Course Outline Introduction to Transportation Highway Users and their Performance Geometric Design Pavement Design
|
|
- Lynn Williams
- 6 years ago
- Views:
Transcription
1 Course Outline Introduction to Transportation Highway Users and their Performance Geometric Design Pavement Design Speed Studies - Project Traffic Queuing Intersections Level of Service in Highways and Intersections 2
2 Previous class Basic Concepts a. Flow Rate b. Spacing c. Headway d. Speed 2 types e. Density Relationships Graphs 3
3 Relationships q = n t t = n h i i = 1 q = n n i = 1 h i u s = 1 n 1 1 n i= 1 l t n k = = l 1 q = uk ( ) u i q 4
4 Consider a linear relationship between speed and density: p p y = k u u 1 = j f k u u 1 = u u k q 2 = f j u u k q Lecture # 11 Department of Civil and Environmental Engineering 5
5 MODELS OF TRAFFIC FLOW Traffic is rarely uniformly distributed ib t d equal time between arriving vehicles or headways? Must make some assumption for arrival patterns (distribution) 6
6 Poisson Model Approximation of non-uniform flow Where: P ( n ) = ( λt) e n λt n! P(n) = probability of having n vehicles arrive in time t, t = duration of the time interval over which vehicles are counted, λ = average vehicle flow or arrival rate in vehicles per unit time, and e = base of the natural logarithm (e = 2.718). 7
7 Poisson Distribution Example Assume: mean = variance (sd^2) λ = 360 veh/h = 0.1 veh/s t = 20 sec P( n) = n λt ( λ t ) e n! 8
8 Poisson Ideas Probability of exactly 4 vehicles arriving P(n=4) Probability bilit of less than 4 vehicles arriving i P(n<4) = P(0) + P(1) + P(2) + P(3) Probability of 4 or more vehicles arriving P(n 4) = 1 P(n<4) = 1 - P(0) + P(1) + P(2) + P(3) Amount of time between arrival of successive vehicles ( 0 ) = P ( h t ) ( 0 λt λt) e λ t qt 3600 P = = e = 0! 9 e
9 Poisson Model The assumption of Poisson vehicle arrivals also implies a distribution of the time intervals between the arrivals of successive vehicles (time headway). To show this, note that the average arrival rate as: λ = q 3600 Where: λ = average vehicle arrival rate in veh/s, q = flow in veh/h, and 3600 = number of seconds per hour. 10
10 Poisson Model Substituting into P(n) equation: P ( n) = ( qt 3600) n n!! e -qt 3600 The probability of having no vehicles arrive in a time interval of length t (P(0)) is equivalent to the probability of a vehicle headway, h, being greater than or equal to the time interval t. ( 0 ) =P ( h t ) =e -qt 3600 negative exponential or P simply called exponential distribution 11
11 Change of mean and distribution shape 0.25 Mean = 0.2 vehicles/minute Probabilit ty of Occur rance Mean = 0.5 vehicles/minute Arrivals in 15 minutes 12
12 Change of mean and inter-arrival times 1.0 of Exceda ance Probability Mean = 0.2 vehicles/minute Mean = 0.5 vehicles/minute Time Between Arrivals (minutes) 13
13 Limitations of the Poisson Model Mean (average number of cars per time period) must be equal to the variance (variance over all time period) Otherwise use an alternative model (negative binomial, etc.). Department of Civil and Environmental Engineering Lecture # 14
14 Poisson Distribution Let s grab 6 hrs of 5 min aggregated counts from a station in the Portland freeway network (I-5) for one lane Does it match what we expect? mean = veh/5 min/ln (1,014 veh/hr/ln) standard deviation = 9.35 veh variance = veh^2 Observed Data Poisson Model Frequency P(n) in t=5 min N vehicles How do we know if a distribution is a good representation of reality? Any objective test? 15
15 Problem 5.8. An observer has determined that the time headways between successive vehicles, on a section of highway, are exponentially distributed and that 60% of the headways ays between ee vehicles es are 13 seconds or greater. If the observer decides to count traffic in 30-second time intervals, estimate the probability of the observer counting exactly four vehicles in an interval. 16
16 Queuing delays significance Queuing delays can account for up to 90% or more of a driver s total trip travel time. Examples of queuing: Traffic Signals Toll booths Traffic incidents (accidents and vehicle disablements) 17
17 Queueing Theory Objects passing through point with restriction on maximum rate of passage Input + storage area (queue) + restriction + output Customers, arrivals, arrival process, server, service mechanism, departures, discipline (FIFO) Input Storage Output Restriction 18
18 Queueing Theory: Study of Congestion Phenomena Applications: Airplane waiting for takeoff, toll gate, wait for elevator, taxi stand, ships at a port, grocery store, telecommunications, circuits Interested in: maximum queue length, typical queueing times. SERVICE LEVEL! Input Storage Output Restriction 19
19 Queueing Theory: common assumptions Arrival patterns (λ, in vehicles per unit time): equal time intervals (derived from the assumption of uniform, deterministic arrivals) and exponentially distributed time intervals (derived from the assumption of Poisson-distributed arrivals). Departure patterns (μ, in vehicles per unit time), equal time intervals (derived from the assumption of uniform, deterministic arrivals) and exponentially distributed time intervals (derived from the assumption of Poisson-distributed arrivals). 20
