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.
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.
This document provides an introduction to queuing theory. It discusses how queues form due to an imbalance between customer arrivals and service capabilities. Common examples where queues occur include buses, movie theaters, and service stations. Key terms are defined, such as customers, service stations, waiting time, and queue length. The elements that make up a queuing system are described as the arrival pattern of customers, the service mechanism, the queue discipline for selecting the next customer, and the output of the queue. First-come, first-served is provided as a common queue discipline.
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 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.
This document provides an overview of queuing theory, which is used to model waiting lines. It discusses key concepts like arrival processes, service systems, queuing models and their characteristics. Some examples where queuing theory is applied include telecommunications, traffic control, and manufacturing layout. Common elements of queuing systems are customers, servers and queues. The document also presents examples of single and multiple channel queuing models.
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.
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.
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.
This document provides an introduction to queuing theory. It discusses how queues form due to an imbalance between customer arrivals and service capabilities. Common examples where queues occur include buses, movie theaters, and service stations. Key terms are defined, such as customers, service stations, waiting time, and queue length. The elements that make up a queuing system are described as the arrival pattern of customers, the service mechanism, the queue discipline for selecting the next customer, and the output of the queue. First-come, first-served is provided as a common queue discipline.
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 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.
This document provides an overview of queuing theory, which is used to model waiting lines. It discusses key concepts like arrival processes, service systems, queuing models and their characteristics. Some examples where queuing theory is applied include telecommunications, traffic control, and manufacturing layout. Common elements of queuing systems are customers, servers and queues. The document also presents examples of single and multiple channel queuing models.
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.
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.
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 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 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.
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted
This document provides an introduction to queuing theory, which analyzes systems where customers wait in line for service. It discusses the key elements of a queuing model, including the arrival process, service system, and queue structure. Common assumptions are that arrivals and service times follow Poisson and exponential distributions respectively. Key metrics analyzed include the average number of customers in queue and in the system, as well as the average waiting times. The M/M/1 queuing model with a Poisson arrival process and exponential service times at a single server is presented.
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.
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.
Queueing theory studies waiting lines and delays. It can be used to analyze systems like emergency rooms with patients waiting for service. Models represent the input of customers, the queue where they wait, and the service process. In an M/M/1 model, arrivals and service times follow exponential distributions, there is one server, and customers are served in order of arrival. Key metrics include the expected length of the system, length of the queue, and waiting times in both the system and queue. Queueing theory provides insights to reduce delays by adjusting staffing levels or other factors.
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.
Queueing Theory is the mathematical study of waiting lines in systems where demand for service exceeds the available resources. A pioneer in the field was Agner Krarup Erlang who applied its principles to telecommunications. The document discusses key concepts in queueing theory including arrival and service processes, queue configurations, performance measures and examples of real-world applications. It also covers limitations of classical queueing models in fully representing complex real systems.
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.
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
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.
Queuing theory describes the analysis of waiting lines in customer service systems. It examines issues like optimal staffing levels and expected wait times. The key components of a queuing system include the input source (customers), the service system (servers), and queue discipline (order of service). Common configurations include single or multiple servers with single or multiple queues. Service can be characterized by rate (customers served per time unit) or time (time to serve each customer). The M/M/1 model assumes arrivals follow a Poisson process, service times are exponentially distributed, and there is one server following a first-come, first-served queue discipline. This model provides formulas to calculate statistics like average wait time based on arrival and service rates
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.
This document provides an overview of Network Simulator 2 (NS-2) and how to use it to simulate computer networks. It discusses:
- The basic design of NS-2, which uses Tcl for scripting and C++ for implementing network objects. Simulation scripts are written in OTcl to set up the network topology and control packet transmissions.
- Common tasks in NS-2 like creating nodes and links, defining traffic sources and sinks, generating traffic patterns, and outputting trace files.
- Example scripts that demonstrate how to initialize a simulation, generate network traffic with different protocols (UDP, TCP), and visualize results using the Network Animator (NAM) tool.
- Key aspects of
Queueing Theory provides tools to analyze relationships between congestion and delay in systems. The total delay experienced by messages in a communication system can be classified into processing, queuing, transmission, propagation, and retransmission delays. Little's Theorem states that the average number of customers (N) in a system equals the average arrival rate (λ) multiplied by the average time spent in the system (T). Key characteristics used to classify queueing systems are the arrival process, service process, and number of servers. The M/M/1 queueing system, with Poisson arrivals and exponentially distributed service times with a single server, is commonly analyzed due to its tractability.
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 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 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.
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted
This document provides an introduction to queuing theory, which analyzes systems where customers wait in line for service. It discusses the key elements of a queuing model, including the arrival process, service system, and queue structure. Common assumptions are that arrivals and service times follow Poisson and exponential distributions respectively. Key metrics analyzed include the average number of customers in queue and in the system, as well as the average waiting times. The M/M/1 queuing model with a Poisson arrival process and exponential service times at a single server is presented.
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.
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.
Queueing theory studies waiting lines and delays. It can be used to analyze systems like emergency rooms with patients waiting for service. Models represent the input of customers, the queue where they wait, and the service process. In an M/M/1 model, arrivals and service times follow exponential distributions, there is one server, and customers are served in order of arrival. Key metrics include the expected length of the system, length of the queue, and waiting times in both the system and queue. Queueing theory provides insights to reduce delays by adjusting staffing levels or other factors.
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.
Queueing Theory is the mathematical study of waiting lines in systems where demand for service exceeds the available resources. A pioneer in the field was Agner Krarup Erlang who applied its principles to telecommunications. The document discusses key concepts in queueing theory including arrival and service processes, queue configurations, performance measures and examples of real-world applications. It also covers limitations of classical queueing models in fully representing complex real systems.
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.
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
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.
Queuing theory describes the analysis of waiting lines in customer service systems. It examines issues like optimal staffing levels and expected wait times. The key components of a queuing system include the input source (customers), the service system (servers), and queue discipline (order of service). Common configurations include single or multiple servers with single or multiple queues. Service can be characterized by rate (customers served per time unit) or time (time to serve each customer). The M/M/1 model assumes arrivals follow a Poisson process, service times are exponentially distributed, and there is one server following a first-come, first-served queue discipline. This model provides formulas to calculate statistics like average wait time based on arrival and service rates
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.
