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.
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.
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.
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 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.
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
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.
Queuing theory is the mathematical study of waiting lines in service systems where customers arrive for service. Some key concepts include arrival and service rates, queues, queue lengths, waiting times, and models like the M/M/1 model. The M/M/1 model describes a system with Poisson arrivals, exponential service times, and a single server. Formulas are provided to calculate values like the probability of no customers, average queue length, and average waiting times in both the queue and system for the M/M/1 model. Several examples demonstrate how to apply the M/M/1 model and formulas to calculate performance measures for queuing systems.
Queuing theory 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
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.
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.
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 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.
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
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.
Queuing theory is the mathematical study of waiting lines in service systems where customers arrive for service. Some key concepts include arrival and service rates, queues, queue lengths, waiting times, and models like the M/M/1 model. The M/M/1 model describes a system with Poisson arrivals, exponential service times, and a single server. Formulas are provided to calculate values like the probability of no customers, average queue length, and average waiting times in both the queue and system for the M/M/1 model. Several examples demonstrate how to apply the M/M/1 model and formulas to calculate performance measures for queuing systems.
Queuing theory 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
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
Queuing theory is the mathematical study of waiting lines and delays. It examines properties like average wait time, number of servers, arrival and service rates. Queues form when demand for a service exceeds capacity. The simplest queuing system has two components - a queue and server - with attributes of inter-arrival and service times. Queuing models use Kendall notation to describe systems, and the M/M/1 model is commonly used to analyze average queue length, wait times, and probability of overflow for single server queues. Queuing theory has applications in fields like telecommunications, healthcare, and computer networking.
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.
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 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.
The document discusses queuing theory and waiting lines. Queuing theory uses a mathematical approach to analyze waiting lines. The goal is to minimize the sum of customer waiting costs and service capacity costs. Key factors of queuing systems include arrival and service patterns, population source, number of servers, and queue discipline. Performance is measured by average wait time, number of customers, utilization, and probability of waiting. The Poisson distribution can model customer arrivals, and service times are often exponentially distributed.
Queueing theory is the study of waiting lines and systems. A queue forms when demand exceeds the capacity of the service facility. Key components of a queueing model include the arrival process, queue configuration, queue discipline, service discipline, and service facility. Common queueing models include the M/M/1 model (Poisson arrivals, exponential service times, single server), and the M/M/C model (Poisson arrivals, exponential service times, multiple servers). These models provide formulas to calculate important queueing statistics like expected wait time, number of customers in system, and resource utilization.
Queuing theory is the mathematical study of waiting lines in systems like 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 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.
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.
Opersea report waiting lines and queuing theoryIsaac Andaya
Queuing theory is the study of waiting lines. It analyzes systems with three components: arrivals, a waiting line, and service facilities. Key aspects studied include the patterns and behaviors of arrivals and the characteristics of waiting lines and service. The Poisson distribution is often used to model random arrivals. Performance is measured by factors like average wait times and system utilization. A common queuing model involves a single server, unlimited queue, FIFO queueing, independent Poisson arrivals, and negative exponential service times. This model is useful for analyzing systems like telecommunications and traffic control.
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.
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.
This presentation introduces discrete-event simulation software. It discusses what discrete-event simulation is, how it models systems as sequences of events over time. It covers the basic constructs like entities, resources, control elements and operations. It explains how simulation execution advances by processing the next event. It discusses entity states like active, ready, time-delayed and conditional-delayed. It also summarizes different implementations in discrete-event modeling languages and tools like Arena and AutoMod.
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 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.
Talks about what is Queuing and its application, practical life usage, with a complex problem statement with its solution. Pre-emptive and non-preemptive queue models and its algorithm.
The document provides an introduction to queuing theory, which deals with problems involving waiting in lines or queues. It discusses key concepts such as arrival and service rates, expected queue length and wait times, and the utilization ratio. Common applications of queuing theory include determining the number of servers needed at facilities like banks, restaurants, and hospitals to minimize customer wait times. The summary provides the essential information about queuing theory and its use in analyzing waiting line systems.
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
Queuing theory is the mathematical study of waiting lines and delays. It examines properties like average wait time, number of servers, arrival and service rates. Queues form when demand for a service exceeds capacity. The simplest queuing system has two components - a queue and server - with attributes of inter-arrival and service times. Queuing models use Kendall notation to describe systems, and the M/M/1 model is commonly used to analyze average queue length, wait times, and probability of overflow for single server queues. Queuing theory has applications in fields like telecommunications, healthcare, and computer networking.
