Discussion

The approach for statistical static resource management presented in this section is mainly based on three assumption. It will be discussed in the following, in which cases they meet and how violations of them will affect the results.

Statistical Independent Random Variables of the Stochastic Process Yi

The assumption of statistical independence of the random variables Y_i of Y_i(t) has already been discussed in Section 5.3.4. The same is assumed again two times within this section to trade off CPU time against performance. The cases in which this assumption is actually valid have been discussed before. Hence, it will be only pointed out in this section how missing independence will impact the results.

First, statistical independence is required to ensure not to exceed the allowed probability of performances losses specified in SLOs in a given time frame (cf. Section 5.4.2). The worst case occurs, when all random variables that describe the demand behavior within the time frame are completely positively correlated. If in this case performance losses occur, they will occur during the whole time frame because of the positive correlations. The SLO will be violated.

Correlations must be explicitly modeled to deal with them in this case. These models must be characterized using external knowledge. Simply observing resource demand will not provide

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5.4 Static Scheduling

any information about correlations because at each time tonly one sample can be observed.

This means that only one realization of each random variable can be observed, which is quite not enough to derive any information about correlations between them.

In addition, statistical independence of two successive random variables of the stochastic process Yi(t) is required to deal with interdependencies between resource demand and pro-vided resource capacity (cf. Section 5.4.3). It was suggested to use convolution to derive the probability distribution of the sum of the residual demandiand the initial demandYi. Incor-rect probability distributions are calculated, if this assumption is violated. The real demand is underestimated in case of positive correlations. Less resource capacity than actually required is provided, which can lead to SLO violations.

A second option to deal with correlations is to pessimistically overestimate the resulting probability distribution of two added random variables the way presented in [58]. This method helps to prevent any underestimates, when the demand behavior modelY_i⁺(t) is derived from the initial one Yi(t) as described in Section 5.4.3.

Interdependence between Performance Loss and User Interaction

It has been shown in Section 5.4.1 that the demand behavior of a VM is influenced by provided resource capacity. Mainly residual resource demand that could not yet be satisfied by the capacity at a certain time t remains and hence increases the demand at time t+ 1. This interdependence was addressed in Section 5.4.3.

The presented approach works only under the assumption that the initial resource demand behavior described byYi(t) does not depend on the provided resource capacity. This is actually not true in any case because resource shortages will slow down the response time of requests.

A sequence of requests and answers is typically sent between a client and the service. An increased response time of the service can delay the following requests of the client. This shifts the respective resource demand caused by the sequence. Such shifts can lead to reduced but also to increased resource demand in the future depending on the overall workload behavior.

Detailed information about the service deployed in the VM are required to analyzing this kind of user influence. This interdependence is not pursued any deeper within this thesis, since the scope of the thesis is limited to a black box view of the service only. Future work must address this issue.

Using Negative Correlations Requires Periodic Demand Behavior

The idea of trace based resource management has been picked up in previous section to take advantage of negatively correlated resource demand of the VMs. Resource savings are achieved by putting VMs together at the same server that show complementary resource demand be-havior caused by negative structural or temporal correlations.

5 Statistical Static Resource Management

Known approaches, such as presented in [43, 95], try to derive correlations from the demand behavior of the VMs observed in the past. It has been simply implied that negatively correlated resource demand observed in the past leads to negatively correlated resource demand in the future as well. Real negative correlations (temporal or structural) are assumed behind the observed ones. The concept presented in Section 5.4.5 is based on the same assumption as well. Hence, in the following it will be discussed under which condition this assumption is actually valid.

It has been distinguished between structural and temporal correlation earlier in this thesis.

Structural correlations are caused by dependencies between VMs. Temporal correlations are caused by time depended workload variations.

Structural correlations mainly depend on the kind of requests processed by the VMs. Some requests involve different servers others do not. Conventionally, this behavior does not change over time. Requests that require the work of two different VMs will require the work of both VMs in the future as well. Hence, possible structural correlations observed will be still valid in the future, if during the characterization phase all possible requests and sequences of requests ever expected in the future have been observed.

The situation is more difficult for temporal correlations since they depend on time. It has been assumed that all kinds of requests that could possibly occur in the future have already been observed in the past to deal with structural correlations. This is definitely not given for the time and temporal correlations. One point in time observed in the past will never occur in the future again. The same is true for intervals of time. Hence, one can not derive correlations in the future from the ones observed in the past without any restriction.

An example can be simply constructed. Two VMs show strongly daytime dependent work-load behavior. Over night both have nothing to do. At about noon both have their peak resource demand. One of the VMs shows this behavior every day in a week. The demand behavior of the second one differs during the weekend. After both VMs have been observed for a few days, one would say that their workload is strongly positively correlated. But the demand behavior turns out not to be that positively correlated, when the weekend is reached.

This fact has been completely neglected in [95]. The authors of [43] instead suggest to perform trace based resource management only on VMs that show periodic resource demand behavior. A period of the resource demand time series repeats over and over again in this case (e.g. the resource demand during a week is repeating every week). Hence, different instances of the period observed in the past can be used to predict the demand behavior expected in the future. The same is true for possible correlations of the demand behavior of different VMs.

All VMs without any periodic behavior need to be treated pessimistically by the resource management by reserving A^max_i all the time. It is unknown when in the future they will demand their maximal required resources.

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