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Spring 2003 Homework 2 Solution
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM 501 Homework 2 Solutions Problem 3.11(b) Let A, A1 , and A2 as in the hint. Let I be the interarrival interval of A2 and consider the number of arrivals of A1 that lie in I. The probability that this number is n is the probability of n successive arr

Tcom501final02
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis April 26, 2002 Answer all problems. Good Luck! Problem 1 [30 points]: Consider a ring network with nodes 1,2,K. In this network, a customer that completes service at n

Tcom501midterm02sol
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501 Networking: Theory & Fundamentals Midterm Examination Professor Yannis A. Korilis March 6, 2002 Answer all problems. Good Luck! Problem 1 [15 points]: 1. State the PASTA theorem. [5 points] 2. Provide an analytical proof of the PASTA theorem. [10

Tcom501midterm02
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501 Networking: Theory & Fundamentals Midterm Examination Professor Yannis A. Korilis March 6, 2002 Answer all problems. Good Luck! Problem 1 [15 points]: 1. State the PASTA theorem. [5 points] 2. Provide an analytical proof of the PASTA theorem. [10

Tcom501midterm03sol
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501 Networking: Theory & Fundamentals Midterm Examination Professor Yannis A. Korilis March 5, 2003 Problem 1 [20 points]: Consider an M/M/1 queue that can accommodate at most M customers in the system (queued or in service), and suppose that a custom

Tcom501midterm03
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501 Networking: Theory & Fundamentals Midterm Examination Professor Yannis A. Korilis March 5, 2003 Answer all problems. Good Luck! Problem 1 [20 points]: Consider an M/M/1 queue that can accommodate at most K customers in the system (queued or in ser

Lecture4 5
School: UPenn
TCOM501: NetworkingTheory&Fundamentals Lectures4&5 February5and12,2003 Prof.YannisA.Korilis 1 4&52 Topics MarkovChains M/M/1Queue PoissonArrivalsSeeTimeAverages M/M/*Queues IntroductiontoSojournTimes 4&53 TheM/M/1Queue Arrivalprocess:Poissonwithrate Servi

Lecture6
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lecture 6 February 19, 2003 Prof. Yannis A. Korilis 1 62 Topics TimeReversal of Markov Chains Reversibility Truncating a Reversible Markov Chain Burkes Theorem Queues in Tandem 63 TimeReversed Markov Chains c

Tcom501midterm02
School: UPenn
TCOM501 Networking: Theory & Fundamentals Midterm Examination Professor Yannis A. Korilis March 6, 2002 Answer all problems. Good Luck! Problem 1 [15 points]: 1. State the PASTA theorem. [5 points] 2. Provide an analytical proof of the PASTA theorem. [10

Lecture10
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lectures 9 & 10 M/G/1 Queue Prof. Yannis A. Korilis 1 102 Topics M/G/1 Queue PollaczekKhinchin (PK) Formula Embedded Markov Chain Observed at Departure Epochs PollaczekKhinchin Transform Equation Queues with

Tcom501final02sol
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis April 26, 2002 Problem 1 [30 points]: Consider a ring network with nodes 1,2,K. In this network, a customer that completes service at node i exits the network with pro

Hw6sol
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501 Networking: Theory & Fundamentals Professor Yannis A. Korilis Assignment #6 Due April 16, 2003 Problem 1: Consider the embedded Markov chain cfw_L j : j = 1, 2, . , that is obtained from the M/G/1 queue observed at departure epochs. As shown in th

Spring 2003 Homework 4 Solution
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM 501 Homework 4 Solutions Problem 3.53 (a) If customers are served in the order they arrive then given that a customer departs at time t for queue 1, the arrival time of that customer at queue 1 (and therefore the time spent at queue 1), is independen

Spring 2003 Homework 6 Solution
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501 Networking: Theory & Fundamentals Professor Yannis A. Korilis Assignment #6 Due April 16, 2003 Problem 1: Consider the embedded Markov chain cfw_L j : j = 1,2,. , that is obtained from the M/G/1 queue observed at departure epochs. As shown in the

Spring 2003midterm Exam 3solution
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501 Networking: Theory & Fundamentals Midterm Examination Professor Yannis A. Korilis March 5, 2003 Problem 1 [20 points]: Consider an M/M/1 queue that can accommodate at most M customers in the system (queued or in service), and suppose that a custom

