Performance Analysis for Large IaaS Clouds - Large Scale and Big Data: Processing and Management - page 571

and cold, respectively) can accept a job is computed by VM provisioning sub-
models. The RPDE submodel uses these probabilities as input parameters. Output
measures such as rejection probability due to buffer full ( P block ), rejection prob-
ability due to insufficient capacity ( P drop ), and their sum ( P reject ) are obtained from
the RPDE submodel. Observe that P block computed in the RPDE submodel is used
as an input parameter in VM provisioning submodels. Also, outputs from VM
provisioning submodels ( P h , P w , P c ) are needed as input parameters to solve the
RPDE submodel. Hence, there is a cyclic dependency among the submodels. Such
dependency is resolved using fixed-point iteration [14,17]. Proof of existence of a
solution can be shown for such fixed-point iteration [8]. Apart from existence, two
other important issues with fixed-point iteration are uniqueness of solution and
rate convergence. By trying different initial guesses, we have never found mul-
tiple solutions, that is, the final solution remains unique. Also, in all the scenarios
investigated, the maximum number of iterations required was 4, which indicates
reasonably fast convergence.
18.4 ANALYTIC-NUMERIC RESULTS
Using SHARPE [20] software package, we solve the interacting submodels to com-
pute: (1) job rejection probability and (2) mean response delay. As shown in Figure
18.8a, for a fixed arrival rate (1000 jobs/hour) and given number of PMs in each pool
(e.g., 80 PMs in each pool), job rejection probability increases with longer mean
service time. Also, at a given value of mean service time, if the PM capacity in each
pool is increased, job rejection probability reduces. Similar effects on mean response
delay is shown in Figure 18.8b. With increasing mean service time, mean response
delay increases for a fixed number of PMs in each pool. To explain these effects we
define a term called marginal gain. When all other input parameter values are kept
unchanged, marginal gain is the amount of reduction in job rejection probability or
mean response delay with increasing PM capacity. Notice that the marginal gains in
Figure 18.8b, change with increasing mean service time. Marginal gain increases for
gradual increase in PM capacity from 70 to 100 in each pool. In the example scenario
investigated, marginal gain is maximum when the mean service time is around 1000
minutes. We provide the following arguments to explain this behavior. For a given
number of PMs, mean response delay has three components: (i) mean queuing delay
in front of RPDE, (ii) mean decision delay, and (iii) conditional mean provisioning
delay. With increasing mean service time, marginal gain changes depending upon
a dominant component of the overall delay. For a low mean service time (100-300
minutes), gain due to addition of more PM is almost insignificant. This is because
jobs quickly leave the cloud, making room for new requests. As a result, small num-
ber of PMs is sufficient to keep the overall mean response delay low. When the mean
service time of jobs increases (say 1000 minutes), for low capacity systems (e.g., 70
PMs in each pool for our example), the mean queuing delay in front of RPDE starts
increasing. Hence, the benefit of adding more PMs is reflected by having a lower
mean queuing delay in front of RPDE. If the mean service time is further increased
to 1800 minutes, the mean queuing delay in front of RPDE increases even for larger
capacity systems (e.g., 100 PMs in each pool for our example). So, the marginal