This document discusses key concepts in queuing theory. It defines queuing theory as applying to situations where arrival and service rates are unpredictable. Queuing theory aims to determine the optimal level of service that minimizes the costs of offering service and customer wait times. The document outlines components of a queuing system including the calling population, queuing process, queue discipline, and service process. It provides examples of different queue disciplines and discusses concepts like arrival patterns, inter-arrival times, finite vs infinite sources, and balking.
This document provides an introduction to queuing models and simulation. It discusses key characteristics of queuing systems such as arrival processes, service times, queue discipline, and performance measures. Common queuing notations are also introduced, including the widely used Kendall notation. Examples of queuing systems from various applications are provided to illustrate real-world scenarios that can be modeled using queuing theory.
This document summarizes a student project on queuing theory using the M/M/1 model. It includes an acknowledgements section thanking those who guided the project. It then covers queuing theory concepts like characteristics, assumptions, and formulas. The document describes simulating arrival and service data and comparing results to theoretical values. It finds the simulated values match theory. It also analyzes how changing arrival and service rates could improve the system's performance.
This document discusses P, NP and NP-complete problems. It begins by introducing tractable and intractable problems, and defines problems that can be solved in polynomial time as tractable, while problems that cannot are intractable. It then discusses the classes P and NP, with P containing problems that can be solved deterministically in polynomial time, and NP containing problems that can be solved non-deterministically in polynomial time. The document concludes by defining NP-complete problems as those in NP that are as hard as any other problem in the class, in that any NP problem can be reduced to an NP-complete problem in polynomial time.
This document discusses queueing theory and queuing networks. It begins by defining a queue as a model where arrivals come at random times and require random amounts of service from one or more servers. A queuing network can then be modeled as interconnected queues. Key inputs for analyzing a queue include the arrival and service processes, number of servers, and queueing rules. Additional inputs are needed for queueing networks, such as the interconnections between queues and routing strategies. Queues can be open, with arrivals from outside and departures, or closed, with a fixed number of jobs circulating. The document outlines analytical approaches for studying queues and networks through equilibrium analysis, focusing on obtaining mean performance parameters.
This document discusses key concepts in queuing theory. It defines queuing theory as applying to situations where arrival and service rates are unpredictable. Queuing theory aims to determine the optimal level of service that minimizes the costs of offering service and customer wait times. The document outlines components of a queuing system including the calling population, queuing process, queue discipline, and service process. It provides examples of different queue disciplines and discusses concepts like arrival patterns, inter-arrival times, finite vs infinite sources, and balking.
This document provides an introduction to queuing models and simulation. It discusses key characteristics of queuing systems such as arrival processes, service times, queue discipline, and performance measures. Common queuing notations are also introduced, including the widely used Kendall notation. Examples of queuing systems from various applications are provided to illustrate real-world scenarios that can be modeled using queuing theory.
This document summarizes a student project on queuing theory using the M/M/1 model. It includes an acknowledgements section thanking those who guided the project. It then covers queuing theory concepts like characteristics, assumptions, and formulas. The document describes simulating arrival and service data and comparing results to theoretical values. It finds the simulated values match theory. It also analyzes how changing arrival and service rates could improve the system's performance.
This document discusses P, NP and NP-complete problems. It begins by introducing tractable and intractable problems, and defines problems that can be solved in polynomial time as tractable, while problems that cannot are intractable. It then discusses the classes P and NP, with P containing problems that can be solved deterministically in polynomial time, and NP containing problems that can be solved non-deterministically in polynomial time. The document concludes by defining NP-complete problems as those in NP that are as hard as any other problem in the class, in that any NP problem can be reduced to an NP-complete problem in polynomial time.
This document discusses queueing theory and queuing networks. It begins by defining a queue as a model where arrivals come at random times and require random amounts of service from one or more servers. A queuing network can then be modeled as interconnected queues. Key inputs for analyzing a queue include the arrival and service processes, number of servers, and queueing rules. Additional inputs are needed for queueing networks, such as the interconnections between queues and routing strategies. Queues can be open, with arrivals from outside and departures, or closed, with a fixed number of jobs circulating. The document outlines analytical approaches for studying queues and networks through equilibrium analysis, focusing on obtaining mean performance parameters.
Queuing theory is the mathematical study of waiting lines and delays. It examines properties like average wait time, number of servers, arrival and service rates. Queues form when demand for a service exceeds capacity. The simplest queuing system has two components - a queue and server - with attributes of inter-arrival and service times. Queuing models use Kendall notation to describe systems, and the M/M/1 model is commonly used to analyze average queue length, wait times, and probability of overflow for single server queues. Queuing theory has applications in fields like telecommunications, healthcare, and computer networking.
