Queuing theory is the mathematical study of waiting lines in systems where demand for service exceeds the available resources. Key concepts include the patterns of arrivals and service, number of servers, queue discipline, size of the waiting area, and population size. Queuing models use notation like M/M/1 to describe characteristics like exponentially distributed inter-arrival and service times with a single server. Important metrics include the average number of customers in line and in the system, the average wait time, probability of idle servers, and server utilization. Little's law relates these metrics for systems in equilibrium.
The document describes a Monte Carlo simulation process for modeling uncertainty. It provides examples of simulating daily demand for a bakery and a car rental company using random numbers and probability distributions. For the bakery, the average daily demand over 5 days was calculated to be 17 units. For the car rental company, the average number of trips per week over 10 weeks was calculated to be 2.8 trips. The document demonstrates how Monte Carlo simulation can be used to model systems with uncertain variables and calculate average outcomes.
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.
A Monte Carlo simulation involves modeling a system with random variables to estimate outcomes. It repeats calculations using randomly generated values for the variables and averages the results. The document discusses using Monte Carlo simulations to model demand in business situations with uncertain variables. Examples show generating random numbers to simulate daily product demand over multiple days and calculating the average demand from the results.
This document discusses simulation examples of queueing systems. It describes three key steps in simulations: 1) determining input characteristics, 2) constructing a simulation table, and 3) generating input values and evaluating responses over repetitions. It then provides details on simulating a single-channel queue, including modeling arrivals and services as probability distributions and tracking events in a simulation table. The document concludes with an example simulation of customers arriving at and being served by a single checkout counter.
This document discusses methods for solving the travelling salesman problem, specifically focusing on the Hungarian method. It provides an overview of the travelling salesman problem and describes several algorithms to solve it, including genetic algorithms, branch and bound, and dynamic programming. It then explains the Hungarian method in more detail, outlining the algorithm's steps to find an optimal assignment with minimum cost by drawing lines through rows and columns to cover zero entries in the cost matrix. The advantages of the Hungarian method for solving travelling salesman problems are highlighted, along with its time complexity of O(n^3).
Applications of simulation in Business with ExamplePratima Ray
Simulation is modeling a real-world system on a computer to understand its behavior and evaluate strategies. It allows experimenting with a model instead of the real system. Some key uses of simulation include handling complex problems without optimal solutions, risky or costly real-world experiments, and answering "what if" questions. Simulation can be static or dynamic, deterministic or stochastic, discrete or continuous. Monte Carlo simulation uses random numbers to model uncertainty and is useful for decision-making under risk. Businesses apply simulation to areas like stock analysis, pricing, marketing, and cash flow forecasting. An example is using simulation to analyze a university health clinic's queuing system and improve operations.
The document describes a Monte Carlo simulation process for modeling uncertainty. It provides examples of simulating daily demand for a bakery and a car rental company using random numbers and probability distributions. For the bakery, the average daily demand over 5 days was calculated to be 17 units. For the car rental company, the average number of trips per week over 10 weeks was calculated to be 2.8 trips. The document demonstrates how Monte Carlo simulation can be used to model systems with uncertain variables and calculate average outcomes.
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.
A Monte Carlo simulation involves modeling a system with random variables to estimate outcomes. It repeats calculations using randomly generated values for the variables and averages the results. The document discusses using Monte Carlo simulations to model demand in business situations with uncertain variables. Examples show generating random numbers to simulate daily product demand over multiple days and calculating the average demand from the results.
This document discusses simulation examples of queueing systems. It describes three key steps in simulations: 1) determining input characteristics, 2) constructing a simulation table, and 3) generating input values and evaluating responses over repetitions. It then provides details on simulating a single-channel queue, including modeling arrivals and services as probability distributions and tracking events in a simulation table. The document concludes with an example simulation of customers arriving at and being served by a single checkout counter.
This document discusses methods for solving the travelling salesman problem, specifically focusing on the Hungarian method. It provides an overview of the travelling salesman problem and describes several algorithms to solve it, including genetic algorithms, branch and bound, and dynamic programming. It then explains the Hungarian method in more detail, outlining the algorithm's steps to find an optimal assignment with minimum cost by drawing lines through rows and columns to cover zero entries in the cost matrix. The advantages of the Hungarian method for solving travelling salesman problems are highlighted, along with its time complexity of O(n^3).
Applications of simulation in Business with ExamplePratima Ray
Simulation is modeling a real-world system on a computer to understand its behavior and evaluate strategies. It allows experimenting with a model instead of the real system. Some key uses of simulation include handling complex problems without optimal solutions, risky or costly real-world experiments, and answering "what if" questions. Simulation can be static or dynamic, deterministic or stochastic, discrete or continuous. Monte Carlo simulation uses random numbers to model uncertainty and is useful for decision-making under risk. Businesses apply simulation to areas like stock analysis, pricing, marketing, and cash flow forecasting. An example is using simulation to analyze a university health clinic's queuing system and improve operations.
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 is a lab manual for the Compiler Design lab course at Swami Keshvanand Institute of Technology Management & Gramothan. It provides an overview of the course, including the list of experiments, lab instructions, assessment criteria, and details of each experiment. The experiments cover topics such as lexical analysis using Lex, syntax analysis using Yacc, symbol tables, and parsing context-free grammars. Background information on C language tokens and lexical elements is also presented to introduce students to the concepts covered in the course.
This document provides an overview of Chapter 10 from the textbook "Quantitative Analysis for Management" which covers transportation and assignment models. The chapter objectives are to teach students how to structure and solve linear programming problems using transportation and assignment models. It introduces the transportation model for distributing goods from suppliers to customers and the assignment model for allocating resources to tasks. The document outlines the chapter and provides examples of setting up and solving a transportation problem using methods like the northwest corner rule and stepping stone method.
implementation of travelling salesman problem with complexity pptAntaraBhattacharya12
This document discusses the travelling salesman problem and its implementation using complexity analysis algorithms. It introduces the travelling salesman problem, which aims to find the shortest route for a salesman to visit each city once and return to the starting point. It describes using graphs and dynamic programming to model and solve the problem. An algorithm is presented that uses dynamic programming to solve the travelling salesman problem in polynomial time by breaking it down into subproblems. Applications including routing software for delivery vehicles are discussed.
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
The document contains solutions to 3 problems involving generating random numbers using different linear congruential methods. The first problem uses the linear congruential method to generate 3 two-digit random integers: 63, 51, and 55. The second problem uses the multiplicative congruential method to generate 4 three-digit random integers: 333, 319, 717, and 831. The third problem uses the mixed congruential method to generate 3 two-digit random numbers: 88, 45, and 44.
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 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.
Outsourcing of Supply Chain Management VINAY KENKERE
The document discusses outsourcing and third-party logistics. It describes how outsourcing can replace entire business functions and has become a major strategy. There are six steps to selecting outsourcing partners and making the right decision is important for business success. Third-party logistics providers offer various services and range from basic to highly integrated with customer operations. Fourth-party logistics providers focus on allocating resources and integrating supply chains.
S T Cargo Services offers global logistics services including warehousing, packaging, freight forwarding for air, ocean and other types of cargo. They have over 65,000 square meters of multi-user warehouse space across India connected to their global freight management network. The company provides various logistics services customized to customer needs including storage, packaging and transportation of general and specialized containers.
