16:330:543 Communication Networks I Midterm Exam November 7, 2005
|
|
- Roberta Sherman
- 6 years ago
- Views:
Transcription
1 l l l l l l l l 1 3 np n = ρ 1 ρ = λ µ λ. n= T = E[N] = 1 λ µ λ = 1 µ 1. 16:33:543 Communication Networks I Midterm Exam November 7, 5 You have 16 minutes to complete this four problem exam. If you know what you re doing, you should be able to complete the exam in half the allotted time. The point value of each problem is indicated. Put your name and your Rutgers netid (but no part of your SSN) on each exam book (1 points). Please read both sides of the exam carefully and ask the instructor if you have any questions points Data packets arrive at the input buffer of a transmission link as a Poisson process of rate 1 packets per second. The packets have exponential service requirements with mean 1/µ seconds.sketchamarkovchainforn, the number of buffered packets. Under what conditions on the service rate µ does the Markov chain have a stationary distribution? For those cases, find the stationary probabilities p n. What is the minimum service rate µ such that the average system time T is no more than msec? This problem is a gift. The system is an M/M/1 queue with arrival rate λ = 1 packets/sec, service rate µ. Here is the usual Markov chain for N: In terms of the offered load ρ = λ/µ = 1/µ, the stationary distribution is The average number of packets in the system is p n =(1 ρ)ρ n n =, 1,,... (1) E[N] = By Little s Law, the average system time is The requirement that T ms corresponds to µ 1 1/(.) = 5. Thus the system must have service rate µ 15 packets per second.. 5 points A collection of sensors transmit fixed unit-length packets over a shared multiaccess channel to data collector node. Packets are generated by the collection of sensors as a Poisson process of rate λ per unit time. The number of sensors is large enough so that the probability a sensor has more than one packet in its buffer is negligible. When a packet arrives, the sensor immediately begins to transmit that packet to the data collector. If all other sensors are silent during the transmission of a packet, then the transmitted packet is successfully received by the data collector. After a packet is transmitted, whether successful or not, the transmitting sensor discards that packet. At time t =, the system has been running for a long time as we begin to measure the performance of the system. 1
2 (a) Compare this system to unslotted Aloha. How is this system the same? How is this system different? The system is similar to slotted Aloha in that packets are transmitted immediately when they arrive; the system is unslotted; the model assumes that the number of terminals are high so each terminal has only one packet and thus a terminal never queues packets; overlapping transmission alays cause collisions. The system differs from slotted Aloha in that a packet is discarded when it collides with another packet transmission; there is no mechanism for backoff and retry; the system has no instability problem caused by backlogged packets; the system also has no need for feedback regarding transmitted packets. (b) What is the PMF of N(t), the number of packets generated in the interval [,t]? This is also a gift. For a Poisson process of rate λ, the number of packets in the interval [,t] has the Poisson PMF { (λt) P N(t) (n) = n e λt /n! n =, 1,,... otherwise (c) What is the probability q that a transmitted packet is successfully received by the data collector? A packet transmitted at time t is successfully received if there are no packets that start transmission either in the interval [t 1,t) or in the interval (t, t +1].Let N denote the number of arrivals over these two intervals. Since the probability of an arrival (really a second arrival) exactly at time t is zero, N is Poisson with expected value λ q = P {N =} =(λ) e λ /! = e λ. () (d) Let R(t) be the number of packets successfully received at the data collector over the interval [,t]. What is the expected success rate r = lim t E[R(t)]/t? What value of λ maximizes r? Suppose we divide the interval [,t] into t/ small intervals of size. Let I n denote an indicator variable that equals 1 if a packet is successfully transmitted in interval n. The probability of a successful transmission being initiated in interval n is E[I n ]=qλ. (3) since λ is the probability of a packet transmission in the interval and q is the probability the transmission is successful. The total number of successes is R(t) = t/ n=1 I n.