Queueing theory studies waiting line systems where customers arrive for service but servers have limited capacity. This document outlines components of queueing models including: arrival processes, queue configurations, service disciplines, service facilities, and analytical solutions. Key points are that customers wait in queues when demand exceeds server capacity, and queueing formulas provide expected wait times and number of customers in the system based on arrival and service rates.
Queuing theory is the mathematical study of waiting lines in systems like transportation, banks, and stores. It was developed in 1903 and is used to predict system performance and determine costs. Queuing models make assumptions like customers arriving randomly and service times being exponentially distributed. They can be applied to situations involving customers like restaurants or manufacturing. The models provide metrics like expected wait times that are used to optimize staffing and inventory levels.
Queuing theory is the mathematics of waiting lines and is useful for predicting system performance. It models processes where customers arrive, wait for service, are serviced, and leave. Key elements include the arrival process, queue structure, and service system. Common applications include telecommunications, traffic control, and health services. Characteristics like arrival patterns, queue discipline, and service times are analyzed. Models can be deterministic or probabilistic and include metrics like average wait times, number of customers in line, and server utilization. Managing queues effectively requires understanding customer wait times and segmenting customer flows.
The presentation describes Measures of Information, entropy, source coding, source coding theorem, huffman coding, shanon fano coding, channel capacity theorem, capacity of a discrete and continuous memoryless channel, Error Free Communication over a Noisy Channel
Queuing theory and simulation are important tools for analyzing systems with random variability. Key components of queuing systems include the arrival and service processes, queue configuration, and service mechanisms. The exponential distribution is commonly used to model interarrival and service times due to its memoryless property. A Poisson process can be used to model arrivals when interarrival times are independent and exponentially distributed. Important metrics in queuing analysis include utilization and queue length. Steady state analysis focuses on the long-run behavior of queueing systems.
1) The document discusses key concepts in information theory such as entropy, information rate, and Shannon-Fano coding. Entropy is a measure of uncertainty in a random variable and is highest when all messages are equally likely. Information rate is the product of entropy and message rate.
2) Shannon-Fano coding assigns variable length binary codewords to messages based on their probabilities such that more likely messages get shorter codes. This achieves the optimal average code length of entropy.
3) Entropy is maximized when all messages are equally likely according to the inequality relating entropy to message probabilities. Equiprobable messages result in the highest uncertainty.
Queueing theory is the study of waiting lines and systems. A queue forms when demand exceeds the capacity of the service facility. Key components of a queueing model include the arrival process, queue configuration, queue discipline, service discipline, and service facility. Common queueing models include the M/M/1 model (Poisson arrivals, exponential service times, single server), and the M/M/C model (Poisson arrivals, exponential service times, multiple servers). These models provide formulas to calculate important queueing statistics like expected wait time, number of customers in system, and resource utilization.
Queuing theory is the mathematical study of waiting lines in systems like traffic networks, telephone systems, and more. It examines elements like arrival and service rates to predict system performance. The document outlines key concepts in queuing systems such as customers, servers, applications, and components like arrival processes, queue configuration, service disciplines, and service facilities. Special delay studies models are also discussed, including models for merging delays and peak flow delays.
This document discusses queuing theory, which is the mathematical study of waiting lines in systems where demand for service exceeds the available capacity. It covers the key characteristics of queuing systems including arrival patterns, service mechanisms, queue discipline, and number of service channels. Common configurations like single server-single queue and multiple server-multiple queue systems are described. Software used for queuing simulations is discussed along with the Kendall notation for representing queuing models. Limitations of queuing theory are noted.
This document discusses queueing theory and queueing models. It defines key queue parameters like arrival and service processes, the number of servers, and queue discipline. Common distributions for these parameters are presented along with examples of single and multiple server, single and multiple stage queue configurations. Specific queueing models are introduced including the fundamental M/M/1 model. Finally, key performance measures and formulas are provided for analyzing the M/M/1 model like probability of n customers, average wait times, and more.
Lecture 1 introduction and signals analysistalhawaqar
This document provides an introduction to communication systems and signal analysis. It discusses key components of a communication system including the information source, transmitter, channel, receiver and information user. It also describes different types of communication channels and various analog and digital modulation techniques. The document further discusses noise sources in communication channels including natural and man-made noise. It introduces concepts of time and frequency domains and Fourier analysis which are important for signal analysis in communication systems.