Queueing Theory- Waiting Line Model, Heizer and RenderAi Lun Wu
I HOPE IT IS HELPFUL FOR YOU> BUT PLS IWANT CREDITS> OR ADD ME AND MESSAGE ME THANKS
THERE IS A NOTE FOR PRESENTERS VIEW
HAVE A GOOD DAY
KEEP CALM AND DRINK ON
NAME: Ellen Magalona
GNDR: FML
BRTHDY: FEB. 1998
@ellenmaaee
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 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.
Types of layout algorithms include construction algorithms like ALDEP and CORELAP as well as improvement algorithms like pairwise exchange methods. ALDEP is a construction algorithm that builds a layout by successively placing departments based on their relationship information and minimum department preferences. It requires inputs like department areas, a REL chart, and sweep width. CORELAP is also a construction algorithm that determines department placement order based on total closeness ratings and places departments to maximize placing ratings, resulting in an optimal layout.
This presentation introduces the simulation software Arena. It discusses how Arena allows users to model complex systems through discrete event simulation. Some key points:
- Arena uses a graphical interface to model systems as processes involving entities and their movement between queues. It simulates operations without disrupting the real system.
- Example applications include manufacturing, business, military, and transportation systems. Arena is useful for answering "what if" questions and identifying bottlenecks.
- The presentation demonstrates building an Arena model of a railway station to simulate passenger flow, using modules like Generate, Assign, Decide, and Dispose. Running the model produces results like resource utilization and queue lengths.
- In summary, Arena is a flexible
The document discusses the Hungarian method for solving assignment problems. It begins by defining an assignment problem as minimizing the cost of completing jobs by assigning workers to tasks, where each job is assigned to exactly one worker. It then outlines the steps of the Hungarian method, which involves constructing a cost matrix, subtracting rows and columns to find zeros, and using the zeros to determine the optimal assignment. Finally, it provides an example and lists some applications of the Hungarian method like assigning machines, salespeople, contracts, teachers, and accountants.
The Traveling Salesman Problem (TSP) involves finding the minimum cost tour that visits each customer exactly once and returns to the starting depot. Key heuristics to solve the TSP include nearest neighbor, insertion methods, and 2-opt exchanges. The Vehicle Routing Problem (VRP) extends the TSP by routing multiple vehicles of limited capacity from a central depot to serve customer demands. Common heuristics for the VRP include savings algorithms and sweep methods.
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.
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 models systems with queues that form due to demand for services exceeding the system's capacity. It was originally developed to model telephone traffic congestion. Key components of queueing systems include the arrival process, queue discipline, service mechanism, and outlet. Common models include the M/M/1 queue with a single server and Poisson arrivals, and the M/M/m queue with multiple servers. Queueing theory aims to minimize waiting times and costs by understanding the tradeoff between service levels and waiting.
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 is a lab manual for the Compiler Design lab course at Swami Keshvanand Institute of Technology Management & Gramothan. It provides an overview of the course, including the list of experiments, lab instructions, assessment criteria, and details of each experiment. The experiments cover topics such as lexical analysis using Lex, syntax analysis using Yacc, symbol tables, and parsing context-free grammars. Background information on C language tokens and lexical elements is also presented to introduce students to the concepts covered in the course.
This document provides an overview of Chapter 10 from the textbook "Quantitative Analysis for Management" which covers transportation and assignment models. The chapter objectives are to teach students how to structure and solve linear programming problems using transportation and assignment models. It introduces the transportation model for distributing goods from suppliers to customers and the assignment model for allocating resources to tasks. The document outlines the chapter and provides examples of setting up and solving a transportation problem using methods like the northwest corner rule and stepping stone method.
implementation of travelling salesman problem with complexity pptAntaraBhattacharya12
This document discusses the travelling salesman problem and its implementation using complexity analysis algorithms. It introduces the travelling salesman problem, which aims to find the shortest route for a salesman to visit each city once and return to the starting point. It describes using graphs and dynamic programming to model and solve the problem. An algorithm is presented that uses dynamic programming to solve the travelling salesman problem in polynomial time by breaking it down into subproblems. Applications including routing software for delivery vehicles are discussed.
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
The document contains solutions to 3 problems involving generating random numbers using different linear congruential methods. The first problem uses the linear congruential method to generate 3 two-digit random integers: 63, 51, and 55. The second problem uses the multiplicative congruential method to generate 4 three-digit random integers: 333, 319, 717, and 831. The third problem uses the mixed congruential method to generate 3 two-digit random numbers: 88, 45, and 44.
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 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.
Outsourcing of Supply Chain Management VINAY KENKERE
The document discusses outsourcing and third-party logistics. It describes how outsourcing can replace entire business functions and has become a major strategy. There are six steps to selecting outsourcing partners and making the right decision is important for business success. Third-party logistics providers offer various services and range from basic to highly integrated with customer operations. Fourth-party logistics providers focus on allocating resources and integrating supply chains.
S T Cargo Services offers global logistics services including warehousing, packaging, freight forwarding for air, ocean and other types of cargo. They have over 65,000 square meters of multi-user warehouse space across India connected to their global freight management network. The company provides various logistics services customized to customer needs including storage, packaging and transportation of general and specialized containers.
Queueing Theory- Waiting Line Model, Heizer and RenderAi Lun Wu
I HOPE IT IS HELPFUL FOR YOU> BUT PLS IWANT CREDITS> OR ADD ME AND MESSAGE ME THANKS
THERE IS A NOTE FOR PRESENTERS VIEW
HAVE A GOOD DAY
KEEP CALM AND DRINK ON
NAME: Ellen Magalona
GNDR: FML
BRTHDY: FEB. 1998
@ellenmaaee
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 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.
Types of layout algorithms include construction algorithms like ALDEP and CORELAP as well as improvement algorithms like pairwise exchange methods. ALDEP is a construction algorithm that builds a layout by successively placing departments based on their relationship information and minimum department preferences. It requires inputs like department areas, a REL chart, and sweep width. CORELAP is also a construction algorithm that determines department placement order based on total closeness ratings and places departments to maximize placing ratings, resulting in an optimal layout.
This presentation introduces the simulation software Arena. It discusses how Arena allows users to model complex systems through discrete event simulation. Some key points:
- Arena uses a graphical interface to model systems as processes involving entities and their movement between queues. It simulates operations without disrupting the real system.
- Example applications include manufacturing, business, military, and transportation systems. Arena is useful for answering "what if" questions and identifying bottlenecks.
- The presentation demonstrates building an Arena model of a railway station to simulate passenger flow, using modules like Generate, Assign, Decide, and Dispose. Running the model produces results like resource utilization and queue lengths.
- In summary, Arena is a flexible
The document discusses the Hungarian method for solving assignment problems. It begins by defining an assignment problem as minimizing the cost of completing jobs by assigning workers to tasks, where each job is assigned to exactly one worker. It then outlines the steps of the Hungarian method, which involves constructing a cost matrix, subtracting rows and columns to find zeros, and using the zeros to determine the optimal assignment. Finally, it provides an example and lists some applications of the Hungarian method like assigning machines, salespeople, contracts, teachers, and accountants.
The Traveling Salesman Problem (TSP) involves finding the minimum cost tour that visits each customer exactly once and returns to the starting depot. Key heuristics to solve the TSP include nearest neighbor, insertion methods, and 2-opt exchanges. The Vehicle Routing Problem (VRP) extends the TSP by routing multiple vehicles of limited capacity from a central depot to serve customer demands. Common heuristics for the VRP include savings algorithms and sweep methods.
