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saboteurs in a colluding group, the colluded saboteurs access to a third party,
e.g. a colluding server, and obtain the value which should be set to the incorrect
result. Thus, saboteurs can communicate with each other indirectly (via the col
luding server) without knowing informations of other saboteurs. This method
has a possibility of allowing all saboteurs to collude together, and increasing
error rate of a computation dramatically.
2.3.3.
Defection Model
While sabotagetolerance is a mandatory issue on the reliability of computation, VC also
has another issue; that is, the stability of computation [35]. The basic model assumes that
workers execute allocated jobs and return their results without fails; however, workers in
real VC are never coerced into sustained execution of jobs since they participate in VC
systems and provide their resources as volunteers. Even if the allocated job is in execution
and not finished, workers can defect from the VC system for any time by terminating the
execution and discarding the job. During their execution, workers may have problem due to
hardware/software failures (e.g. thermorunaway) or network disconnection, each of which
results in defection from the VC system. In addition, after the problem is solved, those
workers may rejoin to the VC system. Therefore, for modeling more realistic VCs than the
basic model, it must consider those dynamic activities of workers,
Actually, workers in real VC frequently defect and rejoin to the system, as shown in the
real traces of VC projects [33, 34]. However, no adequate mathematicalmodel is provided
since the analysis of activities is an extremelycomplex problem. In this chapter, we pro
vide a simple mathematicalmodel, referred to as “random defection model”, for discussion
about the performance of VC systems with workers' defection. The details of this model
are described as follows.
•
Two states are defined for each worker, “up” and “down” states. The “up” state
represents the worker is active and is able to work for the VC project, whereas the
“down” represents the worker is inactive and is not able to work.

W
n
gets “up” state at
T
i
with probability
P
up
(T
i
)
.

W
n
gets “down” state at
T
i
with probability
P
down
(T
i
)
.
•
In each time step, the state of a worker
W
n
changes as shown in Figure 4.

If the state is “up”, it changes from “up” to 'down” with probability
utod
n
.

If the state is “down”, it changes from “down” to “up” with probability
dtou
n
.
P
up
(T
i
)
and
P
down
(T
i
)
are given by following recurrence formulas.
P
up
(T
i
)=(1− utod
n
) × P
up
(T
i−1
)+dtou
n
× P
down
(T
i−1
)
(1)
P
down
(T
i
)=(1− dtou
n
) × P
down
(T
i−1
)+utod
n
× P
up
(T
i−1
)
(2)
Because
W
n
gets either up or down state, the sum of
P
up
(T
i
)
and
P
down
(T
i
)
is
1
in every
time step. From this and the recurrence formulas, the general expressions of
P
up
(T
i
)
and