The document summarizes key concepts about queuing systems and simple queuing models. It discusses:
1) Components of a queuing system including the arrival process, service mechanism, and queue discipline.
2) Performance measures for queuing systems such as average delay, waiting time, and number of customers.
3) The M/M/1 queuing model where arrivals and service times follow exponential distributions with a single server. Expressions are given for performance measures in this model.
4) How limiting the queue length to a finite number affects performance measures compared to an infinite queue system.
This document provides an introduction to basic teletraffic concepts such as traffic intensity and characterization of telephone call traffic as a stochastic process. It discusses modeling traffic generated by multiple users, the Poisson distribution for modeling an infinite number of users, and the Erlang-B formula for calculating blocking probability with a finite number of telephone channels. The Erlang-B formula is fundamental for telecommunications network planning by allowing engineers to dimension systems to meet target grades of service for users.
This document summarizes a student project on queuing theory using the M/M/1 model. It includes an acknowledgements section thanking those who guided the project. It then covers queuing theory concepts like characteristics, assumptions, and formulas. The document describes simulating arrival and service data and comparing results to theoretical values. It finds the simulated values match theory. It also analyzes how changing arrival and service rates could improve the system's performance.
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
Georgia Tech: Performance Engineering - Queuing Theory and Predictive ModelingBrian Wilson
This is one lecture in a semester long course \'CS4803EPR\' I put together and taught at Georgia Tech, entitled "Enterprise Computing Performance Engineering"
----
Performance Engineering Overview - Part 2…
Queuing Theory Overview
Early life-cycle performance modeling
Simple Distributed System Model
Sequence Diagrams
Bounds for overlapping interval join on MapReduceShantanu Sharma
This document summarizes a paper on mapping interval join problems to MapReduce. It presents algorithms for assigning intervals to reducers in different scenarios: unit-length and equally-spaced intervals, variable-length but equally-spaced intervals, and a general case of variable-length intervals. The algorithms aim to respect reducer capacity while ensuring overlapping intervals are assigned to a common reducer. Proofs are given that the algorithms achieve near-optimal upper bounds on data replication compared to theoretical lower bounds.
Digital data communication techniques require synchronization between transmitters and receivers. There are two main solutions:
1) Asynchronous transmission synchronizes on a per-character basis, resynchronizing with each character.
2) Synchronous transmission synchronizes transmitters and receivers at the bit level or block level, preventing timing drift through synchronized clocks.
3) Error detection and correction techniques like parity checks and cyclic redundancy checks are used to detect errors, while forward error correction allows detection and correction of errors.
To achieve unnecessary control over data communication link
A logic is added above the physical interface which is referred as Data Link Control/ Protocol
To see the need of data link control, some of the requirements and objectives for data communication are listed as follows
Frame synchronization- start and end should be recognizable
Flow control- the sender and receiver data rate
Error control- check the error and correct it
Addressing- identify the address
Control and Data on same link- should have control information
Link Management- the initiation, maintenance and termination
We will study the three mechanisms that are a part of data link control
Flow control, error detection and error control
A presentation prepared by my friend's friend. I have done no editing at all, I'm just uploading the presentation as it is.
- The document discusses methods for estimating traffic matrices, which describe the flow of traffic between origin-destination pairs in a network.
- Early methods relied on direct measurements, which are computationally intensive. Recent approaches use inference based on link measurements and routing information.
- Current research looks at techniques like principal component analysis, Kalman filtering, and incorporating additional data like access link measurements to improve estimates while reducing measurement needs. Hybrid methods combining analysis and some direct measurements are also promising.
This paper investigates fairness among network sessions that use the Multiplicative Increase Multiplicative Decrease (MIMD) congestion control algorithm. It first studies how two MIMD sessions share bandwidth in the presence of synchronous and asynchronous packet losses. It finds that rate-dependent losses lead to fair sharing, while rate-independent losses cause unfairness. The paper also examines fairness between sessions using MIMD (e.g. Scalable TCP) versus Additive Increase Multiplicative Decrease (AIMD, e.g. standard TCP). Simulations show the AIMD sessions converge to equal throughput, while MIMD sessions' throughput depends on initial conditions. Adding rate-dependent losses can achieve fairness between
This document discusses several graph algorithms:
- Minimum spanning tree algorithms like Prim's and parallel formulations.
