Cloud based large-scale online services are faced with regionally distributed stochastic demands for various resources. With multiple regional cloud data centers, a crucial problem that needs to be settled is how to properly place resources to satisfy massive stochastic demands from many different regions. For the general stochastic demands oriented cross region resource placement problem, the time complexity of existing optimal algorithm is linear to total amount of resources and thus may be inefficient when dealing with a large number of resources. To end this, we propose an efficient algorithm, named discrete function based unbound resource placement (D-URP). Experiments show that in scenarios with general settings, D-URP can averagely achieve at least 97% revenue of optimal solution, with reducing time by three orders of magnitude. Moreover, due to the generality of problem setting, it can be extended to get efficient solution for a broad range of similar problems under various scenarios with different constraints. Therefore, D-URP can be used as an effective supplement to existing algorithm under time-tense scheduling scenarios with large number of resources.

Large-scale online services (such as YouTube [

For instance, the cloud based content distribution network can be viewed as an example that falls under above paradigm, where a number of geographically dispersed cloud data centers are selected to be close to end users. Under this scenario, user’s requests are preferred to be served by a local data center, and service provider would like to predistribute content replicas at each data center so as to meet as much content demand as possible in the most efficient way. Another example is cloud based media streaming, in which service provider faces the problem of where to place virtual machine instances, so as to satisfy time-varying bandwidth demands from users spreading across various geographical regions.

In above paradigm, online services are faced with stochastic demands which are represented by demand distribution for each resource type in each region. Since we do not suppose stochastic demands to follow a specific distribution, a variety of scenarios can be covered:

Summarized from above scenarios, the general resource placement problem can be formulated as follows: given a global resource budget, how to satisfy as many requests as possible by placing resources optimally over regions. Since each request can be satisfied either by the data center in same region with high QoS (i.e., high revenue) or by the data center from a different region with low QoS (i.e., low revenue), algorithm’s aim is to maximize expected revenue of resources.

By considering stochasticity of demands, algorithm can be more efficient than prior mean demand based ones. Consider the scenario of placing

However, stochastic demands will significantly increase computation complexity. It is challenging to efficiently solve a resource allocation problem that combines combinatorial aspects and arbitrary stochastic demand distribution in time-tense scheduling scenarios. To the best of our knowledge, only one SSP (successive shortest path) based optimal algorithm was proposed for a similar stochastic demand based problem [

To reduce the computational complexity, one can efficiently optimize servicing revenue by utilizing the inner structure of the problem. We develop an efficient placement algorithm accordingly, suitable for time-tense scheduling in large-scale systems. Our paper mainly makes three contributions.

One part of related works was cloud based resource scheduling for multiple services [

There was also a substantial research effort being made about similar problems of resource consolidation [

Other similar works were cloud based content delivery [

Some related works in P2P (peer-to-peer) systems also investigated similar problems [

Most of existing works were carried out only based on mean demands of resource placement. However, a more realistic scenario is that resource demands from massive users are dynamic and stochastic. The main challenge posed by this paradigm is how to deal with an arbitrary multidimensional (high-dimensionality) stochastic demand. A method of resource placement for similar problem is proposed in [

We consider a system consisting of

Let

Denote

Generally, the problem formulated by (

distributions.

(1)

(2)

(3)

(4)

(5)

(6)

However, the traversing algorithm is of high complexity; for example, in a typical cloud based scenario, there is a very large data space to search. And in each iteration, the computation of

Firstly, the problem to be solved by lines (3)–(5) in Algorithm

And the problem to be solved by lines (2)–(6) in Algorithm

It can be easily observed that the solution of Subproblem 1 can contribute to solution of Subproblem 2, and Algorithm

To investigate the properties of solution to Subproblem 1, we introduce some notations. Denote by

For placement

Given type-placement

Therefore, we can get Subproblem 1.2.

For any continuous nonnegative random variable

The proof of lemmas, corollaries, and theorems in the paper can be found in Appendix. Then the closed form of

To investigate the properties of solutions to Subproblem 1.2, the following theorem is introduced first.

For

Note that in Theorem

Given type-placement

According to Corollary

Additionally, for a given

For a given global resource constraint

According to Theorem

Given

In Corollary

The above continuous functions can be approximated by discrete ones and the optimal placement for global resource constraint

In our implementation of D-URP, each of these discrete functions is recorded in hash map data structure, and its time complexity of querying is

demand distribution

and

region

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

In this section, we evaluate D-URP algorithm to study the following subjects:

Similar to the experiment settings of prior works [

First we compare the effect of proportional mean method, S-URP, and D-URP under different scenarios, where in method of proportional mean, the number of type-

From Figures

Expected revenue of mean, S-URP, and D-URP with different Zipf values under deficit scenarios (

Expected revenue of mean, S-URP, and D-URP with different Zipf values under balanced scenarios (

Expected revenue of mean, S-URP, and D-URP with different Zipf values under surplus scenarios (

It is observed from Figures

As shown in Figures

Expected revenue of mean, S-URP, and D-URP with different number of resource types under deficit scenarios (

Expected revenue of mean, S-URP, and D-URP with different number of resource types under balanced scenarios (

Expected revenue of mean, S-URP, and D-URP with different number of resource types under surplus scenarios (

It is shown from Figures

In brief, from Figures

Since the result of S-URP is optimal, we then investigated the effect of D-URP using the revenue ratio of D-URP to S-URP. In Figures

Revenue ratio of D-URP to S-URP with different Zipf values under deficit, balanced, and surplus scenarios.

Revenue ratio of D-URP to S-URP with different number of resource types under deficit, balanced, and surplus scenarios.

In Figure

Average revenue ratio of D-URP to S-URP with different D-URP cycle counts under scenarios with different expected number of requests.

In brief, the following is shown in above experiments.

D-URP is lightly affected by the popularity distribution of resources (Zipf) and number of resource types (

D-URP is affected by the scarcity of resources. It is observed that it is better in surplus scenario or with more total available resources. By inspecting the data, we find that since the optimal solution is rounded to integers, the revenue under smaller

In D-URP, near optimal result can be calculated with much lower time complexity. The average effect of D-URP can still achieve at least 97% of optimal solution in all considered scenarios. Note the time complexity of S-URP and D-URP is

In this paper, we propose a fast algorithm D-URP for resource placement in cloud platform considering stochastic demands with multidimensional distribution. Our analysis and the preliminary experiments indicate that D-URP can achieve at most 99% revenue of optimal solution, and compared with existing algorithms, its time complexity is reduced by nearly three orders of magnitude in our scenarios with general settings. Therefore, D-URP can be an effective supplement to existing algorithms under time-tense scheduling scenarios.

Our future interesting works may include more precise approximation of continuous functions using discrete ones and efficient algorithms to variant of resource placement problem with more constraints.

Note that for given constant

Denote random variable

According to [

Denote

We define

Therefore, for given type-placement

Firstly

Then

According to Corollary

Since

And

Therefore, for given

The authors declare that there is no conflict of interests regarding the publication of this paper. The authors have no financial and personal relationships with other people or organizations that can inappropriately influence our work; there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in or the review of this paper.

This paper is supported by the National Natural and Science Foundation of China (nos. 61003052 and 61103007), Natural Science Research of the Education Department of Henan Province (nos. 2010A520008, 13A413001, and 14A520018), Henan Provincial Key Scientific and Technological Plan (no. 102102210025), Program for New Century Excellent Talents of Ministry of Education of China (no. NCET-12-0692), Doctor Foundation of Henan University of Technology (nos. 2012BS011 and 2013BS003), and Plan of Nature Science Fundamental Research in Henan University of Technology.