The document introduces dynamic programming as a technique for making optimal decisions over multiple time periods. It discusses how dynamic programming breaks large problems into smaller subproblems and solves each in order, working backwards from the last period. The document provides an example of using dynamic programming to find the shortest route between two cities by breaking the problem into stages and working backwards from the final destination.
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.
The document provides an introduction to queuing theory, which deals with problems involving waiting in lines or queues. It discusses key concepts such as arrival and service rates, expected queue length and wait times, and the utilization ratio. Common applications of queuing theory include determining the number of servers needed at facilities like banks, restaurants, and hospitals to minimize customer wait times. The summary provides the essential information about queuing theory and its use in analyzing waiting line systems.
Lecture Notes on Adaptive Signal Processing-1.pdfVishalPusadkar1
Adaptive filters are time-variant, nonlinear, and stochastic systems that perform data-driven approximation to minimize an objective function. The chapter discusses adaptive filter applications like system identification, inverse modeling, linear prediction, and noise cancellation. It also covers stochastic signal models, optimum linear filtering techniques like Wiener filtering, and solutions to the Wiener-Hopf equations. Numerical techniques like steepest descent are discussed for minimizing the mean square error function in adaptive filters. Stability and convergence analysis is presented for the steepest descent approach.
Digital control systems (dcs) lecture 18-19-20Ali Rind
This document discusses digital control systems and related topics such as difference equations, z-transforms, and mapping between the s-plane and z-plane. It begins with an outline of topics to be covered including difference equations, z-transforms, inverse z-transforms, and the relationship between the s-plane and z-plane. Examples are provided to illustrate difference equations, z-transforms, and mapping poles between the two planes. Standard z-transforms of discrete-time signals like the unit impulse and sampled step are also defined.
The document provides an overview of queuing theory and queuing models. It discusses key concepts such as arrival and service processes, queuing disciplines, classification of queuing models using Kendall's notation, and solutions of queuing models. Specific queuing models discussed include the M/M/1 model with Poisson arrivals and exponential service times. The document also covers probability distributions for arrivals, service times, and inter-arrival times as well as the pure birth and pure death processes.
The document summarizes key concepts about queuing systems and simple queuing models. It discusses:
1) Components of a queuing system including the arrival process, service mechanism, and queue discipline.
2) Performance measures for queuing systems such as average delay, waiting time, and number of customers.
3) The M/M/1 queuing model where arrivals and service times follow exponential distributions with a single server. Expressions are given for performance measures in this model.
4) How limiting the queue length to a finite number affects performance measures compared to an infinite queue system.
Queuing theory is the mathematical study of waiting lines and delays. It examines properties like average wait time, number of servers, arrival and service rates. Queues form when demand for a service exceeds capacity. The simplest queuing system has two components - a queue and server - with attributes of inter-arrival and service times. Queuing models use Kendall notation to describe systems, and the M/M/1 model is commonly used to analyze average queue length, wait times, and probability of overflow for single server queues. Queuing theory has applications in fields like telecommunications, healthcare, and computer networking.
Queueing theory studies waiting line systems where customers arrive for service but servers have limited capacity. This document outlines components of queueing models including: arrival processes, queue configurations, service disciplines, service facilities, and analytical solutions. Key points are that customers wait in queues when demand exceeds server capacity, and queueing formulas provide expected wait times and number of customers in the system based on arrival and service rates.
Queuing theory is the mathematical study of waiting lines in systems like transportation, banks, and stores. It was developed in 1903 and is used to predict system performance and determine costs. Queuing models make assumptions like customers arriving randomly and service times being exponentially distributed. They can be applied to situations involving customers like restaurants or manufacturing. The models provide metrics like expected wait times that are used to optimize staffing and inventory levels.
Queuing theory is the mathematics of waiting lines and is useful for predicting system performance. It models processes where customers arrive, wait for service, are serviced, and leave. Key elements include the arrival process, queue structure, and service system. Common applications include telecommunications, traffic control, and health services. Characteristics like arrival patterns, queue discipline, and service times are analyzed. Models can be deterministic or probabilistic and include metrics like average wait times, number of customers in line, and server utilization. Managing queues effectively requires understanding customer wait times and segmenting customer flows.
The presentation describes Measures of Information, entropy, source coding, source coding theorem, huffman coding, shanon fano coding, channel capacity theorem, capacity of a discrete and continuous memoryless channel, Error Free Communication over a Noisy Channel
Queuing theory and simulation are important tools for analyzing systems with random variability. Key components of queuing systems include the arrival and service processes, queue configuration, and service mechanisms. The exponential distribution is commonly used to model interarrival and service times due to its memoryless property. A Poisson process can be used to model arrivals when interarrival times are independent and exponentially distributed. Important metrics in queuing analysis include utilization and queue length. Steady state analysis focuses on the long-run behavior of queueing systems.
