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On the other hand, hybrid multicast streaming architectures present unique challenges to
media server design due to the use of both periodic and aperiodic media retrievals. In the
next chapter we address the issues in designing efficient media streaming server that supports
hybrid multicast streaming algorithms.
Appendix
In this Appendix, we derive the mean waiting time for Type-2 users, denoted by
W
2
(
). The
complication is due to length biasing as a Type-2 user is more likely to observe a longer Type-1
wait than a shorter Type-1 wait. First, we compute the waiting time distribution for Type-1
users, denoted by
f
C
(
t
),
as observed by a Type-2 user
using the results from Kleinrock [1]:
δ
tf
C
(
t
)
W
C
(
f
C
(
t
)
=
(19.20)
δ
)
where
f
C
(
t
), and
W
C
(
E
[
f
C
(
t
)] is the actual waiting time distribution and mean waiting
time of Type-1 users respectively. Let
W
C
(
δ
)
=
) be the mean of
f
C
(
t
):
δ
∞
W
C
(
tf
C
(
t
)
dt
δ
)
=
(19.21)
−∞
Substituting equation (19.20) into equation (19.21) we then have:
∞
t
2
f
C
(
t
)
W
C
(
W
C
(
δ
)
=
)
dt
(19.22)
δ
−∞
We note that the waiting time can only range from zero to (
T
R
−
2
δ
), so we can rewrite
equation (19.22) as:
T
R
−
2
δ
t
2
f
C
(
t
)
W
C
(
W
C
(
δ
)
=
)
dt
(19.23)
δ
0
Motivated by simulation results, we assume that
f
C
(
t
) is truncated exponentially distributed:
(1
)
−
1
e
−
(
T
R
−
2
δ
)
t
W
C
(
−
=
−
δ
f
C
(
t
)
)
W
C
(
e
(19.24)
W
C
(
δ
)
δ
)
Substituting equation (19.24) into equation (19.23) we have
T
R
−
2
δ
−
t
W
C
(
t
2
e
δ
)
W
C
(
δ
)
=
dt
(19.25)
e
−
(
T
R
−
2
δ
)
0
(1
−
)
W
C
(
δ
)
2
W
C
(
δ
)
Solving the integral and after a series of simplifications equation (19.25) becomes
)
1
1
(
T
R
−
+
(
T
R
−
2
δ
)
/
2
W
C
(
δ
)
2
δ
)
e
−
(
T
R
−
2
δ
)
W
C
(
δ
)
=
2
W
C
(
δ
−
(19.26)
W
C
(
δ
)
e
−
(
T
R
−
2
δ
)
W
C
(
δ
)
1
−
W
C
(
δ
)
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