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.
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 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.
The document discusses queuing theory and waiting lines. Queuing theory uses a mathematical approach to analyze waiting lines. The goal is to minimize the sum of customer waiting costs and service capacity costs. Key factors of queuing systems include arrival and service patterns, population source, number of servers, and queue discipline. Performance is measured by average wait time, number of customers, utilization, and probability of waiting. The Poisson distribution can model customer arrivals, and service times are often exponentially distributed.
Queueing theory is the study of waiting lines and systems. A queue forms when demand exceeds the capacity of the service facility. Key components of a queueing model include the arrival process, queue configuration, queue discipline, service discipline, and service facility. Common queueing models include the M/M/1 model (Poisson arrivals, exponential service times, single server), and the M/M/C model (Poisson arrivals, exponential service times, multiple servers). These models provide formulas to calculate important queueing statistics like expected wait time, number of customers in system, and resource utilization.
Queuing theory is the mathematical study of waiting lines in systems like 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 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.
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.
Opersea report waiting lines and queuing theoryIsaac Andaya
Queuing theory is the study of waiting lines. It analyzes systems with three components: arrivals, a waiting line, and service facilities. Key aspects studied include the patterns and behaviors of arrivals and the characteristics of waiting lines and service. The Poisson distribution is often used to model random arrivals. Performance is measured by factors like average wait times and system utilization. A common queuing model involves a single server, unlimited queue, FIFO queueing, independent Poisson arrivals, and negative exponential service times. This model is useful for analyzing systems like telecommunications and traffic control.
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.
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.
This presentation introduces discrete-event simulation software. It discusses what discrete-event simulation is, how it models systems as sequences of events over time. It covers the basic constructs like entities, resources, control elements and operations. It explains how simulation execution advances by processing the next event. It discusses entity states like active, ready, time-delayed and conditional-delayed. It also summarizes different implementations in discrete-event modeling languages and tools like Arena and AutoMod.
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 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.
Talks about what is Queuing and its application, practical life usage, with a complex problem statement with its solution. Pre-emptive and non-preemptive queue models and its algorithm.
The document provides an introduction to queuing theory, which deals with problems involving waiting in lines or queues. It discusses key concepts such as arrival and service rates, expected queue length and wait times, and the utilization ratio. Common applications of queuing theory include determining the number of servers needed at facilities like banks, restaurants, and hospitals to minimize customer wait times. The summary provides the essential information about queuing theory and its use in analyzing waiting line systems.
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.
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.
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 is the mathematical study of waiting lines in systems like customer service lines. Key aspects of queuing systems include the arrival and service processes, the number of servers, and the queue capacity and discipline. Little's Law relates the average number of customers in the system, the arrival rate, and the average time a customer spends in the system. Common queuing models include M/M/1 for Poisson arrivals and exponential service times with one server.
The document discusses different queuing models for analyzing efficiency at railway ticket windows. It summarizes four models: 1) M/M/1 queue with infinite capacity, 2) M/M/1 queue with finite capacity N, 3) M/M/S queue with infinite capacity, and 4) M/M/S queue with finite capacity N. The document provides sample data of arrival and service times over 1 hour and outlines the methodology and assumptions used, including Poisson arrivals and exponential service times. It then shows the manual calculations and Java code for the M/M/1 infinite queue model to find values like average number of customers and waiting times.
Queuing theory is used to determine optimal service levels by using mathematical formulas and simulations to calculate variables like wait times and average service times based on arrival rates and facility capacities. It was originally developed for telephone systems but is now used in various contexts like manufacturing, transportation, and more. Key aspects of queuing systems include arrival patterns, service disciplines, number of service channels, service times, system capacities, and customer abandonment behavior. Common queuing models include the single-channel M/M/1 model and multiple-channel M/M/s models.
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.
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.
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.
Models have been developed to help managers make better decisions about waiting line operations. Waiting line models use mathematical formulas to determine performance measures like average wait times and number of customers in the system. Burger Dome wants to study its single-server waiting line to reduce wait times and improve service. Waiting line models can indicate when improvements, like adding servers, are needed. Spreadsheets can easily implement waiting line models to analyze systems.