- Single-source and all-pairs shortest path algorithms like Dijkstra's and Floyd-Warshall. Parallel formulations are described.
- Other graph algorithms like connected components, transitive closure. Parallel formulations using techniques like merging forests are summarized.
This document discusses various sorting algorithms that can be used on parallel computers. It begins with an overview of sorting and comparison-based sorting algorithms. It then covers sorting networks like bitonic sort, which can sort in parallel using a network of comparators. It discusses how bitonic sort can be mapped to hypercubes and meshes. It also covers parallel implementations of bubble sort variants, quicksort, and shellsort. For each algorithm, it analyzes the parallel runtime and efficiency. The document provides examples and diagrams to illustrate the sorting networks and parallel algorithms.
This document summarizes a research paper that analyzes the performance of a single-server queuing system under denial-of-service attacks through simulation. The simulation models flooding and complexity attacks and measures their impact on queue growth rate and average response time. The simulation results match the analytical models in the research paper and show that both attack types significantly degrade performance but complexity attacks have a higher impact on response time. Future work is proposed to analyze more complex scenarios.
This document discusses dynamic programming and provides examples of serial and parallel formulations for several problems. It introduces classifications for dynamic programming problems based on whether the formulation is serial/non-serial and monadic/polyadic. Examples of serial monadic problems include the shortest path problem and 0/1 knapsack problem. The longest common subsequence problem is an example of a non-serial monadic problem. Floyd's all-pairs shortest path is a serial polyadic problem, while the optimal matrix parenthesization problem is non-serial polyadic. Parallel formulations are provided for several of these examples.
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The document discusses algorithm analysis and complexity. It defines a priori and a posteriori analysis, and explains that algorithm analysis deals with running time. There are two main complexity measures: time complexity, which describes how time scales with input size, and space complexity, which describes how memory usage scales with input size. Time complexity can be best-case, average-case, or worst-case. Asymptotic notation like Big-O, Big-Omega, and Big-Theta are used to describe these complexities. Common loop types like linear, logarithmic, quadratic, and dependent quadratic loops are covered along with their time complexities.
GEOframe-NewAge: documentation for probabilitiesbackward componentMarialaura Bancheri
This document provides information about the ProbabilitiesBackward component in OMS 3, which computes backward probability density functions (pdfs) of residence time, travel time, and evapotranspiration time given actual time and input data. The component solves an ordinary differential equation to obtain the pdfs as tridimensional matrices. It also calculates the mean travel and evapotranspiration times by integrating the output matrices over injection time. Details are provided on the component's inputs like rainfall, storage, evapotranspiration, and outputs including the various pdfs and mean times.
The document discusses sampling and the Hilbert transform. It begins by defining sampling as the reduction of a continuous-time (CT) signal to a discrete-time (DT) signal using a sampler. It describes the Nyquist sampling theorem which specifies the minimum sampling rate to reconstruct the original signal. It then discusses different types of sampling including impulse, natural, and flat top sampling. The document also covers aliasing, the Hilbert transform, and properties and examples of using the Hilbert transform including on bandpass signals and for system representation.
This document proposes a new algorithmic framework called Cache-Oblivious Wavefront (COW) for parallelizing recursive dynamic programming algorithms. COW aims to improve parallelism without sacrificing cache efficiency. It does so by scheduling tasks for execution as soon as their real dependency constraints are satisfied, while still using the same recursive divide-and-conquer strategy as cache-optimal algorithms to maintain optimal cache performance. The document shows that COW can theoretically reduce the span of several important dynamic programming algorithms like Floyd-Warshall's algorithm and longest common subsequence, while keeping the total work and cache complexity optimal. Experimental results on real machines demonstrate a 3-5x speedup in running time and 10-20x improvement
This document provides an overview of queuing theory and its operational and decision-making applications. It discusses key characteristics of waiting lines like arrivals, queue length and discipline, and service characteristics. It also covers measuring queue performance through metrics like average wait time and utilization. Examples given include using online check-in and self-service kiosks at airports, and fast-pass systems at Disney, to improve customer wait times.
