UNIFORM LARGE DEVIATIONS FOR HEAVY-TAILED QUEUES UNDER HEAVY TRAFFIC

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1 Bol. Soc. Mat. Mexicana (3 Vol. 19, 213 UNIFORM LARGE DEVIATIONS FOR HEAVY-TAILED QUEUES UNDER HEAVY TRAFFIC JOSE BLANCHET AND HENRY LAM ABSTRACT. We provide a complete large and moderate deviations asymptotic for the steady-state waiting time of a class of subexponential M/G/1 queues under heavy traffic. The asymptotic is uniform over the positive axis, and reduces to heavy-traffic asymptotics and heavy-tail asymptotics on two ends, both of which are known to be valid over restricted asymptotic regimes. The link between these two well-known asymptotics is a transition term that is expressible as a convolution-type integral. The class of service times that we consider includes regularly varying and Weibull tails in particular. It is our pleasure to contribute to this special issue dedicated to the International Year of Statistics. In response to the request of the editors of this special issue we briefly overview the research topics that we have investigated recently. Our research group has pursued several themes in recent years. All of them lie under the scope of applied probability. Some of our projects deal with computational probability. In this context, our goal is to enable efficient computation in stochastic systems using (and often developing theory of probability to inform the design of algorithm that are optimal and robust in certain sense (see Blanchet and Glynn (28. Most of the computations that we study relate to stochastic simulation (also known as Monte Carlo methods (see Blanchet and Lam (212. Other projects that we pursue relate to classical analysis in probability, such as asymptotic approximations, large deviations, and heavy-traffic limits (Blanchet and Glynn (26 and Lam et al (211. All of our research efforts are motivated by models and problems in areas such as: Finance, Insurance, Operations Research, and Statistics. Here we shall study a class of asymptotic results that lie at the intersection of large deviations and heavy-traffic limit theory. We use a classical model in queueing theory to illustrate these types of results, namely, the classical M/G/1 queue. Despite its apparent tractability, most of the asymptotics for the steadystate waiting time of the M/G/1 queue that have been proposed in the literature are only provably valid in restricted regimes. Among them are the well-known heavy-traffic or Kingman asymptotic (see Kingman (1961 and the heavy-tail or Pakes-Veraberbeke asymptotic (see for example Embrechts and Veraverbeke (1982. More precisely, in heavy traffic (i.e. when the long-run proportion of time the server is utilized, ρ, is close to 1 one approximates the distribution of the steady-state waiting time in spatial scales of size 1/(1 ρ by the steady-state distribution of reflected Brownian motion (which is exponential. On the other hand, the heavy-tail asymptotic assumes fixed traffic intensity while the tail parameter 21 Mathematics Subject Classification: 6F1. Keywords and phrases: uniform large deviations, queueing systems, heavy-tails, heavy-traffic. 183

