(Problem 12 is not in the assignment but can help you understand the solution of 13)
13. Condition on the initial toss:
12
Pcfw_win Pcfw_win | toss iPcfw_toss i
i 2
1, i 7or11
Pcfw_win | toss i 0, i 2or 3or12
Pcfw_toss i
Pcfw_i _ occurs _ before7
, i o
Overview of Last Class
Busy periods of M/G/1/
P0 =
E[I]
E[I]+E[B]
E[I] =
1
Average number of customers in service = a E[S]
M/G/1 with random-sized batch
[
V = [N ] E[S]WQ +
E[S 2 ]
2
]
WQ = V + E[WB ]
Conditioning technique to calculate E[WB ]
Prior
Overview of Last Class
Renewal Reward Process
() =
=1 ()
lim () = lim [()] =
[]
[]
Wireless network throughput analysis
Queueing Theory
Evaluate , , , .
Cost Equation:
Average rate at which the system earns
= average amount an entering customer pays.
Overview of last class
M/G/1/ system
Theoretical foundation: cost equation
Average rate at which the system earns = a average amount a customer pays
Dene cost rule: each customer pays y/unit-time when the remain service time is y
E[S 2 ]
V = a E[S
+
2
Overview of Last Class
Renewal process: a counting process ( ) with . interarrival
= []
=
=1
Basic Property:
nite
( )
() nite
Distribution of ()
cfw_ () = = () +1()
Mean value function
() = [ ()] =
<
=1 ()
() <
Renewal Equation
() = () + 0
Review of Markov Chains
Transforming a process into a Markov chain: weather forecasting depends on two
days
Limiting probabilities:
irreducible, ergodic Markov chain has limiting probabilities
unique solutions of
j = i Pij
i=0
j=0 j
=1
matrix techniq
Overview of Last Class
Time reversibility of a DTMC
Embedded DTMC with transition probabilities Pij
i : proportion of transitions taking the process into state i
Pi : proportion of time the process in state i
Pi =
i 1
vi 1
j j v
j
consider the seque
Review of Last Class
State transition probability
C-K equation
Pij (t + s) = Pik (t)Pkj (s)
k=0
Transition rates
vi ,
qij = vi Pij
Lemma 6.2
ii
lim 1Ph (h) = vi
h0
Pij (h)
h0 h
lim
Prove: P cfw_one transition within h = vi h + o(h)
P cfw_two or moretra
Review of last class
CTMC
Continuous time Markov property
cfw_( + ) = () = , () = (), 0
= cfw_( + ) = () =
The state transition are in accordance with an embedded DTMC.
The state residence time is exponential distributed.
The residence time at diff
Review of last class
Poisson denition 3: Interarrival time exponential ()
The arrival of the nth event: Gamma(n, )
Conditional arrival
Given n events happened, the arrival time of each event s1 , , sn , considered as
unordered R.V.s, are independently u
Review of last class
Exponential Distribution
Exponential distribution
cfw_
f (x) =
ex
x0
0
x<0
CDF, MGF
Memoryless property
P cfw_X > s + t | X > t = P cfw_X > s
Failure rate
r(t) =
f (t)
1 F (t)
constant
Summation of independent and identically expo
Overview of Last Class
M/G/1/ with server break down
New service distribution
=
+
=1
[ 2]
=
2(1 [ ])
Conditioning technique to calculate [ ], [ 2]
G/M/1
Embedded DTMC at arrival moment
+1 = + 1
Transition probabilities ; solve = (1 ) based on
r
Overview of Last Week
Open Intro
Importance of Analytic Performance Evaluation
analysis, simulation, testbed
Fundamental knowledge
R.V., Random Process, Queueing Theory
Review of the Probability
Sample space, event, union/intersection/partition
Deni
i 1
36 , i 2,., 7
Pcfw_sum i
13 i , i 8,.,12
36
Pcfw_diff i
6 i
, i 5, 4,., 4,5
36
n 1 r 1
nr
p (1 p ) p
r 1
31. pcfw_ X i e
i
i!
, np
i
ii
iii
iv
Correct binomial prob. Poisson approximation
0.1488
0.1438
0.3151
0.1300
0.3487
0.3679
0.0661
43. Refer to Example 3.10 and 3.18
2
2
mean 12 andvariance 1 2 2 12
Therefore,
Cov( X , Y ) E[ XY ] E[ X ][Y ]
E[ E[ XY | X ] E[ X ]E[ E[Y | X ]
E[ XE[Y | X ] E[ X ]E[ E[Y | X ]
Cov( X , E[Y | X ])
(refer to exercise 1.12)
1. (a) is the contrapositive of Eq. 7.2 and is true;
(b) is the contrapositive of (c) and both of them are false.
7. Once every five months. (Refer to Example 7.5)
10. (a) Yes
(b)
p
value is T=2.
Overview of Last Class
Network Queues
Two-server open system
state(n,m): two-dimensional transition diagram
Productive joint distribution; independence by time reversibility
P (n, m) = (
n
) (1 ) ( )m (1 )
1
1 2
2
A general open system
ri to queue i
Review of Last Class
Steady-State Probabilities
PASTA: Poisson arrival always sees time average
M/M/1:
Time reversible Markov chain
Pn = ( )n (1 )
Littles formula illustrated
W : the amount of time a customer spends in the systenm
P cfw_N = n|W =
Overview of Last Class
An Application of Bayes Formula
Independent Events
P (A B) = P (A)P (B)
physically unrelated
P (A|B) = P (A) or P (B|A) = P (B)
disjoint / pairwise-independence / independence
Chapter 2 Random Variables
R.V.
X : S RX R
Cumulative
Review of last class
Class of states: communicate, recurrent
n
In |X0 = i] =
Pii
E[
n=0
cfw_
n=0
=
<
recurrent
transient
All states in a nite-state irreducible Markov chain are recurrent
If i is recurrent/transient & i j, j is recurrent/transient
At l
Introduction to the Course
Network Performance
Computer
.
Router
C
A router modeled as a queue
A network modeled as a network of queues
Performance measures
Delay
Packet loss rate
Throughput
Best-effort Internet: performance not guaranteed
ECE541: Per
Chapter 1. Introduction to Probability Theory
1.1 Introduction
Probabilistic model: Any realistic model of a real-world phenomenon must take into
account the possibility of randomness
1.2 Sample Space and Events
Sample space (): the set of all possible ou
Review of Markov Chains
Transforming a process into a Markov chain: weather forecasting depends on two
days
Limiting probabilities:
irreducible, ergodic Markov chain has limiting probabilities
unique solutions of
=
=0
=0
=1
matrix techniques to com
Overview of Last Class
Discrete R.V.
PMF
p(xi ) = P cfw_X = xi
PMF CDF
Examples: Bernoulli, Binomial, Uniform, Geometric, Poisson
Continuous R.V.
P cfw_X B =
dF (a)
a
B
f (x)dx
= f (a)
Examples: Uniform, Exponential, Normal
X N(, 2 )
Y = X + N( + ,
Review of last class
Class of states: communicate, recurrent
[
0 = ] =
=0
=0
cfw_
=
<
recurrent
transient
All states in a finite-state irreducible Markov chain are recurrent
If is recurrent/transient & , is recurrent/transient
At least one state in a