3 Note that the I n are dependent. When is small, I n =1implies I n 1 = I n+1 =. However, since the expected value of the sum equals the sum of the expeced values, no matter if they are dependent, t/ E[R(t)] = E[I n ]= n=1 t/ n=1 qλ = t qλ =qλt. It follows that R(t) r = lim = lim qλ = qλ = λe λ. t t t The maximum success rate is found by dr dλ = e λ λe λ =. This yields λ =1/, which balances setting λ high to increase the number of attempts against setting λ low to minimize the probability of a collision. (e) Is R(t) a Poisson process? Make sure to justify your answer. Successful packet transmissions represent arrivals for the R(t) counting process. However, the nature of the collisions is that if there is an arrival at time t, the next arrival cannot occur until after time t +1. Thus the process is not memoryless and is not Poisson points Packets arrive as a Poisson process of rate λ at a network. The packets proceed through a series of m unreliable communication links L 1,L,...,L m.oneach link, a transmission error can occur with probability q, independent of an error on any other link. However, error checking occurs only on an end-to-end basis; a packet error is detected at the output of link L m if an error occurs on any of the m links. If a packet is received in error, a feedback channel is used to transmit a NAK message back to the server at link L 1. Fortunately, the NAK signal is always received without error. Reception of the NAK at L 1 initiates a retransmission of the packet through links L 1,...,L m. Transmissions at each link (including the feedback link) require, on average, one unit of time for transmission. (a) Model the system consisting of links L 1,...,L m and the feedback link as a Jackson network. Sketch the network of queues. We model the system as a Jackson network with m +1 queues for the links L 1,...,L m and the feedback queue. For convenience, we call the feedback queue L. Since packets pass though each queue L j,andthen,withsomeprobability, require retrasmission: we model the system as the Jackson network: r L 1 L L m p L 1-p 3
4 The probability a packet is successfully transmitted through all m links and leaves the system is p =(1 q) m. The arrival rate at link L 1,satisfiesr = λ +(1 p)r, which implies r = λ p = λ (1 q) m. Moreover, since a packet that passes through link L 1 passes through links L,...,L m, the arrival rate is r at each of these links. (b) For what arrival rates λ is the network of queues stable? At each link L j, 1 j m, the arrival rate is r and the service rate is µ =1. Thus the offered load is ρ j = r µ = r, 1 j m. We conclude that each queue L j is stable if and only if r<1, orλ<(1 q) m. Also, we note that at the feedback link L, the arrival rate is r(1 p) <r.since the service rate on the feedback queue is also 1, the feedback queue has load ρ = r(1 p) =r[1 (1 q) m ]. The condition λ<(1 q) m is also sufficient to ensure that ρ < 1 and the feedback queue is stable. (c) Find the stationary probabilities when the queueing network is stable. Under the Jackson network model, the stationary distribution for the network has the form of m +1 independent M/M/1 queues: p(n,...,n m )= m (1 ρ i )ρ n i In this case, using the loads ρ i from the previous part, we obtain the stationary distribution i= p(n,...,n m )=[1 r(1 p)][r(1 p)] n (1 r) m r n 1+ +n m, n i. (4) (d) Identify any additional assumptions, whether physically reasonable or not, required for correctness the Jackson network model. For the Jackson network model, we make the assumption (unstated in the problem description) that the packets have exponential service times. In addition, we have to assume that the service times of a packet are independent from link to link. If the links have fixed data rates, this is equivalent to assuming the packet gets a new independent length at each queue. Note that this is not the same thing as the Kleinrock independence assumption in which we assume (or make the modeling assumption) that queues are independent. Rather, once we satisfy the assumptions of the Jackson network model, its a mathematical fact that the stationary distribution is the same as it would be if the queues were independent (despite the fact that the individual queue processes are not actually independent.) 4 i.