This document provides an overview of Network Simulator 2 (NS-2) and how to use it to simulate computer networks. It discusses:
- The basic design of NS-2, which uses Tcl for scripting and C++ for implementing network objects. Simulation scripts are written in OTcl to set up the network topology and control packet transmissions.
- Common tasks in NS-2 like creating nodes and links, defining traffic sources and sinks, generating traffic patterns, and outputting trace files.
- Example scripts that demonstrate how to initialize a simulation, generate network traffic with different protocols (UDP, TCP), and visualize results using the Network Animator (NAM) tool.
- Key aspects of
Queueing Theory provides tools to analyze relationships between congestion and delay in systems. The total delay experienced by messages in a communication system can be classified into processing, queuing, transmission, propagation, and retransmission delays. Little's Theorem states that the average number of customers (N) in a system equals the average arrival rate (λ) multiplied by the average time spent in the system (T). Key characteristics used to classify queueing systems are the arrival process, service process, and number of servers. The M/M/1 queueing system, with Poisson arrivals and exponentially distributed service times with a single server, is commonly analyzed due to its tractability.
This document discusses queueing models and their analysis. It begins by outlining key characteristics of queueing systems such as customers, servers, arrival processes, and service times. It then introduces common performance measures used to evaluate queueing systems like average time in system and number of customers. The document presents Kendall notation for describing queueing models and discusses estimating long-run performance measures through simulation. It provides examples of queueing systems and models to illustrate the concepts.
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.
This document discusses queueing models and their characteristics. It defines key elements like customers, servers, arrival and service processes. It introduces common queueing notation and performance measures like utilization, wait times and number of customers. Specific queueing systems are examined like the M/M/1 model. The conservation equation relating arrival rate, utilization and wait times is also covered. In summary, it provides an overview of fundamental queueing theory concepts.
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 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 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.
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 a collection of mathematical models used to analyze systems with random variability in demand and service times, like customer arrival times at service facilities. Queues form when demand exceeds service capacity, meaning customers must wait for service rather than being served immediately. Queuing models help determine optimal levels of service, like the number of cash registers or repair technicians needed. Key components of queuing systems include the input source (size, arrival patterns), service system configuration (number of servers/queues), and service speed (rate or time).
This document discusses queuing analysis and its applications. Queuing theory models systems with queues and servers that process items. It is useful for analyzing network and system performance when load or design changes are expected. The document outlines different analysis methods and key metrics like arrival rate, service time, waiting time, number of items, and utilization. It also covers important assumptions like Poisson arrivals, service time distributions, Little's Law, and example applications like database servers and multi-processor systems.
Queuing theory is used to model waiting lines in systems where demand fluctuates. It can be used to optimize resource allocation to minimize costs associated with customer wait times and unused service capacity. The key elements of a queuing system include arrivals, a queue or waiting line, service channels, and a service discipline for determining order of service. Customers arrive according to a Poisson distribution and service times follow an exponential distribution. The goal of queuing analysis is to determine the number of service channels needed to balance wait time costs and idle resource costs.
This document discusses discrete event simulation and queueing systems. It provides definitions and explanations of key concepts in discrete event simulation including: entities and attributes, events, activities, system state, and components of discrete event simulation models. It also defines concepts related to queueing systems such as arrival processes, service times, queue disciplines, and how to simulate single and multiple server queueing systems. Simulation is presented as an important tool for analyzing complex, stochastic queueing systems when mathematical analysis is not possible.
This document discusses discrete event simulation. It provides definitions of key concepts including entities, attributes, events, activities, and delays. It explains that discrete event simulation is appropriate for systems where state variables change discretely over time. It also discusses components of discrete event simulation like the system, model, state variables, and entities. Examples of queueing system simulation and steps to simulate customer arrivals and services are provided.
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….
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 summarizes key aspects of electrocardiogram (ECG) analysis including rhythm, heart rate variability, axis determination, key waves and intervals like P, QRS, and QT. It also discusses common artifacts that can appear on ECGs like power line interference, motion artifacts, and baseline drift. Rhythm is analyzed and heart rate variability is categorized from very high to poor. Axis is determined using methods like the quadrant or three lead analyses. Intervals like PR and QT are examined.
Undergraduate Topics in Computer Science, Concise Computer Vision Reinhard Klette An Introduction
into Theory and Algorithms:
FEATURE DETECTION AND OBJECT DETECTION - Localization, Classification, and Evaluation - Descriptors, Classifiers and Learning
Image Processing, Facial Expression
Implicit and explicit methods are used to insert null or empty values into columns. Relational operators compare values and return true or false based on equal, not equal, greater than, less than comparisons. Logical operators return true if conditions joined by AND, OR, or NOT are met. Values can be sorted in ascending order based on their type as either lowest to highest numeric, alphabetical character, or earliest to latest date.
The document describes creating two partitioned tables. The first table book_17 is partitioned by range of the bid column into three partitions - p1 for bids less than 200, p2 for bids between 200 and 300, and p3 for bids between 300 and 500. The second table stud_89 is partitioned by list on the department column into two partitions - s1 for the departments of Maths and Phy, and s2 for the department of BCA. Some sample data is inserted into book_17 to demonstrate the partitions, with one row failing to insert as its value falls outside the partitions.
The document discusses how to create and modify Oracle tables using SQL. It describes how to use the CREATE TABLE statement to define a table's structure, columns, constraints, and storage parameters. It also explains how to view table metadata using data dictionary views, add comments to tables and columns, and alter existing tables by adding or modifying columns while following certain restrictions.
The document provides information on data definition language (DDL) concepts in Oracle databases including naming rules and conventions for tables and columns, data types, constraints, and default values. Some key points summarized:
- Table and column names can be 1-30 characters long using letters, numbers, $, _, and # but not spaces or hyphens. Names are stored in uppercase.
