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to obtain the average waiting time for a dynamic multicast channel:
C
A
+
T
S
C
S
E
C
(
N
D
,
u
)
W
C
(
δ
)
=
(19.10)
N
D
(1
−
ρ
)
2
where
C
A
=
1 is the coefficient of variation for Poisson process,
2
)
2
(
T
R
−
2
δ
2
T
R
−
1
3
C
S
=
=
(19.11)
12
2
δ
is the coefficient of variation for uniformly-distributed service time, and
T
S
is the average
service time, given by
T
R
−
2
δ
T
S
=
(19.12)
2
Additionally,
u
=
λ
S
T
S
is the traffic intensity,
ρ
=
u
/
N
D
is the server utilization, and
E
C
(
N
D
,
u
) is the Erlang-C function:
u
N
D
/
N
D
!
E
C
(
N
D
,
u
)
=
(19.13)
u
k
k
!
N
D
−
k
=
0
1
u
N
D
/
N
D
!
+
(1
−
ρ
)
Since the traffic intensity depends on the average waiting time, and the traffic intensity is
needed to compute the average waiting time, equation (19.10) is in fact recursively defined. Due
to equation (19.13), equation (19.10) does not appear to be analytically solvable. Therefore,
we apply numerical methods to solve for
W
C
(
δ
) in computing the numerical results presented
in Section 19.4.
Now that we have obtained the waiting time for a
START
request, we can proceed to compute
the average waiting time for dynamically-admitted client requests. Specifically, we assume the
waiting time for a
START
request is exponentially distributed with mean
W
C
(
). We classify
client requests into two types. AType-1 request is the first request that arrives at the beginning of
the admission cycle. Type-2 requests are the other requests that arrive after a Type-1 request.
For example, request 1 in Figure 19.8 is a Type-1 request, and request 2 and 3 are Type-2
requests.
We first derive the average waiting time for Type-2 requests. Let
W
2
(
δ
) be the average
waiting time for Type-2 requests which can be shown to be (please refer to the Appendix):
δ
)
1
1
(
T
R
−
+
(
T
R
−
2
δ
)
/
2
W
C
(
δ
)
2
δ
)
e
−
(
T
R
−
2
δ
)
W
2
(
δ
)
=
W
C
(
δ
−
(19.14)
W
C
(
δ
)
e
−
(
T
R
−
2
δ
)
W
C
(
δ
)
−
1
W
C
(
δ
)
Next for Type-1 requests, the average waiting time, denoted by
W
1
(
δ
), is simply equal to
W
C
(
δ
). Therefore, the overall average waiting time, denoted by
W
D
(
δ
), can be computed from
a weighted average of Type-1 and Type-2 requests:
W
1
(
δ
)
+
M
2
(
δ
)
W
2
(
δ
)
W
D
(
δ
)
=
(19.15)
1
+
M
2
(
δ
)
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