5 (e) Is this queueing network a time-reversible system? Explain. One approach to answering this question is to find the reverse time transition probabilities and show they are not the same as the transition probabilities of the ordinary forward time system. That s pretty tedious however. A simpler observation is to note that in reverse time, external packets will arrive at link L m and proceed through links m 1 down to link 1. At that point, a packet will either leave the system or go through the feedback queue back to the queue at link m. This process is easily distringuished from the forward time process in which the packet circulation is reversed. Hence the network is not reversible points Computation jobs arrive at a processor as a Poisson process of rate λ. For each job, the processor performs two calculations. These calculations require processing times X and Y with PDFs f X (x) = { 1/ x otherwise f Y (y) = { 1/4 y 4 otherwise The processing times required for each calculation, whether in the same job or different jobs, are independent. Given a job, a single processor performs the two calculations sequentially so the service time of a job is X + Y. A job is queued if it arrives when the processor is busy. (a) For what values of the arrival rate λ is the job queue stable? For the single processor, the service time of a packet is U = X + Y.theresulting queue is an M/G/1 queue with service time PDF f U (u) which is the convolution of the service time PDFs f X (x) and f Y (y). We can use the P-K formula to find the average waiting time E[W ]= λe[ U ] (1 λe[u]). In this problem, the appropriate definition for stability would be that E[W ] is finite since this would imply that the expected number of queued customers is finite. In this case, we would have two conditions: E [ U ] <, λe[u] < 1. Because X and Y 4, we know that U 6 and thus U 36. ThusE [ U ] is always finite. In addition, since X is uniform (, ) and Y is uniform (, 4), E[U] =E[X]+E[Y ]=1+=3. Thus the condition λe[u] < 1 simplifies to λ<1/3. (b) What is the expected waiting time E[W ]ofajob? To use the P-K formula, we need to find E [ U ]. We recall (or you can derive) that a uniform (a, b) random variable has variance (b a) /1. Since the calculation times X and Y are independent, σ U = σ X + σ Y = ( ) 1 + (4 ) 1 =
6 Since E[U] =3, U has second moment E [ U ] = σ U +(E[U]) = 5 3 Thus, by the P-K formula, for λ<3, +9= 3 3. E[W ]= λe[ U ] (1 λe[u]) = 16λ 3(1 3λ). (c) What is the expected number E[N] of jobs in the system? The expected system time of a job is E[T ]=E[W ]+E[U] = 3λ 9 11λ +3= 6(1 3λ) 3(1 3λ) By Little s Law, the expected number of jobs in the system is E[N] =λe[t ]= λ(9 11λ) 3(1 3λ). (d) points Suppose now that the processor is replaced by a dual processor system that can perform the two calculations in parallel. The processing times X and Y for each calculation remain independent and each has the same distribution as in the single processor system. A job is finished when both calculations are completed. For what range of arrival rates λ is the queue stable? What is the expected waiting time E[W ]ofajob? In this case, the service time is V =max(x, Y ) since the job is completed when both calculations are completed. The queue is still M/G/1, and by the P-K formula, the expected waiting time is E [ W ] = λe[ V ] (1 λe[v ]). However, we need to solve a probability problem to find E[V ] and E [ V ]. First we find the CDF F V (v) and then the PDF f V (v). Since V is the maximum of X and Y and since X and Y are independent, F V (v) =P {V v} = P {X x, Y v} = P {X v}p {Y v} = F X (v) F Y (v). Now we observe from the PDFs f X (x) and f Y (y) that x<, x<, F X (x) = x/ x, F Y (y) = y/4 y 4, 1 <x, 1 4 <y. This implies v< (v/)(v/4) v, F V (v) =F X (v) F Y (v) = v/4 v 4, 1 v>4. 6
7 Taking derivatives, V has PDF f V (v) = df V (v) dv = v/4 v, 1/4 v 4, otherwise. Thus the moments of V are and E[V ]= E [ V ] = v(v/4) dv + 4 v (v/4) dv + v(1/4) dv = v3 1 4 From the P-K formula, it follows that + v 8 v (1/4) dv = v4 16 E[W ]= λe[ V ] (1 λe[v ]) = 17λ 6 13λ. 4 = = v3 4 1 = We conclude that the queue is stable for arrival rates λ<6/13. Quite a few people guessed tht the queue was stable for rates λ<1/, since the longer calculation required expected time E[Y ]=. However, since the calculation times are random, sometimes X>Y. Averaged over all possible outcomes, this results in E[V ]=13/6 >. 7
