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Columbia University
Exercise 14
. Letting the number of cars in the station be the state variable,
we have a birthdeath process with
λ
0
=
λ
1
=
λ
2
=20
,μ
1
=
μ
2
=12
λ
i
=0
,i>
2
.
Hence the limiting probabilities must satisfy, as the equations on p. 380371
12
P
1
=2
0
P
0
12
P
2
=2
0
P
1
12
P
3
=2
0
P
2
P
0
+
P
1
+
P
2
+
P
3
=1
which leads to
P
0
=
Ã
1+
5
3
+
∙
5
3
¸
2
+
∙
5
3
¸
3
!
−
1
=
27
272
.
(1) The fraction of the attendant’s time spent servicing cars is equal to the
fraction of time there are cars in the system and is therefore 1
−
P
0
= 245
/
272
.
(2) The fraction of potential customers that are lost is equal to the fraction
of customers that arrive when there are 3 cars in the station, and is therefore
P
3
=
μ
5
3
¶
2
P
0
=
125
272
.
Exercise 15
.L
e
t
X
(
t
) denote the number of customers in the service
center at time
t
,th
en
{
X
(
t
)
,t
≥
0
}
is a birth and death process with state
space
{
0
,
1
,
2
,
3
}
and rates
λ
0
=
λ
1
=
λ
2
=3
μ
1
=2
,μ
2
=
μ
3
=4
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 Spring '09

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