Statistical Sciences 4654a / 9654a Assignment 3
Due Wednesday Nov. 27th @ 3:30 p.m. NB! Start of class time!
All of these questions concern an M/G/1 queue with
.
1. Compare the mean waiting times before service assuming that the mean service time in all c
Textbook Section 4.1
A Stochastic Process:
A stochastic process is defined in your text as
a collection of random variables (p. 84).
A better definition is that a stochastic process is this:
A stochastic process X(t) is a collection (or family) of
rando
Textbook Section 6.2
6.2: Q: How to define a CTMC?
Our verbal definition of a Markov chain stipulated that
the probability of the future given the present, is
independent of the past.
Now however, our past includes a continuous record
of where the MC ha
Textbook Sections 4.3, 4.4
The reason to classify states
As we have seen through many examples, it is possible
for there to be a number of possible outcomes when
one investigates as n gets large.
It is possible that there is a limiting matrix, all of
wh
Textbook Section 4.2
4.1 Definition of Markov chain
Let us assume that the stochastic process assumes discrete
values, and that time occurs in discrete steps. Instead of
X(t) we tend to write for time n=0,1,2,
The Markov property, in words can be stated
Textbook Section 6.2,6.4
6.2: Q: How to define a CTMC?
Our verbal definition of a Markov chain stipulated that
the probability of the future given the present, is
independent of the past.
Now however, our past includes a continuous record
of where the M
Textbook Section 6.2,6.4
What have we learned about CTMCs?
The textbook definition of a CTMC is the condition
+ = = , = , 0 < =
[ + = | = ] = (t)
For all states , ; , 0. From this we have seen:
1) The length of time spent in state i is exponentially
dis
Textbook Sections 4.7, 4.8
4.7 Branching Processes
A branching process is one model to study whether a
famil line survives or eventually dies out.
One starts with a single individual, who produces j 0
offspring with probability . Each of these offspring
Section 4.9
Background
Suppose that you are interested not so much in terms
of a particular random variable X, but rather some
function h(X) of that random variable.
Suppose further that it is the expected outcome
E[h(X)]; denote this expected outcome by
Textbook Section 4.1
A Stochastic Process:
A stochastic process is defined in your text as
a collection of random variables (p. 84).
A better definition is that a stochastic process is this:
A stochastic process X(t) is a collection (or family) of
rando
Textbook Section 6.2,6.4
6.2: Q: How to define a CTMC?
Our verbal definition of a Markov chain stipulated that
the probability of the future given the present, is
independent of the past.
Now however, our past includes a continuous record
of where the M
Statistical Sciences 4654a / 9654a Assignment 1
Due Thursday Sept. 29th @ 4:30 p.m.
1. (30 marks) This question asks you to determine the probability of winning or losing at the game
called craps, using a Markov chain. The rules for craps are as follows:
Section 4.9
Background
Suppose that you are interested not so much in terms
of a particular random variable X, but rather some
function h(X) of that random variable.
Suppose further that it is the expected outcome
E[h(X)]; denote this expected outcome b
Phase-type distributions
Definition:
Consider a CTMC with m transient states (let us
number them 1,2,.,m) and a single recurrent state
(say, state 0). Since all the other states are transient,
eventually the CTMC will end up in state 0, and so we
call it
Statistical Sciences 4654a / 9654a Assignment 2
Due Tuesday Oct. 29th @ 4:30 p.m.
1. (30 marks) A rural community has m households, each with its own landline telephone. When a
phone is idle calls to it plus from it occur according to a Poisson process at
Poisson Arrivals See Time Averages
Prepared by Chuan H. Foh and M. Zukerman
Let N (t ) be the number of customers in the queue at time t.
The probability that the queue carries k customer(s) is expressed as Prcfw_N (t ) = k .
Let Pk be the steady state p
Statistical Sciences 4654a / 9654a Assignment 4
Due Friday December 6th @ 4:30 p.m.
All of these questions concern an M/G/1 queue with
.
1. Determine the probability of waiting in excess of 3 minutes in an M/ /1 queue where the
mean service time is 1 minu
Birth and Death Queues
The M/M/s queue
Average Delay Chart from Systems Analysis
for Data Transmission by J. Martin
Some queueing terminology:
= average time waiting for service to commence
E S = average or mean service time
= + E S = average time in s
The M/G/1 queue
Distribution of number in system
Solving the M/G/1 Queue
In yesterdays lecture, we derived what is called the
Pollaczek-Khintchine (P-K) formula for the mean
queue length:
= + 2 cfw_ 2 /cfw_2(1- ), where = < 1.
From it we obtained the P-
The University of Western Ontario
Actuarial Science 9654a/4654a Test 1
Prof. D.A. Stanford
4:30-6 p.m.
October 20, 2011
INSTRUCTIONS: Print you Name, Student Number, & Course Number
on the booklet. No programmable calculators allowed.
1. [10 marks total]
Only the Mean value material is in Textbook, p. 539
Q: What are the consequences of removing
the exponential service requirement?
Since CTMC models require the residency time in a state to
be exponentially distributed, we can no longer use a CTMC
model i
The University of Western Ontario
Statistical Science 9654a/4654a Test 2
Prof. D.A. Stanford
4:306 p.m.
November 10, 2011
INSTRUCTIONS: Print you Name, Student Number, & Course Number
on the booklet. No programmable calculators allowed.
1. [5+5+10+10=30 m
Statistical Sciences 4654a / 9654a Assignment 1
1. (30 marks) This question asks you to determine the probability of winning or losing at the game
called craps, using a Markov chain. The rules for craps are as follows:
a) You roll two dice. You win right
The University of Western Ontario
Statistical Sciences 9654a/4654a Final
Prof. D.A. Stanford 9 a.m. to Noon
December 19, 2011
INSTRUCTIONS: Print you Name, Student Number, 85 Course Number
on the booklet. No programmable calculators allowed.
1. [3+6+4+2:1
Textbook Sections 4.3, 4.4
The reason to classify states
As we have seen through many examples, it is possible
for there to be a number of possible outcomes when
one investigates as n gets large.
It is possible that there is a limiting matrix, all of
wh