J. Virtamo
38.3143 Queueing Theory / Queueing networks
1
QUEUEING NETWORKS
A network consisting of several interconnected queues Network of queues Examples Customers go form one queue to another in post oce, bank, supermarket etc Data packets traverse a n
J. Virtamo
38.3143 Queueing Theory / Stochastic processes
1
STOCHASTIC PROCESSES Basic notions
Often the systems we consider evolve in time and we are interested in their dynamic behaviour, usually involving some randomness. the length of a queue the temp
J. Virtamo
38.3143 Queueing Theory / Probability Theory
1
ESSENTIALS OF PROBABILITY THEORY Basic notions
Sample space S S is the set of all possible outcomes e of an experiment. Example 1. In tossing of a die we have S = cfw_1, 2, 3, 4, 5, 6. Example 2. T
IEOR 6711: Stochastic Models I, Fall 2003, Professor Whitt Final Exam: Thursday, December 18, Chapters 4,5 and 9 in Ross
Below are six questions with several parts. Do as much as you can. Show your work. 1. Two-Pump Gas Station (12 points) Potential custo
Homework Assignment 1 Queuing Theory: 56 645558 01 Problem 2 - Sum of Random Variables
Tony Smaldone tony@sntco.com http:/www.sntco.com/tony/masters.html July 26, 2001
Problem 2 - Sum of Random Variables Find the distribution (density) of Y = X1 + X2 for
Math331, Fall 2008 Instructor: David Anderson
2nd Markov Chain Homework
1. Consider a Markov chain transition matrix 1 / 2 1/ 3 1/ 6 P = 3 / 4 0 1/ 4 . 0 1 0 (a) Show the this is a regular Markov chain. (b) If the process is started in state 1, nd the pro
HOMEWORK 4 1. A Markov chain with state space cfw_1, 2, 3 has transition probability matrix 1 / 3 1/ 3 1/ 3 P = 0 1/2 1/2 0 0 1 Show that state 3 is absorbing and, starting from state 1, nd the expected time until absorption occurs. 2. Smith is in jail an
HOMEWORK 4: SOLUTIONS 1. A Markov chain with state space cfw_1, 2, 3 has transition probability matrix 1 / 3 1/ 3 1/ 3 P = 0 1/2 1/2 0 0 1 Show that state 3 is absorbing and, starting from state 1, nd the expected time until absorption occurs. Solution. L
Large Sample Theory Ferguson Exercises, Section 5, Central Limit Theorems. 1. (a) Using a Chebyshevs-Inequality-like argument, show that (assuming the expectations exist) E|X |2+ t E[X 2 I(|X | t)] for all > 0 and t > 0. (b) Using part (a) and Lindeberg,
One Hundred1 Solved2 Exercises3 for the subject:
Stochastic Processes I4
Takis Konstantopoulos5 1. In the Dark Ages, Harvard, Dartmouth, and Yale admitted only male students. Assume that, at that time, 80 percent of the sons of Harvard men went to Harvard
J. Virtamo
38.3143 Queueing Theory / Markov processes
1
Markov processes (Continuous time Markov chains)
Consider (stationary) Markov processes with a continuous parameter space (the parameter usually being time). Transitions from one state to another can
J. Virtamo
38.3143 Queueing Theory / Continuous Distributions
1
CONTINUOUS DISTRIBUTIONS Laplace transform (Laplace-Stieltjes transform)
Denition The Laplace transform of a non-negative random variable X 0 with the probability density function f (x) is de
Midterm Exam I ECE534 There are a total of ve problems You are allowed one sheet (two pages) of notes; no calculators. Please put your NAME here: 1. (a) Let X be a uniformly distributed random variable on [0, 1]. Find the characteristic function of X. (b)
IEOR 6711, HMWK 3, Professor Sigman
1. Consider a positive recurrent Markov chain with limiting stationary distribution . We know that i is, by denition, the long-run proportion of time that the chain moves into 1 state i; i = limn n n=1 I cfw_Xk = i. It
IEOR 6711, HMWK 3 Solutions, Professor Sigman
1. Consider a positive recurrent Markov chain with limiting stationary distribution . We know that i is, by denition, the long-run proportion of time that the chain moves into 1 state i; i = limn n n=1 I cfw_X
5. Continuous-time Markov Chains
Many processes one may wish to model occur in continuous time (e.g. disease transmission events, cell phone calls, mechanical component failure times, . . .). A discrete-time approximation may or may not be adequate. cfw_
IEOR 6711: Stochastic Models I, Fall 2003, 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 Station (12 points) Potential customers arrive at
ECE 534 September 16, 2009
Fall 2009
Probability Quiz Solutions
1. (8 pts) Warm-up (a) Consider the probability space (, F , P ). If A F is such that P (A) = 1, then prove that for any C F , P (A C ) = P (C ). (Note: you cannot assume that A = .)
Ans: P (
J. Virtamo
38.3143 Queueing Theory / Birth-death processes
1
Birth-death processes
General A birth-death (BD process) process refers to a Markov process with - a discrete state space - the states of which can be enumerated with index i=0,1,2,. . . such th
J. Virtamo
38.3143 Queueing Theory / Discrete Distributions
1
DISCRETE DISTRIBUTIONS Generating function (z-transform)
Denition Let X be a discrete r.v., which takes non-negative integer values, X cfw_0, 1, 2, . . .. Denote the point probabilities by pi p
Solutions to Quiz Problem 1 (6 points) Let X have the pdf fX (x) =
sin(x) 2
ECE 534 Fall 2005
x [0, ] 0 else (a) Find the cumulative distribution function FX . In particular, what is FX (2 )? (b) Compute E [sin(X )].
(a) The support of fX 8 <R x is the in