Department of Management- Queuing Theory
Queue is formed because:-
The service facility is limited & the arrivals are infinite.
The mismatch between service facility & arrivals
TERMINOLOGY
QUEUEING SYSTEM
ARRIVAL PROCESS
CATEGORIES OF CUSTOMERS
SERVICE PROCESS
REPRESENTATION OF QUEUEING SYSTEM
NOTATION
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.
The document outlines queuing theory and various queuing models. It discusses characteristics of waiting line systems such as arrival patterns, queue characteristics, and service characteristics. Four common queuing models are described: single-channel (M/M/1), multiple-channel (M/M/S), constant-service-time (M/D/1), and limited-population models. Key concepts like Little's Law and metrics for measuring queue performance are also covered. Examples are provided to demonstrate how to apply the single-channel queuing model.
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 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
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.
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.
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.
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.
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 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.
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.
1. The document discusses queueing theory and the M/M/1 queue model. The M/M/1 queue refers to a single server queue where inter-arrival times and service times are exponentially distributed.
2. It provides the equations that define the steady-state probabilities (Pn) for each number of customers (n) in the M/M/1 queue. These equations balance the rates of entering and leaving each state.
3. The limiting probabilities are then used to calculate key metrics like the average number of customers in the system (L) and average wait time (W).
- Queuing theory deals with waiting lines that occur in many real-life situations like airports, banks, etc. It aims to minimize costs from customer wait times and idle facilities.
- A queuing system has four main components: the arrival source, service system, queue, and service discipline. The arrival and service distributions are often exponential or Poisson.
- Common characteristics include arrival and service rates, utilization, queue length, and waiting times. Models calculate these values to analyze a queuing system like a new bank branch with one teller.
Analysis of single server fixed batch service queueing system under multiple ...Alexander Decker
This document analyzes a single server queueing system with fixed batch service, multiple vacations, and the possibility of catastrophes. The system uses a Poisson arrival process and exponential service times. The server provides service in batches of size k. If fewer than k customers remain after service, the server takes an exponential vacation. If a catastrophe occurs, all customers are lost and the server vacations. The document derives the generating functions and steady state probabilities for the number of customers when the server is busy or vacationing. It also provides closed form solutions for performance measures like mean number of customers and variance. Numerical studies examine these measures for varying system parameters.
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 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.
This document provides an overview of elementary queuing theory and single server queues. It defines key characteristics of queuing systems such as the arrival process, service process, number of servers, system capacity, and queue discipline. Common distributions for arrivals (Poisson) and service times (exponential) are described. Performance measures of queuing systems like delay, queue length, throughput and utilization are introduced. Other concepts covered include PASTA properties, Kendall's notation, traffic intensity, Little's Law, Markov chains, and transition probability matrices. The document serves as a lecture on introductory queuing theory concepts.
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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 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: 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….
The document discusses transportation problems and assignment problems in operations research. It provides:
1) An overview of transportation problems, including the mathematical formulation to minimize transportation costs while meeting supply and demand constraints.
2) Methods for obtaining initial basic feasible solutions to transportation problems, such as the North-West Corner Rule and Vogel's Approximation Method.
3) Techniques for moving towards an optimal solution, including determining net evaluations and selecting entering variables.
4) The formulation and algorithm for solving assignment problems to minimize assignment costs while ensuring each job is assigned to exactly one machine.
The document discusses time and space complexity analysis of algorithms. Time complexity measures the number of steps to solve a problem based on input size, with common orders being O(log n), O(n), O(n log n), O(n^2). Space complexity measures memory usage, which can be reused unlike time. Big O notation describes asymptotic growth rates to compare algorithm efficiencies, with constant O(1) being best and exponential O(c^n) being worst.
The oc curve_of_attribute_acceptance_plansAnkit Katiyar
The document discusses operating characteristic (OC) curves, which describe the probability of accepting a lot based on the lot's quality level. The typical OC curve has an S-shape, with the probability of acceptance decreasing as the percent of nonconforming items increases. Sampling plans can approach the ideal step-function OC curve as the sample size and acceptance number increase. Specific points on the OC curve correspond to acceptance quality limits and rejection quality limits.
