Performance Analysis for Large IaaS Clouds - Large Scale and Big Data: Processing and Management - page 573

We compare the scalability and accuracy of our approach w.r.t. a one-level mono-
lithic model [8]. Such a monolithic model is constructed using a variant of stochastic
Petri net called stochastic reward net (SRN). Stochastic Petri Net Package (SPNP)
[11] is used to solve the monolithic SRN model. In Table 18.1, we compare the solu-
tion times for both the monolithic model and interacting submodels. When the num-
ber of PMs in each pool increases beyond 3 and the number of VMs per PM increases
beyond 38, monolithic model runs into a memory overflow problem. Even for a small
number of PMs and VMs, state space size of the monolithic model increases quickly
and becomes too large to construct the reachability graph. Also, for a given number
of PMs and VMs, the nonzero elements in the infinitesimal generator matrix of the
underlying CTMC of monolithic model, are hundreds to thousands of orders of mag-
nitude larger compared with interacting submodels [8]. In the interacting submodels,
a reduced number of states and nonzero entries leads to concomitant reduction in
solution time needed. As shown in Table 18.1, solution time for monolithic model
increases almost exponentially with the increase in model size, while the solution
time for interacting submodels remains almost constant with the increase in model
size. This analysis shows that the proposed approach is scalable and tractable com-
pared with the one-level monolithic model.
Next, we compare the accuracy of interacting submodels approach w.r.t. mono-
lithic modeling approach. Specifically, we compare the values of two performance
measures: (i) job rejection probability ( P reject ) and (ii) mean number of jobs in RPDE
( E [ N RPDE ]). In Figure 18.9a and b, we show that as we change the arrival rate and
maximum number of VMs per PM, outputs obtained from both the modeling
approaches are near similar. Hence, the errors introduced by the decomposition of

TABLE 18.1
Comparison of Model Solution Times (in seconds)
(#PMs per pool, #VMs per PM)

Monolithic Model Interacting Submodels

(1, 1) 0.124 0.066 (n.i. = 4)

(1, 2) 0.196 0.074 (n.i. = 4)

(1, 4) 0.556 0.088 (n.i. = 3)

(1, 8) 2.998 0.141 (n.i. = 3)

(1, 16) 79.563 0.208 (n.i. = 3)

(1, 32) 188.174 0.346 (n.i. = 3)

(1, 38) 293.672 0.399 (n.i. = 3)

(1, 39) m.o. 0.406 (n.i. = 3)

(2, 1) 2.024 0.063 (n.i. = 3)

(3, 1) 166.704 0.062 (n.i. = 3)

(4, 1) m.o. 0.060 (n.i. = 2)

(500, 64) m.o. 0.055 (n.i. = 1)

Note: m.o., memory overflow; n.i., number of iterations.