Information Technology Reference
In-Depth Information
k
−
1
to state
k
rate is
λ
P
The figure shows: the shift from state
, from state
−
1
k
to state
k
−
1
kP
μ
, when the system is in stable state, each state
of pass in and out of subscribers is equal, so we can get the differential equations are
as follows:
transfer rate is
kk
λμ
λμ
PP
kPk P Pkn
PnP
=
⎧
⎪
0
1
1
(
)
(
)
+
=
+
1
μ
+
λ
<
⎨
⎪
(5)
k
k
k
++
11
k
k
−
1
λ
=
μ
k n
=
⎩
k
−
1
n n
It is available from the above recursive equation:
⎧
2
k
λ
P
λ
P
λ
P
P
=
=
=
...
=
k
−
1
k
−
2
0
⎪
k
(
)
k
μ
k k
−
1
μ μ
k
!
μ μ
...
μ
⎪
⎨
k
k
k
−
1
k
k
−
1
1
(6)
−
1
⎛
k
⎞
⎡
k
⎤
n
n
λ
n
λ
⎪
⎩
∑
∑
∑
1
1
1
PP
=
+
=
⇒
P
=+
⎜
⎟
⎢
⎥
⎪
k
0
0
k
!
μμ
...
μ
k
!
μμ
...
μ
⎝
⎠
⎣
⎦
k
=
0
k
=
1
k
=
1
kk
−
1
1
kk
−
1
1
So get probability formula when system states is k
k
λ
P
λ
k
P
=
0
=
k
k
!
μμ
...
μ
⎛
n
λ
k
⎞
(7)
∑
kk
−
1
1
k
!
μμ
...
μ
1
+
⎜
⎟
kk
−
1
1
k
!
μμ
...
μ
⎝
⎠
k
=
1
kk
−
1
1
In theory, if the number of users k is not more than system limit. It has a little in-
fluence on the Internet time, the average service rate of reception desk is unchanged,
and
μμ
====
...
μμ
.
In practical application, along with the increase of the number of users, the entire
system service efficiency will gradually decrease, the user average Internet time has
been gradually made longer, make each reception average service rate
k
k
1
1
μ
less than
k
theoretical value
, When k increases to a certain extent, the system service effi-
ciency will be dropped dramatically, this will produce greatly influences on user's
psychological.
According to because the user is sensitive to the network speed, so they would
have written off the net, leaving the service system. The number of user enters and
leave the system began to tend to a new equilibrium, the number of system users will
no longer be increased, thus constraints:
μ
n
μλ
≥
.
n
μ
μ
=
ω
Calculation can be assumed
, in which
is the user-sensitive factor,
k
ω
and
1
<<
1.1
, then:
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