This document discusses conceptual problems in statistics, testing, and experimentation in cognitive psychology. It identifies three main sources of variability in psychological data: (1) participant interest and motivation, (2) individual differences, and (3) potentially stochastic cognitive mechanisms. Addressing this variability poses challenges for developing normative and descriptive models of cognition and for making inferences from group-level data to individuals. The document also discusses approaches like individual differences research and modeling heterogeneous groups to help address these challenges.
Scatter diagrams and correlation and simple linear regresssionAnkit Katiyar
The document discusses scatter diagrams, correlation, and linear regression. It defines key terms like predictor and response variables, positively and negatively associated variables, and the correlation coefficient. It also describes how to calculate the linear correlation coefficient and interpret it. The document shows an example of using least squares regression to fit a line to productivity and experience data. It provides formulas to calculate the slope and intercept of the regression line and how to make predictions with the line. However, predictions should stay within the scope of the observed data used to fit the model.
This document provides an introduction to queueing theory, covering basic concepts from probability theory used in queueing models like random variables, generating functions, and common probability distributions. It then discusses fundamental queueing models and relations, including Kendall's notation for describing queueing systems and Little's Law relating average queue length and waiting time. Specific queueing models are analyzed like the M/M/1, M/M/c, M/Er/1, M/G/1, and G/M/1 queues.
This document provides an introduction to queueing theory. It discusses key concepts such as random variables, probability distributions, performance measures, Little's law and the PASTA property. It then examines several common queueing models including the M/M/1, M/M/c, M/Er/1, M/G/1 and G/M/1 queues. For each model it derives the equilibrium distribution and discusses measures like mean queue length and waiting time. The goal is to provide the fundamental mathematical techniques for analyzing queueing systems.
This document provides an introduction to queueing theory. It discusses key concepts such as random variables, probability distributions, performance measures, Little's law and the PASTA property. It then examines several common queueing models including the M/M/1, M/M/c, M/Er/1, M/G/1 and G/M/1 queues. For each model it derives the equilibrium distribution and discusses measures like mean queue length and waiting time. The goal is to give an overview of basic queueing theory concepts and common single-server and multi-server queues.
Probability mass functions and probability density functionsAnkit Katiyar
This document discusses probability mass functions (pmf) and probability density functions (pdf) for discrete and continuous random variables. A pmf fX(x) gives the probability of a discrete random variable X taking on the value x. A pdf fX(x) defines the probability that a continuous random variable X falls within an interval via its cumulative distribution function FX(x). The pdf must be non-negative and have an area/sum of 1 under the curve/over all x values.
The document outlines a lesson on basic statistical concepts for comparative studies. It covers terminology used in comparative studies including factors, levels, treatments, response variables and experimental units. It discusses topics like randomization to avoid confounding, Simpson's paradox, and the difference between experiments and observational studies. Factorial experiments involving multiple factors are also introduced.
This document discusses histograms and stem-and-leaf plots for analyzing and visualizing the distribution of a single set of numerical data. It provides examples using yearly precipitation data from New York City to demonstrate how to create histograms and stem-and-leaf plots in R. Histograms partition data into bins to show the frequency or relative frequency of observations in each bin, while stem-and-leaf plots list the "stems" and "leaves" of values to show their distribution.
This document discusses inventory management for multiple items and locations. It introduces the concepts of:
1) Setting aggregate inventory policies to meet system-wide objectives when dealing with multiple items and locations.
2) Using exchange curves to analyze the tradeoffs between total inventory levels and other factors like number of replenishments and service levels. These curves allow setting parameters like order costs and carrying costs.
3) Determining optimal reorder quantities, cycle stock, and safety stock levels across an inventory system using techniques like exchange curves. This helps allocate limited inventory budgets across items to maximize performance.
The document summarizes the economic production quantity (EPQ) model and its extensions. It discusses:
1) The EPQ model balances fixed ordering costs and inventory holding costs to determine optimal production/order quantities and intervals.
2) The economic order quantity (EOQ) model is a special case where production rate is infinite and demand is met through ordering.
3) Sensitivity analysis shows how the optimal solutions change with different parameters like production rate and setup costs.