1) The document discusses key concepts in information theory such as entropy, information rate, and Shannon-Fano coding. Entropy is a measure of uncertainty in a random variable and is highest when all messages are equally likely. Information rate is the product of entropy and message rate.
2) Shannon-Fano coding assigns variable length binary codewords to messages based on their probabilities such that more likely messages get shorter codes. This achieves the optimal average code length of entropy.
3) Entropy is maximized when all messages are equally likely according to the inequality relating entropy to message probabilities. Equiprobable messages result in the highest uncertainty.
Queueing theory is the study of waiting lines and systems. A queue forms when demand exceeds the capacity of the service facility. Key components of a queueing model include the arrival process, queue configuration, queue discipline, service discipline, and service facility. Common queueing models include the M/M/1 model (Poisson arrivals, exponential service times, single server), and the M/M/C model (Poisson arrivals, exponential service times, multiple servers). These models provide formulas to calculate important queueing statistics like expected wait time, number of customers in system, and resource utilization.
Queuing theory is the mathematical study of waiting lines in systems like traffic networks, telephone systems, and more. It examines elements like arrival and service rates to predict system performance. The document outlines key concepts in queuing systems such as customers, servers, applications, and components like arrival processes, queue configuration, service disciplines, and service facilities. Special delay studies models are also discussed, including models for merging delays and peak flow delays.
This document discusses queuing theory, which is the mathematical study of waiting lines in systems where demand for service exceeds the available capacity. It covers the key characteristics of queuing systems including arrival patterns, service mechanisms, queue discipline, and number of service channels. Common configurations like single server-single queue and multiple server-multiple queue systems are described. Software used for queuing simulations is discussed along with the Kendall notation for representing queuing models. Limitations of queuing theory are noted.
This document discusses queueing theory and queueing models. It defines key queue parameters like arrival and service processes, the number of servers, and queue discipline. Common distributions for these parameters are presented along with examples of single and multiple server, single and multiple stage queue configurations. Specific queueing models are introduced including the fundamental M/M/1 model. Finally, key performance measures and formulas are provided for analyzing the M/M/1 model like probability of n customers, average wait times, and more.
Lecture 1 introduction and signals analysistalhawaqar
This document provides an introduction to communication systems and signal analysis. It discusses key components of a communication system including the information source, transmitter, channel, receiver and information user. It also describes different types of communication channels and various analog and digital modulation techniques. The document further discusses noise sources in communication channels including natural and man-made noise. It introduces concepts of time and frequency domains and Fourier analysis which are important for signal analysis in communication systems.
The document introduces dynamic programming as a technique for making optimal decisions over multiple time periods. It discusses how dynamic programming breaks large problems into smaller subproblems and solves each in order, working backwards from the last period. The document provides an example of using dynamic programming to find the shortest route between two cities by breaking the problem into stages and working backwards from the final destination.
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.
The document provides an introduction to queuing theory, which deals with problems involving waiting in lines or queues. It discusses key concepts such as arrival and service rates, expected queue length and wait times, and the utilization ratio. Common applications of queuing theory include determining the number of servers needed at facilities like banks, restaurants, and hospitals to minimize customer wait times. The summary provides the essential information about queuing theory and its use in analyzing waiting line systems.
Lecture Notes on Adaptive Signal Processing-1.pdfVishalPusadkar1
Adaptive filters are time-variant, nonlinear, and stochastic systems that perform data-driven approximation to minimize an objective function. The chapter discusses adaptive filter applications like system identification, inverse modeling, linear prediction, and noise cancellation. It also covers stochastic signal models, optimum linear filtering techniques like Wiener filtering, and solutions to the Wiener-Hopf equations. Numerical techniques like steepest descent are discussed for minimizing the mean square error function in adaptive filters. Stability and convergence analysis is presented for the steepest descent approach.
Digital control systems (dcs) lecture 18-19-20Ali Rind
This document discusses digital control systems and related topics such as difference equations, z-transforms, and mapping between the s-plane and z-plane. It begins with an outline of topics to be covered including difference equations, z-transforms, inverse z-transforms, and the relationship between the s-plane and z-plane. Examples are provided to illustrate difference equations, z-transforms, and mapping poles between the two planes. Standard z-transforms of discrete-time signals like the unit impulse and sampled step are also defined.