- Common data types include VARCHAR2, CHAR, NUMBER, DATE, LONG, and LOB types like CLOB and BLOB. Each has specific size and storage characteristics.
- Constraints like PRIMARY KEY, FOREIGN KEY, UNIQUE, CHECK, and NOT NULL are used to enforce data integrity rules and validate column values.
- Oracle is a popular client/server database management system based on the relational database model. It is capable of supporting thousands of users simultaneously and storing terabytes of data.
- Oracle Corporation is the second largest software company in the world. Their flagship product is the Oracle database, which is widely used by organizations for mission-critical applications.
- Oracle software can run in stand-alone, client/server, or multi-tier architectures. The database component provides high availability, fault tolerance, security and management tools.
The document discusses key concepts related to database management systems including data, entities, entity sets, relationships, and database management systems. It defines data as information that has been translated into a form that is efficient for movement or processing. Entities are described as people, places, objects, events or items and entity sets are collections of related entities. Relationships describe interactions between entity sets and can be one-to-one, one-to-many, many-to-one, or many-to-many. A database management system (DBMS) is software that allows users to define, create, maintain and control access to the database, and is responsible for data management, transactions, data independence and security.
This document discusses inventory problems and the economic order quantity (EOQ) model. It covers:
- Types of inventory costs and notations used in EOQ models.
- The basic EOQ model which assumes constant demand rate and minimizes total inventory costs. This model is used to determine the optimal order quantity.
- Extensions of the basic EOQ model which relax some assumptions, such as models allowing for finite replenishment rates, shortages, quantity discounts, etc.
- Examples demonstrating how to apply the EOQ model to determine optimal order quantity and total costs for different inventory situations.
Unit-II B discusses game theory and mixed strategies. Mixed strategies involve assigning probabilities to each player's available strategies. This determines the expected payoffs for each player. For a 2x2 game without a saddle point, the method of oddments can be used to find the optimal mixed strategies. This involves calculating the differences between values in the payoff matrix, known as oddments. The oddments are then used to determine the probabilities with which each player should select their available strategies. The value of the game is equal to the expected payoff for either player using their optimal mixed strategy.
This document outlines the key concepts in game theory. It introduces two-person zero-sum games and discusses optimal strategies, including the maximin and minimax principles. A game has a saddle point solution when the maximin value for one player equals the minimax value for the other player, indicating their optimal strategies and the game's value.
Why Operations Research?
Introduction
Origin of operations research
Definition of operations research
Characteristics of operations research
Role of operations research in decision-making
Methods of solving operations research problem
Phases in solving operations research problems
Typical problems in operations research
Scope of operations research
Why to study operations research
Nita H.Shah Ravi M. Gor Hardik Soni
Why Operations Research?
Introduction
Origin of operations research
Definition of operations research
Characteristics of operations research
Role of operations research in decision-making
Methods of solving operations research problem
Phases in solving operations research problems
Typical problems in operations research
Scope of operations research
Why to study operations research
Web Technology under CSS - Introduction, Advantages, Adding CSS, Browser Compatibility, CSS and Page Layout finally Selectors all are referred with Uttam K. Roy
The document discusses various hashing techniques for storing data in a list. It describes how hashing works by using a hashing algorithm to map a key to an address in the list. Common hashing methods include direct hashing, modulo division, digit extraction, and pseudorandom generation. The document also discusses collision resolution techniques like open addressing, linked lists, linear probing, and bucket hashing. Real-world hashing algorithms may combine multiple steps and techniques to optimize storage and retrieval of data.
The document discusses different types of searching algorithms for lists. It describes linear/sequential search which searches the entire list sequentially to find a target. Binary search requires an ordered list and works by dividing the list in half on each step to search for the target. The document provides pseudocode for linear search and binary search algorithms. It also discusses variations of linear search like sentinel search and search of ordered lists. Binary search has better efficiency of O(log n) compared to O(n) for linear search.
Standard Generalized Markup Language (SGML) is a metalanguage used to define markup languages like HTML and XML. It requires defining document structure rules through a Document Type Definition (DTD). While powerful, SGML is complex. HTML simplified SGML and does not require a DTD. The World Wide Web Consortium (W3C) develops standards to ensure the long-term growth of the web, including HTML versions like HTML5 which adds audio/video embedding and new elements.
This document provides an overview of data structures and algorithms. It discusses pseudo code, abstract data types, atomic and composite data, data structures, algorithm efficiency using Big O notation, and various searching algorithms like sequential, binary, and hashed list searches. Key concepts covered include pseudo code structure and syntax, defining algorithms with headers and conditions, and analyzing different search algorithms.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
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CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
2. Syllabus
• Introduction
• Queuing System
• Classification of Queuing models
• Distribution of Arrivals (The Poisson Process): Pure Birth Process
• Distribution of Inner-arrival Time
• Distribution of Departures (Pure Death Process)
• Distribution of Service Time
• Solution of Queuing models
• Model 1 (M/M/1) : (∞/FCFS): Birth and Death Model
3. Introduction
• In 1903, A.K. Erlang, a Swedish engineer started theoretical analysis of
waiting line problem in telephone calls.
• Later, in 1951, D.G. Kendall provided a systematic and mathematical
approach to waiting line problem.
• Queue is a waiting line or act of joining a line.
• Example: Ration shop, airport runways, traffic signal, etc.
• Waiting has become an integral part of out day-to-day life.
• The objective is to formulate the system in such a manner that the
average waiting time of the customers in queue is minimized.
• The utilization of server is maximized.
4. Queuing System
• The input or arrival pattern (customers)
• The service mechanism (service pattern)
• The ‘queue discipline’
• Customer’s behavior
6. Input process or Arrival pattern
• This process is usually called arrival process and arrivals are called
customers.
• No more than one arrival can occur at any given time. If more than
one arrival can occur at any given time, we call it as bulk arrival.
• In general, customers arrive in a more or less random fashion. The
arrival pattern can be described in terms of probabilities and
consequently the probability distribution for inter-arrival time.
• There may be many arrival patterns but Poisson, Exponential and
Erlang distributions are most commonly considered.