Multiaccess Communication
Information Networks p. 1 Multiaccess Communication Satellite systems, radio networks (WLAN), Ethernet segment The received signal is the sum of attenuated transmitted signals from a set of other nodes,
More informationrequests/sec. The total channel load is requests/sec. Using slot as the time unit, the total channel load is 50 ( ) = 1
Prof. X. Shen E&CE 70 : Examples #2 Problem Consider the following Aloha systems. (a) A group of N users share a 56 kbps pure Aloha channel. Each user generates at a Passion rate of one 000-bit packet
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 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 informationBirth-Death Processes
Birth-Death Processes Birth-Death Processes: Transient Solution Poisson Process: State Distribution Poisson Process: Inter-arrival Times Dr Conor McArdle EE414 - Birth-Death Processes 1/17 Birth-Death
More informationChapter 5. Elementary Performance Analysis
Chapter 5 Elementary Performance Analysis 1 5.0 2 5.1 Ref: Mischa Schwartz Telecommunication Networks Addison-Wesley publishing company 1988 3 4 p t T m T P(k)= 5 6 5.2 : arrived rate : service rate 7
More informationQueuing Networks: Burke s Theorem, Kleinrock s Approximation, and Jackson s Theorem. Wade Trappe
Queuing Networks: Burke s Theorem, Kleinrock s Approximation, and Jackson s Theorem Wade Trappe Lecture Overview Network of Queues Introduction Queues in Tandem roduct Form Solutions Burke s Theorem What
More informationCPSC 531 Systems Modeling and Simulation FINAL EXAM
CPSC 531 Systems Modeling and Simulation FINAL EXAM Department of Computer Science University of Calgary Professor: Carey Williamson December 21, 2017 This is a CLOSED BOOK exam. Textbooks, notes, laptops,
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 informationLink Models for Packet Switching
Link Models for Packet Switching To begin our study of the performance of communications networks, we will study a model of a single link in a message switched network. The important feature of this model
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 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 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 informationThe exponential distribution and the Poisson process
The exponential distribution and the Poisson process 1-1 Exponential Distribution: Basic Facts PDF f(t) = { λe λt, t 0 0, t < 0 CDF Pr{T t) = 0 t λe λu du = 1 e λt (t 0) Mean E[T] = 1 λ Variance Var[T]
More informationChapter 3 Balance equations, birth-death processes, continuous Markov Chains
Chapter 3 Balance equations, birth-death processes, continuous Markov Chains Ioannis Glaropoulos November 4, 2012 1 Exercise 3.2 Consider a birth-death process with 3 states, where the transition rate
More informationGI/M/1 and GI/M/m queuing systems
GI/M/1 and GI/M/m queuing systems Dmitri A. Moltchanov moltchan@cs.tut.fi http://www.cs.tut.fi/kurssit/tlt-2716/ OUTLINE: GI/M/1 queuing system; Methods of analysis; Imbedded Markov chain approach; Waiting
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 informationMarkov Chain Model for ALOHA protocol
Markov Chain Model for ALOHA protocol Laila Daniel and Krishnan Narayanan April 22, 2012 Outline of the talk A Markov chain (MC) model for Slotted ALOHA Basic properties of Discrete-time Markov Chain Stability
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 informationAnswers to the problems from problem solving classes
Answers to the problems from problem solving classes Class, multiaccess communication 3. Solution : Let λ Q 5 customers per minute be the rate at which customers arrive to the queue for ordering, T Q 5
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262 Discrete Stochastic Processes Midterm Quiz April 6, 2010 There are 5 questions, each with several parts.