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.
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 models systems with queues that form due to demand for services exceeding the system's capacity. It was originally developed to model telephone traffic congestion. Key components of queueing systems include the arrival process, queue discipline, service mechanism, and outlet. Common models include the M/M/1 queue with a single server and Poisson arrivals, and the M/M/m queue with multiple servers. Queueing theory aims to minimize waiting times and costs by understanding the tradeoff between service levels and waiting.
The document discusses queuing theory and its application at Rao Dental Clinic. Queuing theory deals with analyzing systems where customers arrive for service and may need to wait in a queue. Observations were made of arrival and service times at the dental clinic. With one server, the utilization rate was too high, leading to long wait times. Adding more servers would reduce wait times by lowering the utilization rate. The document provides calculations to determine expected numbers of customers in the system and queue, as well as average wait times, under different service rates.
Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...IOSR Journals
Waiting lines and service systems are important parts of the business world. In this article we describe several common queuing situations and present mathematical models for analyzing waiting lines following certain assumptions. Those assumptions are that (1) arrivals come from an infinite or very large population, (2) arrivals are Poisson distributed, (3) arrivals are treated on a FIFO basis and do not balk or renege, (4) service times follow the negative exponential distribution or are constant, and (5) the average service rate is faster than the average arrival rate. The model illustrated in this Bank for customers on a level with service is the multiple-channel queuing model with Poisson Arrival and Exponential Service Times (M/M/S). After a series of operating characteristics are computed, total expected costs are studied, total costs is the sum of the cost of providing service plus the cost of waiting time. Finally we find the total minimum expected cost.
Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.
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.
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: 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….
Delays were occurring at DDM International Airport due to increasing flight volumes that were exceeding the airport's capacity in good weather conditions. The airport had 3 runways that could handle 120 flights per hour during good weather, but this was being exceeded by 45-60 arrivals per hour. Delays were increasing costs for airlines and passengers. During inclement weather, capacity was reduced, further increasing delays and costs. Building an additional runway for $1 million was recommended to increase capacity and reduce delays and associated costs, which were estimated to save nearly $2 million per year.
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.
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 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.
The document describes applying a multi-server queue model (M/M/C) to analyze waiting lines at bank ATMs. Data was collected on customer arrival and service times at 5 ATMs over 5 days. The M/M/C model was used to calculate performance metrics like average wait times and server utilization. It was found that the busy time was 2.6 hours while idle time was 7.4 hours, showing efficient service. The utilization rate of 26% indicated no need for additional servers. The study demonstrated how queue models can help banks evaluate ATM performance and waiting lines.
Queuing theory analyzes systems where customers wait in line for services. The key components of a queuing system are the input source (customers arriving), the service system (facilities providing service), and the queue discipline (order customers are served in). Common examples include lines at banks, grocery stores, and gas stations. Queuing models can have single or multiple servers and queues, and examine how changing factors like service rates, number of servers, or arrival patterns impact wait times.
The document discusses queuing systems and simulation of queuing systems. It provides examples of real-world queuing situations and defines key elements of queuing systems including arrivals, queues, servers, and outputs. It also describes measures used to evaluate system performance such as average number of customers, average wait times, and system utilization. Simulation is presented as an important technique for analyzing queuing systems when theoretical analysis is difficult.
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.
A Study on Queuing Theory and its real Life Applications.pptxmathematicssac
This document discusses the application of queuing theory to traffic management. It presents results from a study of traffic intensity at four intersections in Victoria Island, Lagos during morning, afternoon and evening peak periods. Queuing models were used to determine arrival and service rates, from which traffic intensity was calculated. Morning and evening sessions saw the highest intensities, especially on two roads. The analysis confirms queuing theory can help minimize congestion by optimizing traffic light times during peak periods. Proper road design considering separate lanes, flyovers and parking restrictions can also help ensure smooth traffic flow.
This document discusses queue management and queuing models. It describes the components of a queuing system including arrivals, servers, waiting lines, and exit. It provides suggestions for managing queues such as determining acceptable wait times and informing customers. It also presents four queuing models: single channel infinite, single channel constant, multichannel infinite, and single/multi finite. Examples are given to demonstrate how to use the models to calculate performance measures like utilization, wait times, and number of customers in the system or line.
Synergizing Forces Unveiling the Power of Data Mining, AI, Visualization, and...Erandika Gamage
Explore the transformative fusion of Data Mining, Artificial Intelligence (AI), Visualization, and Expert Knowledge in this insightful presentation. Uncover the strategic interplay of these forces and their role in crafting Intelligent Business Intelligence (BI) systems. Real-world examples, including a deep dive into Starbucks' success, illuminate how organizations leverage these elements for operational efficiency, consumer insights, and strategic decision-making. This presentation promises to demystify the complexities of Intelligent BI, providing actionable insights for data enthusiasts and industry professionals alike.
This presentation discusses the relationship between overconfidence and unethical behavior. It provides definitions of ethics and describes common reasons for unethical choices. An experiment is described that surveyed male undergraduates about road rule violations to analyze the relationship between confidence and ethical levels. The results showed an inverse relationship, with more confident participants more likely to choose unethical options. The conclusion is that overconfident people have a higher possibility of acting unethically. Comparisons are drawn to other studies that also found links between overconfidence and unethical leadership or decision making.
How to determine a firm’s cost of equity capital, How to determine a firm’s cost of debt, How to determine a firm’s overall cost of capital, How to correctly include flotation costs in capital budgeting projects, Some of the pitfalls associated with a firm’s overall
cost of capital & what to do about them
This document discusses architectural innovation, which involves changing the way components of a product are linked together while leaving the core design concepts and underlying components unchanged. It provides examples of architectural innovations like the desktop photocopier and multi-core processors. The document notes that architectural innovation can be difficult for firms to adopt as it requires changing established ways of thinking while preserving existing component knowledge. However, it can benefit organizations by improving communication and capabilities, and customers through multifunctional and convenient products.
Industry 4.0 Is your ERP system ready for the digital era?.pptxErandika Gamage
The document discusses Industry 4.0 and whether ERP systems are ready for the digital era. It covers the demands that Industry 4.0 places on ERP systems, including data storage, exchange, use and visualization. It assesses SAP technologies and how they support increased flexibility. Specific SAP solutions that enable vertical data exchange are described. Example use cases demonstrating successful Industry 4.0 ERP implementations are provided. The conclusion is that new ERP systems like SAP S/4HANA are an important first step but do not fully meet all Industry 4.0 requirements.
This document discusses strategies for tracking customer behavior in real-time to improve recommendations and personalization. It emphasizes leveraging customer service interactions and social media engagement to better understand customers' tastes and preferences. Specifically, it recommends identifying customer segments, focusing recommendations on browsing history and past purchases, and investing in social media to build customer loyalty. The overarching goal is to use data on customer behavior to offer more relevant products and drive higher revenue.
Open Source Contributions to Postgres: The Basics POSETTE 2024ElizabethGarrettChri
Postgres is the most advanced open-source database in the world and it's supported by a community, not a single company. So how does this work? How does code actually get into Postgres? I recently had a patch submitted and committed and I want to share what I learned in that process. I’ll give you an overview of Postgres versions and how the underlying project codebase functions. I’ll also show you the process for submitting a patch and getting that tested and committed.
