... D
A
= 3 + D
E
D
E
= 4 + D
G
D
G
= 3 + D
B
= 3
D
D
= 2 + D
G
D
C
= 1 + D
D
D
F
= 6 + D
B
= 6
AB = A-E-G-B
B
A
C
D
E
F
G
3
4
2
7
3
6
4
1
4
34
Complexity of Floyd-Warshall
... After N-1 iterations, the algorithm terminates, and
G' contains N nodes and N-1 arcs.
{},n
′
′
=
=∅NA
:{},:{(,)}jij
′′ ′′
=∪ =∪NN AA
10
Minimum Weight Spanning Tree
5
2...
... times and service times
Exponentially distributed service times
Network model: Jackson network
“Product-Form” stationary distribution
1
TCOM 501:
Networking Theory & Fundamentals
Lecture ... packet is randomly selected each time it is transmitted over a
network link
3. Service times and interarrival times: independent
Assumption has been validated with experimental and...
... Discrete-time birth-death processes are reversible,
since they satisfy the DBE
ππ, , 0,1,
iij j ji
PPij
=
=
7 -3 2
Queues in Tandem
Theorem 11: Network consisting of K single-server queues in tandem. ... E
p
ρρ
−
∈
=
∑
State diagram for B =2
7 -3 3
Queues in Tandem: State-Dependent Service Rates
Theorem 12: Network consisting of K queues in tandem. Service times
at queue...
... lim
i
i
tt
Tt
pPNti
t
→∞ →∞
===
4& 5-1 8
PASTA Theorem: Intuitive Proof
t
a
and
t
r
: randomly selected arrival and observation times,
respectively
The arrival processes prior to
t
a
and
t
r
respectively ... Functions and Independence:
X; Y : independent ) M
X+Y
(t)=M
X
(t)M
Y
(t)
The opposite isnottrue.
4& 5-1 9
Arrivals that Do not See Time-Averages
Example 1: No...
... … and packets that arrive at a network switch
{} , 0,1,2,
!
k
PX k e k
k
λ
λ
−
== =
1
TCOM 501:
Networking Theory & Fundamentals
Lecture 2
January 22, 20 03
Prof. Yannis A. Korilis
2-8
Queue ... Modeling Arrival and Service Statistics
2 -3 3
Probabilistic Form of Little (cont.)
In principle, we can find the probability distribution of the delay
T
i
for customer...
... to wait in queue for the first avail-
able server if, and only if, all the servers are busy, i.e., there are precisely
m
39
TCOM 501: Networking Theory and Fundamentals
University of Pennsylvania
Class ... can now be grandly deployed to determine the average de-
lay per customer
T = N/λ in the steady state.
30
Class Notes
Santosh S. Venkatesh
c
1997
TCOM 501: Networking Theory...
... discrete random variable taking values 0, 1, 2,…, and let
{}
n
p
PX n
=
=
. The z-
transform is well-defined for
||1z
<
:
23
01 2 3
0
()
n
Xn
n
Gz p zp zp zp pz
∞
=
=+ + + +=
∑
Z-transform ... ]
2
E
REX=λ
P-K Formula:
2
[] [ ]
[]
12(1)
E
REX
EW
λ
==
−ρ
−
ρ
1
TCOM 501:
Networking Theory & Fundamentals
Lectures 9 & 10
M/G/1 Queue
Prof. Yannis A. Koril...
... filters
Tt
)(
1
ts
Tt
)(
2
ts
T
T
0
0
T
A
T
A−
T
A−
T
A
0
0
Lecture 3 5
Example of M-ary PAM
-B
B
T
‘01’
3B
T
T
-3 B
T
‘00’
‘10’
‘1’
A.
T
‘0’
T
-A.
Assuming real time tr. and equal energy per tr. data bit for
binary-PAM and 4-ary PAM:
• 4-ary: T=2Tb and ... TR
=
mRR
b
=
Lecture 3 6
Example of M-ary PAM …
0 Tb 2Tb 3Tb 4Tb 5Tb 6Tb
0 Ts 2Ts
0 T 2T 3T
2.2762 V 1 .36 57 V
1 1 0...