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 informationCDA5530: Performance Models of Computers and Networks. Chapter 4: Elementary Queuing Theory
CDA5530: Performance Models of Computers and Networks Chapter 4: Elementary Queuing Theory Definition Queuing system: a buffer (waiting room), service facility (one or more servers) a scheduling policy
More informationECE 3511: Communications Networks Theory and Analysis. Fall Quarter Instructor: Prof. A. Bruce McDonald. Lecture Topic
ECE 3511: Communications Networks Theory and Analysis Fall Quarter 2002 Instructor: Prof. A. Bruce McDonald Lecture Topic Introductory Analysis of M/G/1 Queueing Systems Module Number One Steady-State
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 informationM/G/1 and M/G/1/K systems
M/G/1 and M/G/1/K systems Dmitri A. Moltchanov dmitri.moltchanov@tut.fi http://www.cs.tut.fi/kurssit/elt-53606/ OUTLINE: Description of M/G/1 system; Methods of analysis; Residual life approach; Imbedded
More informationECE 302 Division 1 MWF 10:30-11:20 (Prof. Pollak) Final Exam Solutions, 5/3/2004. Please read the instructions carefully before proceeding.
NAME: ECE 302 Division MWF 0:30-:20 (Prof. Pollak) Final Exam Solutions, 5/3/2004. Please read the instructions carefully before proceeding. If you are not in Prof. Pollak s section, you may not take this
More informationIEOR 6711: Stochastic Models I, Fall 2003, Professor Whitt. Solutions to Final Exam: Thursday, December 18.
IEOR 6711: Stochastic Models I, Fall 23, Professor Whitt Solutions to Final Exam: Thursday, December 18. Below are six questions with several parts. Do as much as you can. Show your work. 1. Two-Pump Gas
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 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 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 informationUNIVERSITY OF YORK. MSc Examinations 2004 MATHEMATICS Networks. Time Allowed: 3 hours.
UNIVERSITY OF YORK MSc Examinations 2004 MATHEMATICS Networks Time Allowed: 3 hours. Answer 4 questions. Standard calculators will be provided but should be unnecessary. 1 Turn over 2 continued on next
More informationExercises Stochastic Performance Modelling. Hamilton Institute, Summer 2010
Exercises Stochastic Performance Modelling Hamilton Institute, Summer Instruction Exercise Let X be a non-negative random variable with E[X ]
More informationSolutions to Homework Discrete Stochastic Processes MIT, Spring 2011
Exercise 6.5: Solutions to Homework 0 6.262 Discrete Stochastic Processes MIT, Spring 20 Consider the Markov process illustrated below. The transitions are labelled by the rate q ij at which those transitions
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 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 informationStatistics 253/317 Introduction to Probability Models. Winter Midterm Exam Friday, Feb 8, 2013
Statistics 253/317 Introduction to Probability Models Winter 2014 - Midterm Exam Friday, Feb 8, 2013 Student Name (print): (a) Do not sit directly next to another student. (b) This is a closed-book, closed-note
More informationECEn 370 Introduction to Probability
RED- You can write on this exam. ECEn 370 Introduction to Probability Section 00 Final Winter, 2009 Instructor Professor Brian Mazzeo Closed Book Non-graphing Calculator Allowed No Time Limit IMPORTANT!
More informationRandom Access Protocols ALOHA
Random Access Protocols ALOHA 1 ALOHA Invented by N. Abramson in 1970-Pure ALOHA Uncontrolled users (no coordination among users) Same packet (frame) size Instant feedback Large (~ infinite) population
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 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 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 informationBasics of Stochastic Modeling: Part II
Basics of Stochastic Modeling: Part II Continuous Random Variables 1 Sandip Chakraborty Department of Computer Science and Engineering, INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR August 10, 2016 1 Reference
More informationEECS 126 Probability and Random Processes University of California, Berkeley: Fall 2014 Kannan Ramchandran November 13, 2014.