20 Queueing Theory: Some definitions D/D/1 G/G/m Deterministic arrivals General arrivals Deterministic departures General departures 1 channel departures Multi-channel departures Graphical solution easiest SIMULATION M/D/1 Exponential arrivals Deterministic departures 1 channel departures Mathematical solution M/M/1 Exponential arrivals Exponential departures 1 channel departures Mathematical solution 21
21 Queuing discipline first-in-first-out (FIFO), indicating that the first vehicle to arrive is the first vehicle to depart, and last-in-first-out (LIFO), indicating that the last vehicle to arrive is the first to depart. For virtually all traffic-oriented queues, the FIFO queuing discipline is the more appropriate of the two. 22
22 D/D/1 Queuing deterministic i ti arrivals and departures with one departure channel (D/D/1 queue) D/D/1 queue lends itself to a graphical or mathematical solution. 23
23 Queueing Theory : Conservation Principle Customers don t disappear Arrival times of customers completely characterizes arrival process. Time/accumulation axes Uniform arrivals/departures N(x,t) 3 A=l(t) 2 D=m(t) 1 t 1 t 2 t 3 Time, x 24
24 Queueing Theory: Departure Process Observer records times of departure for corresponding objects to construct D(t). N(x,t) A(t) t 1 t 2 t 3 t 1 t 4 t 2 t 3 t 4 D(t) Time, x 25
25 Queueing Theory: Analysis If system empty at t=0: Vertical distance is queue length at time t: Q(t)=A(t)-D(t) For FIFO horizontal distance is waiting time for jth customer. N(x,t) Uniform arrivals/departures 3 A=l(t) 2 Q(t) D=m(t) 1 W j t 1 t 2 t 3 Time, x 26
26 Queueing Theory: Analysis Horizontal strip of unit height, width W j N(x,t) W 2 A(t) t 1 t 2 t 3 t 1 t 4 t 2 t 3 t 4 D(t) Time, x 27
27 Queueing Theory: Analysis Add up horizontal strips total delay Total time spent in system by some number of vehicles (horizontal strips) N(x,t) Total Delay=Area A(t) t 1 t 2 t 3 t 1 t 4 t 2 t 3 t 4 D(t) Time, x 28
28 Queueing Theory Total delay = W Average time in queue: w = W/n Average number in queue: Q = W/T N(x,t) A(t) t 1 t 2 t 3 t 1 t 4 t 2 t 3 t 4 D(t) Time, x 29
29 EXAMPLE 5.7 Vehicles arrive at an entrance to a recreational park. There is a single gate (at which all vehicles must stop), where a park attendant distributes a free brochure. The park opens at 8: A.M.,,at which time vehicles begin to arrive at a rate of 480 veh/h. After 20 minutes, the arrival flow rate declines to 120 veh/h and continues at that level for the remainder of the day. If the time required to distribute the brochure is 15 seconds, and assuming D/D/1 queuing, describe the operational characteristics of the queue. 30
30 EXAMPLE SOLUTION Begin by putting arrival and departure rates into common units of vehicles per minute. 480 veh/h λ = = 8 veh/min for t 20 min 60 min/h 120 veh/h λ = = 2 veh/min for t > 20 min 60 min/h 60 s/min μ = = 4 veh/min for all t 15 s/veh 31
31 EXAMPLE SOLUTION Begin by putting arrival and departure rates into common units of vehicles per minute. 8 t for t 20 min 160 ( t 20) for t 20 min + 2 > the number of vehicle departures is: 4 t for all t 32
32 EXAMPLE SOLUTION A=l(t)=2(t) Equation of line = (t-20) 80 veh A=l(t)=8(t) D=m(t)=4(t) 33
33 EXAMPLE SOLUTION When the arrival curve is above the departure curve, a queue condition will exist. The point at which the arrival curve meets the departure curve is the moment when the queue dissipates (no more queue exists). The point of queue dissipation can be determined by equating appropriate arrival and departure equations, that is ( t 20) = t Solving for t gives t = 60 minutes. 34
34 EXAMPLE SOLUTION Thus the queue that began to form at 8:00 A.M. will dissipate 60 minutes later (9:00 A.M.), at which time 240 vehicles will have arrived and departed (4 veh/min 60 min). Individual vehicle delay: Under FIFO queuing discipline, the delay of an individual vehicle is given by the horizontal distance between arrival and departure curves. So, by inspection of Fig. 5.7, the 160 th vehicle to arrive will have the longest delay of 20 minutes (the longest horizontal distance between arrival and departure curves) 35
35 The total length of the queue is given by the vertical distance between arrival and departure curves at that time. The longest queue (longest vertical distance between arrival and departure curves) will occur at t = 20 minutes and is 80 vehicles long Total vehicle delay, defined as the summation of the delays of each individual vehicle, is given by the total area between arrival and departure curves 36
36 In this example, the areas between arrival and departure curves can be determined by summing triangular areas, giving total delay, Dt, as 1 D t = + 2 ( ) ( ) = 2400 veh - min Because 240 vehicles encounter queuing-delay (as previously determined), the average delay per vehicle is 10 minutes (2400 veh-min/240 veh), and dthe average queue length is 40 vehicles (2400 veh-min/60 min). 37
37 Problem Vehicles begin to arrive at a parking lot at 6:00 A.M. at a rate 8 per minute. Due to an accident on the access highway, no vehicles arrive from 6:20 to 6:30 A.M. From o 6:30 A.M. on, vehicles es arrive at a rate of 2 per minute. The parking lot attendant processes incoming vehicles (collects parking fees) at a rate of 4 per minute throughout the day. Assuming D/D/1 queuing, determine total vehicle delay. 38