Orchestrating the Future: Navigating Today's Data Workflow Challenges with Ai...Kaxil Naik
Navigating today's data landscape isn't just about managing workflows; it's about strategically propelling your business forward. Apache Airflow has stood out as the benchmark in this arena, driving data orchestration forward since its early days. As we dive into the complexities of our current data-rich environment, where the sheer volume of information and its timely, accurate processing are crucial for AI and ML applications, the role of Airflow has never been more critical.
In my journey as the Senior Engineering Director and a pivotal member of Apache Airflow's Project Management Committee (PMC), I've witnessed Airflow transform data handling, making agility and insight the norm in an ever-evolving digital space. At Astronomer, our collaboration with leading AI & ML teams worldwide has not only tested but also proven Airflow's mettle in delivering data reliably and efficiently—data that now powers not just insights but core business functions.
This session is a deep dive into the essence of Airflow's success. We'll trace its evolution from a budding project to the backbone of data orchestration it is today, constantly adapting to meet the next wave of data challenges, including those brought on by Generative AI. It's this forward-thinking adaptability that keeps Airflow at the forefront of innovation, ready for whatever comes next.
The ever-growing demands of AI and ML applications have ushered in an era where sophisticated data management isn't a luxury—it's a necessity. Airflow's innate flexibility and scalability are what makes it indispensable in managing the intricate workflows of today, especially those involving Large Language Models (LLMs).
This talk isn't just a rundown of Airflow's features; it's about harnessing these capabilities to turn your data workflows into a strategic asset. Together, we'll explore how Airflow remains at the cutting edge of data orchestration, ensuring your organization is not just keeping pace but setting the pace in a data-driven future.
Session in https://budapestdata.hu/2024/04/kaxil-naik-astronomer-io/ | https://dataml24.sessionize.com/session/667627
Codeless Generative AI Pipelines
(GenAI with Milvus)
https://ml.dssconf.pl/user.html#!/lecture/DSSML24-041a/rate
Discover the potential of real-time streaming in the context of GenAI as we delve into the intricacies of Apache NiFi and its capabilities. Learn how this tool can significantly simplify the data engineering workflow for GenAI applications, allowing you to focus on the creative aspects rather than the technical complexities. I will guide you through practical examples and use cases, showing the impact of automation on prompt building. From data ingestion to transformation and delivery, witness how Apache NiFi streamlines the entire pipeline, ensuring a smooth and hassle-free experience.
Timothy Spann
https://www.youtube.com/@FLaNK-Stack
https://medium.com/@tspann
https://www.datainmotion.dev/
milvus, unstructured data, vector database, zilliz, cloud, vectors, python, deep learning, generative ai, genai, nifi, kafka, flink, streaming, iot, edge
Beyond the Basics of A/B Tests: Highly Innovative Experimentation Tactics You...Aggregage
This webinar will explore cutting-edge, less familiar but powerful experimentation methodologies which address well-known limitations of standard A/B Testing. Designed for data and product leaders, this session aims to inspire the embrace of innovative approaches and provide insights into the frontiers of experimentation!
2. o What is a Queue?
o Queue is a linear arrangement of items waiting to be served
o Queuing Theory is the Mathematical Study of Waiting Lines/Queues
Ex: • Waiting in line at a bank for a teller
• Waiting for a customer service representative to answer a call
• Waiting for a train/bus to come
• Waiting for a computer to perform a task or respond
• Waiting for an automated car wash to clean a line of cars
By Ms. Erandika Gamage
3. • People - Bus queue, Cinema queue
• Items - Vehicle queue, Queue of applications
• Events - Queue of telephone calls, Queue of births
o Queues/Waiting Lines are formed when people or items come faster than
they can be served (due to limited resources for providing a service and
resources and demand mismatched)
By Ms. Erandika Gamage
5. o At its core, a queuing situation involves two parts.
o Someone or something that receive the service—People, Items, Events
o Someone or something that provides the service—Server
Ex: • At a bank - the customers are people seeking to deposit or withdraw money,
and the servers are the bank tellers.
• When looking at the queuing situation of a printer, the customers are the
requests that have been sent to the printer, and the server is the printer.
By Ms. Erandika Gamage
6. o Those that receive service would like to know
§ How long the queue will be ?
§ How long will I have to wait in the queue ?
o Those who provides service would like to know
§ How many hours the server idle ?
§ Should we have two servers ?
By Ms. Erandika Gamage
7. § To find the best level of service that a firm should provide
Ex: • Supermarket
Should decide how many cash register checkout position should be opened
• Petrol station
How many pumps should be opened and how many attendants should be on
duty
• Bank
Should decide how many teller windows to keep open to serve customers
during various hours of the day
By Ms. Erandika Gamage
8. Population Arrival Units in Queue/Waiting Line
Unit currently
being served
Server
Departure
Units in Queuing System
By Ms. Erandika Gamage
9. System Customers Servers
Reception Desk People Receptionist
Hospital Patients Nurses
Airport Airplanes Runway
Road network Cars Traffic light
Grocery Shoppers Checkout
Station Computer Jobs CPU, disk, CD
By Ms. Erandika Gamage
10. o Telecommunications
o Traffic Control
o Determining the sequence of computer operations
o Predicting computer performance
o Healthcare (hospital bed assignments)
o Airport traffic, airline ticket sales
o Layout of manufacturing system
By Ms. Erandika Gamage
11. 1) Pattern of arrivals
2) Pattern of service provision
3) Number of servers
4) Queue discipline
5) Size of waiting room
6) Size of population
By Ms. Erandika Gamage
12. Population Arrival Units in Queue/Waiting Line
Unit currently
being served
Departure
Units in Queuing System
Server
o Arrival Characteristics
• Pattern of arrival
• Size of population
o Waiting Line Characteristics
• Size of waiting room
• Queue discipline
o Service Characteristics
• Pattern of service provision
• Number of servers
By Ms. Erandika Gamage
13. There are different pattern of arrivals
I. Poisson Distribution
II. Uniform Distribution
III. Earlang Distribution
By Ms. Erandika Gamage
14. I. Poisson Distribution
Arrivals are considered random when they are independent of one another and
their occurrence can’t be predicted exactly
Probability of “x” occurrences/arrivals within a given unit of time or space “t” is
given by
e!"# λt $
x!
𝜆 – Average rate of arrivals
Ex: 20 customers per hour
10 lorries per day
5 items per minute
By Ms. Erandika Gamage
15. Ex:
The average number of telephone calls received per hour is 2.What is the
probability of receiving 3 telephone calls within the next hour?
𝜆 = 2 per hour x = 3 t = 1 hour
=
!!"# "# $
$!
=
!!% & &
'!
By Ms. Erandika Gamage
16. There are different pattern of service provision
I. Poisson Distribution
II. Uniform Distribution
III. Earlang Distribution
By Ms. Erandika Gamage
17. I. Poisson Distribution
In many cases it can be assumed that random service times are described by the
negative exponential distribution ( µe!%#)
Probability of “x” service completions within a given unit of time or space “t” is
given by
e!%# µt $
x!
µ – Average rate of service provision
Ex: 12 customers served per hour
10 vehicles serviced per day
5 items assembled per minute
By Ms. Erandika Gamage
18. o In a queuing system, the pattern of service completion can be expressed in 2 ways
I. Service rate (average rate of service provision)
Ex: 2 vehicles serviced per hour
II. Service time ( time taken to serve one unit)
Ex: average time taken to service a vehicle is 30 min.