2 184 JOSE BLANCHET AND HENRY LAM increases. It states that for service time with so-called stationary excess distribution (in the tradition of renewal theory B (x lying in the class, S, of subexponential distribution (see, for example, Embrechts et. al. (23 and Asmussen (21, 23, the probability that the steady-state waiting time is larger than x is asymptotically (ρ/(1 ρ B (x. In this paper we provide a uniform large deviations asymptotic of the steadystate waiting time distribution for heavy-tailed M/G/1 under heavy traffic. Our results can handle the case when heavy traffic is present but the tail parameter level is moderate, which is covered by neither heavy-traffic or heavy-tail asymptotics. Our results in this paper extend and unify previous work by Olvera-Cravioto et. al. (211 and Olvera-Cravioto and Glynn (211. In these two papers, the authors studied first the regularly varying M/G/1 queue and showed that heavy-traffic and heavy-tail asymptotics remain valid on regimes that are respectively smaller and larger than an explicitly identified transition point. Then, a separate argument is given in Olvera-Cravioto and Glynn (211 in order to deal with Weibull type distributions. Our framework here provides means to develop a unified theory of transitions from heavy-traffic to heavy-tailed asymptotics that covers both regularly varying and Weibullian tails at once. In addition, and in contrast to Olvera-Cravioto et. al. (211, for regularly varying distributions we provide an explicit asymptote for the behavior of the tail of the steady-state waiting time in the M/G/1 queue right at the transition point in complete generality. We shall use the machinery developed by Rozovskii (1989, 1993 for large and moderate deviations of random walks. Related papers that develop similar methods include Nagaev (1969 and Borokov and Borovkov (21. A central argument to obtaining these deviations results is finding a suitable truncation level depending on p. In Section 2 we outline this truncation argument and we provide the details of the proofs in Section Statement of Result and Outline of Argument Let (X i : i 1 be a sequence of non-negative i.i.d. r.v. s (independent and identically distributed random variables and define S n X X n, with S. Let M be a geometrically distributed random variable with parameter p > and independent of the X i s. In particular, P (M k pq k for k,1,..., where q 1 p. The random variable S M is said to be a geometric sum. It turns out (see Asmussen (23, that the steady-state waiting time of an M/G/1 queue can be represented as geometric sum, is p 1 ρ >. Throughout the rest of the paper we use F(x to denote the distribution of X i and we write X to denote a generic copy of X i. We are interested in P(S M > x as p and x x(p. We use the following notation. Given non-negative functions f 1 (, f 2 (, we write f 1 (x f 2 (x if f 1 (x o(f 2 (x as x (i.e. f 1 has smaller order than f 2 and f 1 (x f 2 (x if f 2 (x o(f 1 (x (i.e. f 1 has larger order than f 2. Also we use " and " to denote having order smaller than or equal to and having order larger than or equal to", respectively. For example, f 1 (x f 2 (x means that f 1 (x c f 2 (x for some c >. Lastly, we use " to denote asymptotically equivalent or the same order" (i.e. f 1 (x/f 2 (x 1.

3 UNIFORM LARGE DEVIATIONS FOR We shall consider X i in class S of subexponential distribution (i.e. P(X 1 + X 2 > x/ F(x 2 together with the assumption F(x : P(X i > x e g(x, where g( is clearly non-decreasing, and g(x as x. We also assume that g( is differentiable, and that g(x/x δ and is eventually decreasing for some < δ < 1. We further assume that EX µ <, EX 2 1 and EX 2+ɛ < for some ɛ > (we sometimes drop the subscript i in X i for convenience. We assume that (2 + ɛlog x g(x x. In addition, we assume that h(x g(x + µ 2log x is eventually non-decreasing and goes to, which is intuitive given that EX 2+ɛ <. We also assume that h (x/h(x (δ+η/x for some η > eventually. Finally, we also assume that X i is strongly non-lattice, in the sense that inf 1 χ(ω > ω >υ for any υ >, where χ(ω Ee iωx is the characteristic function of X. Set 1/p and define for x > and p (,1 [ (1.1 Γ(x, p e θ x + ( 1 p F(x + ( 1 p + x y µ ] e θ (x y df(y I(x, where θ is the solution to the equation E[e θx ; X ] 1/q. Our main result is the following: THEOREM (1.2. Let 1/p. We have uniformly over x > that lim sup P (S M > x p 1 Γ(x, p. x> Note that we can as well choose to be any quantity having the same order as 1/p. Blanchet and Glynn (27 shows that θ admits a Taylor series type expansion θ p/µ + c 2 p 2 +. The expansion is valid up to the order of the moment of X i. Thus if EX 2 < we can ensure that θ can be expanded up to the second order of p. Note that this gives e θ x e px/µ for x 1/p 2, which coincides with Kingman s asymptotic. On the other hand, the second term in (1.1 is the heavy-tail asymptotic. It can be shown that the first term is dominant for small order of x (with respect to p while the second term is dominant for large order. The third term can be the dominant component in a neighborhood of the transition between the first and the second. These observations are apparent through the following example. EXAMPLE (1.3 (Regularly Varying Tail. Suppose X i has density L(x/x 1+α, x > where α > 2 and L( is a slowly varying function, so F(x L(x/x α. We are interested in computing the third term of (1.1. First we have 1 p e θ (x y L(y y 1+α d y 1 p e θ x L(x x α 1 /x e θ xu 1 u 1+α L(ux L(x du 1 p e θ x L(x x α 1 /x e θ xu du. u1+α