38 EXAMPLE 5.8 After observing arrivals and departures at a highway toll booth over a 60-minute time period, the observer notes that the arrival and departure rates (or service rates) are deterministic but, instead of being uniform, they change over time according to a known function. The arrival rate is given by the function (t) = t t 2 and the departure rate is given by (t) = t, where t is in minutes after the beginning of the observation period and (t) and (t) are in vehicles per minute. Determine the total vehicle delay at the toll booth and the longest queue assuming D/D/1 queuing. 39
39 M/D/1 Queuing exponentially distributed times between the arrivals of successive vehicles (Poisson arrivals) deterministic departures, and one departure channel Obvious example Traffic Signals 40
40 M/D/1 Basic relationship: ρ = λ μ Where: ρ = traffic intensity, and is unit-less, λ = average arrival rate in vehicles per unit time, and μ = average departure rate in vehicles per unit time. 41
41 M/D/1 assuming that ρ <1, for an M/D/1: average length of queue in vehicles: Q = 2 ρ 2 1 ( ρ ) average waiting time in the queue (for each vehicle): ρ w = 2μ 1 ( ρ) average time spent in the system (summation of average queue waiting time + average departure time (service time)): 2 ρ t = 2μ 1 ρ 42 ( )
42 M/D/1 NOTE! that t the traffic intensity it is less than one (ρ <1), the D/D/1 queue will predict NO queue formation. Models with random arrivals or departures, such as the M/D/1 queuing model, will predict queue formations! Q = ρ ( ρ) w = 2 ρ μ ( 1 ρ) t = 2 2μ ρ ( 1 ρ) 43
43 M/M/1 Queuing exponentially distributed times between the arrivals of successive vehicles (Poisson arrivals) exponentially distributed departure time patterns in addition to exponentially distributed arrival times one departure channel Traffic applications: Toll booth where some arriving drivers have the correct toll and can be processed quickly, and others may not have the correct toll, thus producing a distribution of departures about some mean departure rate. 44
44 M/M/1 λ = arrival rate μ = departure (service) rate λ μ ρ = ρ < 1.0 Average length of queue Q ρ 2 = 1 ( ρ ) Average time waiting in queue 1 λ w = μ μ λ Average time spent in system 1 t = μ λ 45
45 M/M/N Queuing Applications: M/M/N queuing is a reasonable assumption at toll booths on turnpikes or at toll bridges where there is often more than one departure channel available (more than one toll booth open). M/M/N queuing is also frequently encountered in non-traffic but transportation applications such as security checks at airports, vessel queueing at ports/airports and so on. Other non-transportation : checkout lines at retail stores, call centers, etc. 46
46 M/M/N Multi-channel λ ρ = ρ N < 1.0 μ Average length of queue Average time waiting in queue Q Average time spent in system +1 P N 0ρ 1 N! N 1 ρ N = 2 w t = = ( ) ρ + Q 1 λ μ ρ + Q λ 47
47 M/M/N Probability of having no vehicles = 1 P0 N 1 n N ρ c ρ + n = 0 nc!! N! 1 ρ N c ( ) Probability of having n vehicles n n ρ P = 0 ρ P Pn for n N P = 0 for n n! N! n n N N N Probability of being in a queue P n > N = N + 1 P0 ρ N! N 1 ρ ( N ) 48
48 Problem 5.35 Vehicles leave an airport parking facility (arrive at parking fee collection booths) at a rate of 500 veh/h (the time between arrivals is exponentially distributed). The parking facility has a policy that the average time a patron spends in a queue while waiting to pay for parking is not to exceed 5 seconds. If the time required to pay for parking is exponentially distributed with a mean of 15 seconds, what is the fewest number of payment processing booths that must be open to keep the average time spent in a queue less than 5 seconds. 49
2.2t t 3 =1.2t+0.035' t= D, = f ? t 3 dt f ' 2 dt
5.5 Queuing Theory and Traffic Flow Analysis 159 EXAMPLE 5.8 After observing arrivals and departures at a highway toll booth over a 60-minute tim e period, an observer notes that the arrival and departure
More informationCE351 Transportation Systems: Planning and Design
CE351 Transportation Systems: Planning and Design TOPIC: Level of Service (LOS) at Traffic Signals 1 Course Outline Introduction to Transportation Highway Users and their Performance Geometric Design Pavement
More informationPBW 654 Applied Statistics - I Urban Operations Research
PBW 654 Applied Statistics - I Urban Operations Research Lecture 2.I Queuing Systems An Introduction Operations Research Models Deterministic Models Linear Programming Integer Programming Network Optimization
More informationCumulative Count Curve and Queueing Analysis
Introduction Traffic flow theory (TFT) Zhengbing He, Ph.D., http://zhengbing.weebly.com School of traffic and transportation, Beijing Jiaotong University September 27, 2015 Introduction Outline 1 Introduction
More informationEE 368. Weeks 3 (Notes)
EE 368 Weeks 3 (Notes) 1 State of a Queuing System State: Set of parameters that describe the condition of the system at a point in time. Why do we need it? Average size of Queue Average waiting time How
More informationQueueing Theory I Summary! Little s Law! Queueing System Notation! Stationary Analysis of Elementary Queueing Systems " M/M/1 " M/M/m " M/M/1/K "