The unit of measurement for pattern of arrival and pattern of service completion
should be equal.
Ex: A TV repairman finds that the average time spent on his jobs is 30 min per TV set and
is negative exponential.TV sets arrive in a Poisson fashion at the rate of 10 per eight hour
day.
Average rate of arrival = 10 per eight hour day
Average rate of service completion = 30 min per TV set
= ½ hours per TV set * 2 *8
= 16 per eight hour day
By Ms. Erandika Gamage
19. o Single Server Queue
• Single Server Sigle Queue
• Single Server Multiple Queues
o Multi Server Queue
• Multi Server Single Queue
• Multi Server Multiple Queues
Server Server 2
Server 3
Server 1
Server 2
Server 3
Server 1
Server
By Ms. Erandika Gamage
20. o Single Server Queue
• Single Server Sigle Queue
Ex:
§ Single Server Multiple Queues
Ex:
o Multi Server Queue
• Multi Server Single Queue
Ex:
§ Multi Server Multiple Queues
Ex:
• Students arriving at a library counter
• Family doctor’s office
• Different cash counters in an electricity
office
• Different boarding pass encounters at an
airport
• Booking at a service station
By Ms. Erandika Gamage
21. o Single Server Queue
§ Single Server Sigle Queue
o Multi Server Queue
§ Multi Server Single Queue
Server Server 2
Server 3
Server 1
By Ms. Erandika Gamage
22. o This is the rule in which units in the queue are being selected for service
I. First Come First Served (FCFS) / First in First Out (FIFO)
Ex: Payment counter at shops
II. Last Come First Served (LCFS)/ Last in First Out (LIFO)
Ex: Elevator
III. Service in Random Order (SIRO)
Ex: Drawing tickets out of a pool of tickets for service
IV. Priority Service
Ex: Hospital Emergency Room (patients who are critically injured will move ahead in
treatment)
By Ms. Erandika Gamage
23. o This is the rule in which units in the queue are being selected for service
I. First Come First Served (FCFS) / First in First Out (FIFO)
Ex: Payment counter at shops
II. Last Come First Served (LCFS)/ Last in First Out (LIFO)
Ex: Elevator
III. Service in Random Order (SIRO)
Ex: Drawing tickets out of a pool of a pool of tickets for service
IV. Priority Service
Ex: Hospital Emergency Room (patients who are critically injured will move ahead in
treatment)
By Ms. Erandika Gamage
24. o Maximum allowable size/number of units in the queue
I. Size of waiting room is infinity
Ex:Tollbooth serving arriving vehicles
II. Limited size of waiting room
Ex: A small restaurant has 10 tables and can serve no more than 50 customers
Server
Waiting Room
Server
Waiting Room
By Ms. Erandika Gamage
25. o This is the size of population eligible to receive the service. (Total number of
eligible units outside the queuing system)
I. Infinite Population
Ex: all people of a city or state (and others) could be the potential customers at a
milk parlor.
II. Finite Population
Ex: Customers at University Base Canteen
• In this event the rate of arrival is considered to be proportional to the size of
population
By Ms. Erandika Gamage
26. § Standard system of notation used to describe and classify the queueing
model that a queueing system corresponds to.
§ The notation was first suggested by D. G. Kendall in 1953
Pattern
of
Arrivals
Pattern of
Service
Provision
Number
of
Servers
Queue
Discipline
Size of
Waiting
Room
Size of
Population
By Ms. Erandika Gamage
27. § Ex:Think of an ATM
It can serve: one customer at a time; in a first-in-first-out order; with a randomly-
distributed arrival process and service distribution time; unlimited queue capacity;
and unlimited number of possible customers.
(M / M / 1) : ( FIFO / ∞ / ∞ )
M –Markovian, Poisson process (Poisson distribution for arrival and Negative
Exponential distribution for service provision
By Ms. Erandika Gamage
28. 1. (M / M / 1) : ( FIFO / ∞ / ∞ )
2. (M / M / 1) : ( FIFO / L / ∞ )
3. (M / M / S) : ( FIFO / ∞ / ∞ )
4. (M / M / S) : ( FIFO / L / ∞ )
5. (M / M / 1) : ( FIFO / ∞ / N )
6. (M / M / S) : ( FIFO / ∞ / N )
7. Queuing Networks
By Ms. Erandika Gamage
29. The system starts with empty and idle condition in the beginning (a supermarket just
open early in morning and no customers yet)
Then it gradually go in to one or more peak time where the number of customers in
the system reach the highest level and gradually reduces.
At the end of the service hour (in night before closing the supermarket) either the
arrival of the customers are cutoff or there is really no more customers.
The assumption of steady state in the queuing theory doesn’t represent the reality
But in queuing theory we assume that a queue will acquire steady state
condition(system doesn’t change anymore) after few time, which means queue has
reached equilibrium.
Queue has reached equilibrium means the probability that the system is in a given
state is not time dependent
By Ms. Erandika Gamage
31. Server
Waiting Room
Rate of arrival ( 𝜆 )
Rate of service
provision (µ)
Average time spent by unit in queuing system ( Ws )
Avg number of units in queuing system ( Ls )
Avg number of units in queue ( Lq )
Average time spent by unit in queue ( Wq )
By Ms. Erandika Gamage
32. Probability of “n” units in queuing system
P(n) = 𝜃 P(n - 1)
P(n) = 𝜃)P(0)
P(n) = 𝜃!
1 − 𝜃
By Ms. Erandika Gamage
33. P(n) = θn (1 – θ)
P (n) is a measure of probability and cannot be negative
P (n) = θn (1 – θ) cannot be negative
(1- θ) > 0 (since if θ >1 then θn (1 – θ) will be negative)
θ < 1
λ / μ < 1
λ < μ
Condition for equilibrium is that, λ < μ
(1)
(*)
By Ms. Erandika Gamage
34. Probability that the queuing system is empty
P(n) = θn (1 – θ)
If queuing system is empty then n = 0
Hence,
P(0) = θ0 (1 – θ)
P(0) = (1 – θ)
Probability that the queuing system is empty = (1 – θ)
(1)
(4)
By Ms. Erandika Gamage
35. Probability that the server is idle
Server will idle only if the queuing system is empty.
Hence,
Probability that the server is idle = (1 – θ)
Number of hours server idle per day
Suppose a working day has H hours,
Number of hours server idle per day = H (1 – θ)
(5)
(6)
By Ms. Erandika Gamage
36. Average number of units in queuing system ( Ls )
Ls = θ / (1 – θ)
Average number of units in queue ( Lq )
Lq = θ2 / (1 – θ)
By Ms. Erandika Gamage
37. § First proven by mathematician John Little in 1961
§ Long-term average number of units (L) in a queuing system is equal to the long-term
average rate of arrival (λ) multiplied by the average time that a unit spends in the
system (W)
L = λW
Note:
§ Little’s law assumes that the system is in “equilibrium”
§ Valid for any queuing model
By Ms. Erandika Gamage
38. Average time spent by unit in queuing system (Ws )
According to Little’s Law,
L = λW
Hence,
Ls = λ Ws
Average time spent by unit in queue (Wq )
According to Little’s Law,
L = λW
Hence,
Lq = λ Wq
(9)
(10)
By Ms. Erandika Gamage
39. § Average time spent by unit in queuing system ( Ws )
Ls = λ Ws
§ Average time spent by unit in queue ( Wq )
Lq = λ Wq
Variables :
λ = Rate of arrival of units
μ = Rate of service provision
θ = λ / μ
H = Number of working hours per day
By Ms. Erandika Gamage
40. Ex:
A TV repairman finds that the average time spent on his jobs is 30 min per TV set and
is negative exponential.TV sets arrive in a Poisson fashion at the rate of 10 per eight
hour day.
a) What is the repairman idle time each day?
b) How many jobs are ahead of the set just taken for repairs?
c) How long on the average must a TV set be kept with the repairman?