7. The service mechanism
• It includes the distribution of time to service a customer, number of
servers is more than one, example is parallel counters for providing
service and arrangement of servers.
• Queues in tandem service is to be provided in multistage in
sequential order.
• Service time is a random variable with the same distributions for all
arrivals.
• It follows ‘negative exponential distribution’
8. The queue discipline
• It refers to the order or manner in which customers join queues.
• It describes the method used to determine the order in which
customers are served.
• First In First Out (FIFO) or First Come First Served (FCFS): The
customers are served in the order in which they join in the queue.
• Example: cinema ticket windows, bank counters etc.
• Last in First Out (LIFO) or Last Come First Served (LCFS): In this
system, the element arrived last will have a chance of getting service
first.
9. • Service in Random Order (SIRO): A service in a random order
irrespective of their arrival order in the system.
• Example: Government offices, etc.
• Service by Priority or Priority Selection: Priority disciplines are those
where any arrival is chosen for service ahead of some other
customers already in queue.
• Example: in a doctor’s chamber, doctor may ask the patient with
stomach pain to wait for sometime and give attention to the heart
patient.
10. Customer’s behavior
• It is assumed that the arrivals in the system are one by one.
• Patient customer: If a customer, on arriving at the service system,
waits in the queue until served and does not switch between waiting
lines.
• Impatient customer: Machines arrived at the maintenance shop are
examples of patient customers.
• The customer who waits for a certain time in the queue and leaves
the service system without getting service due to certain reasons.
• Bulk arrival: Customers may arrive in groups.
• This is observed when groups of ladies arrive at shopping malls during
afternoon.
11. • Balking: A customer may not like to join the queue due to long
waiting line and has no time to wait or there is not sufficient space for
waiting.
• Reneging: A customer may leave the queue after waiting for
sometime due to impatience.
• Priorities: In some circumstances, some customers are served before
others regardless of their order of arrival. These customers have
priority over others.
• Jockeying: When there are a number of queues, a customer may
move from one queue to another with the hope of receiving the
service quickly.
• Collusion: Several customers may cooperate and only one of them
may stand in the queue.
12. Queue model needs a question
• What probability distribution is followed by arrival and service
mechanism?
• What is the busy period distribution?
• How much time is spent by a customer in the queue before his
service starts and total time spent by him in system in terms of
waiting time and service time?
13. • Busy period: Server is free when customer arrives, latter will be
served immediately during this time some more customers arrive,
they will be served in their turn. It continues until no customer is left
unserved and server becomes free.
• Idle period: No customer is present in the system.
• Busy cycle: Comprises a busy period and idle period following it.
• A system is said to ‘transient state’ its operating characteristics are
dependent on time.
• The system is said to be ‘steady or prevail state’ when its behavior
becomes independent on time.
14. • Let Pn(t) denote the probability that there are n units in the system at
time t and P’n(t) be change in Pn(t) with respect to time t.
• Pn(t) Pn as t ∞ and P’n(t) 0 as t ∞
• Explosive state: If arrival rate of system is greater than its service rate,
a steady-state may not be reached regardless of long run-time and
queue length will increase with time.
15. Only Steady-state queuing system. The mathematical
models are developed by using this notations.
• N Number of customers in the system
• C Number of servers in the system
• Pn(t) Transient state probability that exactly n-customers of units are
in the system at time t
• Pn Steady state probability of having n-customers of units in the
system
• P0 Probability of having no customers in the system
• Lq Average number of customers waiting in the queue
16. • Ls Average number of customers waiting in the system
• Wq Average waiting time of customers in the queue
• Ws Average waiting time of customers in the system
• λ Average arrival rate of customers
• μ Average service rate of server
• ρ Traffic intensity or utilization factor of the server defined as the
ration of λ and μ (λ < μ)
• M Poisson arrival
• N Maximum number of customers allowed in the system
• GD General discipline for services
17. Classification of Queuing models
• Using kendall’s notation, the queuing model can be defined by
(a/b/c) : (d/e)
• where
• A = arrival rate distribution
• B = service rate distribution
• C = number of servers
• D = capacity of the system
• E = service discipline
• Example: (M/M/1) : (∞/FCFS) means arrival follow poisson distribution.
• Poisson service rate, single server, system can accommodate infinite number
of customers and service discipline is FCFS.
18. Solution of Queuing models
• Step 1: To derive the system of steady-state equations governing the
queue.
• Step 2: To solve these equations for finding the probability distribution
of the queue length.
• Step 3: To obtain probability density function for waiting time
distribution.
• Step 4: To find busy period distribution.
• Step 5: To derive formula for Ls, Lq, Ws, Wq, and var{n} or v{n}.
• Step 6: To obtain the probability of arrival during service time of any
customer.
19. Distribution of Arrivals (The poisson process):
Pure Birth Process
• Pure Birth Model: The arrival pattern of customers varies from one
system to another and it is random too, mathematically, have a
poisson distribution and only arrivals are counted and also no
departure takes place.
20. 1. Assume that there are n-units in the system at time t, and the
probability that exactly one arrival occur during small interval ∆t be
given by λ∆t + 0(∆t), where λ is the arrival rate independent of time
t and 0(∆t) contains the terms of higher powers of ∆t.
2. Under the assumption that ∆t is very small, the probability of more
than one arrival in time ∆t is negligible.
3. The number of arrivals in non-overlapping intervals is mutually
independent.
The probability of n-arrivals in time t. Denote it by Pn(t), (n > 0).
The difference-differential equations governing the process in two
different situations.
Order to derive the arrival distribution in queues
21. • Case 1: For n > 0, there are two mutually exhaustive events of having
n-units at time (t+ ∆t) in system.
• There are n-units in system at time t and no arrival takes place during
time interval ∆t.
• So at time (t + ∆t), there will be n-units in system.
n-units n-units
t t + ∆t
no-arrivals
Figure 2
22. • The probability of these two combined events will be
= Probability of n-units at time t x Probability of no arrivals during ∆t
= Pn(t) (1 - λ∆t) ______________ (1)
• There are (n – 1) units in the system at time r and one arrival takes place
during time interval ∆t.