Queueing Theory I Summary Little s Law Queueing System Notation Stationary Analysis of Elementary Queueing Systems " M/M/1 " M/M/m " M/M/1/K " Little s Law a(t): the process that counts the number of arrivals
More informationCHAPTER 3. CAPACITY OF SIGNALIZED INTERSECTIONS
CHAPTER 3. CAPACITY OF SIGNALIZED INTERSECTIONS 1. Overview In this chapter we explore the models on which the HCM capacity analysis method for signalized intersections are based. While the method has
More information1.225 Transportation Flow Systems Quiz (December 17, 2001; Duration: 3 hours)
1.225 Transportation Flow Systems Quiz (December 17, 2001; Duration: 3 hours) Student Name: Alias: Instructions: 1. This exam is open-book 2. No cooperation is permitted 3. Please write down your name
More informationSignalized Intersection Delay Models
Chapter 35 Signalized Intersection Delay Models 35.1 Introduction Signalized intersections are the important points or nodes within a system of highways and streets. To describe some measure of effectiveness
More informationCEE 320 Midterm Examination (50 minutes)
CEE 320 Midterm Examination (50 minutes) Fall 2009 Please write your name on this cover. Please write your last name on all other exam pages This exam is NOT open book, but you are allowed to use one 8.5x11
More informationSolutions to COMP9334 Week 8 Sample Problems
Solutions to COMP9334 Week 8 Sample Problems Problem 1: Customers arrive at a grocery store s checkout counter according to a Poisson process with rate 1 per minute. Each customer carries a number of items
More informationAdvanced Computer Networks Lecture 3. Models of Queuing
Advanced Computer Networks Lecture 3. Models of Queuing Husheng Li Min Kao Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Spring, 2016 1/13 Terminology of
More informationTraffic Flow Theory and Simulation
Traffic Flow Theory and Simulation V.L. Knoop Lecture 2 Arrival patterns and cumulative curves Arrival patterns From microscopic to macroscopic 24-3-2014 Delft University of Technology Challenge the future
More informationNetworking = Plumbing. Queueing Analysis: I. Last Lecture. Lecture Outline. Jeremiah Deng. 29 July 2013
Networking = Plumbing TELE302 Lecture 7 Queueing Analysis: I Jeremiah Deng University of Otago 29 July 2013 Jeremiah Deng (University of Otago) TELE302 Lecture 7 29 July 2013 1 / 33 Lecture Outline Jeremiah
More informationIntroduction to queuing theory
Introduction to queuing theory Claude Rigault ENST claude.rigault@enst.fr Introduction to Queuing theory 1 Outline The problem The number of clients in a system The client process Delay processes Loss
More informationQueueing Review. Christos Alexopoulos and Dave Goldsman 10/25/17. (mostly from BCNN) Georgia Institute of Technology, Atlanta, GA, USA
1 / 26 Queueing Review (mostly from BCNN) Christos Alexopoulos and Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 10/25/17 2 / 26 Outline 1 Introduction 2 Queueing Notation 3 Transient
More informationQueueing Review. Christos Alexopoulos and Dave Goldsman 10/6/16. (mostly from BCNN) Georgia Institute of Technology, Atlanta, GA, USA
1 / 24 Queueing Review (mostly from BCNN) Christos Alexopoulos and Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 10/6/16 2 / 24 Outline 1 Introduction 2 Queueing Notation 3 Transient
More informationSlides 9: Queuing Models
Slides 9: Queuing Models Purpose Simulation is often used in the analysis of queuing models. A simple but typical queuing model is: Queuing models provide the analyst with a powerful tool for designing
More informationQueuing Theory. The present section focuses on the standard vocabulary of Waiting Line Models.
Queuing Theory Introduction Waiting lines are the most frequently encountered problems in everyday life. For example, queue at a cafeteria, library, bank, etc. Common to all of these cases are the arrivals
More informationProbability and Statistics Concepts
University of Central Florida Computer Science Division COT 5611 - Operating Systems. Spring 014 - dcm Probability and Statistics Concepts Random Variable: a rule that assigns a numerical value to each
More informationSignalized Intersection Delay Models
Signalized Intersection Delay Models Lecture Notes in Transportation Systems Engineering Prof. Tom V. Mathew Contents 1 Introduction 1 2 Types of delay 2 2.1 Stopped Time Delay................................
More informationIntroduction to Queueing Theory with Applications to Air Transportation Systems
Introduction to Queueing Theory with Applications to Air Transportation Systems John Shortle George Mason University February 28, 2018 Outline Why stochastic models matter M/M/1 queue Little s law Priority
More informationPerformance Evaluation of Queuing Systems
Performance Evaluation of Queuing Systems Introduction to Queuing Systems System Performance Measures & Little s Law Equilibrium Solution of Birth-Death Processes Analysis of Single-Station Queuing Systems
More informationChapter 10. Queuing Systems. D (Queuing Theory) Queuing theory is the branch of operations research concerned with waiting lines.
Chapter 10 Queuing Systems D. 10. 1. (Queuing Theory) Queuing theory is the branch of operations research concerned with waiting lines. D. 10.. (Queuing System) A ueuing system consists of 1. a user source.