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2014 Kannan Ramchandran November 13, 2014 Midterm Exam 2 Last name First name SID Rules. DO NOT open the exam until instructed
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 informationContinuous-time Markov Chains
Continuous-time Markov Chains Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ October 23, 2017
More informationMidterm Exam 1 Solution
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2015 Kannan Ramchandran September 22, 2015 Midterm Exam 1 Solution Last name First name SID Name of student on your left:
More informationQ = (c) Assuming that Ricoh has been working continuously for 7 days, what is the probability that it will remain working at least 8 more days?
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2005, Professor Whitt, Second Midterm Exam Chapters 5-6 in Ross, Thursday, March 31, 11:00am-1:00pm Open Book: but only the Ross
More information2905 Queueing Theory and Simulation PART III: HIGHER DIMENSIONAL AND NON-MARKOVIAN QUEUES
295 Queueing Theory and Simulation PART III: HIGHER DIMENSIONAL AND NON-MARKOVIAN QUEUES 16 Queueing Systems with Two Types of Customers In this section, we discuss queueing systems with two types of customers.
More informationName of the Student:
SUBJECT NAME : Probability & Queueing Theory SUBJECT CODE : MA 6453 MATERIAL NAME : Part A questions REGULATION : R2013 UPDATED ON : November 2017 (Upto N/D 2017 QP) (Scan the above QR code for the direct
More informationPart I Stochastic variables and Markov chains
Part I Stochastic variables and Markov chains Random variables describe the behaviour of a phenomenon independent of any specific sample space Distribution function (cdf, cumulative distribution function)
More informationChapter 8 Queuing Theory Roanna Gee. W = average number of time a customer spends in the system.
8. Preliminaries L, L Q, W, W Q L = average number of customers in the system. L Q = average number of customers waiting in queue. W = average number of time a customer spends in the system. W Q = average
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 information6 Solving Queueing Models
6 Solving Queueing Models 6.1 Introduction In this note we look at the solution of systems of queues, starting with simple isolated queues. The benefits of using predefined, easily classified queues will
More informationPower Laws in ALOHA Systems
Power Laws in ALOHA Systems E6083: lecture 8 Prof. Predrag R. Jelenković Dept. of Electrical Engineering Columbia University, NY 10027, USA predrag@ee.columbia.edu March 6, 2007 Jelenković (Columbia University)
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 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 informationProbability Models in Electrical and Computer Engineering Mathematical models as tools in analysis and design Deterministic models Probability models
Probability Models in Electrical and Computer Engineering Mathematical models as tools in analysis and design Deterministic models Probability models Statistical regularity Properties of relative frequency
More informationAnalysis of random-access MAC schemes
Analysis of random-access MA schemes M. Veeraraghavan and Tao i ast updated: Sept. 203. Slotted Aloha [4] First-order analysis: if we assume there are infinite number of nodes, the number of new arrivals
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 informationMultiaccess Problem. How to let distributed users (efficiently) share a single broadcast channel? How to form a queue for distributed users?
Multiaccess Problem How to let distributed users (efficiently) share a single broadcast channel? How to form a queue for distributed users? z The protocols we used to solve this multiaccess problem are
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 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 informationBernoulli Counting Process with p=0.1
Stat 28 October 29, 21 Today: More Ch 7 (Sections 7.4 and part of 7.) Midterm will cover Ch 7 to section 7.4 Review session will be Nov. Exercises to try (answers in book): 7.1-, 7.2-3, 7.3-3, 7.4-7 Where
More information(b) What is the variance of the time until the second customer arrives, starting empty, assuming that we measure time in minutes?
IEOR 3106: Introduction to Operations Research: Stochastic Models Fall 2006, Professor Whitt SOLUTIONS to Final Exam Chapters 4-7 and 10 in Ross, Tuesday, December 19, 4:10pm-7:00pm Open Book: but only
More informationCHAPTER 4. Networks of queues. 1. Open networks Suppose that we have a network of queues as given in Figure 4.1. Arrivals
CHAPTER 4 Networks of queues. Open networks Suppose that we have a network of queues as given in Figure 4.. Arrivals Figure 4.. An open network can occur from outside of the network to any subset of nodes.