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 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.
This document discusses queueing simulation. It begins with an introduction to queueing models and their use in simulation to analyze systems. It then describes the key elements of a queuing system, including the calling population, system capacity, arrival process, queue behavior/discipline, service times/mechanism. An example of simulating a single-server queue and multi-server queue are also mentioned.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
This document summarizes a study on queuing processes and their application to customer service delivery at an ATM service point of Fidelity Bank in Maiduguri, Nigeria. Data was collected over 10 days on customer arrival times and service times. The study found that the traffic intensity (ρ) was 0.96, indicating the system was busy 95% of the time. This M/M/1 queuing model found a high utilization of the system. The document discusses queuing theory concepts like arrival patterns, service mechanisms, system capacity, and characteristics that can help banks minimize customer wait 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….
Comparative analysis between traditional aquaponics and reconstructed aquapon...bijceesjournal
The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
Applications of artificial Intelligence in Mechanical Engineering.pdfAtif Razi
Historically, mechanical engineering has relied heavily on human expertise and empirical methods to solve complex problems. With the introduction of computer-aided design (CAD) and finite element analysis (FEA), the field took its first steps towards digitization. These tools allowed engineers to simulate and analyze mechanical systems with greater accuracy and efficiency. However, the sheer volume of data generated by modern engineering systems and the increasing complexity of these systems have necessitated more advanced analytical tools, paving the way for AI.
AI offers the capability to process vast amounts of data, identify patterns, and make predictions with a level of speed and accuracy unattainable by traditional methods. This has profound implications for mechanical engineering, enabling more efficient design processes, predictive maintenance strategies, and optimized manufacturing operations. AI-driven tools can learn from historical data, adapt to new information, and continuously improve their performance, making them invaluable in tackling the multifaceted challenges of modern mechanical engineering.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
Design and optimization of ion propulsion dronebjmsejournal
Electric propulsion technology is widely used in many kinds of vehicles in recent years, and aircrafts are no exception. Technically, UAVs are electrically propelled but tend to produce a significant amount of noise and vibrations. Ion propulsion technology for drones is a potential solution to this problem. Ion propulsion technology is proven to be feasible in the earth’s atmosphere. The study presented in this article shows the design of EHD thrusters and power supply for ion propulsion drones along with performance optimization of high-voltage power supply for endurance in earth’s atmosphere.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELijaia
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Software Engineering and Project Management - Introduction, Modeling Concepts...Prakhyath Rai
Introduction, Modeling Concepts and Class Modeling: What is Object orientation? What is OO development? OO Themes; Evidence for usefulness of OO development; OO modeling history. Modeling
as Design technique: Modeling, abstraction, The Three models. Class Modeling: Object and Class Concept, Link and associations concepts, Generalization and Inheritance, A sample class model, Navigation of class models, and UML diagrams
Building the Analysis Models: Requirement Analysis, Analysis Model Approaches, Data modeling Concepts, Object Oriented Analysis, Scenario-Based Modeling, Flow-Oriented Modeling, class Based Modeling, Creating a Behavioral Model.
Gas agency management system project report.pdfKamal Acharya
The project entitled "Gas Agency" is done to make the manual process easier by making it a computerized system for billing and maintaining stock. The Gas Agencies get the order request through phone calls or by personal from their customers and deliver the gas cylinders to their address based on their demand and previous delivery date. This process is made computerized and the customer's name, address and stock details are stored in a database. Based on this the billing for a customer is made simple and easier, since a customer order for gas can be accepted only after completing a certain period from the previous delivery. This can be calculated and billed easily through this. There are two types of delivery like domestic purpose use delivery and commercial purpose use delivery. The bill rate and capacity differs for both. This can be easily maintained and charged accordingly.
1. Chapter - 4
Queueing Models
System Simulation and Modeling
Lecture Notes
From Banks, Carson, Nelson & Nicol
Discrete-Event System Simulation
2. 2
Introduction
Each one of us has spent a great deal of time waiting
in lines.
One example in the Cafeteria.
Other examples of queues are
Printer queue
Customers in front of a cashier
Calls waiting for answer by a technical support
3. 3
Purpose
Simulation is often used in the analysis of queueing models.
A simple but typical queueing model:
Queueing models provide the analyst with a powerful tool for
designing and evaluating the performance of queueing systems.