The document provides an overview of queuing theory and queuing models. It discusses key concepts such as arrival and service processes, queuing disciplines, classification of queuing models using Kendall's notation, and solutions of queuing models. Specific queuing models discussed include the M/M/1 model with Poisson arrivals and exponential service times. The document also covers probability distributions for arrivals, service times, and inter-arrival times as well as the pure birth and pure death processes.
The document summarizes key concepts about queuing systems and simple queuing models. It discusses:
1) Components of a queuing system including the arrival process, service mechanism, and queue discipline.
2) Performance measures for queuing systems such as average delay, waiting time, and number of customers.
3) The M/M/1 queuing model where arrivals and service times follow exponential distributions with a single server. Expressions are given for performance measures in this model.
4) How limiting the queue length to a finite number affects performance measures compared to an infinite queue system.
Queuing theory analyzes systems where customers arrive for service and may need to wait if service is not immediate. A queuing system consists of an arrival process, queue configuration, service mechanism, and queue discipline. Common examples include banks, restaurants, and computer networks. The M/M/1 model assumes arrivals follow a Poisson process and service times are exponentially distributed. It can be used to calculate average queue length, wait time, and resource utilization. Little's theorem relates average queue length, arrival rate, and wait time. Queuing delay at routers depends on packet arrival and service rates.
Analysis of single server fixed batch service queueing system under multiple ...Alexander Decker
This document analyzes a single server queueing system with fixed batch service, multiple vacations, and the possibility of catastrophes. The system uses a Poisson arrival process and exponential service times. The server provides service in batches of size k. If fewer than k customers remain after service, the server takes an exponential vacation. If a catastrophe occurs, all customers are lost and the server vacations. The document derives the generating functions and steady state probabilities for the number of customers when the server is busy or vacationing. It also provides closed form solutions for performance measures like mean number of customers and variance. Numerical studies examine these measures for varying system parameters.
The document provides an introduction to queuing theory, covering key concepts such as queues, stochastic processes, Little's Law, and types of queuing systems. It discusses topics like arrival and service processes, the number of servers, system capacity, and service disciplines. Common variables in queueing analysis are defined. Relationships among variables for G/G/m queues are described, including the stability condition, number in system vs. number in queue, number vs. time relationships, and time in system vs. time in queue. Different types of stochastic processes like discrete-state, continuous-state, Markov, and birth-death processes are introduced. Properties of Poisson processes are outlined. The document concludes by noting some applications of queuing
The document discusses different queuing models for analyzing efficiency at railway ticket windows. It summarizes four models: 1) M/M/1 queue with infinite capacity, 2) M/M/1 queue with finite capacity N, 3) M/M/S queue with infinite capacity, and 4) M/M/S queue with finite capacity N. The document provides sample data of arrival and service times over 1 hour and outlines the methodology and assumptions used, including Poisson arrivals and exponential service times. It then shows the manual calculations and Java code for the M/M/1 infinite queue model to find values like average number of customers and waiting times.
Queuing theory is the mathematical study of waiting lines in systems like customer service lines. Key aspects of queuing systems include the arrival and service processes, the number of servers, and the queue capacity and discipline. Little's Law relates the average number of customers in the system, the arrival rate, and the average time a customer spends in the system. Common queuing models include M/M/1 for Poisson arrivals and exponential service times with one server.
Queuing theory is the mathematical study of waiting lines in systems like customer service lines. The document discusses the M/M/c queuing model, which models systems with exponential arrival and service times and c parallel servers. Key measures calculated by queuing models include expected wait times, number of customers, and server utilization. An example analyzes a hospital emergency room's performance with 1 or 2 doctors. With 2 doctors, average wait times drop significantly while more patients can be served.
Queuing theory is the mathematical study of waiting lines in systems like customer service lines. It enables the analysis of processes like customer arrivals, waiting times, and service times. The document discusses the M/M/c queuing model, which assumes arrivals and service times follow exponential distributions and there are c parallel servers. It provides the steady state probabilities and performance measures like expected number of customers in the system and in the queue for the M/M/c model. An example applies the M/M/1 model to analyze whether a hospital should hire a second doctor based on arrival and service rates.
The document discusses queueing theory and stochastic processes. It defines key concepts in queueing systems such as interarrival times, service times, traffic intensity, and queueing notations. The document also summarizes the steady state behavior of the M/M/1 queue where arrivals and services times follow exponential distributions. It presents the balance equations and rate equality principle to derive the geometric distribution for the steady state probabilities of the number of customers in the system.
