ORIE3510
Introduction to Engineering Stochastic Processes
Spring 2010
Homework 7: CTMCs and the Poisson Process
Not to be handed in.
1. (a) Let
S
n
be the time that the
n
th rider departs, then
S
n
is the sum of
n
independent
exponentials with rate
λ
and thus it has a gamma distribution with parameters
n
and
λ
. This is given by
f
S
n
(
t
) =
λe

λt
(
λt
)
n

1
(
n

1)!
for
t
≥
0.
(b) Recall that if
{
N
(
t
)
,t
≥
0
}
is a Poisson process with rate
λ
and
points
{
S
i
,i
≥
0
}
, then
{
(
S
1
,S
2
,...,S
n
)

N
(
t
) =
n
)
}
is equal in distribution to
{
U
(1)
,U
(1)
,...,U
(
n
)
}
. Therefore, each of these riders will still be walking at time
t
with
probability p =
R
t
0
e

μ
(
t

s
)
ds
t
=. Hence the probability that none of the riders are walking
at time
t
is (1

p
)
n

1
.
2.
X
n
= Number of customers in the system immediately before the nth arrival
Y
n
= Number of customers that remain the system when the
n
th customer departs
If
X
n
= i then
X
n
+1
∈ {
i
+ 1

j
;
j
≥
0
}
(j is the number of departures between the arrivals)
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 Spring '10
 LEWIS
 Exponential Function, Exponential distribution, Poisson process, yn, Exponentials

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