By Ms. Erandika Gamage
41. Ex: Answers
a) Rate of arrival of TV sets = 10 (per eight hour day)
Average time take to repair one set = 30 min = 1 / 2 hours
Number of sets repaired per day = 8 / (1/2) = 16 (per eight hour day)
Rate of service provision = 16 (per eight hour day)
λ= 10 μ= 16 θ= λ / μ = 10/16 = 5/8
Number of working hours per day = H = 8
Repairman idle time per day = H (1-θ)
= 8 (1-5 / 8)
= 3 hrs
b) Number of sets ahead of the set just taken for repair (LQ)
LQ = θ2/ (1-θ) = (5/8)2/ (1-5/8) = 1 ½4
By Ms. Erandika Gamage
42. Ex: Answers
c) How long must a TV set be kept with the repairman relates to WS,
LS = λ WS
WS = LS /λ
But LS = θ / (1-θ) = (5/8) / [1- (5/8)] = 5 /3
WS = ( 5 /3 ) * ( 1 /10 ) = 1 /6 (eight hour day)
= 1/6 * 8 = 1 1/3 hrs
By Ms. Erandika Gamage
43. Server
Waiting Room
Rate of arrival ( 𝜆 )
Rate of service
provision (µ)
Average time spent by unit in queuing system ( Ws )
Avg number of units in queuing system ( Ls )
Avg number of units in queue ( Lq )
Average time spent by unit in queue ( Wq )
By Ms. Erandika Gamage
44. Probability of “n” units in queuing system
If n <= L ,
P(n) = 𝜃 P(n - 1)
P(n) = 𝜃&P(0)
Probability that queuing system is empty
P (0) = (1-θ)/ (1- θL+1)
If n > L ,
P(n) = 0
By Ms. Erandika Gamage
45. Probability that the server is idle
Server will idle only if the queuing system is empty.
Hence,
Probability that the server is idle = (1-θ)/ (1- θL+1)
Number of hours server idle per day
Suppose a working day has H hours,
Number of hours server idle per day = H (1-θ)/ (1- θL+1)
By Ms. Erandika Gamage
46. Average number of units in queuing system ( Ls )
LS =
θ
(1− θL+1) {
'
'!(
- [LθL +
θL
'!(
] }
Average number of units in queue ( Lq )
Lq = LS - [1- P(0)]
By Ms. Erandika Gamage
47. Average time spent by unit in queuing system (Ws )
Ls = λ Ws
Average time spent by unit in queue (Wq )
Lq = λ Wq
Variables :
λ = Rate of arrival of units
μ = Rate of service provision
θ = λ / μ
L = Size of waiting room
H = Number of working hours per day
By Ms. Erandika Gamage
48. Ex:
At a barber shop customers arrive in a Poisson fashion at the rate of 14 per hour. There is
only one barber who takes 4mins per hair cut. There are five chairs for waiting customers.
When a customer arrives if he finds that all the waiting chairs are occupied he proceeds
to another barber shop.
a) What is the probability that the barber is idle?
b) What is the probability that there are three customers at the barber shop?
c) What is the probability that a customer that arrives turns back and proceeds to
another barber shop?
d) How many customers on the average will he lose on a eight hour working day on
account of having only five chairs.
e) If the cost of a hair cut is Rs.50 then on the average how much more would he earn
per day if he has five more chairs.
By Ms. Erandika Gamage
49. Ex: Answers
Rate of arrival of customers (λ) = 14 per hour
Time taken for one hair cut = 4 mins
Number of haircuts completed in one hour = 60 / 4 = 15
Rate of service provision (μ) =15 per hour
Size of waiting room (L)= (5+1) = 6
θ = λ / μ =14 /15
(a) Probability that barber is idle = (1-θ) / (1- θL+1)
= (1-14/15) / [1- (14/15)7]
= (0.06669)/ (0.38304)
=0.174
By Ms. Erandika Gamage
50. Ex: Answers
(b) P (n) = θn (1-θ) / (1- θL+1)
If there are three customers then n=3
P (3) = θ3 (1-θ) / (1-θ7)
= (14/15)3 * (1-(14/15) / [1-(14/15)7]
= (14/15)3 * 0.174
=0.14146
By Ms. Erandika Gamage
51. Ex: Answers
(c) A customer arrive will leave and proceed to another barber shop only if the barber shop
is full.That is only if there are six customers in the shop.
So probability that a customer who arrives will leave for another barber shop is P (6)
P (6) = θ6 (1- θ) / (1- θL+1)
= (14/15)6 * [1-(14/15)] / [1-(14/15)7]
= (14/15)6 * 0.174
= 0.115
(d) Rate of arrival of customers = λ = 14 per hour
Number of customers that arrive on an 8 hour working day = 8*14 = 112
Probability of losing a customer = 0.115
Therefore number of customers lost per day = 112*0.115 = 12.88
By Ms. Erandika Gamage
52. Ex: Answers
(e) If there are five more chairs then all together there will be 10 waiting chairs and it will be a
limited waiting room size queue with size of waiting room as 11.
A customer will leave for another barber shop if the number of customers at the shop is 11.
Prob. that a customer arriving will leave for another shop = θ11 (1- θ) / (1- θL+1)
= θ11 (1- θ) / (1- θ12)
= (14/15)11 * [1-(14/15)] / [1-(14/15)12]
= 0.4681705 * (0.0666666) / (0.5630409)
= 0.0554334
Number of customers lost per day when (L =11) = 0.0554334 * 112
= 6.21
Number of customers lost per day when (L =6) = 112*0.115
= 12.88
Number of customers saved per day = 12.88 – 6.21 = 6.67
The increasing profit = Rs.6.67*50 = Rs. 333.00
By Ms. Erandika Gamage
53. µ Server 2
Waiting Room
Rate of arrival ( 𝜆 )
Average time spent by unit in queuing system ( Ws )
Avg number of units in queuing system ( Ls )
Avg number of units in queue ( Lq )
Average time spent by unit in queue ( Wq )
µ
µ
Rate of service
provision (3µ)
Server 1
Server 3
By Ms. Erandika Gamage
54. Case I : n <= S
§ Rate of arrival = λ
§ Rate of service completion = n μ
Case II : n > S
§ Rate of arrival = λ
§ Rate of service completion = S μ
Condition for Equlibrium
Condition for equilibrium, λ ≤ Sμ
Variables :
n = Number of units in the queuing system
S = Number of servers
By Ms. Erandika Gamage
55. § Probability of “n” units in queuing system
Case I : n <= S
P (n) = (θn / n!) P (0)
P (S) = (θS / S!) P (0)
§ Evaluate P(0)
1 / P (0) = [ ∑&)*
+!'
