ESTADO LÍMITE DE SERVICIO

Li(ǫ) = max x≥0 Fi x+Fi−1(ǫ) −Fi(x) ǫ ,

we show that RBA _{is asymptotically (1−1/e) competitive if Li(ǫ)ǫ→ asǫ→0. In particular, if}

ǫLi(ǫ) decreases to 0 as strongly asO(ǫη) for someη >0, then we have a polynomial convergence rate of ˜O(c−

η 1+η

i ). We show that this holds for many common IFR families (Chapter 2 of [BP96]). An approximation forL(ǫ)ǫthat eases calculations is given by{F−1(ǫ) maxx≥0f(x)}and F(x)→

O(1)xf(x) for smallx.

• For exponential, uniform and IFR families with non-increasing density: it is easy

to see that Li(ǫ) = 1 and thus,η = 1. This implies a ˜O(c−i 0.5) convergence.

• Weibull distributions: This family is characterized by two non-negative parameters λ, k

with c.d.f. given by, F(x) = 1₋e(−x/λ)k

for x _≥0. The family is IFR only for k _≥1. It is easy to see that,

L(ǫ)ǫ=O(1)ǫ1/k.

Hence, for any finiteλ, k we have ˜O(c−

1 1+k

i ) convergence to (1−1/e).

• Gamma distributions: This family is given by non-negative parametersk, θ such that the

c.d.f. isF(x) = _Γ(1_k)γ(k, x/θ). Here Γ, γare the upper and lower incomplete gamma functions resp.. The family is IFR only for k≥1. It can be shown that for small ǫ,L(ǫ)ǫ→ O(1)ǫ1/k. Thus, we have ˜O(c−

1 1+k

i ) convergence for any finite set of parameter values.

• Modified Extreme Value Distributions: This family is characterized by the density

function,f(x) = _λ1e−ex−λ1+xwhere parameterλ >0. For smallǫwe have thatL(ǫ)ǫ→O(1)ǫ, for any finite λ. Giving a a ˜O(c−_i 0.5) convergence.

• Truncated Normal: For truncated normal distribution with finite mean µ, finite variance

σ2_{, and support [0, b] where b could → ∞, it is easy to see that L(ǫ) = O(1), leading to a}

˜

O(c−_i 0.5) convergence. Note that if the support is given by [a, b] for some a > 0, then we do not have convergence asL(ǫ)ǫ_→O(1).

B

Sufficiency of Static Rewards

Lemma 22. Given an algorithm (online or clairvoyant) for allocation that does not know the realizations of usage durations, the total expected reward of the algorithm is the same if we replace dynamic usage duration dependent rewards functions ui(·) with their static (finite) expectations

ri =Edi∼fi[ui(di)], for every resource.

Proof. Consider arbitrary algorithm A as described in the lemma statement. Let B denote an algorithm that mimics the decisions of A but receives static rewards ri,∀i ∈ I instead. Now, supposeAsuccessfully allocates resourceito arrival ton some sample pathω(t) observed thus far. Then, conditioned on observing ω(t), the expected reward from this allocation is exactly ri. More generally, using the linearity of expectation it follows that the total expected reward of B is the same as that ofA.

Note that for algorithms that also know the realization of usage durations in advance, the decision of allocation can depend on usage durations and conditioned on observed sample path

ω(t), the expected reward from successful allocation of ito tneed not be ri.

C

LP matches Clairvoyant for Large Inventory

Consider the LP upper bound for assortments [DSSX18, BM19, FNS19], max X S,t X i∈S ri·φ(i, S)·yt(S) s.t. X S|i∈S τ X t=1 [1₋Fi(a(τ)−a(t))]·φ(i, S)·yt(S)≤ci ∀τ ∈ {1, . . . , T},∀i∈I X S yt(S)≤1 ∀t∈T

0_≤yt(S)≤1 ∀t∈T, S ∈ F (The collection of feasible assortments). (19) For an interpretation of the constraints see Appendix E. Clearly, OPT _≤ OPT_{(LP) and the}

allocations generated by any algorithm (offline or online) can be converted into a feasible solution for the LP, regardless ofci. Perhaps surprisingly, we show that for large ci, the solution to this LP can be turned into a randomized clairvoyant algorithm (that does not know the realizations of usage in advance) with nearly the same expected reward, implying that the LP gives a tight asymptotic bound and moreover, all asymptotic competitive ratios shown against the clairvoyant also hold against the LP. For the purpose of this proof we assume that we can offer an assortment with some resources that may not be available. Whenever a customer chooses an unavailable resource it is equivalent to the outside option and we gain no reward.