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 informationSOLUTIONS IEOR 3106: Second Midterm Exam, Chapters 5-6, November 8, 2012
SOLUTIONS IEOR 3106: Second Midterm Exam, Chapters 5-6, November 8, 2012 This exam is closed book. YOU NEED TO SHOW YOUR WORK. Honor Code: Students are expected to behave honorably, following the accepted
More informationQueueing Theory II. Summary. ! M/M/1 Output process. ! Networks of Queue! Method of Stages. ! General Distributions
Queueing Theory II Summary! M/M/1 Output process! Networks of Queue! Method of Stages " Erlang Distribution " Hyperexponential Distribution! General Distributions " Embedded Markov Chains M/M/1 Output
More informationA Study on Performance Analysis of Queuing System with Multiple Heterogeneous Servers
UNIVERSITY OF OKLAHOMA GENERAL EXAM REPORT A Study on Performance Analysis of Queuing System with Multiple Heterogeneous Servers Prepared by HUSNU SANER NARMAN husnu@ou.edu based on the papers 1) F. S.
More informationContinuous Time Processes
page 102 Chapter 7 Continuous Time Processes 7.1 Introduction In a continuous time stochastic process (with discrete state space), a change of state can occur at any time instant. The associated point
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 informationPage 0 of 5 Final Examination Name. Closed book. 120 minutes. Cover page plus five pages of exam.
Final Examination Closed book. 120 minutes. Cover page plus five pages of exam. To receive full credit, show enough work to indicate your logic. Do not spend time calculating. You will receive full credit
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 informationIEOR 3106: Second Midterm Exam, Chapters 5-6, November 7, 2013
IEOR 316: Second Midterm Exam, Chapters 5-6, November 7, 13 SOLUTIONS Honor Code: Students are expected to behave honorably, following the accepted code of academic honesty. You may keep the exam itself.
More informationSlides 8: Statistical Models in Simulation
Slides 8: Statistical Models in Simulation Purpose and Overview The world the model-builder sees is probabilistic rather than deterministic: Some statistical model might well describe the variations. An
More informationTime Reversibility and Burke s Theorem
Queuing Analysis: Time Reversibility and Burke s Theorem Hongwei Zhang http://www.cs.wayne.edu/~hzhang Acknowledgement: this lecture is partially based on the slides of Dr. Yannis A. Korilis. Outline Time-Reversal
More informationThe Transition Probability Function P ij (t)
The Transition Probability Function P ij (t) Consider a continuous time Markov chain {X(t), t 0}. We are interested in the probability that in t time units the process will be in state j, given that it
More informationChapter 5. Statistical Models in Simulations 5.1. Prof. Dr. Mesut Güneş Ch. 5 Statistical Models in Simulations
Chapter 5 Statistical Models in Simulations 5.1 Contents Basic Probability Theory Concepts Discrete Distributions Continuous Distributions Poisson Process Empirical Distributions Useful Statistical Models
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 informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262 Discrete Stochastic Processes Midterm Quiz April 6, 2010 There are 5 questions, each with several parts.
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 informationFinal Examination December 16, 2009 MATH Suppose that we ask n randomly selected people whether they share your birthday.