The Kano Model classifies customer needs into three categories - threshold, performance, and excitement - based on their effect on customer satisfaction. Threshold attributes are basic needs whose absence causes dissatisfaction. Performance attributes directly improve satisfaction as implementation increases. Excitement attributes unexpectedly delight customers when implemented. The model is useful for identifying needs, setting requirements, concept development, and analyzing competitors to maximize performance attributes while including excitement attributes.
This document provides an overview of basic probability and statistics concepts. It covers variables, descriptive statistics like mean and standard deviation, frequency distributions through histograms, the normal distribution, linear regression, and includes a practice test in the appendices. Key topics are qualitative and quantitative data, parameters versus statistics, measures of central tendency and dispersion, and generating frequency tables and histograms from data sets.
Conceptual foundations statistics and probabilityAnkit Katiyar
This document provides guidance for a 6th grade statistics and probability unit of study. It outlines key concepts students should understand, including developing questions that anticipate variability, understanding data distributions in terms of center, spread and shape, and summarizing and describing distributions using various graphs such as dot plots, histograms and box plots. Students learn to analyze subgroups within data sets and how to match statistical questions to the appropriate graph. The document emphasizes interpreting and constructing dot plots, histograms and box plots to display and analyze numerical data.
This document provides an overview of basic statistical concepts including populations, samples, parameters, statistics, and sampling methods. It defines key terms like population, sample, parameter, statistic, and discusses sampling methods like simple random sampling and stratified sampling. It also covers sampling variability, estimation, hypothesis testing, prediction, and issues around representative vs non-representative samples.
The document outlines 5 axioms of probability:
1) Probabilities are non-negative
2) Probabilities of mutually exclusive events add
3) The probability of the sample space is 1
It then proves 5 theorems about probability:
1) The probability of an event equals 1 minus the probability of its complement
2) The probability of the impossible event (the empty set) is 0
3) The probability of a subset is less than or equal to the probability of the larger set it is contained within
4) A probability is between 0 and 1
5) The addition law - for two events the probability of their union equals the sum of their probabilities minus the probability of their intersection
Applied statistics and probability for engineers solution montgomery && rungerAnkit Katiyar
This document is the copyright page and preface for the book "Applied Statistics and Probability for Engineers" by Douglas C. Montgomery and George C. Runger. The copyright is held by John Wiley & Sons, Inc. in 2003. This book was edited, designed, and produced by various teams at John Wiley & Sons and printed by Donnelley/Willard. The preface states that the purpose of the included Student Solutions Manual is to provide additional help for students in understanding the problem-solving processes presented in the main text.
A hand kano-model-boston_upa_may-12-2004Ankit Katiyar
This document introduces the Kano Model, a framework used to classify product features based on their impact on customer satisfaction. It explains that some features are "basic" and expected, while others provide linear satisfaction proportional to quality or performance. Some "excitement" features unexpectedly delight customers. The document outlines a process to apply the Kano Model to user experience design including researching customer needs, analyzing data, plotting features on the Kano diagram, and strategizing priorities with clients. It provides an example workshop applying the model to a fictional business and discusses extending the model with personas and use cases.
1. QUT
SCHOOL OF MECHANICAL, MANUFACTURING &
MEDICAL ENGINEERING
MEN170: SYSTEMS MODELLING
AND SIMULATION
7. SIMPLE QUEUING MODELS:
7.1 INTRODUCTION: A queuing system consists of one or more servers that provide
service of some sort to arriving customers. Customers who arrive to find all servers busy
generally join one or more queues (lines) in front of the servers, hence the name queuing
systems. There are several everyday examples that can be described as queuing systems,
such as bank-teller service, computer systems, manufacturing systems, maintenance
systems, communications systems and so on.
Components of a Queuing System: A queuing system is characterised by three
components:
- Arrival process
- Service mechanism
- Queue discipline.
Arrival Process
Arrivals may originate from one or several sources referred to as the calling population.
The calling population can be limited or 'unlimited'. An example of a limited calling
population may be that of a fixed number of machines that fail randomly. The arrival
process consists of describing how customers arrive to the system. If Ai is the inter-
arrival time between the arrivals of the (i-1)th and ith customers, we shall denote the
mean (or expected) inter-arrival time by E(A) and call it (λ ); = 1/(E(A) the arrival
frequency.
