Performance Analysis for Large IaaS Clouds - Large Scale and Big Data: Processing and Management - page 564

18.3.3 CtmC For vm P rovisioning
We construct one CTMC submodel for a hot, warm, and cold PM respectively.
Overall VM provisioning submodel of a pool is modeled by a set of independent
submodels that represent each PM in the pool. These submodels keep track of num-
ber of VMs running on a PM.

18.3.3.1 Hot PM CTMC
Figure 18.3 shows the VM provisioning submodel for a hot PM. State index of this
model is denoted by ( i, j, k ), where, i denotes the number of jobs in queue, j denotes
number of provisioning VMs, and k denotes number of deployed VMs. The model
input parameters are: (i) effective job arrival rate (λ h ), (ii) VM provisioning rate (β h ),
(iii) service rate (μ), (iv) size of buffer ( L h ), and (v) maximum number of VMs that
can run in parallel ( m ). The value of j is 0 or 1, since we assume that the VMs are
provisioned one at a time. With n h PMs in the hot pool, λ h is given by

λ ( 1 − P

n

)

block

λ = (18.22)

h

h

Notice that P block in Equation 18.22 is computed from RPDE submodel. All other
input parameters can be measured. As shown in Figure 18.3, after a job arrival sub-
model moves from state (0,0,0) to state (0,1,0), with rate λ h , mMean VM provisioning
time on a hot PM is 1/β h . Submodel moves from state (0,1,0) to state (0,0,1) with rate
β h . The mean service completion time is 1/μ. When the service finishes, VM instance
is removed and the submodel moves from state (0,0,1) to state (0,0,0) with rate μ. The
rest of the submodel can be described in a similar manner. Notice that by using a

λ h λ h λ h

…

0,0,0 0,1,0 L h ,1,0

µ

µ

β h µ

β h

µ

λ h λ h

λ h

…

0,0,1 ( L h − 1),1,1 L h ,1,1

β h β h

2µ

2µ

2µ

( m - 1)µ

β h

β h β h

( m - 1)µ ( m - 1)µ ( m - 1)µ

…

λ h

λ h

0,0,( m − 1) 0,1,( m − 1)

( L h − 1),1,( m − 1) L h ,1,( m − 1)

λ h

m µ

β h

β h

β h β h

λ h

m µ

m µ

…

0,0, m 1,0, m

L h ,0, m

λ h λ h

λ h

FIGURE 18.3 VM provisioning submodel for each hot PM. λ h is an effective job arrival rate
to each hot PM, β h is rate of VM provisioning on hot PM, μ is job service rate, L h is buffer size,
m is the maximum number of VMs on each PM.