Typical measures of system performance:
Server utilization, length of waiting lines, and delays of
customers
For relatively simple systems, compute mathematically
For realistic models of complex systems, simulation is
usually required.
4. 4
Outline
Discuss some well-known models (not development
of queueing theories):
General characteristics of queues,
Meanings and relationships of important
performance measures,
Estimation of mean measures of performance.
Effect of varying input parameters,
Mathematical solution of some basic queueing
models.
5. 5
Characteristics of Queueing Systems
Key elements of queueing systems:
Customer: refers to anything that arrives at a
facility and requires service, e.g., people, machines,
trucks, emails.
Server: refers to any resource that provides the
requested service, e.g., repairpersons, retrieval
machines, runways at airport.
6. 6
Calling Population
[Characteristics of Queueing System]
Calling population: the population of potential
customers, may be assumed to be finite or infinite.
Finite population model: if arrival rate depends on
the number of customers being served and waiting.
e.g., model of one corporate jet, if it is being
repaired, the repair arrival rate becomes zero.
Infinite population model: if arrival rate is not
affected by the number of customers being served and
waiting.
e.g., systems with large population of potential
customers.
7. 7
System Capacity
[Characteristics of Queueing System]
System Capacity: a limit on the number of customers
that may be in the waiting line or system.
Limited capacity, e.g., an automatic car wash only
has room for 10 cars to wait in line to enter the
mechanism.
Unlimited capacity, e.g., concert ticket sales with
no limit on the number of people allowed to wait to
purchase tickets.
8. 8
Arrival Process
[Characteristics of Queueing System]
For infinite-population models:
In terms of interarrival times of successive customers.
Random arrivals: interarrival times usually characterized by a probability
distribution.
Most important model: Poisson arrival process (with rate λ), where An
represents the interarrival time between customer n-1 and customer n, and
is exponentially distributed (with mean 1/λ).
Scheduled arrivals: interarrival times can be constant or constant plus or
minus a small random amount to represent early or late arrivals.
e.g., patients to a physician or scheduled airline flight arrivals to an airport.
At least one customer is assumed to always be present, so the server is
never idle, e.g., sufficient raw material for a machine
9. 9
Arrival Process
[Characteristics of Queueing System]
For finite-population models:
Customer is pending when the customer is outside the queueing system,
e.g., machine-repair problem: a machine is “pending” when it is
operating, it becomes “not pending” the instant it demands service form
the repairman.
Runtime of a customer is the length of time from departure from the
queueing system until that customer’s next arrival to the queue, e.g.,
machine-repair problem, machines are customers and a runtime is time
to failure.
Let A1
(i), A2
(i), … be the successive runtimes of customer i, and S1
(i), S2
(i)
be the corresponding successive system times:
10. 10
Queue Behavior and Queue Discipline
[Characteristics of Queueing System]
Queue behavior: the actions of customers while in a queue
waiting for service to begin, for example:
Balk: leave when they see that the line is too long,
Renege: leave after being in the line when its moving too slowly,
Jockey: move from one line to a shorter line.
Queue discipline: the logical ordering of customers in a queue
that determines which customer is chosen for service when a
server becomes free, for example:
First-in-first-out (FIFO)
Last-in-first-out (LIFO)
Service in random order (SIRO)
Shortest processing time first (SPT)
Service according to priority (PR).
11. 11
Service Times and Service Mechanism
[Characteristics of Queueing System]
Service times of successive arrivals are denoted by S1, S2, S3.
May be constant or random.
{S1, S2, S3, …} is usually characterized as a sequence of
independent and identically distributed random variables,
e.g., exponential, Weibull, gamma, lognormal, and truncated
normal distribution.
A queueing system consists of a number of service centers and
interconnected queues.
Each service center consists of some number of servers, c,
working in parallel, upon getting to the head of the line, a
customer takes the 1st available server.
12. 12
Service Times and Service Mechanism
[Characteristics of Queueing System]
Example: consider a discount warehouse where customers
may:
Serve themselves before paying at the cashier:
13. 13
Service Times and Service Mechanism
[Characteristics of Queueing System]
Wait for one of the three clerks:
Batch service (a server serving several customers
simultaneously), or customer requires several servers
simultaneously.