Service Mechanism
The service mechanism of a queuing system is specified by the number of servers
(denoted by s), each server having its own queue or a common queue and the probability
2. distribution of customer's service time. let Si be the service time of the ith customer, we
shall denote the mean service time of a customer by E(S) and µ = 1/(E(S) the service rate
of a server.
Queue Discipline
Discipline of a queuing system means the rule that a server uses to choose the next
customer from the queue (if any) when the server completes the service of the current
customer. Commonly used queue disciplines are:
FIFO - Customers are served on a first-in first-out basis.
LIFO - Customers are served in a last-in first-out manner.
Priority - Customers are served in order of their importance on the basis of their service
requirements.
Measures of Performance for Queuing Systems:
There are many possible measures of performance for queuing systems. Only some of
these will be discussed here.
Let,
Di be the delay in queue of the ith customer
Wi be the waiting time in the system of the ith customer = Di + Si
Q(t) be the number of customers in queue at time t
L(t) be the number of customers in the system at time t = Q(t) + No. of customers being
served at t
Then the measures,
(if they exist) are called the steady state average delay and the steady state average
waiting time in the system. Similarly, the measures,
3. (if they exist) are called the steady state time average number in queue and the steady
state time average number in the system. Among the most general and useful results of
a queuing system are the conservation equations:
Q =(λ ) d and L = (λ ) w
These equations hold for every queuing system for which d and w exist. Another
equation of considerable practical value is given by,
w = d + E(S)
Other performance measures are:
- the probability that any delay will occur.
- the probability that the total delay will be greater than some pre-determined value
- that probability that all service facilities will be idle.
- the expected idle time of the total facility.
- the probability of turn-aways, due to insufficient waiting accommodation.
7.2: Notation for Queues.
Since all queues are characterised by arrival, service and queue and its discipline, the
queue system is usually described in shorten form by using these characteristics. The
general notation is:
[A/B/s]:{d/e/f}
Where,
A = Probability distribution of the arrivals
B = Probability distribution of the departures
s = Number of servers (channels)
d = The capacity of the queue(s)
e = The size of the calling population
f = Queue ranking rule (Ordering of the queue)
There are some special notation that has been developed for various probability
distributions describing the arrivals and departures. Some examples are,
M = Arrival or departure distribution that is a Poisson process
E = Erlang distribution
G = General distribution
GI = General independent distribution.
4. Thus for example, the [M/M/1]:{infinity/infinity/FCFS} system is one where the
arrivals and departures are a Poisson distribution with a single server, infinite queue
length, calling population infinite and the queue discipline is FCFS. This is the simplest
queue system that can be studied mathematically. This queue system is also simply
referred to as the M/M/1 queue.
7.3 Single Channel Queuing Theory
7.3.1: [M/M/1]:{//FCFS} Queue System.
A Arrival Time Distribution.
The simple model assumes that the number of arrivals occurring within a given interval
of time t, follows a Poisson distribution. with parameter (λ )t. This parameter (λ )t is the
average number of arrivals in time t which is also the variance of the distribution. If n
denotes the number of arrivals within a time interval t, then the probability function p(n)
is given by,
The arrival process is called Poisson input.
The probability of no(zero) arrival in the interval [0,t] is,
Pr(zero arrival in [0,t]) = e- λ t = p(0)
also,
P(zero arrival in [0,t]) = P(next arrival occurs after t)
= P(time bet. two successive arrivals exceeds t)
From this it can be shown that the probability density function of the inter-arrival times
is given by,
e- λ t for t >= 0
This called the negative exponential distribution with parameter λ or simply
exponential distribution. The mean inter-arrival time and standard deviation of this
distribution are both 1/ (λ ) where, (λ ) is the arrival rate.
NOTE: At first glance this distribution seems unrealistic. But it turns out that this is
an extremely robust distribution and approximates closely a large number of arrival
and breakdown patterns in practice.