Theorem 23. Let cmin= mini∈Ici. Then,

OPT_(LP)1−O r logcmin cmin ≤OPT_≤OPT_(LP). Hence, for cmin →+∞, OPT→OPT(LP).

Proof. We focus on the lower bound and more strongly show that every feasible solution of the LP can be turned into a clairvoyant algorithm with nearly the same objective value. Let_{yt(S)}t∈T,S∈2I be a feasible solution for the LP. Consider the clairvoyant algorithm,

When tarrives, offer set Y(t) w.p. yt(Y(t))/(1 + 2δ).

Hereδ=

q

logcmin

cmin . Since the LP does not use usage durations, the above algorithm doesn’t either. The critical element to be argued is that the expected reward of this algorithm is roughly the same as the objective value for the feasible solution. For this we rely on appropriate usage of Chernoff bounds. Note that the marginal probability of offering any resource is the same as that given by the LP solution (within a factor of (1₋2δ)), except that in the algorithm if the customer chooses a resource that is unavailable we gain no reward. Therefore, it suffices to show that any assortment offered by the algorithm contains resources that are available w.h.p.. To this end, the probability

of the event that at arrival τ, a given resource i (with yτ(S) > 0 for some set S containing i) is available, is lower bounded by the following probability,

P X

t|1≤t≤τ;i∈St

1(dit> a(τ)−a(t))·1(i, St)·1(St)≤ci−1

.

The LHS in the inequality is a lower bound on the number of units of ithat are in use atτ. The mean of the LHS is upper bounded byci/(1 + 2δ). This follows from the fact thatyt(S) is a feasible solution to the LP and the three indicator random variables are independent. Now, by using the Chernoff bound we have,

P X t|1≤t≤τ;i∈St 1(dit> a(τ)−a(t))·1(i, St)·1(St)>(1 +δ)ci/(1 + 2δ) ≤e−δ 2 2+δci.

Therefore, given our choice ofδ and for largeci we have,Pt|1≤t≤τ;i∈St1(dit > a(τ)−a(t))·1(i, St)· 1(St)≤(1+δ)ci/(1+ 2δ)< ci−1, w.p. at least 1−1/ci. Thus, at any arrivalτ the algorithm makes expected reward at least (1₋O(δ))_·ri·P_S|i∈Syτ(S)·φ(i, S) from every resource. Therefore, the expected reward made by the algorithm at arrival τ is at least (1−O(δ)) times the contribution to the LP objective at τ.

D

Upper Bound for Deterministic Non-stationary Usage Dura-

tions

Lemma 24. Suppose the usage duration is allowed to depend on the arrivals such that a resourcei

matched to arrivaltis used for deterministic durationdit(revealed whentarrives). Then there is no

online algorithm with a constant competitive ratio bound when comparing against offline algorithms that know all arrivals and durations in advance.

Proof. Using Yao’s minimax, it suffices to show the bound for deterministic online algorithms over a distribution of arrival sequences. For simplicity, suppose we have a single unit of a single resource and a family of arrival sequences A(j) for j ∈[n] (the example can be naturally extended to the setting of large inventory). The arrivals in the sequences will be nested so that all arrivals inA(j) also appear in A(j+ 1). A(1) consists of a single arrival with usage duration of 1. Suppose this vertex arrives at time 0. A(2) additionally consists of two more arrivals, each with usage duration of 1/2−ǫ, arriving at timesǫand 1/2 respectively. More generally, sequenceA(j) is best described using a balanced binary tree where every node represents an arrival and the depth of the node determines the usage duration. Each child node has less than half the usage duration (d/2₋ǫ) of its parent (d). If the parent arrives at time t, one child arrives at timet+ǫ and the other at

t+d/2. The depth of the tree for sequence A(j) is j (where depth 1 means a single node). Note that the maximum number of arrivals that can be matched in A(j) is 2j−1_.

Let Z = Pn

j=12j. Now, consider a probability distribution over A(j), where probability pj of sequence A(j) occurring is 2n−j_Z+1. Clearly, an offline algorithm that knows the full sequence in advance can match 2j−1 _{arrivals on sequence A(j) and thus, has revenue n2}n_{/Z = n/2. It is} not hard to see that the best deterministic algorithm can do no better (in expectation over the random arrival sequences) than trying to match all arrivals with a certain time duration. Any such deterministic algorithm has revenue at mostZ/Z = 1. Therefore, we have a competitive ratio upper bound of n/2.