This document discusses stochastic processes and stochastic control. It defines stochastic processes as collections of random variables that represent the evolution of some random value over time, as opposed to deterministic processes where the trajectory can be known with certainty. Stochastic control aims to design controlled variables over time that perform a desired task with minimum cost despite the presence of random noise affecting the system. It provides examples of stochastic processes like routes taken to work and discusses how stochastic models incorporate probability and uncertainty compared to deterministic models.
Queuing theory is the mathematical study of waiting lines. It is commonly used to model systems where customers arrive for service, such as at cafeterias, banks, and libraries. The key components of queuing systems include arrivals, service times, queues, and servers. Common assumptions in queuing theory include Poisson arrivals and exponential service times. Formulas can be used to calculate values like average queue length, waiting time, and number of customers in the system. Queuing models help analyze real-world systems and identify ways to reduce waiting times.
Queuing theory is the mathematical study of waiting lines in service systems where customers arrive for service. Some key concepts include arrival and service rates, queues, queue lengths, waiting times, and models like the M/M/1 model. The M/M/1 model describes a system with Poisson arrivals, exponential service times, and a single server. Formulas are provided to calculate values like the probability of no customers, average queue length, and average waiting times in both the queue and system for the M/M/1 model. Several examples demonstrate how to apply the M/M/1 model and formulas to calculate performance measures for queuing systems.
Queuing theory and traffic analysis in depthIdcIdk1
This document provides a summary of concepts in queuing theory and network traffic analysis. It discusses queuing theory concepts like Little's Law, M/M/1 queues, and Kendall's notation. It then covers an empirical study of router delay that models delays using a fluid queue and reports on busy period metrics. Finally, it discusses the concept of network traffic self-similarity found in measurements of Ethernet LAN traffic.
This document discusses key concepts in telecommunication switching including:
- Statistical parameters and how they are used to summarize data from populations or samples.
- Random/stochastic processes and how they represent evolving systems over time, including discrete-time and discrete-state processes commonly used in telecommunications.
- Concepts of ergodicity, stationarity, and pure chance traffic in modeling telephone call patterns.
- Markov processes and how they model systems that change states based only on the previous state, including continuous-time vs discrete-time Markov chains and birth-and-death processes.
This document summarizes key aspects of queueing theory and its application to analyzing bank service systems. It discusses queuing models like the M/M/1 and M/M/s models. The purpose is to measure expected queue lengths and wait times to improve efficiency. Variables like arrival rate, service rate, and utilization are defined. Different queue disciplines and customer behaviors are also outlined. The document aims to simulate queue performance and compare single and multiple queue models to provide estimated solutions for optimizing bank service systems.
This document provides an overview of elementary queuing theory and single server queues. It defines key characteristics of queuing systems such as the arrival process, service process, number of servers, system capacity, and queue discipline. Common distributions for arrivals (Poisson) and service times (exponential) are described. Performance measures of queuing systems like delay, queue length, throughput and utilization are introduced. Other concepts covered include PASTA properties, Kendall's notation, traffic intensity, Little's Law, Markov chains, and transition probability matrices. The document serves as a lecture on introductory queuing theory concepts.
This document provides an overview of queuing systems and their analysis. It discusses key concepts like arrival and service processes, performance measures, steady-state analysis using Little's Law, and birth-death processes. An example M/M/1 queue is analyzed to find the steady-state probabilities and performance metrics like expected number in the system and average wait times. The methodology of setting up balance equations, solving for the steady-state distribution, and applying it to derive performance measures is demonstrated.
Similar to basics of stochastic and queueing theory (20)
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
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Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Create a More Engaging and Human Online Learning Experience
basics of stochastic and queueing theory
1. 1. APOORVA GUPTA(11972)
2. JYOTI ( 11973)
M.Sc. (Mathematical Sciences- STATISTICS)
2014-2015
Under the supervision of
Dr. Gulab Bura
Department of Mathematics and Statistics,
AIM & ACT
Basic Terminologies related
to Queuing theory
2. Content
• Stochastic process
• Markov process
• Markov chain
• Poisson process
• Birth death process
• Introduction to queueing theory
• History
• Elements of Queuing System
• A Commonly Seen Queuing Model
• Application
• Future plan
• References
3. Stochastic process
Definition : A stochastic process is family of time
indexed random variable where t belongs to
index set . Formal notation , where I is an
index set that is subset of R.
Examples :
• No. of telephone calls are received at a switchboard.
Suppose that is the r.v. which represents the
number of incoming calls in an interval (0,t) of
duration t units.
• Outcomes of X(t) are called states .