14. 14
Queueing Notation
[Characteristics of Queueing System]
A notation system for parallel server queues: A/B/c/N/K
A represents the interarrival-time distribution,
B represents the service-time distribution,
c represents the number of parallel servers,
N represents the system capacity,
K represents the size of the calling population.
Common symbols for A and B include M (exponential or Markov), D (constant
or deterministic), G (arbitrary or general).
For example, M /M /1/ ∞ / ∞ indicates a single-server system that has unlimited
queue capacity and an infinite population of potential arrivals. The interarrival
times and service times are exponentially distributed.
When N and K are infinite, they may be dropped from the notation. For
example, M/M/1 / ∞ / ∞ is often shortened to M/M/1 .
15. 15
Queueing Notation
[Characteristics of Queueing System]
Primary performance measures of queueing systems:
Pn: steady-state probability of having n customers in system,
Pn(t): probability of n customers in system at time t,
λ : arrival rate,
λe: effective arrival rate,
μ : service rate of one server,
ρ server utilization,
An: interarrival time between customers n-1 and n,
Sn: service time of the nth arriving customer,
Wn: total time spent in system by the nth arriving customer,
Wn
Q: total time spent in the waiting line by customer n,
L(t): the number of customers in system at time t,
LQ(t): the number of customers in queue at time t,
L: long-run time-average number of customers in system,
LQ: long-run time-average number of customers in queue,
w: long-run average time spent in system per customer,
wQ: long-run average time spent in queue per customer.
16. 16
Time-Average Number in System L
[Characteristics of Queueing System]
There are two types of estimators: an ordinary sample average,
and a time-integrated (or time-weighted) sample average
Consider a queueing system over a period of time T,
Let Ti denote the total time during [0,T] in which the system
contained exactly i customers, the time-weighted-average
number in a system is defined by:
Consider the total area under the function is L(t), then,
The long-run time-average # in system, with probability 1:
00
1ˆ
i
i
i
i
T
T
iiT
T
L
T
i
i dttL
T
iT
T
L
0
0
)(
11ˆ
TLdttLL
T
as)(
1ˆ
17. Time-Average Number in System L
[Characteristics of Queueing System]
The time-weighted-average number in queue is:
G/G/1/N/K example: consider the results from the queueing system.
17
TLdttL
T
iT
T
L Q
T
Q
i
Q
iQ as)(
11ˆ
0
0
18. 18
Time-Average Number in System L
[Characteristics of Queueing System]
1if,1)(
0if,0
)(
L(t)tL
L(t)
tLQ
customers3.0
20
)1(2)4(1)15(0ˆ
QL
cusomters15.120/23
20/)]1(3)4(2)12(1)3(0[ˆ
L
19. 19
Average Time Spent in System Per
Customer w [Characteristics of Queueing System]
The average time spent in system per customer, called the
average system time, is:
where W1, W2, …, WN are the individual times that each of the N
customers spend in the system during [0,T].
For stable systems:
If the system under consideration is the queue alone:
G/G/1/N/K example (cont.): the average system time is
Nww asˆ
N
i
iW
N
w
1
1
ˆ
1
1
ˆ as
N
Q
Q i Q
i
w W w N
N
unitstime6.4
5
)1620(...)38(2
5
...
ˆ 521
WWW
w
20. Average Time Spent in System Per
Customer w [Characteristics of Queueing System]
20
21. 21
The Conservation Equation
[Characteristics of Queueing System]
Conservation equation (a.k.a. Little’s law)
It says that the average number of customers in the system at an
arbitrary point in time is equal to the average number of arrivals
per time unit, times the average time spent in the system.
Holds for almost all queueing systems or subsystems (regardless of the
number of servers, the queue discipline, or other special circumstances).
G/G/1/N/K example (cont.): On average, one arrival every 4 time units
and each arrival spends 4.6 time units in the system. Hence, at an
arbitrary point in time, there is (1/4)(4.6) = 1.15 customers present on
average.
wL ˆˆˆ
NTwL andas
Arrival rate
Average
System time
Average # in
system
22. 22
Server Utilization
[Characteristics of Queueing System]
Definition: the proportion of time that a server is busy.
Observed server utilization, , is defined over a specified
time interval [0,T].
Long-run server utilization is ρ .