More informationCPSC 531: System Modeling and Simulation. Carey Williamson Department of Computer Science University of Calgary Fall 2017
CPSC 531: System Modeling and Simulation Carey Williamson Department of Computer Science University of Calgary Fall 2017 Motivating Quote for Queueing Models Good things come to those who wait - poet/writer
More informationAn Analysis of the Preemptive Repeat Queueing Discipline
An Analysis of the Preemptive Repeat Queueing Discipline Tony Field August 3, 26 Abstract An analysis of two variants of preemptive repeat or preemptive repeat queueing discipline is presented: one in
More informationChapter 6 Queueing Models. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter 6 Queueing Models Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Purpose Simulation is often used in the analysis of queueing models. A simple but typical queueing model: Queueing
More informationNon Markovian Queues (contd.)
MODULE 7: RENEWAL PROCESSES 29 Lecture 5 Non Markovian Queues (contd) For the case where the service time is constant, V ar(b) = 0, then the P-K formula for M/D/ queue reduces to L s = ρ + ρ 2 2( ρ) where
More informationλ λ λ In-class problems
In-class problems 1. Customers arrive at a single-service facility at a Poisson rate of 40 per hour. When two or fewer customers are present, a single attendant operates the facility, and the service time
More informationSystems Simulation Chapter 6: Queuing Models
Systems Simulation Chapter 6: Queuing Models Fatih Cavdur fatihcavdur@uludag.edu.tr April 2, 2014 Introduction Introduction Simulation is often used in the analysis of queuing models. A simple but typical
More informationQueueing systems. Renato Lo Cigno. Simulation and Performance Evaluation Queueing systems - Renato Lo Cigno 1
Queueing systems Renato Lo Cigno Simulation and Performance Evaluation 2014-15 Queueing systems - Renato Lo Cigno 1 Queues A Birth-Death process is well modeled by a queue Indeed queues can be used to
More informationTraffic Progression Models
Traffic Progression Models Lecture Notes in Transportation Systems Engineering Prof. Tom V. Mathew Contents 1 Introduction 1 2 Characterizing Platoon 2 2.1 Variables describing platoon............................
More informationLink Models for Circuit Switching
Link Models for Circuit Switching The basis of traffic engineering for telecommunication networks is the Erlang loss function. It basically allows us to determine the amount of telephone traffic that can
More informationSignalized Intersection Delay Models
Transportation System Engineering 56. Signalized Intersection Delay Models Chapter 56 Signalized Intersection Delay Models 56.1 Introduction Signalized intersections are the important points or nodes within
More informationLecture 7: Simulation of Markov Processes. Pasi Lassila Department of Communications and Networking
Lecture 7: Simulation of Markov Processes Pasi Lassila Department of Communications and Networking Contents Markov processes theory recap Elementary queuing models for data networks Simulation of Markov
More informationEngineering Mathematics : Probability & Queueing Theory SUBJECT CODE : MA 2262 X find the minimum value of c.
SUBJECT NAME : Probability & Queueing Theory SUBJECT CODE : MA 2262 MATERIAL NAME : University Questions MATERIAL CODE : SKMA104 UPDATED ON : May June 2013 Name of the Student: Branch: Unit I (Random Variables)
More informationFigure 10.1: Recording when the event E occurs
10 Poisson Processes Let T R be an interval. A family of random variables {X(t) ; t T} is called a continuous time stochastic process. We often consider T = [0, 1] and T = [0, ). As X(t) is a random variable
More information2905 Queueing Theory and Simulation PART IV: SIMULATION
2905 Queueing Theory and Simulation PART IV: SIMULATION 22 Random Numbers A fundamental step in a simulation study is the generation of random numbers, where a random number represents the value of a random
More informationChapter 5: Special Types of Queuing Models
Chapter 5: Special Types of Queuing Models Some General Queueing Models Discouraged Arrivals Impatient Arrivals Bulk Service and Bulk Arrivals OR37-Dr.Khalid Al-Nowibet 1 5.1 General Queueing Models 1.
More informationMAT SYS 5120 (Winter 2012) Assignment 5 (not to be submitted) There are 4 questions.
MAT 4371 - SYS 5120 (Winter 2012) Assignment 5 (not to be submitted) There are 4 questions. Question 1: Consider the following generator for a continuous time Markov chain. 4 1 3 Q = 2 5 3 5 2 7 (a) Give
More informationChapter 2 Queueing Theory and Simulation
Chapter 2 Queueing Theory and Simulation Based on the slides of Dr. Dharma P. Agrawal, University of Cincinnati and Dr. Hiroyuki Ohsaki Graduate School of Information Science & Technology, Osaka University,
More informationQueuing Analysis. Chapter Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Queuing Analysis Chapter 13 13-1 Chapter Topics Elements of Waiting Line Analysis The Single-Server Waiting Line System Undefined and Constant Service Times Finite Queue Length Finite Calling Problem The
More informationSince D has an exponential distribution, E[D] = 0.09 years. Since {A(t) : t 0} is a Poisson process with rate λ = 10, 000, A(0.