4 186 JOSE BLANCHET AND HENRY LAM where the first equality follows by a substitution y xu, and the equivalence relation follows from the property of slowly varying function that L(ux/L(x 1 uniformly over compact set as x. If θ x O(1, which implies x C 1 /θ C 1 µ/p for some constant C 1 > (see the proof of Lemma (2.4 for the equivalence θ p/µ, then 1 /x for some C 2 > and so e θ xu 1 p e θ x L(x x α du eθ u1+α 1 /x x 1 e θ xu /x u 1+α du C 2 p 1 u 1+α du C 2e θ x L(x x α x e θ. On the other hand, if θ x, then applying Laplace s method yields and 1 p e θ x L(x x α 1 /x e θ xu u 1+α du 1 p L(x x α 1 θ x 1 p F(x. Now by the same analysis, and noting that θ p/µ, we obtain that for θ x O(1, and x µ x µ 1 µ e θ (x y L(y e θ (x y L(y e θ (x y L(y y 1+α d y 1 µ y 1+α d y C L(x 3 y α d y C L(x 4 e θ xα 1 e θ xα 1 e θ (x y L(y y α for θ x, which implies that x y µ e θ (x y L(y ( 1 y 1+α d y o p Hence we have L(x x α [ P(S M > x e θ x + 1 ] p F(x (1 + o(1 x x d y 1 L(x p x α for any x >. This recovers a basic result in Olvera-Cravioto et. al. (211 and identifies the transition point located at ((α 2/2 log(p/p. We now give a brief outline of our argument leading to Theorem (1.2. Detailed proofs will be provided in the next section. Our method is mainly inspired by Rozovskii (1989, 1993 together with the use of uniform renewal theorem in Blanchet and Glynn (27. We first find an appropriate truncation for X i, so that the geometric sum of the truncated part can be approximated by uniform renewal theorem while the remaining part follows the big-jump asymptotic. Uniform renewal theory then yields Kingman s asymptotic. On the other hand, the heavy-tail component will boil down to calculating a convolution of negative binomial sum with the increment distribution. From now on we will adopt the following notations. Recall that 1/p, and δ satisfies g(x/x δ. This allows us to find δ δ + η < 1 for some η >. For.

5 UNIFORM LARGE DEVIATIONS FOR convenience of development, when g(x log x, we take δ and δ be a small number such that < δ < 1. We then let K p (1/p 2δ e (1 δ g(, and { Bp for M K p C p,m µ + M for M > K p Let us state the result on the split into truncated and remaining part: PROPOSITION (1.4. [ ( P(S M > x P S M > x, max i C p,m 1in ( { } n (1.5 + P S M > x, X i > C p,m,max X j C p,m ](1 + o(1 j i uniformly over x >. i1 Note that we have used a truncation level that remains at for small M but grows in order M for large M. Such level will ensure that the contribution of two or more jumps i.e. X i > C p,m, is negligible for both small and large M. Moreover, as we shall see in Proposition (1.6 below, K p is chosen such that the truncated part is regular enough to invoke uniform renewal theorem. Our argument is finished by recognizing the two components in the right hand side of (1.5 as the terms in (1.1, via the following propositions: PROPOSITION (1.6. ( ( P S M > x, max X i C p,m e θ x + o e θ x + 1 1in p F(xI(x uniformly over x > z(p for any z(p such that z(p as p. PROPOSITION (1.7. ( { } n P S M > x, X i > C p,m,max X j C p,m i1 j i {[ 1 F(x p + ( ] x 1 B p p + x y e θ (x y df(y (1 + o(1 uniformly over x B µ p e θ x uniformly over x < 2. Proofs Note that Theorem (1.2 is uniform over x. The results that follow are obtained uniformly over x z (p as long as z (p as p. Of course, for x z (p < we have that Γ(x, p e θ x and therefore the limit lim sup P (ps M > px p 1 Γ(x, p lim sup P (ps M > u p exp( (θ /pu 1, xpz(pp uz(pp is easily established if z (p p o(1. Consequently, to obtain Theorem (1.2 it suffices to indeed assume x z (p as indicated.