• So at time (t + ∆t), there will be n-units in system.
(n-1)-units n-units
t t + ∆t
One arrivals
Figure 3
23. • The probability of these two combined events will be
• = Probability of (n-1)-units at time t x Probability of no arrivals during
∆t
= Pn-1(t) (1 - λ∆t) ______________ (2)
• Adding (1) and (2) two probabilities, we get probability of n-arrivals at
time t + ∆t as
Pn(t) (1 - λ∆t) = Pn(t) (1 - λ∆t) + Pn-1(t) λ∆t _________ (3)
24. • Case 2: When n=0, that is, there is no customer in the system.
• P0(t + ∆t) = Probability of no units at time t x Probability of no arrivals
during ∆t
= P0(t) (1 - λ∆t) ___________ (4)
Pn(t + ∆t) – Pn(t) = Pn(t)(1 - λ∆t) + Pn-1(t) λ∆t, n > 0 ________ (5)
Pn(t + ∆t) – Pn(t) = Pn(t)(1 - λ∆t), n = 0 ________ (6)
• The above two equations constitute the system of differential-
difference equations.
• Equation 6 can be written as
•
𝑃′
0
(𝑡)
𝑃0
(𝑡)
= - λ
25. • Integrating both sides with respect to t,
• log P0(t) = - λt + A ___________ (7)
• where A is constant of integration. Its value can be computed using
the boundary conditions
• ___________ (8)
• t = 0, P0(0) = 1 and hence A = 0.
• P0(t) = 𝑒−𝑖𝜆𝑡 ___________ (9)
26. • Putting n = 1, in equation (5)
• P′1(t) = -λP1(t) + λP0(t) = -λP1(t) + λ𝑒−𝜆𝑡
• which is linear differential equation of first order.
• ________ (10)
• Equation (8), B =0. Thus, Equation (10) can be written as
• P1(t) = λt𝑒−𝜆𝑡
• Arguing we get
• P2(t) =
(𝜆𝑡)2
2!
𝑒−𝜆𝑡 ……. Pn(t) =
(𝜆𝑡)𝑛
𝑛!
𝑒−𝜆𝑡
27. Distribution of Inter-Arrival Time
• The time T between two consecutive arrivals is called inter-arrival
time.
• Mathematical development is given to show that T follows negative
exponential law.
• Let f(T) be the probability density function of arrivals in time T.
• f(T) = λ𝑒−𝜆𝑡
________ (11)
• where λ is an arrival rate in time t.
28. • Proof: Let t be the instant of an arrival.
• Since there is no arrival during (t, t + T) and (t + T, t + T, T + ∆t), arrival be at
t + T + ∆t.
• Putting n = 0 and t = T + ∆t in Equation (10)
• P0(T + ∆t) = 𝑒−𝜆(𝑇+∆t) = 𝑒−𝜆𝑇[1 −𝜆∆t+О ∆t ]
• But P0(T) = 𝑒−𝜆𝑇
• So P0(T + ∆t) – P0(T) = P0(T)[- 𝜆 ∆t + О ∆t ]
• Dividing both sides by ∆t and taking time ∆t 0,
P’0(T) = -𝜆P0(T) ___________ (12)
• Left Hand Side (LHS) of equation (11) is probability density function of T,
say f(T).
f(T) = 𝜆𝑒−𝜆𝑇
___________ (13)
• which is negative exponential law of probability for T.
29. Markovian property of inter-arrival time
• It states that at any instant of the time until the next arrival occurs is
independent of the time that has elapsed since the occurrence of the
last arrival, i.e., Prob. (T > t1/T > to) = Prob. (0 < T < t1 – to).
0 to t1
Figure 4
31. Distribution of departures (Pure Death Process)
• The model in which only departures occur and no arrival takes place is
called pure death process.
• There are N-customers in the system at time t = 0, no arrivals occur in the
system, and departures occur at a rate 𝝁 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒕𝒊𝒎𝒆.
• The distribution of departures from the system on the basis of three
axioms.
1. The probability of exactly one departure during small interval ∆t be given
by 𝜇 ∆t + 0(∆t ).
2. The term ∆t is so small that the probability of more than one departure
in time ∆t is negligible.
3. The number of departures in non-overlapping intervals is mutually
independent.
32. • Case 1: When 0 < n < N (1 < n < N – 1)
• The probability will be
Pn (t + ∆t) = Pn(t)(1 - 𝜇 ∆t) + Pn+1(t) 𝜇 ∆t ___________ (14)
• Case 2: When n = N, that is, there are N-customers in the system.
PN (t + ∆t) = PN(t)(1 - 𝜇 ∆t) ___________ (15)
• Case 3: When n = 0, that is, there is no customer in the system.
P0 (t + ∆t) = P0(t) + P1(t) 𝜇 ∆t ___________ (16)
• Thus,
P0 (t + ∆t) = P0(t) + P1(t) 𝜇 ∆t, n = 0
Pn (t + ∆t) = Pn(t)(1 - 𝜇 ∆t) + Pn+1(t) 𝜇 ∆t, 0 < n < N
PN (t + ∆t) = PN(t)(1 - 𝜇 ∆t), n = N
33. • Rearranging the above equations, dividing by ∆t and taking limit
∆t0,
P’0(t) = P1(t) 𝜇, n =0 _________ (17)
P’n(t) = -Pn(t) 𝜇 + Pn+1(t) 𝜇, 0 < n < N _________ (18)
P’N(t) = - PN(t) 𝜇, n = N _________ (19)
• The above three equations are the required system of differential-
difference equations for pure death process.
• The solution of equation (19) is
log PN(T) = - 𝜇t + A
• where A is constant of integration.
34. • Its value can be computed using the boundary conditions
• Pn(0) = 0,
1, 𝑛 = 𝑁 ≠ 0
𝑛 ≠ 𝑁
gives A = 0. Therefore, putting n = N – 1 in
equation 18.
