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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
/β
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
/β
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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