IEOR 46: Introduction to Operations Research: Stochastic Models Chapters 5-6 in Ross, Thursday, April, 4:5-5:35pm SOLUTIONS to Second Midterm Exam, Spring 9, Open Book: but only the Ross textbook, the
More informationClassification of Queuing Models
Classification of Queuing Models Generally Queuing models may be completely specified in the following symbol form:(a/b/c):(d/e)where a = Probability law for the arrival(or inter arrival)time, b = Probability
More informationQueueing Theory and Simulation. Introduction
Queueing Theory and Simulation Based on the slides of Dr. Dharma P. Agrawal, University of Cincinnati and Dr. Hiroyuki Ohsaki Graduate School of Information Science & Technology, Osaka University, Japan
More informationQueueing Systems: Lecture 3. Amedeo R. Odoni October 18, Announcements
Queueing Systems: Lecture 3 Amedeo R. Odoni October 18, 006 Announcements PS #3 due tomorrow by 3 PM Office hours Odoni: Wed, 10/18, :30-4:30; next week: Tue, 10/4 Quiz #1: October 5, open book, in class;
More informationNICTA Short Course. Network Analysis. Vijay Sivaraman. Day 1 Queueing Systems and Markov Chains. Network Analysis, 2008s2 1-1
NICTA Short Course Network Analysis Vijay Sivaraman Day 1 Queueing Systems and Markov Chains Network Analysis, 2008s2 1-1 Outline Why a short course on mathematical analysis? Limited current course offering
More informationIntroduction to Queueing Theory
Introduction to Queueing Theory Raj Jain Washington University in Saint Louis Jain@eecs.berkeley.edu or Jain@wustl.edu A Mini-Course offered at UC Berkeley, Sept-Oct 2012 These slides and audio/video recordings
More informationTraffic Flow Theory & Simulation
Traffic Flow Theory & Simulation S.P. Hoogendoorn Lecture 4 Shockwave theory Shockwave theory I: Introduction Applications of the Fundamental Diagram February 14, 2010 1 Vermelding onderdeel organisatie
More informationCHAPTER 2. CAPACITY OF TWO-WAY STOP-CONTROLLED INTERSECTIONS
CHAPTER 2. CAPACITY OF TWO-WAY STOP-CONTROLLED INTERSECTIONS 1. Overview In this chapter we will explore the models on which the HCM capacity analysis method for two-way stop-controlled (TWSC) intersections
More informationQueueing Theory. VK Room: M Last updated: October 17, 2013.
Queueing Theory VK Room: M1.30 knightva@cf.ac.uk www.vincent-knight.com Last updated: October 17, 2013. 1 / 63 Overview Description of Queueing Processes The Single Server Markovian Queue Multi Server
More informationClass 11 Non-Parametric Models of a Service System; GI/GI/1, GI/GI/n: Exact & Approximate Analysis.
Service Engineering Class 11 Non-Parametric Models of a Service System; GI/GI/1, GI/GI/n: Exact & Approximate Analysis. G/G/1 Queue: Virtual Waiting Time (Unfinished Work). GI/GI/1: Lindley s Equations
More information5/15/18. Operations Research: An Introduction Hamdy A. Taha. Copyright 2011, 2007 by Pearson Education, Inc. All rights reserved.
The objective of queuing analysis is to offer a reasonably satisfactory service to waiting customers. Unlike the other tools of OR, queuing theory is not an optimization technique. Rather, it determines
More informationThe Timing Capacity of Single-Server Queues with Multiple Flows
The Timing Capacity of Single-Server Queues with Multiple Flows Xin Liu and R. Srikant Coordinated Science Laboratory University of Illinois at Urbana Champaign March 14, 2003 Timing Channel Information
More informationReview of Queuing Models
Review of Queuing Models Recitation, Apr. 1st Guillaume Roels 15.763J Manufacturing System and Supply Chain Design http://michael.toren.net/slides/ipqueue/slide001.html 2005 Guillaume Roels Outline Overview,
More informationA cork model for parking
A cork model for parking Marco Alderighi U. Valle d Aosta SIET 2013 Venezia Marco Alderighi U. Valle d Aosta () Cork model SIET 2013 Venezia 1 / 26 The problem Many facilities, such as department stores,
More informationNANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION MH4702/MAS446/MTH437 Probabilistic Methods in OR
NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION 2013-201 MH702/MAS6/MTH37 Probabilistic Methods in OR December 2013 TIME ALLOWED: 2 HOURS INSTRUCTIONS TO CANDIDATES 1. This examination paper contains
More informationWaiting Line Models: Queuing Theory Basics. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 1
Waiting Line Models: Queuing Theory Basics Cuantitativos M. En C. Eduardo Bustos Farias 1 Agenda Queuing system structure Performance measures Components of queuing systems Arrival process Service process
More informationThe effect of probabilities of departure with time in a bank
International Journal of Scientific & Engineering Research, Volume 3, Issue 7, July-2012 The effect of probabilities of departure with time in a bank Kasturi Nirmala, Shahnaz Bathul Abstract This paper
More informationName of the Student: Problems on Discrete & Continuous R.Vs
SUBJECT NAME : Probability & Queueing Theory SUBJECT CODE : MA 6453 MATERIAL NAME : Additional Problems MATERIAL CODE : JM08AM1004 REGULATION : R2013 UPDATED ON : March 2015 (Scan the above Q.R code for
More informationSignalized Intersection Delay Models
hapter 56 Signalized Intersection Delay Models 56.1 Introduction Signalized intersections are the important points or nodes within a system of highways and streets. To describe some measure of effectiveness