P’N-1(t) + 𝜇PN-1(t) = 𝜇𝑒−𝜇𝑡
• which is linear differential equation of first order. Its solution is
• PN-1(t) = 𝜇𝑡𝑒−𝜇𝑡
• Putting n = N – 2, N- 3, …… N – n, in equation (18)
Pn-k(t) =
(𝜇𝑡)𝑘
𝑘!
𝑒−𝑖𝜇𝑡 In general, Pn(t) =
(𝜇𝑡)𝑁 − 𝑛
(𝑁 − 𝑛)!
𝑒−𝑖𝜇𝑡
• which is a poisson distribution.
35. Distribution of Service Time
• Let T be the random variable denoting the service time and t be the
possible value of T.
• Let S(t) and s(t) be the cumulative density function and the probability
density function of T respectively.
• To find s(t) for the poisson departure case, it can be observed that the
probability of no service during time (0, t), which means the probability
of having no departures during the same period.
• Prob. (service time T > t) = Prob. (no departure during t) = PN(t)
• where there are N-units in the system and no arrival is allowed after N.
• PN(t) = 𝑒−𝜇𝑡
.
36. • S(t) = Prob. (service time T < t) = 1 – Prob. (service time T > t) = 1 - 𝑒−𝑖𝜇𝑡
• Differentiating both sides with respect to ‘t’
• S(t) = 0,
𝜇𝑒−𝜇𝑡, 𝑡 ≥ 0
𝑡 < 0
• This shows that the service time distribution is exponential with mean
service time 1/𝜇 and variance 1/𝜇2.
37. Model 1(M/M/1): (M/M/1) : (∞/FCFS): Birth and Death Model
• Birth and death model deals with a queuing system having a single
server, Poisson arrival, exponential service and there is no limit on the
system’s capacity while the customers are served on a “FCFS” basis.
• Step 1: Formulation of difference-differential equations
• Let Pn(t) denote the probability of n-customers in the system at time t.
• Then the probability that the system has n-customers at time (t + ∆t)
may be expressed as the sum of the combined probability of following
four mutually exclusive and exhaustive events.
38. • Pn(t + ∆t) = Pn(t) x P(no arrival in ∆t) x P(no service completion in ∆t) +
Pn(t) x P(one arrival in ∆t) x P(one service completion in ∆t) +
Pn+1(t) x P(no arrival in ∆t) x P(one service completion in ∆t) +
Pn-1(t) x P(one arrival in ∆t) x P(no service completion in ∆t)
= Pn(t)(1 - 𝜆∆t) (1 - 𝜇∆t) + Pn(t) 𝜆∆t 𝜇∆t + Pn+1(t) )(1 - 𝜆∆t)
𝜇∆t + Pn−1(t) 𝜆∆t (1 - 𝜇∆t)
or
= Pn(t + ∆t) - Pn(t) = -(𝜆 + 𝜇)Pn(t) ∆t + Pn+1(t) 𝜇∆t + Pn−1(t)𝜆∆t
• Dividing by ∆t and taking limit ∆t 0,
• P’n(t) = -(𝜆 + 𝜇)Pn(t) +𝜇Pn+1(t) +𝜆Pn−1(t), n > 1
• If there is no customer in the system at time (t + ∆t), there will be no service
during ∆t. Then for n = 0,
39. • P0(t + ∆t) = P0(t) x P(no arrival in ∆t) + P1(t) x P(no arrival in ∆t) x
P(one service completion in ∆t)
= P0(t) (1- 𝜆∆t) + P1(t) (1- 𝜆∆t) 𝜇∆t
P0(1 + ∆t) – P0(t) = - 𝜆 P0(t) ∆t + P1(t) (1- 𝜆∆t) 𝜇∆t
• Dividing by ∆t and taking limit ∆t → 0,
• P’0(t) = - 𝜆 P0(t) + 𝜇 P1(t), n = 0
• Thus, the required difference-differential equations are
• P’n(t) = -(𝜆 + 𝜇)Pn(t) +Pn+1(t)𝜇 + Pn−1(t)𝜆, n > 1 ___________ (20)
• P’0(t) = - 𝜆 P0(t) + 𝜇 P1(t), n = 0 ___________ (21)
40. • Step 2: Deriving the steady-state difference-differential equations
• Using equation (20), the steady-state difference-differential equations
are
• 0 = -(𝜆 + 𝜇)Pn + 𝜇Pn+1 + 𝜆Pn−1, n > 1 _________ (22)
• 0 = - 𝜆P0 + 𝜇P1, n = 0 _________ (23)
• Step 3: To solve above equation
• Take P0. Then equation 23 gives P1 = (𝜆/𝜇) P0.
• For n = 1, equation 22 gives P2 = (𝜆/𝜇) P1=(𝜆/𝜇)2 P0.
• In general, Pn=(𝜆/𝜇)n P0 = pnP0.
• To obtain the value of P0, we proceed as follows
41. • 1 = 𝑛=0
∞
𝑃𝑛 = 𝑛=0
∞
𝑝𝑛𝑃0 = P0 𝑛=0
∞
𝑃𝑛 =
𝑃0
1−𝑝
• or P0 = 1 – p _____________ (24)
• and Pn = pn(1 – p) _____________ (25)
• Step 4: Characteristics of model
1. Probability of queue size being greater than or equal to n = pn
• = 𝑘=𝑛
∞
𝑃𝑘 = 𝑘=𝑛
∞
1 − 𝑝 𝑝𝑘 = pn ______________ (26)
2. Average number of customers in the system
• Ls = 𝑛=0
∞
𝑛𝑃𝑛 = 𝑛=0
∞
𝑛(1 − 𝑝)𝑃𝑛
• = p(1 – p) 𝑛=1
∞
𝑛𝑝𝑛_1
44. • where X = 𝑛=1
∞
𝑛2𝑃𝑛_1 , Integrating both sides by p,
• 0
𝑝
𝑋 𝑑 𝑝 = 𝑛=1
∞ 𝑛2
𝑃𝑛
𝑛
=
𝑝
1 −𝑝
2
• Now differentiating both sides with respect to p, we get,
• X = (1 + p)/(1 – p)3
• V[n] =
𝑝(1 −𝑝)(1+𝑝)
(1 – p)3
-
𝑝
1 −𝑝
2 =
𝑝
1 −𝑝 2 ________ (29)
45. Waiting time distribution
• The waiting time of a customer in a system is that time a customer
entering for service immediately upon arrival.