6 188 JOSE BLANCHET AND HENRY LAM Proof of Proposition (1.4. First we write (2.1 P(S M > x P(S M > x, max X i, M K p + P(S M > x, max X i µ + M, M > K p + P(S M > x, max X i >, M K p + P(S M > x, max X i > µ + M, M > K p Note that the first two terms constitute the first term in the right hand side of (1.5, and we shall focus on the last two terms. For the third term, we have (2.2 P(S M > x, max X i >, M K p P(S M > x, exactly one X i >, M K p + P(S M > x, more than one X i >, M K p. We will show that the second term in (2.2 is negligible compared to the first term in (2.1, by following the proof in Lemma 4 of Rozovskii (1993. Using the notation there, we denote Q n k,k (x P(S n > x, X 1, X 2,..., X k >, X k+1,..., X n, A sup y 2Bp I Bp (y/ F(y where y Bp I Bp (y F(y udf(u, and ξ Bp sup y Bp F(y/ F(y+. Now Lemma 4a in Rozovskii (1993, applying to our case, states that for k 2, Q n k,k (x AQ n k+1,k 1 (x + F( Q n k+1,k 1 (x (from equation (64 there, and Q n k+1,k 1 (x ξ Bp Q n k+1,k 1 (x (from equation (65 there. Recognizing that X i is always non-negative, we have (2.3 Q n k,k (x (F( 1 Q n k+1,k (x (F( 1 (AQ n k+1,k 1 (x + F( Q n k+1,k 1 (x (F( 1 (A + F( ξ Bp Q n k+1,k 1 (x H p Q n k+1,k 1 (x where H p (F( 1 (A + F( ξ Bp. Lemma 4c in Rozovskii (1993 yields that (note the slight difference in the definition of g(x between there and here. We denote F(x e g(x while Rozovskii defines F(x e g(x /x 2 for the case of finite variance. Thus the g(x s differs by a term of 2log x. We also note that in Rozovskii (1993 B n n in the case of finite variance. A O ( 1 B 2 p max g(yy 2(1 δ e (1 δg(y y ( exp{δg(bp }, and ξ Bp O. B 2δ p

7 UNIFORM LARGE DEVIATIONS FOR Now let g(x g(x 2log x. Then we have A p 2 (1 δ g(y max( g(y + 2log ye y p 2 (1 δ g(y max C g(ye for y some constantc, by our assumption that g(x (2 + ɛlog x C p 2 g( e (1 δ g( C p 2δ (g( 2log e (1 δg( when p is small enough, and ξ Bp O(p 2δ e δg(. Hence we have for some C >. So ( K p pq n n n Q n k,k (x k n2 K p n2 K p n2 K p n2 K p n2 and hence pq n pq n n k2 n k2 n k2 K p H p C(F( 1 (g( + 1e ηg( n k H k 1 p P(S n > x, X 1 >, max X i (by iterating (2.3 2in K k 1 p H k 1 p P(S n > x, X 1 >, max X i 2in pq n n K ph p P(S n > x, X 1 >, max 1 K p H X i (for p small enough p 2in pq n np(s n > x, X 1 >, max 2in X i P(S M > x, more than one X i >, M K p P(S M > x, max X i, M K p uniformly over x >. We are left to prove that the fourth term in (2.1 is equivalent in order to P(S M > x, exactly one X i > µ + M, M > K p Using directly the result in Lemma 4 of Rozovskii (1993, and denoting X i X i µ and S n n i1 X i as the centered random variables, we get P(S M > x, max X i > µ + M, M > K p pq n P(S n > x, max X i > µ + n pq n P( S n > x nµ, max X i > n pq n P( S n > x nµ, exactly 1 X i > n(1 + g(x, n where the last equality follows from (61 in Rozovskii (1993 and g(x, n uniformly in x as n.

8 19 JOSE BLANCHET AND HENRY LAM Since K p as p, for any ɛ, we have pq n P( S n µ > x nµ, exactly 1 X i > ng(x, n ɛ when p becomes small enough. Therefore P(S M > x, max X i > µ + M, M > K p pq n P( S n µ > x nµ, exactly 1 X i > n (1 + ɛp(s M > x,exactly one X i > µ + M, M > K p for p small enough. Since ɛ is arbitrary, we conclude that P(S M > x, max X i > µ + M, M > K p uniformly over x >. The proof of Proposition (1.4 is complete. Proof of Proposition (1.6. Note that P(S M > x, max X i C p,m P(S M > x,exactly one X i > µ + M, M > K p P(S M > x, max X i, M K p + P(S M > x, max X i µ + M, M > K p P(S M > x, max X i P(S M > x, max X i, M > K p + P(S M > x, max X i µ + M, M > K p The proof of the proposition will be finished by the following two lemmas: LEMMA (2.4. P(S M > x, max X i e θ x (1 + o(1 uniformly over x > z(p for any z(p such that z(p as p. LEMMA (2.5. P(S M > x, max X i, M > K p P(S M > x,max X i µ + M, M > K p e θ x + 1 p F(x uniformly over x > z(p for any z(p such that z(p as p. Proof of Lemma (2.4. Let X be the truncated random variable at level i.e. P( X B P(X B X for any measurable set B. We denote P θ ( as the exponential change of measure P θ ( E[e θ X ψ(θ ; ] where ψ( log φ( is the logarithmic moment generating function of X and φ(θ Ee θ X is the moment generating function of X. Also denote E θ [ ] as the expectation under P θ ( and µ θ E θ X.

