Information Technology Reference
In-Depth Information
1.5.1 Modeling Parent Behavior
A Continuous-Time Markov Chain (CTMC) is used to model the behavior of
parents that are currently forwarding packets to a child peer [58,67]. More
specifically, consider an n -source P2P system where each peer always tries
to maintain connections with n parents and concurrently receives pack-
ets from these parents. Even though a child peer tries to maintain con-
nections with n parents, due to parent departures the number of active
parents of the child peer may not always be n . Every time a parent departs,
the child peer invokes a parent recovery process to replace the departed.
Note that the remaining parents might depart from the system during the
parent recovery process; in that case, the child receiver will invoke another
parent recovery process to find a new parent. Assume that multiple parent
recovery processes can be executed simultaneously (e.g., with a concurrent
technique). Let S i be the state of the CTMC representing that i parents of
a child peer that has left the system, namely, i parent recovery processes
are running. Suppose the potential time that a child peer needs to locate
a new parent and the potential time that a parent (peer) stays in the sys-
tem are exponentially distributed with means 1⁄μ, 1⁄λ, respectively. The
whole system can thus be seen as an M/M/c queuing system with arrival
rate λ and departure rate μ. By the memoryless property of the exponential
distribution, we have the following transition diagram of the system (see
Figure 1.33).
Based on the principle that the rate at which the process enters state S i equals
the rate at which it leaves state S i [59], we have the following equation
state
Rateleave =rateenter
0
1
nP
λµ
=
P
0
1
(
µ
( − = +
+− =− +
+
n
1
))
λ
PnPP
λ µ
2
1
0
2
.
(1.1)
2
(
2
µ
(
n
2
))
λ
P
(
nP µ
1
)
λ
3
P
2
1
3
i
,
0
<<
i
ni
(
µ
+− =−+
(
ni Pni
) )
λ
(
1
)
λ
Pi
+ +
(
1
)
µ
P
i
i
1
i
+
1
n
n PP
µλ
=
n
n
1
n λ
( n - 1)λ
( n - 2)λ
( n - 3)λ
λ
0
. . .
1
2
3
n
µ
n µ
Figure 1.33
The transition diagram of the system.
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