More informationIOE 202: lectures 11 and 12 outline
IOE 202: lectures 11 and 12 outline Announcements Last time... Queueing models intro Performance characteristics of a queueing system Steady state analysis of an M/M/1 queueing system Other queueing systems,
More informationISyE 2030 Practice Test 2
1 NAME ISyE 2030 Practice Test 2 Summer 2005 This test is open notes, open books. You have exactly 75 minutes. 1. Short-Answer Questions (a) TRUE or FALSE? If arrivals occur according to a Poisson process
More informationIntroduction to queuing theory
Introduction to queuing theory Queu(e)ing theory Queu(e)ing theory is the branch of mathematics devoted to how objects (packets in a network, people in a bank, processes in a CPU etc etc) join and leave
More informationThe Queue Inference Engine and the Psychology of Queueing. ESD.86 Spring 2007 Richard C. Larson
The Queue Inference Engine and the Psychology of Queueing ESD.86 Spring 2007 Richard C. Larson Part 1 The Queue Inference Engine QIE Queue Inference Engine (QIE) Boston area ATMs: reams of data Standard
More informationJEP John E. Jack Pflum, P.E. Consulting Engineering 7541 Hosbrook Road, Cincinnati, OH Telephone:
JEP John E. Jack Pflum, P.E. Consulting Engineering 7541 Hosbrook Road, Cincinnati, OH 45243 Email: jackpflum1@gmail.com Telephone: 513.919.7814 MEMORANDUM REPORT Traffic Impact Analysis Proposed Soccer
More informationSandwich shop : a queuing net work with finite disposable resources queue and infinite resources queue
Sandwich shop : a queuing net work with finite disposable resources queue and infinite resources queue Final project for ISYE 680: Queuing systems and Applications Hongtan Sun May 5, 05 Introduction As
More informationCHAPTER 5 DELAY ESTIMATION FOR OVERSATURATED SIGNALIZED APPROACHES
CHAPTER 5 DELAY ESTIMATION FOR OVERSATURATED SIGNALIZED APPROACHES Delay is an important measure of effectiveness in traffic studies, as it presents the direct cost of fuel consumption and indirect cost
More informationIntroduction to Queueing Theory
Introduction to Queueing Theory Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: 30-1 Overview Queueing Notation
More informationTravel and Transportation
A) Locations: B) Time 1) in the front 4) by the window 7) earliest 2) in the middle 5) on the isle 8) first 3) in the back 6) by the door 9) next 10) last 11) latest B) Actions: 1) Get on 2) Get off 3)
More informationBuzen s algorithm. Cyclic network Extension of Jackson networks
Outline Buzen s algorithm Mean value analysis for Jackson networks Cyclic network Extension of Jackson networks BCMP network 1 Marginal Distributions based on Buzen s algorithm With Buzen s algorithm,
More informationc) What are cumulative curves, and how are they constructed? (1 pt) A count of the number of vehicles over time at one location (1).
Exam 4821 Duration 3 hours. Points are indicated for each question. The exam has 5 questions 54 can be obtained. Note that half of the points is not always suffcient for a 6. Use your time wisely! Remarks:
More informationData analysis and stochastic modeling
Data analysis and stochastic modeling Lecture 7 An introduction to queueing theory Guillaume Gravier guillaume.gravier@irisa.fr with a lot of help from Paul Jensen s course http://www.me.utexas.edu/ jensen/ormm/instruction/powerpoint/or_models_09/14_queuing.ppt
More informationDeparture time choice equilibrium problem with partial implementation of congestion pricing
Departure time choice equilibrium problem with partial implementation of congestion pricing Tokyo Institute of Technology Postdoctoral researcher Katsuya Sakai 1 Contents 1. Introduction 2. Method/Tool
More informationQueueing Systems: Lecture 6. Lecture Outline
Queueing Systems: Lecture 6 Amedeo R. Odoni November 6, 2006 Lecture Outline Congestion pricing in transportation: the fundamental ideas Congestion pricing and queueing theory Numerical examples A real
More informationEP2200 Course Project 2017 Project II - Mobile Computation Offloading
EP2200 Course Project 2017 Project II - Mobile Computation Offloading 1 Introduction Queuing theory provides us a very useful mathematic tool that can be used to analytically evaluate the performance of
More informationIntroduction to Markov Chains, Queuing Theory, and Network Performance
Introduction to Markov Chains, Queuing Theory, and Network Performance Marceau Coupechoux Telecom ParisTech, departement Informatique et Réseaux marceau.coupechoux@telecom-paristech.fr IT.2403 Modélisation
More informationComputer Systems Modelling
Computer Systems Modelling Computer Laboratory Computer Science Tripos, Part II Michaelmas Term 2003 R. J. Gibbens Problem sheet William Gates Building JJ Thomson Avenue Cambridge CB3 0FD http://www.cl.cam.ac.uk/
More informationSession-Based Queueing Systems
Session-Based Queueing Systems Modelling, Simulation, and Approximation Jeroen Horters Supervisor VU: Sandjai Bhulai Executive Summary Companies often offer services that require multiple steps on the
More informationSignalized Intersections
Signalized Intersections Kelly Pitera October 23, 2009 Topics to be Covered Introduction/Definitions D/D/1 Queueing Phasing and Timing Plan Level of Service (LOS) Signal Optimization Conflicting Operational
More informationIntro to Queueing Theory