Spring 2003 Homework 3 Solution
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501: Networking: Theory & Fundamentals Prof. Yannis A. Korilis Assignment #3 Solutions Problem 3.23 Let p m = P cfw_the 1st m servers are busy As given by the Erlang B formula. Denote rm = Arrival rate to servers (m+1) and above m = Arrival rate to s

Hw1sol
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM 501 Homework 1 Solutions Problem 3.1 A customer that carries out the order (eat in the restaurant) stays for 5 minutes (25 minutes). Therefore the average customer time in the system is T = 0.5 0.5 + 0.5 25 = 15 minutes Applying Littles Theorem the a

Hw2sol
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM 501 Homework 2 Solutions Problem 3.11(b) Let A, A1 , and A2 as in the hint. Let I be the interarrival interval of A2 and consider the number of arrivals of A1 that lie in I. The probability that this number is n is the probability of n successive arr

Hw3sol
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM501: Networking: Theory & Fundamentals Prof. Yannis A. Korilis Assignment #3 Solutions Problem 3.23 Let p m = P cfw_the 1st m servers are busy As given by the Erlang B formula. Denote rm = Arrival rate to servers (m+1) and above m = Arrival rate to s

Hw4sol
School: UPenn
Course: Networking – Theory & Fundamentals
TCOM 501 Homework 4 Solutions Problem 3.53 (a) If customers are served in the order they arrive then given that a customer departs at time t for queue 1, the arrival time of that customer at queue 1 (and therefore the time spent at queue 1), is independen

HW5Sol
School: UPenn
Course: Networking – Theory & Fundamentals
1 TCOM501: Networking Theory and Fundamentals Professor Yannis A. Korilis Spring 2003 Homework Assignment 5 Problem 3.30: Let p0 denote the probability that the system is empty. This is also the probability that the server is empty. Applying Littles Theor

Hw6
School: UPenn
TCOM501 Networking: Theory & Fundamentals Professor Yannis A. Korilis Assignment #6 Due April 16, 2003 Problem 1: Consider the embedded Markov chain cfw_L j : j = 1,2,. , that is obtained from the M/G/1 queue observed at departure epochs. As shown in the

Hw1
School: UPenn
TCOM 501 Homework 1 Solutions Problem 3.1 A customer that carries out the order (eat in the restaurant) stays for 5 minutes (25 minutes). Therefore the average customer time in the system is T = 0.5 0.5 + 0.5 25 = 15 minutes Applying Little's Theorem the

Tcom501final02
School: UPenn
TCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis April 26, 2002 Answer all problems. Good Luck! Problem 1 [30 points]: Consider a ring network with nodes 1,2,K. In this network, a customer that completes service at n

Tcom501midterm02sol
School: UPenn
TCOM501 Networking: Theory & Fundamentals Midterm Examination Professor Yannis A. Korilis March 6, 2002 Answer all problems. Good Luck! Problem 1 [15 points]: 1. State the PASTA theorem. [5 points] 2. Provide an analytical proof of the PASTA theore

HW5Sol
School: UPenn
1 TCOM501: Networking Theory and Fundamentals Professor Yannis A. Korilis Spring 2003 Homework Assignment 5 Problem 3.30: Let p0 denote the probability that the system is empty. This is also the probability that the server is empty. Applying Litt

Lecture11
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lecture 11 April 16, 2003 Prof. Yannis A. Korilis 1 2 Topics Routing in Data Network Graph Representation of a Network Undirected Graphs Spanning Trees and Minimum Weight Spanning Trees Directed Graphs S

Lecture11
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lecture 11 April 2003 Prof. Yannis A. Korilis 1 112 Topics ShortestPath Algorithms: BellmanFord Dijkstra FloydWarshall Distributed Asynchronous BellmanFord Algorithm 113 ShortestPath Ro

Tcom501final02
School: UPenn
TCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis April 26, 2002 Answer all problems. Good Luck! Problem 1 [30 points]: Consider a ring network with nodes 1,2,K. In this network, a customer that completes serv

Lecture11
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lecture 11 April 16, 2003 Prof. Yannis A. Korilis 1 2 Topics Routing in Data Network Graph Representation of a Network Undirected Graphs Spanning Trees and Minimum Weight Spanning Trees Directed Grap