}:{ ItXt
tX
tX
4. • Discrete time , Discrete state space
• Discrete time , Continuous state space
• Continuous time , Discrete state space
• Continuous time, Continuous state space
Stochastic process is classified in four categories on
the basis of state space and time space
5. EXAMPLES
• Discrete time, Discrete state space : number of
accidents at 9 pm, 10 pm, 11 pm etc.
• Continuous –time ,Discrete state space : Number
of accidents on a highway at an interval of time
say 9 pm to 10 pm.
• Discrete time ,continuous state space :age of
people in a city at a particular year say 2001,
2002, 2003 etc.
• Continuous time , Continuous state space :age of
people from March 2013 to march 2014.
6. MARKOV PROCESS
If is a stochastic process such that ,given the
values of X(t) , t >s ,do not depend on the values of X(u) ,u<s,
then the process is said to be a Markov process.
• A definition of such a process is given as :
If for ,
the process is a markov process .
}),({ TttX
tttt n ...21
})(,...,)(|)(Pr{ 11 nn xtXxtXxtX
})(|)(Pr{ nn xtXxtX
}),({ TttX
)}(|)(Pr{)}(),(|)(Pr{ sXtXuXsXtX
7. Markov chain :
A discrete parameter Markov process is known as
Markov chain.
Definition : The stochastic process is
called a markov chain ,if , for
= It is the probability of transition from the
state ‘j’ at (n-1) th trial to the state k at nth trial .
,......}2,1,0,{ nXn
Njjkj n 11,...,,,
},...,,|Pr{ 10121 nnnn jXjXjXkX
jknn pjXkX }|Pr{ 1
jkp
8. Example of markov chain
Example : random walk between two barriers .
Homogeneous chain:
the transition probability independent of n, the
markov chain said to homogeneous.
Two types of markov chain :
• Discrete time markov chain
• Continuous time markov chain
}|Pr{ 1
jXkXp nnjk
9. • A stochastic process {N(t), t>0} with discrete
state space and continuous time is called a
poisson process if it satisfies the following
postulates:
1. Independence
2. Homogeneity in time
3. Regularity
The Poisson Process
10. 10
• Independence:
N( t+h)-N(t), the number of occurrence in the
interval (t,t+h) is independent of the number of
occurrences prior to that interval.
• Homogeneity in time:
pn(t) depends only on length ‘t’ of the interval and is
independent of where this interval is situated i.e.
pn(t) gives the probability of the number of
occurrences in the interval (t1 , t+t1 ) (which is of
length ‘t’) for every t1 .
Postulates for Poisson Process
11. Regularity In the interval of infinite small length
h, the probability of exactly one occurrence is
λh+o(h) and that of occurrence of more than one is
of o(h).
p1 (h)= λh+o(h)
)(
2
hOhp
k
k
)(10 hohhP
λ= rate of arrival
Probability of zero arrival
12. • Pure birth process is originated from Poisson process
in which the rate of birth depends on number of
persons.
Pure Birth Processes
P1 (t)=λnh+o(h)
Pn (t) = o(h) , k>=2
P0(t)= 1- λnh+o(h),
13. Birth-and-Death Process
• We consider pure birth process where
Here only births are considered, further deaths are also
considered as possible so,
Collectively for n>=k, equations are called as Birth and Death
process
P1 (t)=λn(h)+o(h) , k=1
Pn (t) = o(h), k>=2
P0 (t)= 1- λn(h)+o(h),k=0
q1 (t)=μn(h)+o(h) , k=1
qn (t) = o(h), k>=2
q0 (t)= 1- μn(h)+o(h),k=0
14. • Note: The foundation of many of the most
commonly used queuing models
Birth – equivalent to the arrival of a customer
Death – equivalent to the departure of a served
customer
15. 15
A Birth-and-Death Process Rate Diagram
0 1 n-1 n
0 1 2 n-1 n n+1
1 2 n n+1
n = State n, i.e., the case of n customers in the system
Excellent tool for describing the mechanics of a Birth-and-
Death process
16.
17. Introduction of Queueing theory
• Queueing theory is the mathematical study of
waiting lines, or queues.
• In queueing theory a model is constructed so that
queue lengths and waiting times can be predicted.
• Queueing 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.
18. DEFINITION
• A queue is said to occur when the rate at which the
demand arises exceeds the rate at which service is
being provided.
• It is the quantitative technique which consists of
constructing mathematical models for various types
of queuing systems.
• Mathematical models are constructed so that queue
lengths and waiting times can be predicted which
helps in balancing the cost of service and the cost
associated with customers waiting for service
19. HISTORY
• Queueing theory’s history goes back
nearly 105 years.