5. B. Property of Stationarity and Lack of Memory.
A Poisson input implies that arrivals are independent of one another or the state of the
system. The probability of an arrival in any interval of time h does not depend on the
starting point of the arrival or on the specific history of arrivals preceding it, but depends
only on the length h. Thus the queuing systems with Poisson input can be considered as
Markovian process.(The reason for using M in the notation)
C. Analysis of the System.
In this case both the inter-arrival times and the service times are assumed to be negative
exponential distribution with parameters (λ ) and µ. As with Markovian process, we are
interested only in the long-run behaviour of the system. i.e. steady state or statistical
equilibrium state. It is obvious that if the arrival rate is higher than the service rate the
system will be blocked. Hence, we consider only the analysis of the system where the
arrival rate is less than the service rate.
At any moment in time, the state of the queuing system can be completely described by
the number of units in the system. Thus the state of the process can assume values 0,1,2...
(0 means none in the queue and the service is idle) Unlike Markov process, here the
change of state can occur at any time. However the process will approach a steady state
which is independent of the starting position or state.
Let the steady state probabilities are denoted by Pn, n = 0,1,2,3..... , where n refers to the
number in the system. Pn is the probability that there are n units in the system. By
considering a very small interval of time h, the transition diagram for this system can be
seen as:
If h is sufficiently small, no more than one arrival can occur and no more than one
service completion can occur in that time. Also the probability of observing a service
completion and an arrival in time h is µ(λ ).h2 which is very small (approximately zero)
and is neglected. Thus only the following four events are possible:
1. There are n units and 1 arrival occurs in h
2. There are n units and 1 service is completed in h
3. There are n-1 units and 1 arrival occurs in h
4. There are n+1 units and 1 service is completed in h
6. For n > 1, (because of steady state condition)
Pr(being in state n and leaving it) = Pr(being in other states and entering state n)
= Pr(being in state n-1 or n+1 and entering state n)
Thus,
Pn(λ )* h + Pn*µh = Pn-1(λ )* h + Pn+1* µh
This equation is called steady state balance equation.
For n = 0, only events 1 and 4 are possible,
P0(λ )* h = P1*µh
Therefore,
P0 can be determined by using the fact that the sum of the steady state probabilities must
be 1. Therefore,
This is the sum of a geometric series. Therefore,
Since ρ < 1,
7. The term ρ =(λ ) /µ is called utilisation factor or traffic intensity. This is also equal to
the probability that the service is busy, referred to as Pr(busy period).
Performance measures
The average number of units in the system L can be found from
L =Sum of [n*Pn] for n= 1 to (infinity).
The average number in the queue is
Q = L - (1 - P0)
Sum of[ (n-1)*Pn] for n=1 to (infinity) .
The average waiting time in the system (time in the system) can be obtained from,
An Example: A firm operates a 10-ton truck on a job contracting basis. The job requests
is Poisson distributed with a mean request rate of 1.4 per day. The average service time is
4 hours and is exponentially distributed. Determine all the performance measures and the
probability that there are more than two jobs in the system.
7.3.2 [M/M/1] : {N//FCFS} System (Limited queue length system)
If the queue length is limited to N, then some customers (jobs) will be lost. The
maximum number in the system can only be (N+1). Thus the transition diagram will have
(N+2) states as shown.
The steady state balance equation is the same as before except the first and the last one.
8. The relationship Pn = ρ n. P0 holds true. ρ = (λ )/µ
As before, from the fact that the sum of all steady state probabilities is 1, we can obtain
P0, Thus
It is also evident that the average number in the system is not the same as before. It is
given by,
(Finite number of terms)
The above equations hold for (λ ) <> µ. or ρ < 1
The average queue length is,
The average waiting time in the system is
w = L /(. (1 - PN+1))
(1 - PN+1) is required as we know that there are no more than N+1 units will be in the
system at any time because of the limitation of queue length to N.
It can be seen from the above that, limiting the queue length has the following
consequences:
- Average idle time will increase
- Average queue length will decrease
9. - Average waiting time will decrease
- A portion of the customers will be lost.
EXAMPLE: In the above example suppose the number in the queue is limited to 2.
(Refer pp313 - 337 of Introduction to Operations Research Techniques by Daellenbach &
George
Contine with second part?
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