Optimizing Cloud Use Under Interval Uncertainty

One of the main advantages of cloud computing is that it helps the users to save money: instead of buying a lot of computers to cover all their computations, the user can rent the computation time on the cloud to cover the rare peak spikes of computer need. From this viewpoint, it is important to find the optimal division between in-house and in-the-cloud computations. In this paper, we solve this optimization problem, both in the idealized case when we know the complete information about the costs and the user’s need, and in a more realistic situation, when we only know interval bounds on the corresponding quantities. 1 Formulation of the Problem What is cloud computing. The main idea behind cloud computing (see, e.g., [8, 17, 22, 27]) is that instead of performing all the computations on his/her own computer, a user can sometimes rent computing time from a computer-timerental company. This, in effect, is what is known as cloud computing. Computations that use rented computer time are called computing in the cloud. Renting is usually more expensive than buying and maintaining one’s own computer, so if the user needs the same amount of computations day after day, cloud computing is not a good financial option. However, if a peak need for computing occurs rarely, it is often cheaper to rent the corresponding computation time than to buy a lot of computing power and idle it most of the time. How much computation time should we rent? Once the user knows his/her computational requirements, the proper question is: should we use the cloud at all? if yes, how much computing power should we buy for in-house computations and how much computation time should we rent from the cloud company? how much will it cost? Finally, if a cloud company offers a multi-year deal with fixed rates, should we take it or should we buy computation time on a year-by-year basis? Why this is important. Surprisingly, while the main purpose of cloud computing is to save user’s money, most cloud users are computer folks with little knowledge of economics. As a result, often, they make wrong financial decisions about the cloud use; see, e.g., [28]. It is important to come up with proper recommendations for using cloud computing. What we do in this paper. In this paper, we provide the desired financial recommendations, first under the idealized assumption that we have a complete information, and then, in a more realistic situation of interval uncertainty. 2 How Much Computations to Perform In-House and How Much in Cloud: Case of Complete Information Case of complete information: description. Let us first consider the idealized case when we have complete information about our needs and about all the costs. This means, first, that we know the cost of keeping a certain level of computational ability in-house. Let us pick some time quantum (e.g., day or hour). Then, the overall cost of buying and maintaining the corresponding computers is proportional to these computer’s computational ability – i.e., the number of computing operations (e.g., Teraflops) that these computers can perform in this time unit. Let c0 denote the cost per unit of computations. Then, if we buy computers with computational ability x0, we pay c0 · x0 for these computers. This also means that we know the cost of computing in the cloud. Let us denote this cost by c1. So, if one day, we need to perform x computations in the cloud, we have to pay the amount c1 · x. As we have mentioned, computing in the cloud is usually more expensive than computing in-house. Part of this extra cost is the cost of moving data, another part is the overhead to support the computing staff, marketing staff, etc. As a result, c1 > c0. Complete knowledge also means that we know the user’s needs. This means that for each possible computation need x, we know the probability that one of the days, we will need to perform exactly x computations. These probabilities can be estimated by analyzing the previous needs: if we needed x computations in 10% of the days, this means that the probability of needing x computations is exactly 10%. The probability distribution is usually described either by a cumulative distribution function (cdf) F (x) = Prob(X ≤ x), or by the probability density function (pdf) ρ(x) for which the probability to be within an interval [x, x] is equal to the integral ∫ x x ρ(x) dx, and the overall probability is 1: ∫ ρ(x) dx = 1. The relationship between pdf and cdf is straightforward: • F (x) is the integral of pdf: F (x) = ∫ x 0 ρ(t) dt; • vice versa, the pdf is the derivative of the cdf: ρ(x) = dF dx . What is the cost of buying x0 computational abilities and doing all other computations in the cloud? We want to select the amount x0 of computing power to buys, so that everything in excess of x0 will be sent to the cloud. We want to select this amount so that the expected overall cost of computations is the smallest possible. So, to find the corresponding value x0, let us compute how much it will cost the user to buy x0 equipment and to rent all other computation time. We already know that the cost of buying and maintaining an equipment with capacity x0 is equal to c0 · x0. The expected cost of using the cloud can be obtained by adding the costs multiplied by the corresponding probabilities. We need computations in the cloud when x > x0, For each such value x, we need to rent the amount x − x0 in the cloud. The cost of such renting is c1 ·(x−x0). The probability of needing exactly x computations is proportional to ρ(x). To be more precise, the probability that we need between x and x+∆x computations is equal to c1 ·(x−x0)·ρ(x)·∆x. The expected cost of using the cloud is therefore equal to the sum of such products, i.e., to the value ∑ c1 · (x− x0) · ρ(x) ·∆x. In the limit, when ∆x → 0, this sum tends to the integral ∫ x0 c1 · (x− x0) · ρ(x) dx. Thus, the overall cost is equal to the sum of the in-house and in-the-cloud costs: C(x0) = c0 · x0 + c1 · ∫ x0 (x− x0) · ρ(x) dx. (1) Let us use this cost expression to find the optimal value x0. We want to find the value x0 for which the cost expression (1) attains its smallest possible value. To find this minimizing value, we need to differentiate the expression (1) with respect to x0 and equate the corresponding derivative to 0. To make this differentiation easier, let us transform the expression (1) by using integration by parts ∫ u dv = u · v − ∫ v du. Here, ρ(x) = d(F (x)− 1) dx , so we can take u = x−x0 and v = F (x)−1. The product uv = (x−x0) · (F (x)−1) is equal to 0 on both endpoints x = x0 and x = ∞, so we get C(x0) = c0 · x0 − c1 · ∫