Risk Aware Stochastic Placement of Cloud Services: The Multiple Data Center Case

Abstract

Allocating the right amount of resources to each service in any of the data centers in a cloud environment is a very difficult task. In a previous work we considered the case where only two data centers are available and proposed a stochastic based placement algorithm to find a solution that minimizes the expected total cost of ownership. This approximation algorithm seems to work well for a very large family of overflow cost functions, which contains three functions that describe the most common practical situations. In this paper we generalize this work for arbitrary number of data centers and develop a generalized mechanism to assign services to data centers based on the available resources in each data center and the distribution of the demand for each service. We further show, using simulations based on synthetic data that the scheme performs very well on different service workloads.

This paper was supported in part by the Neptune Consortium, Israel and by the Israeli Ministry of Science, Technology and Space.

A Simulation Results

In this section we present our simulation results for k bins. In all simulations we used 4 bins with bin capacity ratio of $c_{i+1} = 2c_i$. For each cost function, we compared the generalized moving sticks (GMVS) algorithm with two algorithms we call BS (Balanced Spares) and BL (Balanced Load), as in [1]. The BS algorithm goes through the list, item by item, and allocates each item to the bin which has more available space. In this way, the spare capacity is balanced. On the other hand, the BL algorithm goes through the list, item by item, and allocates each item to the bin which is less loaded, i.e., the bin with higher $\frac{\text {available space}}{\text {bin capacity}}$ value. In this way, the bin load is balanced. The BL and BS algorithms are natural benchmarks and also much better than other naive solutions like first-fit and first-fit decreasing.

We show two different results for the GMVS algorithm: Fractional GMVS and Integral GMVS. The fractional GMVS algorithm uses the fractional partition output of the generalized moving sticks algorithm, as described in Sect. 8. The integral GMVS algorithm takes this fractional partition and converts it to an integral partition by moving each stick left to next integral point.³

In the case of SP-MWOP cost function, we also added results for the Fractional MVS and Integral MVS algorithms. As before, the fractional MVS algorithm uses the fractional partition output of the moving sticks algorithm, as described in Sect. 7, and the integral MVS algorithm takes this fractional partition and converts it to an integral partition by moving each stick left to next integral point.

We implemented one straight forward improvement for the runtime of both MVS and GMVS algorithms: before examining a fractional movement of an item, we examine the movement of the whole item at once. In this way, if the entire item can be moved from one bin to the other at a given stage, we move it in one step, instead of fractionally moving it in several consecutive steps. This improvement does not change the results of the algorithms, but it improves the running time. Note that we didn’t check the extent of this improvement, since we only implemented this version of the algorithms.

Fig. 1.

Average cost of the Fractional and Integral GMVS and MVS algorithms, and the BS and BL algorithms for SP-MWOP, SP-MED and SP-MOP with four bins and synthetic normally distributed data. $n = 400$. $\tau = 0.001$. The x axis measures $\frac{c}{\mu }$.

Fig. 2.

The BL and BS algorithm costs divided by the dynamic algorithm cost for SP-MWOP, SP-MED and SP-MOP with four bins and synthetic normally distributed data. $n = 400$. $\tau = 0.001$. The x axis measures $\frac{c}{\mu }$.

Fig. 3.

Average main loop number of iterations (t) divided by number of services (n) in the GMVS and MVS algorithms for the various cost functions with four bins and synthetic normally distributed data. $n = 400$. $\tau = 0.001$. The x axis measures $\frac{c}{\mu }$.

1.1 A.1 Results for Synthetic Normally Distributed Data

The synthetic data for this part was generated exactly as explained in [1] for the 2 bin case, except that we generated only 400 elements instead of 500 elements because executing the dynamic algorithm (whose complexity is $O(n^3)$) takes too long on $n=500$.

Figure 1 shows that for SP-MED and SP-MOP cost functions the Integral GMVS is very close to the Fractional GMVS, as expected. However, for SP-MWOP the integral GMVS and integral MVS are a bit higher that their fractional version. We believe that this gap will be reduced as the number of services is increased (i.e. as n is increased). We also see that for all three cost functions the dynamic algorithm cost is higher that the fractional version of the GMVS and MVS algorithm costs, yet very close to them. Also, the BL and BS algorithm results for the three cost functions are much higher than those of the dynamic algorithm. The integral GMVS, integral MVS, BL and BS algorithm costs divided by the dynamic algorithm cost for the three cost functions is shown in Fig. 2.

To get a feeling of the main loop runtime complexity in both GMVS and MVS algorithms, we calculated the number of main loop iterations (t) in each algorithm run and divided it by the number of services (n). The average results are shown in Fig. 3. We can see that in all cases the number of iterations dose not surpass 0.6n. We also see that the number of iterations increases as the spare capacity increases. When the spare capacity increases, the bins are less full. Therefore, the final bin of each service is more likely to be different than (and more distant from) it’s initial bin, and hence more service movements are required, i.e., more main loop iterations are required. In any case, the process always terminates quickly.