For systems with long-run stability:
assuming that the system has a single server, it can be seen that
the server utilization is = (total busy time)/T
Tasˆ
ˆ
23. 23
Server Utilization
[Characteristics of Queueing System]
For G/G/1/∞/∞ queues:
Any single-server queueing system with average arrival rate λ
customers per time unit, where average service time E(S) = 1/ μ
time units, infinite queue capacity and calling population.
Conservation equation, L = λw, can be applied.
For a stable system, the average arrival rate to the server, λs, must
be identical to λ.
The average number of customers in the server is:
T
TT
dttLtL
T
L
T
Qs
0
0
)()(
1ˆ
24. 24
Server Utilization
[Characteristics of Queueing System]
In general, for a single-server queue:
For a single-server stable queue:
For an unstable queue , long-run server utilization
is 1.
)(and
asˆˆ
sE
TLL ss
1
25. 25
Server Utilization
[Characteristics of Queueing System]
For G/G/c/∞/∞ queues:
A system with c identical servers in parallel.
If an arriving customer finds more than one server idle, the
customer chooses a server without favoring any particular
server.
For systems in statistical equilibrium, the average number of
busy servers, Ls, is: Ls, = E(s) = /.
The long-run average server utilization is:
systemsstableforwhere,
c
cc
Ls
26. Server Utilization [Characteristics of Queueing System]
Example: Customers arrive at random to a license bureau at a rate of 50
customers per hour. Currently, there are 20-clerks, each serving 5
customers per hour on the average. The long-run; or steady-state, average
utilization of a server, given by
26
27. 27
Server Utilization and System Performance
[Characteristics of Queueing System]
System performance varies widely for a given utilization .
For example, a D/D/1 queue where E(A) = 1/ and E(S) =
1/, where:
L = = /, w = E(S) = 1/, LQ = WQ = 0.
By varying and , server utilization can assume any value
between 0 and 1.
Yet there is never any line.
In general, variability of interarrival and service times
causes lines to fluctuate in length.
28. 28
Server Utilization and System Performance
[Characteristics of Queueing System]
Example: A physician who schedules patients every 10 minutes and
spends Si minutes with the ith patient:
Arrivals are deterministic, A1 = A2 = … = -1 = 10.
Services are stochastic, E(Si) = 9.3 min and V(S0) = 0.81 min2.
On average, the physician's utilization = / = 0.93 < 1.
Consider the system is simulated with service times: S1 = 9, S2 = 12,
S3 = 9, S4 = 9, S5 = 9, …. The system becomes:
The occurrence of a relatively long service time (S2 = 12) causes a
waiting line to form temporarily.
1.0yprobabilitwithminutes12
9.0yprobabilitwithminutes9
iS
29. 29
Costs in Queueing Problems
[Characteristics of Queueing System]
Costs can be associated with various aspects of the
waiting line or servers:
System incurs a cost for each customer in the queue, say at a rate
of $10 per hour per customer.
The average cost per customer is:
If customers per hour arrive (on average), the average cost
per hour is:
Server may also impose costs on the system, if a group of c
parallel servers (1 c ∞) have utilization r, each server imposes
a cost of $5 per hour while busy.
The total server cost is: $5*c.
Q
N
j
Q
j
w
N
W
ˆ*10$
*10$
1
Wj
Q is the time
customer j spends
in queue
ˆ
hour/ˆ*10$ˆˆ*10$
customer
ˆ*10$
hour
customerˆ
QQ
Q
Lw
w
30. 30
Steady-State Behavior of Infinite-Population
Markovian Models
Markovian models: exponential-distribution arrival process
(mean arrival rate = ).
Service times may be exponentially distributed as well (M) or
arbitrary (G).
A queueing system is in statistical equilibrium if the probability
that the system is in a given state is not time dependent:
P( L(t) = n ) = Pn(t) = Pn.
Mathematical models in this chapter can be used to obtain
approximate results even when the model assumptions do not
strictly hold (as a rough guide).
Simulation can be used for more refined analysis (more faithful
representation for complex systems).
31. 31
Steady-State Behavior of Infinite-Population
Markovian Models
For the simple model studied in this chapter, the steady-state
parameter, L, the time-average number of customers in the
system is:
Apply Little’s equation to the whole system and to the queue alone:
G/G/c/∞/∞ example: to have a statistical equilibrium, a
necessary and sufficient condition is /(c) < 1.