(θn/n!) + (θS /S!) { 1 / (1-( θ /S)) } ]
§ Probability that queuing system is empty
P (0) = 1/ [ ∑&)*
+!'
(θn /n!) + (θS /S!) { 1 / (1-( θ /S)) } ]
Case II : n > S
P (n) = (SS / S!) * (θ / S)n * P(0)
By Ms. Erandika Gamage
56. § Probability that all the servers are idle
P(0) = 1/ [ ∑&)*
+!'
(θn /n!) + (θS /S!) { 1 / (1-( θ /S)) } ]
§ Probability that “r” servers are idle
P (S-r)
§ Number of hours that servers idle per day
.∑,)*
+
Hr P (S−r)
By Ms. Erandika Gamage
57. § Average number of units in queuing system ( Ls )
Ls = P(0) ∑&)'
+!' θn
&!' !
+
SS
S!
θ
S
.
S
' !
θ
S
+
θ
S
1
' !
θ
S
!
§ Average number of units in queue ( Lq )
Lq =
SS
S!
θ
S
/0'
1
' !
θ
S
1
P(0)
By Ms. Erandika Gamage
58. § Average time spent by unit in queuing system ( Ws )
Ls = λ Ws
§ Average time spent by unit in queue ( Wq )
Lq = λ Wq
Variables :
λ = Rate of arrival of units
nμ = Rate of service provision (if n <= S)
Sμ = Rate of service provision (if n > S)
n = Number of units in the queuing system
S = Number of servers
r = Number of servers idle
H = Number of working hours per day
By Ms. Erandika Gamage
59. Ex:
At a photocopy shop there is only one photocopy machine and customers arrive in a
Poisson fashion at the rate of 14 per hour. The average service time (which is negative
exponential) is 4 minutes.The shop works eight hours a day.
a) How long on the average will a customer have to wait at the photocopying shop?
b) By how much will this waiting time be reduced if they had two photocopying
machines?
Suppose the rate of arrival of customers has suddenly increased to 50 per hour due to the
opening of new university close by,
c) How long on the average will a customer have to wait at the photocopying shop.
d) Number of hours server idle per day.
By Ms. Erandika Gamage
60. Ex: Answers
(a) Rate of arrival of customers (λ) = 14 per hour
Time taken to serve one customer = 4 min
Number of customers served in one hour = 60 / 4 = 15
Rate of service completion (μ) = 15 per hour
θ = λ / μ = 14 /15
LS = θ / (1- θ) = (14 / 15) / ( 1 – (14 /15)) = 14
LS = WS λ
WS = 14 /14
Waiting time (WS) =1 hour
By Ms. Erandika Gamage
62. Ex: Answers
(b) Ls = P(0) ∑&)'
+!' θn
&!' !
+
SS
S!
θ
S
.
S
' !
θ
S
+
θ
S
1
' !
θ
S
!
We have S = 2 , θ = 14 / 15 , (θ / S) = 14 / 30 , P (0) = 0.363636
Ls = 0.363636 ∑&)'
&)' (14 /15) n
&!' !
+
22
2!
14/15
2
1
2
' !
14/15
2
+
14/15
2
1
' !
14/15
2
!
LS = 0.363636 [(14 /15) 1 / 0! + 4 / 2 (14 / 30)2 {(60 / 16) + (14 / 30)(30 / 16)2}]
LS = = 1.1931805
Continued...
By Ms. Erandika Gamage
63. Ex: Answers
(b) WS = LS / λ
WS = 1.1931805 / 14
WS = 0.0852271 hour
WS = 5.1136 min
Waiting time when there is 1 server (WS) =1 hour
Waiting time when there are 2 servers (WS) = 5.1136 min
Waiting time is reduced by (60-5.11) = 54.89 min
By Ms. Erandika Gamage
64. Ex: Answers
(c) Rate of arrival of customers (λ) = 50 per hour
Service time per customer = 4 min
Number of service completions per hour = 60 / 4 = 15
Rate of service completion (μ) = 15
λ > μ
Condition for equilibrium for single server queue is λ ≤ μ
System has not met the equilibrium
Condition for equilibrium for multi server queue is λ ≤ Sμ
S ≥ λ / μ
S ≥ 50 / 15
S ≥ 3.33 (number of servers required for equilibrium)
At least 4 photocopying machines required to provide service
Continued...
By Ms. Erandika Gamage
65. Ex: Answers
(c) Rate of arrival of customers (λ) = 50 per hour
Service time per customer = 4 min
Number of service completions per hour = 60 / 4 = 15
Rate of service completion (μ) = 15
λ > μ
Condition for equilibrium for single server queue is λ ≤ μ
System has not met the equilibrium
Condition for equilibrium for multi server queue is λ ≤ Sμ
S ≥ λ / μ
S ≥ 50 / 15
S ≥ 3.33 (number of servers required for equilibrium)
At least 4 photocopying machines required to provide service
Continued...
By Ms. Erandika Gamage
66. µ Server 2
Waiting Room
Rate of arrival ( 𝜆 )
Average time spent by unit in queuing system ( Ws )
Avg number of units in queuing system ( Ls )
Avg number of units in queue ( Lq )
Average time spent by unit in queue ( Wq )
µ
µ
Rate of service
provision (3µ)
Server 1
Server 3
By Ms. Erandika Gamage
67. Case I : n <= S
§ Rate of arrival = λ
§ Rate of service completion = n μ
Case II : n > S
§ Rate of arrival = λ
§ Rate of service completion = S μ
Variables :
n = Number of units in the queuing system
S = Number of servers
By Ms. Erandika Gamage
68. § Probability of “n” units in queuing system
Case I : n <= S
P (n) = (θn / n!) P (0)
P (S) = (θS / S!) P (0)
Case III : n > L
P(n) = 0
§ Evaluate P(0)
1 / P (0) = ∑2)*
/!'
(θn/n!) + (θS /S!)
(θ / S)S − (θ / S)L+1
1 − θ / S
Case II : S < n <= L
P (n) = (SS / S!) * (θ / S)n * P(0)
By Ms. Erandika Gamage
69. § Probability that queuing system is empty
P (0) = 1/ ∑2)*
/!'
(θn/n!) + (θS /S!)
(θ / S)S − (θ / S)L+1
1 − θ / S
§ Probability that all the servers are idle
P (0) = 1/ ∑2)*
/!'
(θn/n!) + (θS /S!)
(θ / S)S − (θ / S)L+1
1 − θ / S
§ Probability that “r” servers are idle
P (S-r) =
§ Number of hours that servers idle per day
.∑,)*
+
Hr P (S−r)
By Ms. Erandika Gamage
70. § Average number of units in queuing system ( Ls )
Ls = P(0) ∑&)'
+!' θn
&!' !
+
SS
S!
θ
S
.
S
' !
θ
S
+
θ
S
1
' !
θ
S
!