Lecture6
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lecture 6 February 19, 2003 Prof. Yannis A. Korilis 1 62 Topics TimeReversal of Markov Chains Reversibility Truncating a Reversible Markov Chain Burkes Theorem Queues in Tandem 63 TimeReversed

Tcom501midterm03
School: UPenn
TCOM501 Networking: Theory & Fundamentals Midterm Examination Professor Yannis A. Korilis March 5, 2003 Answer all problems. Good Luck! Problem 1 [20 points]: Consider an M/M/1 queue that can accommodate at most K customers in the system (queued or

Lecture7
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lecture 7 February 25, 2003 Prof. Yannis A. Korilis 1 72 Topics Open Jackson Networks Network Flows StateDependent Service Rates Networks of Transmission Lines Kleinrocks Assumption 83 Networks

Tcom501final03
School: UPenn
TCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis May 8, 2003 Answer all problems. Good Luck! Problem 1 [50 points]: Two types of jobs arrive at a service station according to two independent Poisson processes w

Tcom501midterm02
School: UPenn
TCOM501 Networking: Theory & Fundamentals Midterm Examination Professor Yannis A. Korilis March 6, 2002 Answer all problems. Good Luck! Problem 1 [15 points]: 1. State the PASTA theorem. [5 points] 2. Provide an analytical proof of the PASTA theore

HW5Sol
School: UPenn
1 TCOM501: Networking Theory and Fundamentals Professor Yannis A. Korilis Spring 2003 Homework Assignment 5 Problem 3.30: Let p0 denote the probability that the system is empty. This is also the probability that the server is empty. Applying Littles Theor

Lecture8
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lecture 8 March 19, 2003 Prof. Yannis A. Korilis 1 82 Topics Closed Jackson Networks Convolution Algorithm Calculating the Throughput in a Closed Network Arrival Theorem for Closed Networks MeanValue A

Lecture3
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lecture 3 January 29, 2003 Prof. Yannis A. Korilis 1 32 Topics Markov Chains DiscreteTime Markov Chains Calculating Stationary Distribution Global Balance Equations Detailed Balance Equations BirthDeat

Lecture10
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lectures 9 & 10 M/G/1 Queue Prof. Yannis A. Korilis 1 102 Topics M/G/1 Queue PollaczekKhinchin (PK) Formula Embedded Markov Chain Observed at Departure Epochs PollaczekKhinchin Transform Equation Que

Lecture6
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lecture 6 February 19, 2003 Prof. Yannis A. Korilis 1 62 Topics TimeReversal of Markov Chains Reversibility Truncating a Reversible Markov Chain Burke's Theorem Queues in Tandem 63 TimeReversed Mar

Hw3
School: UPenn
TCOM501: Networking: Theory & Fundamentals Prof. Yannis A. Korilis Assignment #3 Problems from the book: 3.23, 3.24, 3.25, 3.26, 3.50 Problem 1: Consider a singleserver queue, with a Poisson arrival rate . The service times are independent identica

Hw1sol
School: UPenn
TCOM 501 Homework 1 Solutions Problem 3.1 A customer that carries out the order (eat in the restaurant) stays for 5 minutes (25 minutes). Therefore the average customer time in the system is T = 0.5 0.5 + 0.5 25 = 15 minutes Applying Little's Theo

Tcom501final02sol
School: UPenn
TCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis April 26, 2002 Problem 1 [30 points]: Consider a ring network with nodes 1,2,K. In this network, a customer that completes service at node i exits the network w

Lecture7
School: UPenn
TCOM 501: Networking Theory & Fundamentals Lecture 7 February 25, 2003 Prof. Yannis A. Korilis 1 72 Topics Open Jackson Networks Network Flows StateDependent Service Rates Networks of Transmission Lines Kleinrocks Assumption 83 Networks of .

Lecture3
School: UPenn
TCOM501: NetworkingTheory&Fundamentals Lecture3 January29,2003 Prof.YannisA.Korilis 1 32 Topics MarkovChains DiscreteTimeMarkovChains CalculatingStationaryDistribution GlobalBalanceEquations DetailedBalanceEquations BirthDeathProcess Gen