A
B
HELSINKI UNIVERSITY OF TECHNOLOGY
Networking Laboratory
1
ch6.ppt
S38.215 – Applied Stochastic Processes – Spring 2004
Lawler (1995)
Chapter 6
Renewal Processes
6.1 Introduction
6.2 Renewal Equation
6.3 Discrete Renewal Processes
6.4 M/G/1 and G/M/1 queues
2
Chapter 6: Renewal Processes
6.1 Introduction
•
Definition
:
– Let
T
1
,
T
2
, …
be independent and identically distributed (i.i.d.) nonnegative
random variables with mean
μ
=
E
[
T
i
]
>
0
. The
renewal process
associated with renewal sequence
T
i
is the process
N
t
with
– Thus,
N
t
denotes the number of events occurred up to time
t
.
•
Definition
:
– Let
T
1
,
T
2
, …
be independent and identically distributed (i.i.d.) nonnegative
random variables with mean
μ
=
E
[
T
i
]
>
0
. Furthermore, let
Y
another
nonnegative random variable indepedndent of the sequence
T
i
. The
renewal process
associated with random variable
Y
and sequence
T
i
is
the process
N
t
with
≥
≤
+
+
=
<
=
1
1
1
},

,
2
,
1
max{
,
0
T
t
t
T
T
n
T
t
N
n
t
K
K
≥
>
+
+
+
=
<
=
Y
t
t
T
T
Y
n
Y
t
N
n
t
},

,
2
,
1
min{
,
0
1
K
K
(6.1)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document3
Chapter 6: Renewal Processes
6.1 Introduction
•
Definitions
:
– Let
N
t
be a renewal process associated with renewal sequence
T
i
. The
age
associated with renewal process
N
t
is the process
A
t
with
–T
h
e
residual life
associated with renewal process
N
t
is the process
B
t
with
h
e
total lifetime
associated with renewal process
N
t
is the process
C
t
with
•
Note
:
– Renewal process
N
t
alone is not a Markov process (unless the interarrival
times
T
i
be exponential) but the pair
(
N
t
,
A
t
)
constitutes a Markov process
≥
+
+

<
=
1
1
1
),
(
,
T
t
T
T
t
T
t
t
A
t
N
t
K
}
:
inf{
1
t
s
t
t
N
t
N
N
s
A
T
B
t
>
=

=
+
+
t
t
N
t
B
A
T
C
t
+
=
=
+
1
4
Chapter 6: Renewal Processes
6.1 Introduction
•
Strong Law of Large Numbers (SLLN) for renewal processes
:
– Let
N
t
be a renewal process associated with renewal sequence
T
i
with
mean
μ
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Mckinley,S
 Probability theory, lim, renewal processes

Click to edit the document details