0n
nnPL
QQ
Q
wL
ww
L
w
1
,
32. 32
M/G/1 Queues [Steady-State of Markovian Model]
Single-server queues with Poisson arrivals & unlimited capacity.
Suppose service times have mean 1/ and variance s2 and = /
< 1, the steady-state parameters of M/G/1 queue:
)1(2
)/1(
,
)1(2
)/1(1
)1(2
)1(
,
)1(2
)1(
1,/
2222
222222
0
s
s
s
s
Q
Q
ww
LL
P
33. 33
M/G/1 Queues [Steady-State of Markovian Model]
No simple expression for the steady-state probabilities P0, P1, …
L – LQ = is the time-average number of customers being
served.
Average length of queue, LQ, can be rewritten as:
If and are held constant, LQ depends on the variability, s2, of the
service times.
)1(2)1(2
222
s
QL
34. M/G/1 Queues [Steady-State of Markovian Model]
Example:
There are two workers competing for a job. Able claims an average
service time that is faster than Baker's, but Baker claims to be more
consistent, even if not as fast. The arrivals occur according to a
Poisson process at the rate A, = 2 per hour (l/30 per minute). Able's
service statistics are an average service time of 24 minutes with a
standard deviation of 20 minutes. Baker's service statistics are an
average service time of 25 minutes, but a standard deviation of only
2 minutes. If the average length of the queue is the criterion for
hiring, which worker should be hired?
34
35. 35
M/G/1 Queues [Steady-State of Markovian Model]
Poisson arrivals at rate = 2 per hour (1/30 per minute).
Able: 1/ = 24 minutes and s2 = 202 = 400 minutes2:
The proportion of arrivals who find Able idle and thus experience no delay is P0
= 1- = 1/5 = 20%.
Baker: 1/ = 25 minutes and s2 = 22 = 4 minutes2:
The proportion of arrivals who find Baker idle and thus experience no delay is
P0 = 1- = 1/6 = 16.7%.
Although working faster on average, Able’s greater service variability
results in an average queue length greater than Baker’s.
customers711.2
)5/41(2
]40024[)30/1( 22
QL
customers097.2
)6/51(2
]425[)30/1( 22
QL
36. 36
M/M/1 Queues [Steady-State of Markovian Model]
Suppose the service times in an M/G/1 queue are
exponentially distributed with mean 1/, then the variance
is s2 = 1/2.
M/M/1 queue is a useful approximate model when service
times have standard deviation approximately equal to their
means.
The steady-state parameters:
)1(
,
)1(
11
1
,
1
1,/
22
Q
Q
n
n
ww
LL
P
37. 37
M/M/1 Queues [Steady-State of Markovian Model]
Example: M/M/1 queue with service rate 10 customers
per hour.
Consider how L and w increase as arrival rate, , increases from 5
to 8.64 by increments of 20%:
If / 1, waiting lines tend to continually grow in length.
Increase in average system time (w) and average number in
system (L) is highly nonlinear as a function of .
5.0 6.0 7.2 8.64 10.0
0.500 0.600 0.720 0.864 1.000
L 1.00 1.50 2.57 6.35 ∞
w 0.20 0.25 0.36 0.73 ∞
38. 38
Effect of Utilization and Service Variability
[Steady-State of Markovian Model]
For almost all queues, if lines are too long, they can be reduced
by decreasing server utilization () or by decreasing the service
time variability (s2).
A measure of the variability of a distribution, coefficient of
variation (cv):
The larger cv is, the more variable is the distribution relative to its
expected value
2
2
)(
)(
)(
XE
XV
cv
39. 39
Effect of Utilization and Service Variability
[Steady-State of Markovian Model]
Consider LQ for any M/G/1
queue:
2
)(1
1
)1(2
)1(
22
222
cv
LQ
s
LQ for M/M/1
queue
Corrects the M/M/1
formula to account
for a non-exponential
service time dist’n
40. 40
Multiserver Queue [Steady-State of Markovian Model]
M/M/c/∞/∞ queue: c channels operating in parallel.
Each channel has an independent and identical exponential
service-time distribution, with mean 1/.
To achieve statistical equilibrium, the offered load (/) must satisfy
/ < c, where /(c) = is the server utilization.
Some of the steady-state probabilities:
L
w
cLP
c
cc
Pc
cL
c
c
cn
P
c
c
cc
n
n
1
)(
)1)(!(
)(
!
1
!
