueing theory, a discipline within the mathematical theory of probability, Little's
result, theorem, lemma, law or formulais a theorem by John Little which states:
The long-term average number of customers in a stable system L is equal to the long-term
average effective arrival rate, λ, multiplied by the (Palm-)average time a customer spends
in the system, W; or expressed algebraically: L = λW.
Although it looks intuitively reasonable, it is quite a remarkable result, as the relationship is
"not influenced by the arrival process distribution, the service distribution, the service order,
or practically anything else."
The result applies to any system, and particularly, it applies to systems within systems. So in
a bank, the customer line might be one subsystem, and each of the tellers another subsystem,
and Little's result could be applied to each one, as well as the whole thing. The only
requirements are that the system is stable and non-preemptive; this rules out transition states
such as initial startup or shutdown.
In some cases it is possible to mathematically relate not only the average number in the
system to the average wait but relate the entire probability distribution (and moments) of the
number in the system to the wait.
In a 1954 paper Little's law was assumed true and used without proof. The form L = λW was first
published by Philip M. Morse where he challenged readers to find a situation where the
relationship did not hold. Little published in 1961 his proof of the law, showing that no such
5. Truba College of Science & Technology, Bhopal Queuing
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Compiled By : Ms. Nandini Sharma Page 5
situation existed. Little's proof was followed by a simpler version by Jewelland another by
Eilon. Shaler Stidham published a different and more intuitive proof in 1972.
Finding Response Time
Imagine an application that had no easy way to measure response time. If you can find the mean
number in the system and the throughput, you can use Little's Law to find the average response
time like so:
MeanResponseTime = MeanNumberInSystem / MeanThroughput
For example: A queue depth meter shows an average of nine jobs waiting to be serviced.
Add one for the job being serviced, so there is an average of ten jobs in the system. Another
meter shows a mean throughput of 50 per second. You can calculate the mean response time
as: 0.2 seconds = 10 / 50 per second. When exploring Little’s law and learning to trust it, be
aware of the common mistakes of using arrivals(work arriving) when throughput(work
completed) is called for and not keeping the units of your measurements the same.
Customers In The Store
Imagine a small store with a single counter and an area for browsing, where only one person
can be at the counter at a time, and no one leaves without buying something. So the system is
roughly:
Entrance → Browsing → Counter → Exit
In a stable system, the rate at which people enter the store is the rate at which they arrive
at the store (called the arrival rate), and the rate at which they exit as well (called the exit
rate). By contrast, an arrival rate exceeding an exit rate would represent an unstable
system, where the number of waiting customers in the store will gradually increase
towards infinity.
Little's Law tells us that the average number of customers in the store L, is the effective
arrival rate λ, times the average time that a customer spends in the store W, or simply:
Assume customers arrive at the rate of 10 per hour and stay an average of 0.5 hour.
This means we should find the average number of customers in the store at any time
to be 5.
Now suppose the store is considering doing more advertising to raise the arrival
rate to 20 per hour. The store must either be prepared to host an average of 10
6. Truba College of Science & Technology, Bhopal Queuing
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Compiled By : Ms. Nandini Sharma Page 6
occupants or must reduce the time each customer spends in the store to 0.25
hour. The store might achieve the latter by ringing up the bill faster or by adding
more counters.
We can apply Little's Law to systems within the store. For example, the counter
and its queue. Assume we notice that there are on average 2 customers in the
queue and at the counter. We know the arrival rate is 10 per hour, so customers
must be spending 0.2 hours on average checking out.
We can even apply Little's Law to the counter itself. The average number of
people at the counter would be in the range (0, 1) since no more than one
person can be at the counter at a time. In that case, the average number of
people at the counter is also known as the utilisation of the counter.
However, because a store in reality generally has a limited amount of space,
it cannot become unstable. Even if the arrival rate is much greater than the
exit rate, the store will eventually start to overflow, and thus any new
arriving customers will simply be rejected (and forced to go somewhere else
or try again later) until there is once again free space available in the store.
This is also the difference between the arrival rate and the effective arrival
rate, where the arrival rate roughly corresponds to the rate at which
customers arrive at the store, whereas the effective arrival rate corresponds to
the rate at which customers enter the store. However, in a system with an
infinite size and no loss, the two are equal.
Estimating parameters
To use Little's law on data formulas must be used to estimate the parameters as the result does
not necessarily directly apply over finite time intervals, due to problems like how to log
customers already present at the start of the logging interval and those who have not yet departed
when logging stop.
7. Truba College of Science & Technology, Bhopal Queuing
Model
Compiled By : Ms. Nandini Sharma Page 7