• Let w be the time spent in the queue and 𝜓𝑤(𝑡) be its cumulative
probability distribution.
• 𝜓𝑤 0 = 𝑃 𝑤 − 0 = 𝑃 𝑛𝑜 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑢𝑝𝑜𝑛 𝑎𝑟𝑟𝑖𝑣𝑎𝑙
• = P0(1 – p) __________ (30)
• We want to find 𝜓𝑤 𝑡 .
• Let there be m-customers in the system upon arrival.
• Then for a customer, in order to go into service at a time between (0, t),
all n-customers must have been served by time t.
46. • Let s1, s2, …..sn denote the service time of n-customers respectively.
• w 𝑖=1
𝑛
𝑠𝑖,
0,
𝑛 ≥1
𝑛=0
_____________ (31)
• Then the probability distribution function of waiting time, w, for a
customer who has to wait is given by
• P(w < t) = P 𝑖=1
𝑛
𝑠𝑖 ≤ 𝑡 , n > 1 ___________ (32)
• Since the service time for each customer is independent, its
probability distribution function is 𝜇𝑒−𝜇𝑡(t > 0) where 𝜇 is the mean
service rate.
47. • 𝜓𝑤(𝑡) = 𝑛=1
∞
𝑃𝑛 𝑥 𝑝[ 𝑛 − 1 − 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 𝑎𝑟𝑒 𝑠𝑒𝑟𝑣𝑒𝑑 𝑖𝑛 𝑡𝑖𝑚𝑒 𝑡]
• x P (one customer is served in time ∆t)
• = 𝑛=1
∞
(1 − 𝑝)𝑃𝑛 𝜇𝑡 𝑛
_
1
𝑛−1 !
𝑒−𝜇𝑡
𝜇 ∆t ___________ (33)
• Hence, the waiting time of a customer who has to wait is given by
• 𝜓 𝑤 =
𝑑
𝑑𝑡
𝜓𝑤(t) = p(1 – p) 𝜇𝑒−𝜇𝑡(1 −𝑝) = 𝜆
𝜇 −𝜆
𝜇
𝑒− 𝜇 −𝜆 𝑡 __ (34)
48. Characteristics of waiting time distribution
1. Average waiting time of a customer in the queue
• 𝜓 𝑤 = 0
∞
𝑡 𝜓 𝑤 dt = 0
∞
𝑡p(1 – p) 𝜇𝑒−𝜇𝑡(1 −𝑝)dt
•
𝑝
𝜇(1 −𝑝)
=
𝜆
𝜇(𝜇−𝜆)
________ (35)
2. Average waiting time of an arrival that has to wait in the system
• Ws =
𝐿𝑞
𝑃(𝑤>0)
P w > 0 = 1 − P w = 0 = 1 − P += 1 − 1 − p = p
• Therefore, Ws =
𝑝
𝜇 1 −𝑝 𝑝
=
1
𝜇−𝜆
_________ (36)
3. The busy period distribution is
• 𝜓
𝑤
𝑤
> 0 =
𝜓 𝑤
𝑃(𝑤>0)
= (𝜇 - 𝜆) 𝑒− 𝜇−𝜆 𝑡 __________ (37)
49. Example 12.1:
• The arrival rate of a customer at a service window of a cinema hall
follows a probability distribution with a mean rate of 45 per hour. The
service rate of the clerk follows poisson distribution with a mean of
60 per hour.
• (a) What is the probability of having no customer in the system?
• (b) What is the probability of having five customers in the system?
• (c) Find Ls, Lq, Ws and Wq
51. Example 12.2:
• An arrival rate at a telephone booth is considered to be poisson with
an average time of 10 minutes and exponential call lengths averaging
3 minutes.
• (a) Find the fraction of a day that the telephone will be busy
• (b) What is the probability that an arrival at the booth will have to
wait?
• (c) What is the probability that an arrival will have to wait more than
10 minutes before the phone is free?
• (d) What is the probability that it will take him more than 10 minutes
altogether to wait for phone and complete his call?
52. • Arrival rate, 𝜆 = 1/10 per minute
• Service rate, 𝜇 = 1/3 per minute
(a) The fraction of the day that phone will be busy,
• p =
𝜆
𝜇
=
1
10
1
3
=
1
10
×
3
1
=
1
3
𝑝𝑒𝑟 𝑚𝑖𝑛𝑢𝑡𝑒.
(b) Probability that an arrival at the booth will have to wait
• = 1 – P0 = 1 – (1 – p) = 1 – (1 – 0.3) = 1 – 0.7 = 0.3
(c) Probability that an arrival will have to wait for more than 10 minutes
before the phone is free
• = 10
∞
1 −
𝜆
𝜇
𝜆𝑒− 𝜇 −𝜆 𝑡
dt = 1 −
1
10
1
3
1
10
𝑒
− (
1
3
) −(
1
10
) 𝑡
• = 0.3𝑒− 0.23 ×𝑡= 0.3𝑒−2.3 = 0.03
53. • (d) Probability of an arrival waiting in the system is greater than or
equal to 10
• = 10
∞
𝜇 − 𝜆 𝜆𝑒− 𝜇 −𝜆 𝑡 dt
• = 10
∞ 1
3
−
1
10
𝑒
− (
1
3
) −(
1
10
)
𝑡 𝑑𝑡
• = e-2.3 = 0.1
54. Example 12.3:
• Vehicles pass through a toll gate a rate of 90 per hour. The average
time to pass through the gate is 36 seconds. The arrival rate and the
service rate follow Poisson distribution. There is a complaint that the
vehicles wait for long duration. The authority is willing to install one
more gate to reduce the average time to pass through the toll gate to
30 seconds if the idle time of the toll gate is less than 10% and the
average queue length at the gate is more than 5 vehicles. Discuss
whether the installation of the second gate is justified or not.
