ISyE 3232
Stochastic Manufacturing and Service Systems
Fall 2011
H. Ayhan
Solutions to Homework 10
1. Let
S
1
and
S
2
be independent and exponentially distributed random variables with means
1
μ
1
= 3 and
1
μ
2
= 6. They corresponds to the service time of the first and the second teller.
(a) Since
A
starts service immediately after arrival,
P
(
T
A
≥
6) =
P
(
S
1
≥
6) =
e

1
2
×
6
=
0
.
0498.
(b) By the same reason,
E
(
T
A
) =
E
(
S
1
) = 3.
(c) This amount, in mathematical expression, will be
E
(
T
A

T
A
>
6)
=
E
(
S
1

S
1
>
6)
=
6 +
E
(
S
1
) // by memoryless property of exponential random variables
=
9
.
(d)
P
(
T
A
< T
B
) =
P
(
S
1
< S
2
) =
μ
1
μ
1
+
μ
2
= 2
/
3
.
(e) The time from noon till a customer leaves is min
{
T
A
, T
B
}
.
Its expectation will be the
same as
E
(min
{
S
1
, S
2
}
) =
1
μ
1
+
μ
2
= 2
.
(f) If a customer leaves, one of the tellers will be able to serve
C
. So it is the same as time
till one departure, which is computed in (e).
(g) The total time
C
spends in the system will be the summation of the waiting time
min
{
T
A
, T
B
}
and its service time. The service time will depend on at which teller the
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 Fall '07
 Billings
 Probability theory, Exponential distribution, TA

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