1. Suppose that we ask n randomly selected people whether they share your birthday. (a) Give an expression for the probability that no one shares your birthday (ignore leap years). (5 marks) Solution:
More informationModeling and Simulation NETW 707
Modeling and Simulation NETW 707 Lecture 6 ARQ Modeling: Modeling Error/Flow Control Course Instructor: Dr.-Ing. Maggie Mashaly maggie.ezzat@guc.edu.eg C3.220 1 Data Link Layer Data Link Layer provides
More informationMultimedia Communication Services Traffic Modeling and Streaming
Multimedia Communication Services Medium Access Control algorithms Aloha Slotted: performance analysis with finite nodes Università degli Studi di Brescia A.A. 2014/2015 Francesco Gringoli Master of Science
More informationRecap. Probability, stochastic processes, Markov chains. ELEC-C7210 Modeling and analysis of communication networks
Recap Probability, stochastic processes, Markov chains ELEC-C7210 Modeling and analysis of communication networks 1 Recap: Probability theory important distributions Discrete distributions Geometric distribution
More informationEECS 126 Probability and Random Processes University of California, Berkeley: Fall 2014 Kannan Ramchandran September 23, 2014.
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2014 Kannan Ramchandran September 23, 2014 Midterm Exam 1 Last name First name SID Rules. DO NOT open the exam until instructed
More informationAnalysis of A Single Queue
Analysis of A Single Queue 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 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 informationWeek 5: Markov chains Random access in communication networks Solutions
Week 5: Markov chains Random access in communication networks Solutions A Markov chain model. The model described in the homework defines the following probabilities: P [a terminal receives a packet in
More informationECE 313: Conflict Final Exam Tuesday, May 13, 2014, 7:00 p.m. 10:00 p.m. Room 241 Everitt Lab
University of Illinois Spring 1 ECE 313: Conflict Final Exam Tuesday, May 13, 1, 7: p.m. 1: p.m. Room 1 Everitt Lab 1. [18 points] Consider an experiment in which a fair coin is repeatedly tossed every
More informationBIRTH DEATH PROCESSES AND QUEUEING SYSTEMS
BIRTH DEATH PROCESSES AND QUEUEING SYSTEMS Andrea Bobbio Anno Accademico 999-2000 Queueing Systems 2 Notation for Queueing Systems /λ mean time between arrivals S = /µ ρ = λ/µ N mean service time traffic
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 informationMath Spring Practice for the final Exam.
Math 4 - Spring 8 - Practice for the final Exam.. Let X, Y, Z be three independnet random variables uniformly distributed on [, ]. Let W := X + Y. Compute P(W t) for t. Honors: Compute the CDF function
More informationCS115 Computer Simulation Project list
CS115 Computer Simulation Project list The final project for this class is worth 40% of your grade. Below are your choices. You only need to do one of them. Project MC: Monte Carlo vs. Deterministic Volume
More informationCS 798: Homework Assignment 3 (Queueing Theory)
1.0 Little s law Assigned: October 6, 009 Patients arriving to the emergency room at the Grand River Hospital have a mean waiting time of three hours. It has been found that, averaged over the period of
More informationCHAPTER 6. 1, if n =1, 2p(1 p), if n =2, n (1 p) n 1 n p + p n 1 (1 p), if n =3, 4, 5,... var(d) = 4var(R) =4np(1 p).
CHAPTER 6 Solution to Problem 6 (a) The random variable R is binomial with parameters p and n Hence, ( ) n p R(r) = ( p) n r p r, for r =0,,,,n, r E[R] = np, and var(r) = np( p) (b) Let A be the event
More informationTCOM 501: Networking Theory & Fundamentals. Lecture 6 February 19, 2003 Prof. Yannis A. Korilis
TCOM 50: Networking Theory & Fundamentals Lecture 6 February 9, 003 Prof. Yannis A. Korilis 6- Topics Time-Reversal of Markov Chains Reversibility Truncating a Reversible Markov Chain Burke s Theorem Queues
More informationDiscrete Random Variables
CPSC 53 Systems Modeling and Simulation Discrete Random Variables Dr. Anirban Mahanti Department of Computer Science University of Calgary mahanti@cpsc.ucalgary.ca Random Variables A random variable is
More informationNorthwestern University Department of Electrical Engineering and Computer Science
Northwestern University Department of Electrical Engineering and Computer Science EECS 454: Modeling and Analysis of Communication Networks Spring 2008 Probability Review As discussed in Lecture 1, probability
More information