55. • Arrival rate of vehicles at toll gate, 𝜆 = 90 per hour
• Departure rate of vehicles through the gate, 𝜇 = 36 seconds =
3600
36
= 100 vehicles
per hour
• P =
𝜆
𝜇
=
90
100
= 0.9
• (a) Waiting number of vehicles in the queue
• Lq =
𝑝2
1 −𝑝
=
0.9 2
1 −0.9
=
0.81
0.1
= 8.1 vehicles
• (b) Expected time taken to pass through the gate = 30 seconds.
• Service rate, 𝜇 = 30 seconds =
3600
30
= 120 vehicles per hour
• So, P =
90
120
= 0.75
• Percent of the idle time of the gate = 1 – p = 1 – 0.75 = 0.25 after percentage = 25%
• Thus, the average waiting number of vehicles in the queue is more than 5 but the
idle time of toll gate is not less than 10%. Hence, the installation of another gate is
not justified.
56. Example 12.3:
• At what average rate must a clerk at a super market work in order to
ensure a probability of 0.90 that the customer will not wait longer
than 12 minutes? It is assumed that there is only one counter, at
which customers arrive in a Poisson fashion at an average rate of 15
per hour. The length of service by the clerk has an exponential
distribution.
57. • Arrival rate , 𝜆 =
15
60
= 0.25 customers per minute
• Departure rate of customers be, 𝜇
• Probability that the customer will not have to wait more than 12
minutes.
• = 1 – 0.9 = 0.1
• Therefore, 0.1 = 12
∞ 𝜆
𝜇
(𝜇 - 𝜆) 𝑒−(𝜇 − 𝜆)dt
• 0.1 =
𝜆
𝜇
𝑒−12(𝜇 − 𝜆)
• 0.4 𝜇 = 𝑒−12(𝜇 − 𝜆) = 𝑒−12𝜇 +12𝜆
=𝑒−12𝜇 +12 𝑥 3
= 𝑒−12𝜇 +3
• 0.4 𝜇 = 𝑒3 −12𝜇 gives
1
𝜇
= 2.48 minutes/customer
58. Example 12.5:
• In a railway Marshall yard, goods trains arrive at a rate of 30 trains per
day. Assuming that the inter-arrival time follows an exponential
distribution and the service distribution is also an exponential with an
average 36 minutes, calculate
• (a) The mean size queue and
• (b) The probability that the queue size exceeds 10
• If the input of trains increases to an average 33 per day, what will be
the changes in (a) and (b)?
59. • Arrival rate , 𝜆 =
30
60 𝑥 24
=
30
1440
=
1
48
trains per minute
• Service rate, 𝜇 =
1
36
= trains per minute
• So, P =
𝜆
𝜇
=
1/48
1/36
=
36
48
= 0.75
• (a) The mean queue size, c. Ls =
𝑝
1 −𝑝
=
0.75
1 −0.75
=
0.75
0.25
= 3 trains
• (b) Probability (queue size > 10 trains) = p10 = 0.5410 = 0.056
• When the input increases to 33 trains per day i.e.,
• 𝜆 =
33
60 𝑥 24
=
33
1440
=
11
480
trains per minute
• p =
𝜆
𝜇
=
11/480
1/36
=
11
480
x
36
1
=
396
480
= 0.83
60. • (a) The mean queue size, c. Ls =
𝑝
1 −𝑝
=
0.83
1 −0.83
=
0.83
0.17
= 4.88 = 5 trains
• (b) Probability (queue size > 10 trains) = p10 = 0.8310 = 0.155
61. Example 12.6:
• In a maintenance shop, the inter-arrival times at tool crib are
exponential with an average time of 10 minutes. The length of the
service (i.e. the amount of time taken by the tool crib operator to
meet the needs of the maintenance man) time is assumed to be
exponentially distributed with a mean 6 minutes.
• (a) The probability that a person arriving at the booth will have to
wait
• (b) The average length of the queue that forms and the average time
that an operator spends in the queuing system
• (c) The probability that an arrival will have to wait for more than 12
minutes for service and to obtain his tools
• (d) The estimate of the fraction of the day that the tool crib operator
will be idle.
62. • (e) The probability that there will be six or more operators waiting for
the service
• (f) The manager of the shop will install a second booth when an
arrival would expect to wait 10 minutes or more for the service. By
how much must the rate of arrival be increased in order to justify a
second booth>
63. • Arrival rate , 𝜆 =
60
10
= 6 per hour
• Departure rate, 𝜇 =
60
6
= 10 per hour
• (a) Probability that the arrival will have to wait = p =
𝜆
𝜇
=
6
10
= 0.6
• (b) Average number of arrivals waiting time in the queue,
• Lq =
𝑝2
1 −𝑝
=
0.6 2
1 −0.6
=
0.36
0.4
= 0.9 and
• Average waiting time in system, Ws =
1
𝜇−𝜆
=
1
10−6
=
1
4
= 0.25 hour
• (c) Probability that the arrival will have to wait more than 12 minutes
• = 12
∞ 𝜆
𝜇
(𝜇 − 𝜆) 𝑒−(𝜇 − 𝜆)dt = 0.6𝑒−(4/5)= 0.6𝑒−0.8 = 0.6 x 0.44 = 0.27
64. • (d) Probability of the tool crib operator to be idle = P0 = 1- p = 1 – 0.6 = 0.4
• Therefore, for 40% of the time the tool crib operator will remain free.
• (e) Probability of six or more operators waiting for service
• = p6 = (o.6)6 = 0.05
• (f) Average waiting time of a customer in the queue
• Wq =
𝜆
𝜇(𝜇−𝜆)
=
6
10(4)
=
6
40
=
3
20
hours = 3 x
60
20
= 9 minutes
• The installation of the second booth will be justified if the arrival rate,
𝜆′
𝑠𝑎𝑦 is greater than the waiting time.
• Wq = 10 minutes =
1
6
hours
•
1
6
=
𝜆′
𝜇(𝜇−𝜆′)
= gives 𝜆′ = 6.25
• Therefore, if the arrival rate exceeds 6.25 per hour, the second booth will be
justified.