(j) Using the notation in parts (a)(c), and the result of part (c), we have
E
[
T

D
] =
t
+
E
[
T
−
t

D
∩
A
]
P
(
A

D
) +
E
[
T
−
t

D
∩
B
]
P
(
B

D
)
=
t
+ 1
·
1
1 +
e
−
2
t
+
1
3
1
−
1
1 +
e
−
2
t
=
t
+
1
3
+
2
3
·
1
1 +
e
−
2
t
.
Solution to Problem 6.15.
(a) The total arrival process corresponds to the merging
of two independent Poisson processes, and is therefore Poisson with rate
λ
=
λ
A
+
λ
B
=
7. Thus, the number
N
of jobs that arrive in a given threeminute interval is a Poisson
random variable, with
E
[
N
] = 3
λ
= 21, var(
N
) = 21, and PMF
p
N
(
n
) =
(21)
n
e
−
21
n
!
,
n
= 0
,
1
,
2
, . . .
.
(b) Each of these 10 jobs has probability
λ
A
/
(
λ
A
+
λ
B
) = 3
/
7 of being of type A, inde
pendently of the others. Thus, the binomial PMF applies and the desired probability
is equal to
10
3
3
7
3
4
7
7
.
(c) Each future arrival is of type A with probability
λ
A
/
(
λ
A
+
λ
B
) = 3
/
7, independently
of other arrivals.
Thus, the number
K
of arrivals until the first type A arrival is
geometric with parameter 3
/
7. The number of type B arrivals before the first type A
arrival is equal to
K
−
1, and its PMF is similar to a geometric, except that it is shifted
by one unit to the left. In particular,
p
K
(
k
) =
3
7
4
7
k
,
k
= 0
,
1
,
2
, . . .
.
(d) The fact that at time 0 there were two type A jobs in the system simply states that
there were exactly two type A arrivals between time
−
1 and time 0. Let
X
and
Y
be
the arrival times of these two jobs.
Consider splitting the interval [
−
1
,
0] into many
time slots of length
δ
.
Since each time instant is equally likely to contain an arrival
and since the arrival times are independent, it follows that
X
and
Y
are independent
uniform random variables. We are interested in the PDF of
Z
= max
{
X, Y
}
. We first
find the CDF of
Z
. We have, for
z
∈
[
−
1
,
0],
P
(
Z
≤
z
) =
P
(
X
≤
z
and
Y
≤
z
) = (1 +
z
)
2
.
By differentiating, we obtain
f
Z
(
z
) = 2(1 +
z
)
,
−
1
≤
z
≤
0
.
(e) Let
T
be the arrival time of this type B job.
We can express
T
in the form
T
=
−
K
+
X
, where
K
is a nonnegative integer and
X
lies in [0,1]. We claim that
X
75