• It is originated by A. K. Erlang in 1909, in
Danish when he is at his work place.
• Agner Krarup Erlang, a Danish engineer
who worked for the Copenhagen
Telephone Exchange, published the first
paper on what would now be called
queueing theory in 1909.
• He modeled the number of telephone
calls arriving at an exchange by a Poisson
process.
20. • Johannsen’s “Waiting Times and Number of Calls”
seems to be the first paper on the subject.
• But the method used in this paper was not
mathematically exact and therefore, from the point
of view of exact treatment, the paper that has
historic importance is A. K. Erlang’s, “The Theory of
Probabilities and Telephone Conversations”
• In this paper he lays the foundation for the place of
Poisson (and hence, exponential) distribution in
queueing theory.
21. • His most important work, Solutions of Some
Problems in the Theory of Probabilities of
Significance in Automatic Telephone Exchanges , was
published in 1917, which contained formulas for loss
and waiting probabilities which are now known as
Erlang’s loss formula (or Erlang B-formula) and delay
formula (or Erlang C-formula), respectively.
Agner Krarup Erlang, 1878–1929
22. Elements of Queuing System
Service process
Queue Discipline
Number of
servers
Elements of Queuing
Models
Arrival process
Customer’s
Behavior
System capacity
23. 1. Arrival Process
Arrivals can be measured as the arrival rate or the interarrival
time (time between arrivals).
• These quantities may be deterministic or stochastic (given by a
probability distribution).
• Arrivals may also come in batches of multiple customers,
which is called batch or bulk arrivals.
• The batch size may be either deterministic or stochastic.
Interarrival time =1/ arrival rate
24. 2.Customer’s Behaviour
• Balking: If a customer enters a system and
decides not to enter the queue since it is too
long is called Balking.
• Reneging: If a customer enters the queue but
after sometimes loses patience and leaves it is
called Reneging.
• Jockeying: When there are 2 or more parallel
queues and the customers move from one
queue to another to change his position is called
Jockeying.
25. 25
3. Service Process
• Service Process determines the customer service
times in the system.
• As with arrival patterns, service patterns may be
deterministic or stochastic. There may also be
batched services.
• The service rate may be state-dependent. (This is the
analogue of impatience with arrivals.)
Note that there is an important difference between
arrivals and services. Services do not occur when
the queue is empty.
26. 26
4. Number of Servers
Single Server Queue:
Multiple Server Queue
28. 5. Queue Discipline
• FIFO- First in First out
Or FCFS- First Come First Serve
• LIFO-Last in First out
• SIRO- Service in Random order
• Priority based
33. 6.System capacity
• The capacity of a system can be finite or infinite.
There are many queuing systems whose capacity is
finite. In such cases problem of balking arises. It is
known as forced balking. In forced balking customer
needs the service but due to the problem of limited
capacity of the system the customer leaves the
system.
34. System Customers Server
Reception desk People Receptionist
Hospital Patients Doctors
Airport Airplanes Runway
Road network Cars Traffic light
Grocery Shoppers Checkout station
Queuing examples
35. A Commonly Seen Queuing Model
• Service times as well as inter arrival times are assumed
independent and identically distributed
– If not otherwise specified
• Commonly used notation principle: A/B/C
– A = The inter arrival time distribution
– B = The service time distribution
– C = The number of parallel servers
• Commonly used distributions
– M = Markovian (exponential) - Memory less
– D = Deterministic distribution
– G = General distribution
• Example: M/M/1
– Queuing system with exponentially distributed service and
inter-arrival times and 1 server
36.
37. Examples of Different Queuing Systems
• Arrival Distribution: Poisson rate (M) tells you to
use exponential probability
• Service Distribution: again the M signifies an
exponential probability
• 1 represents the number of servers
M/M/1: The system consists of only one server. This queuing
system can be applied to a wide variety of problems as any
system with a very large number customers.
M/M/1
38. M/M/1 queueing systems assume a Poisson arrival
process.
This assumption is a very good approximation for
arrival process in real systems that meet the
following rules:
•The number of customers in the system is very
large.
•Impact of a single customer on the performance of
the system is very small, i.e. a single customer
consumes a very small percentage of the system
resources.
•All customers are independent, i.e. their decision to
use the system are independent of other users.
39. Application of Queuing Theory
• Telecommunication.
• Traffic control.
• Determine the sequence of computer
operations.
• Predicting computer performance.
• Health service ( e.g. Control of hospital bed
assignments).
• Airport traffic, airline ticket sales.
• Layout of manufacturing systems.