9 UNIFORM LARGE DEVIATIONS FOR Consider the change of measure of X i with θ satisfying 1 φ(θ qf( which gives (2.6 E[e θx ; X ] 1 q This equation is similar to (33 in Blanchet and Glynn (27, but with a different truncation level. Theirs is a truncation level x while here we use regardless of x. We want to characterize the solution of (2.6, which, as we will see, will give the θ in Theorem (1.2. Suppose θ C p for some C > and p small enough. Write Bp E[e θx ; X ] e θ y df(y Bp (1 + θ y + θ2 y 2 2 eνθ y df(y for some ν ν(θ y 1. The equation is valid by our moment assumptions on X. We then get (2.7 E[e θx ; X ] F( + θm( + R(θ where as p and F( 1 F( 1 e g( m( Bp ydf(y µ(1 + o(1 µ Bp θ 2 y 2 R(θ 2 eνθ y df(y θ2 2 ec D( θ2 2 ec where D( y 2 df(y. Equating (2.7 with 1/q 1 + p + p 2 +, and noting that e g(bp p 2, we get θ p/µ, which also verifies that there is a unique θ that indeed lies in [,C p]. Henceforth we will identify this as our θ. Next we have, letting T be exponentially distributed with rate θ in the fourth equality below, P(S M > x,max X i pq n P(S n > x,max X i n1 p(qf( n P( S n > x pe θ [e θ S n ; S n > x] n1 n1 [ ] p E θ [x < S n < T] pe θ I(x < S n < T n1 n1 p x pe θ x θ e θ y (Ū θ (y Ū θ (xd y θ e θ y (Ū θ (y + x Ū θ (xd y,

10 192 JOSE BLANCHET AND HENRY LAM where Ū θ ( n1 P θ ( S n is the renewal measure of X i under the measure P θ (. We shall now find Ū θ (. More specifically, we will show that uniform renewal theorem, as depicted in Theorem 1, part 2 of Blanchet and Glynn (27, is valid for X i under the exponential family P θ ( for all small enough p. Denote χ θ (ω E θ e iω X qf( Ee (iω+θ X as the characteristic function of X under P θ (. The theorem requires that such family is uniformly strongly non-lattice i.e. inf inf 1 χ θ (ω > pκ ω >υ for small enough κ > and any υ >, and that sup pκ E θ X 2+ɛ <. If these conditions hold then we have (as a weaker conclusion than the stated theorem in Blanchet and Glynn (27 (2.8 sup µ 4 θ Ū θ (t t pκ µ θ E θ X 2 2 µ 2 θ as t. We now check the above conditions. First note that χ θ (ω qf( Ee i(ω+θ X qe[e (iω+θ X ; X ] o(1 q(e[e iωx ; X ] + θ r(θ,ω q(χ(ω E[e iωx ; X > ] + θ r(θ,ω where r(θ,ω E[X e νθ X ; X ] for some ν ν(θ X 1 a.s. second equality is valid by our moment assumptions on X. Note that and the E[e iωx ; X > ] F(, and E[X e (νθ X ; X ] e C µ, for some C >. Since X is non-lattice, given any υ >, we have 1 χ(ω > 1 ζ for some < ζ ζ(υ < 1 for all ω > υ. So for ω > υ, we have ] 1 χ θ (ω 1 (1 p(χ(ω E [e iωx ; X > + θ r(θ,ω 1 χ(ω E[e iωx ; X > ] θ r(θ,ω p( χ(ω + E[e iωx ; X > ] + θ r(θ,ω > 1 ζ for some < ζ < 1 and small enough p. This shows that X under the exponential family P θ ( is uniformly strongly non-lattice. Moreover, we have E θ X 2+ɛ qe[x 2+ɛ e θ X ; X ] e C EX 2+ɛ for some C >. Together with our moment assumption on X, this shows that sup pκ E θ X 2+ɛ <. Hence we can invoke the uniform renewal theorem in Blanchet and Glynn (27. Since P θ ( X > s qe[e θ X ; s < X ] qe C F(s for s < and is otherwise, we note that H1 F(t, HF 2 (t and HF 1 HF 1 (t in Theorem 1, part 2 of Blanchet and Glynn (27 all go to uniformly in our exponential family as t. Note also that µ θ qe[x e θ X ; X ]. Since X e θ X I(X X e C which is integrable, by dominated convergence theorem and that θ p/µ we have E[X e θ X ; X ] µ. Hence µ θ µ + o(1. This concludes, from (2.8, that Ū θ (y + x Ū θ (x y + R(y, x,θ µ θ