)/(
/
2
0
1
1
1
0
0
41. 41
Multiserver Queue [Steady-State of Markovian Model]
Other common multiserver queueing models:
M/G/c/∞: general service times and c parallel server. The
parameters can be approximated from those of the M/M/c/∞/∞
model.
M/G/∞: general service times and infinite number of servers, e.g.,
customer is its own system, service capacity far exceeds service
demand.
M/M/C/N/∞: service times are exponentially distributed at rate m
and c servers where the total system capacity is N c customer
(when an arrival occurs and the system is full, that arrival is turned
away).
42. 42
Steady-State Behavior of Finite-Population
Models
When the calling population is small, the presence of one or
more customers in the system has a strong effect on the
distribution of future arrivals.
Consider a finite-calling population model with K customers
(M/M/c/K/K):
The time between the end of one service visit and the next call for
service is exponentially distributed, (mean = 1/).
Service times are also exponentially distributed.
c parallel servers and system capacity is K.
43. 43
Steady-State Behavior of Finite-Population
Models
Some of the steady-state probabilities:
cLwnPL
Kccn
ccnK
K
cnP
n
K
P
ccnK
K
n
K
P
ee
K
n
n
n
cn
n
n
K
cn
n
cn
c
n
n
/,/,
,...1,,
!)!(
!
1,...,1,0,
!)!(
!
0
0
1
1
0
0
K
n
ne
e
PnK
0
)(
service)xitingentering/e(orqueuetocustomersofratearrivaleffectiverunlongtheiswhere
44. 44
Steady-State Behavior of Finite-Population
Models
Example: two workers who are responsible for10 milling
machines.
Machines run on the average for 20 minutes, then require an
average 5-minute service period, both times exponentially
distributed: = 1/20 and = 1/5.
All of the performance measures depend on P0:
Then, we can obtain the other Pn.
Expected number of machines in system:
The average number of running machines:
065.0
20
5
2!2)!10(
!10
20
510
1
10
2
2
12
0
0
n
n
n
n
n
nn
P
machines17.3
10
0
n
nnPL
machines83.617.310 LK
45. 45
Networks of Queues
Many systems are naturally modeled as networks of single
queues: customers departing from one queue may be routed
to another.
The following results assume a stable system with infinite
calling population and no limit on system capacity:
Provided that no customers are created or destroyed in the
queue, then the departure rate out of a queue is the same as the
arrival rate into the queue (over the long run).
If customers arrive to queue i at rate i, and a fraction 0 pij 1 of
them are routed to queue j upon departure, then the arrival rate
form queue i to queue j is ipij (over the long run).
46. 46
Networks of Queues
The overall arrival rate into queue j:
If queue j has cj < ∞ parallel servers, each working at rate j, then
the long-run utilization of each server is j=j/(cj) (where j < 1
for stable queue).
If arrivals from outside the network form a Poisson process with
rate aj for each queue j, and if there are cj identical servers
delivering exponentially distributed service times with mean 1/j,
then, in steady state, queue j behaves likes an M/M/cj queue with
arrival rate
i
ijijj pa
all
Arrival rate
from outside
the network
Sum of arrival rates
from other queues in
network
i
ijijj pa
all
47. 47
Network of Queues
Discount store example:
Suppose customers arrive at the rate 80 per hour and 40%
choose self-service. Hence:
Arrival rate to service center 1 is 1 = 80(0.4) = 32 per hour
Arrival rate to service center 2 is 2 = 80(0.6) = 48 per hour.
c2 = 3 clerks and 2 = 20 customers per hour.
The long-run utilization of the clerks is:
2 = 48/(3*20) = 0.8
All customers must see the cashier at service center 3, the
overall rate to service center 3 is 3 = 1 + 2 = 80 per hour.
If 3 = 90 per hour, then the utilization of the cashier is:
3 = 80/90 = 0.89
48. 48
Summary
Introduced basic concepts of queueing models.
Show how simulation, and some times mathematical analysis, can
be used to estimate the performance measures of a system.
Commonly used performance measures: L, LQ, w, wQ, , and e.
When simulating any system that evolves over time, analyst must
decide whether to study transient behavior or steady-state behavior.
Simple formulas exist for the steady-state behavior of some queues.
Simple models can be solved mathematically, and can be useful in
providing a rough estimate of a performance measure.