§ Average number of units in queue ( Lq )
Lq =
By Ms. Erandika Gamage
71. § Average time spent by unit in queuing system ( Ws )
Ls = λ Ws
§ Average time spent by unit in queue ( Wq )
Lq = λ Wq
Variables :
λ = Rate of arrival of units
nμ = Rate of service provision (if n <= S)
Sμ = Rate of service provision (if n > S)
n = Number of units in the queuing system
L = Size of waiting room
S = Number of servers
r = Number of servers idle
H = Number of working hours per day
By Ms. Erandika Gamage
72. Server
Waiting Room
Rate of
arrival
(N-n)𝜆
Rate of service
provision (µ)
Average time spent by unit in queuing system ( Ws )
Avg number of units in queuing system ( Ls )
Avg number of units in queue ( Lq )
Average time spent by unit in queue ( Wq )
Population Size (N)
By Ms. Erandika Gamage
74. § Probability of “n” units in queuing system
P(n) =
N!
(N−n)!
θn P(0)
§ Evaluate P(0)
1 / P (0) =∑&)*
3 N!
(N−n)!
θn
**Computer software is used to evaluate ∑!"#
$ $!
$ &! !
θn
§ Probability that queuing system is empty
P (0) = 1/∑&)*
3 3!
3 !& !
θn
By Ms. Erandika Gamage
75. § Probability that the server is idle
P (0) = 1/∑&)*
3 N!
(N−n)!
θn
§ Number of hours that servers idle per day
H P(0)
H * 1/ ∑&)*
3 N!
(N−n)!
θn
By Ms. Erandika Gamage
76. § Average number of units in queuing system ( Ls )
Ls = ∑&)*
3 nN!
(N−n)!
θn P(0)
§ Average number of units in queue ( Lq )
Lq =∑&)'
3 (n−1)N!
(N−n)!
θn P(0)
By Ms. Erandika Gamage
77. § Average time spent by unit in queuing system ( Ws )
Ls = λ Ws
§ Average time spent by unit in queue ( Wq )
Lq = λ Wq
Variables :
λ = Rate of arrival of units
μ = Rate of service provision
θ = λ / μ
N = Size of the population
n = Number of units in the queuing system
H = Number of working hours per day
By Ms. Erandika Gamage
78. µ Server 2
Waiting Room
Average time spent by unit in queuing system ( Ws )
Avg number of units in queuing system ( Ls )
Avg number of units in queue ( Lq )
Average time spent by unit in queue ( Wq )
µ
µ
Rate of service
provision (3µ)
Server 1
Server 3
Rate of
arrival
(N-n)𝜆
Population Size (N)
By Ms. Erandika Gamage
80. Case I : n <= S
§ Rate of arrival = (N-n)λ
§ Rate of service completion = n μ
Case II : n > S
§ Rate of arrival = (N-n)λ
§ Rate of service completion = S μ
Variables :
N = Size of the population
n = Number of units in the queuing system
S = Number of servers
By Ms. Erandika Gamage
81. § Probability of “n” units in queuing system
Case I : n <= S
P (n) =
N!
(N−n)!n!
θn P(0)
P (S) =
N!
(N−S)!S!
θS P(0)
§ Evaluate P(0)
1 = ∑&)*
+!' N!
(N−n)!n!
θn P(0) + ∑&)+
3 N!
(N−n)!
(SS / S!) (θ / S)n P(0)
§ Probability that queuing system is empty
P (0) = 1/ [∑&)*
+!' N!
(N−n)!n!
θn P(0) + ∑&)+
3 N!
(N−n)!
(SS/ S!) (θ / S)n P(0) ]
Case II : n > S
P (n) =
N!
(N−n)!
(SS / S!)(θ / S)n P(0)
By Ms. Erandika Gamage
82. § Probability that all the servers are idle
P (0) = 1/ [∑&)*
+!' N!
(N−n)!n!
θn P(0) + ∑&)+
3 N!
(N−n)!
(SS/ S!) (θ / S)n P(0) ]
§ Probability that “r” servers are idle
P (S-r) =
§ Number of hours that servers idle per day
.∑,)*
+
Hr P (S−r)
By Ms. Erandika Gamage
83. § Average number of units in queuing system ( Ls )
Ls = ∑&)*
3
nP(0)
§ Average number of units in queue ( Lq )
Lq =∑&)'
3
(n−1)P(n)
By Ms. Erandika Gamage
84. § Average time spent by unit in queuing system ( Ws )
Ls = λ Ws
§ Average time spent by unit in queue ( Wq )
Lq = λ Wq
Variables :
(N-n)λ = Rate of arrival of units
nμ = Rate of service provision (if n <= S)
Sμ = Rate of service provision (if n > S)
n = Number of units in the queuing system
N = Size of the population
S = Number of servers
r = Number of servers idle
H = Number of working hours per day
By Ms. Erandika Gamage
85. § A queuing system where the output of one queue is the input to another is called a
queuing network.
There are 3 types of queuing networks
I. Queues in Series
II. Cyclic Queues
By Ms. Erandika Gamage
86. I. Queues in Series
II. Cyclic Queues
Arrival Departure
2 3
By Ms. Erandika Gamage
87. I. Queues in Series
§ Now consider as a M/M/S model
§ Condition for equilibrium, λ ≤ Sμ
2 3
𝜆
µ1 µ2 µ3
Departure
Arrival
µ1
𝜆
By Ms. Erandika Gamage
88. II. Cyclic Queues
A cyclic queue has a fixed number of units rotating in a cycle and in the process receiving
service from each station
In solving the problem,
1. Consider the entire network as one system and list down all possible states
2. Write down the balance equation for each state
3. Write down the equation that describes the sum of the probabilities of all the states is
one
4. Solve the equations
By Ms. Erandika Gamage
89. II. Cyclic Queues
Ex 1.
Possible Situations (Waiting to receive service) State Probability
Station 1 Station 2
0 3 S0 P0
1 2 S1 P1
2 1 S2 P2
3 0 S3 P3
By Ms. Erandika Gamage
90. II. Cyclic Queues
Ex 1.
State Balance Equation
S0 μ2P(0) = μ1P(1)
S1 (μ1+ μ2)P1 = μ2P0+ μ1P2
S2 (μ1+ μ2)P2 = μ1P3+ μ2P1
S3 μ1P3 = μ2P2
P0+ P1 + P2 + P3 = 1 [Sum of probabilities = 1]
We can solve the above equations and find P0,P1,P2 and P3 . once we find these measures
we can find expected number of units waiting to receive service from station 1 as,
(0 × P0+1× P1 +2 ×P2 +3× P3)
By Ms. Erandika Gamage
92. II. Cyclic Queues
Ex 2.
Possible Situations (Waiting to receive service) State Probability
Station 1 Station 2 Station 3
3 0 0 S0 P0
2 1 0 S1 P1
2 0 1 S2 P2
1 2 0 S3 P3
1 1 1 S4 P4
1 0 2 S5 P5
0 3 0 S6 P6
0 2 1 S7 P7
0 1 2 S8 P8
0 0 3 S9 P9
By Ms. Erandika Gamage
93. § Determine an acceptable waiting time for customers
§ Try to divert customer’s attention when waiting
§ Inform customers of what to expect
§ Keep employees not serving the customers out of sight
§ Segment customers
§ Train the servers to be friendly
§ Encourage customers to come during the slack periods
By Ms. Erandika Gamage
94. In a life time, the average person will spend ,
§ SIX MONTHS Waiting at stoplights
§ EIGHT MONTHS Opening junk mail
§ ONE YEAR Looking for misplaced 0bjects
§ TWOYEARS Reading E-mail
§ FOUR YEARS Doing housework
§ FIVE YEARS Waiting in line
§ SIX YEARS Eating
By Ms. Erandika Gamage