11 UNIFORM LARGE DEVIATIONS FOR where sup pκ,y> R(y, x,θ as x. So pe θ x θ e θ y (Ū θ (y + x Ū θ (xd y ( pe θ x θ e θ y y µ θ + R(y, x,θ d y pe θ x µ θ ( 1 θ + o(1 e θ x uniformly over x > z(p for any z(p such that z(p as p. Lemma (2.4 is proved. Proof of Lemma (2.5. The first inequality holds obviously when p is small enough. Thus we will focus on the order relation. As in the proof of Proposition (1.5, we let X i X i µ and S n n i1 X i be the centered random variables and their sum. Let P( X > x e g(x+µ and recall that h(x g(x +µ 2log x satisfies h(x/x δ and is eventually decreasing, and (2.9 h (x h(x δ x for large enough x. Note also that ɛlog x h(x x by construction. By (7 and (71 in Rozovskii (1993 (note that the g(x defined there differs from ours by a term of 2log x and hence h(x here will play the role of g(x in Rozovskii s paper we have (2.1 P( S n > x nµ,max X i n e β(x nµh( n/ n for x nµ + Λ n, where Λ n αh( n n and β < α/2, for large enough n and some constant α >. This will be important for our development. We now define the following functions that will prove useful for our argument. Let l(n nµ+λ n. We extend the domain of the function l to the positive real axis and define l 1 (y inf{x : l(x y}, so l 1 (x x/µ (α/µh( l 1 (x l 1 (x. Also let r(x h(x/x, so 1/x r(x 1/x 1 δ as x. Define r 1 (y inf{x : r(x y}. We then have (2.11 1/y 1/(1 δ r 1 (y 1/y as y. We shall also prove a monotone property concerning the function (whose use will become clear in the argument that follows f (n : β(x nµ h( n + nlog q n We have ( h f ( n (n β(x nµ β 2 2n 1 2 n 3/2 ( x ( n µ h ( n h( n n h( n + βµ h( n + log q n + βµ h( n + log q n By (2.9, we have f (n βµh( n/ n + log q, or f (n is increasing, when n (r 1 ( (log q/(βµ 2.

12 194 JOSE BLANCHET AND HENRY LAM Let R p r 1 ( (log q/(βµ. We write (2.12 P(S M > x, max X i µ + M, M > K p pq n P( S n > x nµ,max X i n R 2 p l 1 (x + + l 1 (x n R 2 p K p +1 n K p l 1 (x +1 pq n P( S n > x nµ,max X i ni( R 2 p l 1 (x > K p pq n P( S n > x nµ,max X i ni( l 1 (x > R 2 p K p pq n P( S n > x nµ,max X i n We will now analyze the terms one by one. Note that n l 1 (x implies x l(n. Hence by (2.1 and our monotone property of f ( we have R 2 p l 1 (x pq n P( S n > x nµ,max X i ni( R 2 p l 1 (x > K p R 2 p l 1 (x l 1 (x pq n e β(x nµr( n I( R 2 p l 1 (x > K p { pql 1 (x exp β(x l 1 (xµr( } l 1 (x R 2 p ( } pqr2 p exp { β(x R 2 p µ log q βµ for l 1 (x R 2 p for l 1 (x > R 2 p (2.13 { l 1 (xpq l 1 (x e βαh2 ( l 1 (x R 2 p pe(x/µlog q for l 1 (x R 2 p for l 1 (x > R 2 p and for l 1 (x > K p. The first part of the last inequality follows by substituting x l(l 1 (x l 1 (x+αh( l 1 (x l 1 (x. We shall prove that in both cases they are of smaller order than e θ x + (1/p F(xI(x. Consider the first case, and suppose K p x R p. Dividing the first part of (2.13 by e θ x gives { pα } (2.14 exp µ h( l 1 (x l 1 (x(1 + o(1 βαh 2 ( l 1 (x + log(l 1 (xp by substituting l 1 (x x/µ (α/µh( l 1 (x l 1 (x and using log q p(1+o(1. Note that x R p implies that r(x (log q/(βµ p/(βµ. Since r( l 1 (x r(x, we have h( l 1 (x (p/(βµ l 1 (x. This gives βαh 2 ( l 1 (x (pα/µh( l 1 (x l 1 (x(1 + o(1. Since h(x ɛlog x, we also have βαh 2 ( l 1 (x log l 1 (x log(l 1 (xp. Thus (2.14 is equal to exp{ βαh 2 ( l 1 (x(1 + o(1} exp{ βαh 2 ( K p (1 + o(1} o(1.

