Database Reference
In-Depth Information
λ h
Node A
Node B
FIGURE 18.4 Hot PM modeled as a two-stage tandem network of queues. The queuing
system consists of two nodes: (i) node A is a M/M/1 queue, with service rate β h and (ii) node
B is a M/M/∞ queue, with service rate of each server being μ.
state-dependent multiplier to the VM provisioning rate β h , our submodel can easily
be extended for parallel deployments of multiple VMs.
It is possible to derive closed-form solutions of the state probabilities when L h → ∞,
and m → ∞. In such a case, it can be shown that the CTMC can be represented as a
two-stage tandem network of queues as shown in Figure 18.4. Observe that node A is
an M/M/1 queue with server utilization ρ A = λ h h , while node B is an M/M/∞ queue
with server utilization ρ B = λ h /μ. For node A , steady-state probability mass function
(pmf) of i jobs ( i > 0) waiting in the queue and j VMs ( j ∈ {0,1}) being provisioned
is given by
ij
+
pij
A
(, )(
=−
1
ρρ
)
where,
ρ
<
1
(18.23)
A
A
A
For node B , steady-state pmf of k VMs ( k > 0) running, is given by
k
ρ
ρ
pk
()
=
B
e
(18.24)
B
B
k
!
Thus, for the hot PM CTMC shown in Figure 18.3, when L h → ∞ and m → ∞, the
steady-state probability of being in state ( i, j, k ) is given by
k
ρ
()
h
=− +
ij B
ρ
φ
(
1
ρ ρ
)
e
(18.25)
B
(, ,)
ijk
AA
k
!
where i ≥ 0, j ∈ {0 ,1}, k ≥ 0, ρ A = λ h h , and ρ B = λ h /μ. The condition for stability of
the system is ρ A < 1.
18.3.3.2 Hot PM Submodel Outputs
The steady-state probability ( B h ) that a hot PM cannot accept a job for VM provision-
ing can be obtained after solving the hot PM submodel:
φ
m
1
()
h
()
h
B h
=
+
φ
(18.26)
(, ,)
Li
1
(, ,)
Lm
0
h
h
i
=
0
 
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