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the basic syntaxes. In the prefix (α, r), r means the
apparent rate of action α. For details of the operation,
one can refer to Hillston (2005) and Bradley et al.
(2008).
Based on the new definition of PEPA, fluid-
flow approximation can be used to convert PEPA
into ODEs. According to Castiglione et al. (2014),
the nature of fluid-flow approximation is that the
large number of discrete states is considered to be of
continuous change. Then some ODEs can be built to
describe the trend of state changes.
Given n classes of components, the ith class of
components is denoted as C
i
. Each class of compo-
nents has a series of derivations, and the jth deriva-
tion of C
i
is called C
ij
. Let N(C
ij
, t) denote the number
of components at time t. Exit(C
ij
) and Enter(C
ij
) rep-
resent the sets of exit and entry activities of a local
derivation, respectively. In a short time Δt, the
changes of an arbitrary derivation C
ij
can be de-
scribed as follows:
Exit( ) Enter( )
Exit activities Enter activities
(, ) (,)
(,()) (,()),
ij ij
ij ij
ij ik
CC
NC t t NC t
CPtt CPtt
(4)
where the component rate ρ
α
(C
ij
, P(t)) captures the
local effect of P(t) on component C
ij
, and k≠j. Then a
real-valued variable v
ij
(t) is used to approximate the
discrete variable N(C
ij
, t), denoted as v
ij
(t)=E[N(C
ij
,
t)]. Furthermore, according to Hayden et al. (2012),
the error between v
ij
(t) and N(C
ij
, t) will tend to 0
while
,
(,) .
ij
ij
NC t
Let Δt be close to 0. We obtain
Exit( ) Enter( )
d()
( , ( )) ( , ( )),
d
1,2, ,| |, 1,2, , . (5)
ij ij
ij
ij ik
CC
i
vt
CPt CPt
t
jCin
For details, one can refer to Tribastone et al. (2012a).
Therefore, the complexity of solving the model is
related only to the number of the types of compo-
nents, rather than the population of all types of
components.
2.2 MCC model based on PEPA
Currently, there are many types of MCC appli-
cations in our daily lives, and the most famous ex-
amples include ‘searching the lost child’ (Satyana-
rayanan, 2011), ‘the translation in museum’ (Huerta-
Canepa and Lee, 2010), ‘disaster relief’ (Fernando et
al., 2013), and ‘the traffic congestion map’ (Rishabh
et al., 2013). These scenes basically have the same
mode, and in this study, the traffic congestion map is
chosen as an example to analyze the service availa-
bility of MCC.
Example 1 (Traffic congestion map) While a
driver is on his/her way to the airport, he/she sends a
request to a cloud system by phone to get a traffic
congestion map and avoid encountering a traffic jam.
Then the cloud gathers the urban traffic status in real
time through a lot of taxis distributed in the city, and
forms a near real-time traffic congestion map after
processing the data collected. Then the map is re-
turned to the driver for choosing the best road.
To establish a model of universal significance,
it is necessary to ignore the details not directly asso-
ciated with the service availability of MCC. Taking
the traffic congestion map as an example, it is easy
to find that an MCC system has the following typical
characteristics:
1. An MCC system can be divided into two
parts, the cloud and a set of mobile devices. The
former is either a traditional cloud computing system
or a set of computing nodes closely related.
2. While the cloud receives a service request
from a user, every mobile device is awakened to col-
lect information.
3. The data collected is submitted to the cloud
through a wireless link and computed by means of
classic cloud computing.
4. The final results will be returned to the users.
As stated above, computation- or data-intensive
tasks in an MCC system are submitted to the cloud
instead of being handled in local devices. This is
usually called ‘offloading computing’ (Ou et al.,
2007). Different from traditional interactive compu-
ting, after submitting a request, a user will not inter-
act with the cloud again until a final result is returned.
So, we primarily describe the MCC system from the
cloud and the mobile device, ignoring user behaviors.
2.2.1 Modeling the cloud
Without loss of generality, we assume that the
cloud part is based on a map-reduce structure whose
main steps include requesting & waiting, splitting,
mapping, shuffling, and reducing (Fig. 1).