1 Intro to Queueing Theory Little s Law M/G/1 queue Conservation Law 1/31/017 M/G/1 queue (Simon S. Lam) 1 Little s Law No assumptions applicable to any system whose arrivals and departures are observable
More informationOther properties of M M 1
Other properties of M M 1 Přemysl Bejda premyslbejda@gmail.com 2012 Contents 1 Reflected Lévy Process 2 Time dependent properties of M M 1 3 Waiting times and queue disciplines in M M 1 Contents 1 Reflected
More information57:022 Principles of Design II Final Exam Solutions - Spring 1997
57:022 Principles of Design II Final Exam Solutions - Spring 1997 Part: I II III IV V VI Total Possible Pts: 52 10 12 16 13 12 115 PART ONE Indicate "+" if True and "o" if False: + a. If a component's
More informationQueuing Theory. Using the Math. Management Science
Queuing Theory Using the Math 1 Markov Processes (Chains) A process consisting of a countable sequence of stages, that can be judged at each stage to fall into future states independent of how the process
More informationComputer Networks More general queuing systems
Computer Networks More general queuing systems Saad Mneimneh Computer Science Hunter College of CUNY New York M/G/ Introduction We now consider a queuing system where the customer service times have a
More informationComputer Systems Modelling
Computer Systems Modelling Computer Laboratory Computer Science Tripos, Part II Lent Term 2010/11 R. J. Gibbens Problem sheet William Gates Building 15 JJ Thomson Avenue Cambridge CB3 0FD http://www.cl.cam.ac.uk/
More informationCS418 Operating Systems
CS418 Operating Systems Lecture 14 Queuing Analysis Textbook: Operating Systems by William Stallings 1 1. Why Queuing Analysis? If the system environment changes (like the number of users is doubled),
More informationQuiz Queue II. III. ( ) ( ) =1.3333
Quiz Queue UMJ, a mail-order company, receives calls to place orders at an average of 7.5 minutes intervals. UMJ hires one operator and can handle each call in about every 5 minutes on average. The inter-arrival
More informationM/G/1 and Priority Queueing
M/G/1 and Priority Queueing Richard T. B. Ma School of Computing National University of Singapore CS 5229: Advanced Compute Networks Outline PASTA M/G/1 Workload and FIFO Delay Pollaczek Khinchine Formula
More informationDelay of Incidents Consequences of Stochastic Incident Duration
Delay of Incidents Consequences of Stochastic Incident Duration Victor L. Knoop 1, Serge P. Hoogendoorn 1, and Henk J. van Zuylen 1 Delft University of Technology & TRIL research School, Stevinweg 1, 68
More informationQueuing Theory. Queuing Theory. Fatih Cavdur April 27, 2015
Queuing Theory Fatih Cavdur fatihcavdur@uludag.edu.tr April 27, 2015 Introduction Introduction Simulation is often used in the analysis of queuing models. A simple but typical model is the single-server
More informationPerformance Analysis of Delay Estimation Models for Signalized Intersection Networks
Performance Analysis of Delay Estimation Models for Signalized Intersection Networks Hyung Jin Kim 1, Bongsoo Son 2, Soobeom Lee 3 1 Dept. of Urban Planning and Eng. Yonsei Univ,, Seoul, Korea {hyungkim,
More information1.225J J (ESD 205) Transportation Flow Systems
1.225J J (ESD 25) Transportation Flow Systems Lecture 9 Simulation Models Prof. Ismail Chabini and Prof. Amedeo R. Odoni Lecture 9 Outline About this lecture: It is based on R16. Only material covered
More information$QDO\]LQJ$UWHULDO6WUHHWVLQ1HDU&DSDFLW\ RU2YHUIORZ&RQGLWLRQV
Paper No. 001636 $QDO\]LQJ$UWHULDO6WUHHWVLQ1HDU&DSDFLW\ RU2YHUIORZ&RQGLWLRQV Duplication for publication or sale is strictly prohibited without prior written permission of the Transportation Research Board
More informationQUEUING SYSTEM. Yetunde Folajimi, PhD
QUEUING SYSTEM Yetunde Folajimi, PhD Part 2 Queuing Models Queueing models are constructed so that queue lengths and waiting times can be predicted They help us to understand and quantify the effect of
More information10.2 For the system in 10.1, find the following statistics for population 1 and 2. For populations 2, find: Lq, Ls, L, Wq, Ws, W, Wq 0 and SL.
Bibliography Asmussen, S. (2003). Applied probability and queues (2nd ed). New York: Springer. Baccelli, F., & Bremaud, P. (2003). Elements of queueing theory: Palm martingale calculus and stochastic recurrences
More informationDiscrete Event and Process Oriented Simulation (2)
B. Maddah ENMG 622 Simulation 11/04/08 Discrete Event and Process Oriented Simulation (2) Discrete event hand simulation of an (s, S) periodic review inventory system Consider a retailer who sells a commodity
More informationConception of effective number of lanes as basis of traffic optimization
Available online at www.sciencedirect.com ScienceDirect Procedia Computer Science 91 (016 ) 94 100 Information Technology and Quantitative Management (ITQM 016) Conception of effective number of lanes
More informationShock wave analysis. Chapter 8. List of symbols. 8.1 Kinematic waves
Chapter 8 Shock wave analysis Summary of the chapter. Flow-speed-density states change over time and space. When these changes of state occur, a boundary is established that demarks the time-space domain
More informationIEOR 6711, HMWK 5, Professor Sigman
IEOR 6711, HMWK 5, Professor Sigman 1. Semi-Markov processes: Consider an irreducible positive recurrent discrete-time Markov chain {X n } with transition matrix P (P i,j ), i, j S, and finite state space.
More information