Performance Analysis for Large IaaS Clouds - Large Scale and Big Data: Processing and Management - page 559

duration between the submission of a request to cloud and the VM made available to
the user is called response delay. We use job rejection probability and response delay
as the two performance metrics for our analysis.

18.3 INTERACTING SUBMODELS FOR PERFORMANCE ANALYSIS

18.3.1 CtmC s ubmoDel For rPDe
Figure 18.2 shows the CTMC model for RPDE. The input parameters for this
submodel are: (i) job arrival rate (λ), (ii) mean searching delays to find a PM in a
hot/warm/cold pool that can be used for resource provisioning (1/δ h , 1/δ w , and
1/δ c , respectively), (iii) probabilities that a hot/warm/cold PM can accept a job for
resource provisioning ( P h , P w , and P c , respectively), and (iv) maximum number of
jobs in RPDE ( N ). All the input parameters can be measured directly except the
probabilities P h , P w , and P c . These probabilities are computed from the outputs of the
VM provisioning submodel as described later.
We briefly describe the CTMC submodel here. Detailed description of the sub-
model can be found in [8]. The state index of the CTMC in Figure 18.2 is denoted
by ( j , x ), where j denotes the number of jobs in the queue and x denotes the type of
pool where the job is undergoing provisioning decision. In state (0, 0), there is no
job in the system. When RPDE is trying to find a PM from the hot pool, x is set to h .
Similarly, x is set to “ w ” (or “ c ”), when RPDE is deciding if any warm (or cold) PM
can accept the job. With the arrival of a job, model moves from state (0, 0) to state
(0, h ) with rate λ. Three possible events can occur in state (0, h ): (a) a job is accepted
in hot pool with probability P h , (b) with probability (1 − P h ), the submodel goes to
state (0, w ), (c) the submodel goes to state (1, h ) with the arrival of a new job. The rest
of the submodel can be followed in the similar manner.
It is possible to derive a closed form solution for the state probabilities on the
model shown in Figure 18.2. Assume that π ( j , x ) denotes steady-state probability of
state ( j , x ), where 0 ≤ j ≤ ( N −1) and x ∈ {0, h , w , c }.

λ λ λ λ

…

N - 1,
h

0, 0

δ h P h

0, h

δ h P h

1, h

δ h P h δ h P h

δ h (1 - P h ) δ h (1 - P h )

δ w P w

δ h (1 - P h )

δ w P w

δ c P c

δ w P w

δ c (1 - P c )

δ w P w

…

λ λ λ

N - 1,
w

0, w

1, w

δ c P c

δ c P c

δ w (1 - P w )

δ w (1 - P w ) δ w (1 - P w )

δ c (1 - P c )

δ c P c

δ c (1 - P c )

δ c (1 - P c )

…

N - 1,
c

0, c 1, c

λ λ λ

FIGURE 18.2 Resource provisioning decision engine submodel.