RAERA Problem

This chapter addresses the problem of auctioning edge resources to achieve both profit and social welfare for the resource seller(s) and the resource buyers. Furthermore, this chapter defines these terms as follows:

• (Edge) Resourcesare resources that could be used to support edge computing. These resources are on the Edge in various forms such as micro data centers, cloudlets, and fog nodes. In this chapter, resources are assumed to be atomicunits (I.e., the smallest unit of resource that could be made available) that could be, for example, a) virtualized resources such as VMs, dockers, and unikernels; b) actual resources

45 4.2 RAERAProblem such as CPU cores (or fractions thereof) and memory; or c) indirect resources such as execution time.

• Sellers are those entities who own edge resources at the edge of a network and are willing to lease them to interested buyers. For example, infrastructure providers such as edge/access/eye-ball ISPs and mobile network providers can be sellers.

• Buyers are those who would like to deploy their services on edge resources to en-hance the user experience. For example, content providers or service providers can be buyers.

Here, dynamic pricing leads to cheaper resources for the buyer and better utilization for the seller [63]. However, it is necessary to address the following challenges:

• Uncertainty about pricing: Due to a lack of ground truth, it is difficult to validate resource valuations. For instance, it is difficult not only to infer but also to validate CPU consumption of multiple VMs sharing the same physical core. Hence, pricing these resources is non-trivial, and sellers are uncertain about the resource prices they should call for. Shiet al.[4] emphasize the need for a new cost model, especially for edge resources, which not only guarantee profitability for sellers but also acceptability from buyers.

• Service guarantee: Currently, service providers cannot guarantee service availabil-ity even with fixed pricing [61]. For instance, allocated amazon spot instances are terminated as soon as the current called-for price exceeds the buyer’s bid. As a re-sult, a buyer has to bid higher to ensure instance availability for critical applications.

Otherwise, procure instances with fixed pricing scheme, which favors the seller of the resources.

• Strategic behavior: Buyers often also have strict requirements on the resources (e.g., if they need to meet SLOs), but most of the sellers are unable to provide service guar-antees within fixed pricing for premium customers [61]. As a result, buyers usually try to receive better service than required by inflating their requirements, incurring unnecessary costs at the sellers’ premises [61].

This chapter addresses these challenges by following a sealed bid auctioning approach.

Here, the buyers submit sealed bids for resources and receive resources only if they are win-ners in the auction. Specifically, auctions can counter the price uncertainty problem, provide service guarantees since items are entirely allocated to the winner [21], andincentive com-patibleauctions encourage truthful bidding. One popular example is theVickrey auction, where the bidder with the highest bid wins the auction but only pays the second highest bid instead of his quoted bid, which has proven to prevent strategic behavior of buyers [100].

In this subsection, we formally define our problem as follows: LetN={1,2, . . . ,n}be a set ofnbuyers indexed byi, i.e., irepresents i^th buyer. Similarly, letM={1,2, . . . ,m}

be the set of edge resources indexed by j, i.e., jrepresents the j^th resource. Each buyer ihas a private valuation for each resource. In our model, vi j represents the valuation of buyerifor the j^th resource. The valuation profile or vector of a buyeriis represented by vi = (v_i1,v_i2, . . . ,vim). Each buyerisubmits a bid represented bybi = (b_i1, . . . ,bim)for m resources. Letβ∈Rⁿbe the bid profile or vector of all the users, i.e.,β= (b1,b2, . . . ,bn).

Letb_−ibe the bid vector of all users excepti, i.e.,b_−i= (b₁,b₂, . . . ,b_i−1,b_i+1, . . . ,b_n). Also, β = (bi,b−i).

Let x(·):Rⁿ→N be the allocation function of edge resources to users based on their submitted bids.

Ifνis a bid profile of successful resource allocation, thenx(ν) ={x(ν)_{i j}},∀i∈N,j∈M wherex(ν)i j is the amount of resource ja bidderireceives when the profile is ν. In our model, x(ν)i j is discrete to represent an atomic resource, i.e., the allocationx(ν)i j is an integer since a fractional value would represent a fractional unit and thus does not have any practical significance.

Similarly, let p(·):Rⁿ →R be the payment function which maps a bid profile to an n-dimensional real value which is a monetary payment, i.e., p(ν) = (p₁(ν), . . . ,p_n(ν). In-formally, pi(ν)is the monetary payment of buyerito the seller.

Letui be the utility derived by thei^th buyer. Informally, a utility is the satisfaction ob-tained by consuming goods. We assume the utilityu_i is quasilinear. Quasilinearity implies the risk neutrality for buyers [62], i.e., the more money a buyer spends, the greater the utility. Formally,ui(vi,x(b),p(b)) =vixi(b)−pi(b).

LetU_j∈Rⁿbe the uncertainty set (i.e., data and constraints are allowed to change within the set without affecting optimality) for the j^th resource. Bid vectors for the j^th resource are drawn from this uncertainty set. The uncertainty set for an auction is constructed based on historical bid data. According to the central limit theorem, a distribution with a meanµ and varianceσ²will converge to standard normal distribution when the sample sizen→∞.

Based on [101], we define the uncertainty setUjfor the j^thresource as follows:

U_j=

For a standard normal distributionZ, the confidence interval is 95% and 99% for|Z| ≤2 and |Z| ≤3 respectively, i.e., P(|Z| ≤2)≈0.95 and P(|Z| ≤3)≈0.99. Hence, in Eq.

(4.2.1),Γis chosen between 2 and 3 to reflect the desired confidence interval. Also,(n−k) can be regarded as the window of recent historical values. Ifk=0, then all the historical values are considered. Furthermore, U denotes the uncertainty set formresources, i.e.,

47 4.3 RAERAAlgorithm U =U₁× · · · ×Um.

In our model, all bids and valuation vectors are drawn from the uncertainty setU, i.e.,

∀i∈N,bi,v_i∈U. LetW be thebreak-even optimal (optimal revenue in presence of to-tal uncertainty [41]) of the service provider. The goal of this chapter is to maximizeW satisfying following properties:

• Individual Rationality (IR): The users should not lose money when they bid truth-fully, i.e.,ui≥0.

• Incentive compatibility (IC): The user has to derive higher utility for truthful bid-ding. Let bb_i be a non-truthful bid profile and x(bb_i) and p(bb_i) corresponding allo-cation and payment. Then, this property implies that ui(vi,x(bi,b−i),p(bi,b−i))≥ ui(vi,x(bbi,b−i),p(bbi,b_−i)).

We formulate our goal as the following linear optimization problem.

maximize

Here, constraint (i) ensures a non-negative profit for the auctioneer, (ii) implies allocation of every resource, (iii) and (iv) ensure the IC and IR property, respectively; (v) implies the atomic resource allocation toi^th buyer. Furthermore, the notations are summarized in Table 4.1.