13 UNIFORM LARGE DEVIATIONS FOR Now suppose x R p and l 1 (x R 2 p. Note that for any x R p, one can always find a a(p arbitrarily slowly as p, such that x ar p. Dividing the first part of (2.13 by (1/p F(x gives (2.15 exp{l 1 (xlog q βαh 2 ( l 1 (x + log(l 1 (xp + h(x µ + 2log x + log p}. Note that x ar p implies r(x r(x/a p/(βµ and hence pl 1 (x h(x µ, by using l 1 (x x/µ (α/µh( l 1 (x l 1 (x. By substituting y R p in (2.11 we have par p 1/p δ /(1 δ log p which implies px/µ log x for x ar p. Hence (2.15 is equal to exp{ pl 1 (x(1 + o(1} exp{ pl 1 (ar p (1 + o(1} o(1. We now proceed to the second part of (2.13. But l 1 (x R 2 p implies x R p, since h 2 ( l 1 (x/l 1 (x. Hence by the same argument px/µ h(x and px/µ log x log R p. Hence dividing the expression by (1/p F(x gives { } x exp µ log q + 2log R p + log p + h(x µ + 2log x + log p { exp px } { (1 + o(1 exp µ pr2 p (1 + o(1 µ } o(1 We now analyze the second term of (2.12. We have, for l 1 (x > R 2 p K p, l 1 (x n R 2 p K p +1 pq n P( S n > x nµ,max X i n (2.16 l 1 (x n R 2 p +1 l 1 (x n R 2 p +1 pq n e β(x nµr( n l 1 (x n R 2 p +1 log q βµ p(qe βµ l n e βxr( 1 (x l 1 (xpe βxr( l 1 (x p(qe βµr( n n e βxr( n l 1 (x n R 2 p +1 pe βxr( l 1 (x where the third inequality holds because R p r 1 ( log q/(βµ and r(x is eventually decreasing. Note that xr( l 1 (x µ l 1 (xh( l 1 (x h( l 1 (x. Also, since ɛlog x h(x x, we have xr l 1 (x l 1 (xh( l 1 (x log x. Hence dividing (2.16 by (1/p F(x gives exp{ βxr( l 1 (x + log(l 1 (xp + h(x µ + 2log x + log p} exp{ βxr( l 1 (x(1 + o(1} exp{ βr 2 p r( l 1 (R 2 p(1 + o(1} o(1 We now analyze the final term of (2.12. We have (2.17 n K p l 1 (x +1 { pq n P( S n > x nµ,max X i n q l 1 (x K p +1 { q K p/a +1 for l 1 (x K p /a q l 1 (x+1 for l 1 (x K p /a e pk p/a for l 1 (x K p /a e pl 1 (x for l 1 (x K p /a where a a (p as p at a rate that will be chosen later on.

14 196 JOSE BLANCHET AND HENRY LAM For the first part, dividing by e θ x yields { exp p K } { p a + θ x exp p K } p (1 + o(1 o(1 a For the second part observe that ( K p r a h((1/a 2δ e (1 δ g( Ca 1 δ p 2δ(1 δ p (1/a 2δ e (1 δ g( e (1 δ 2 g( for some constant C >, for a suitably chosen a, since h(x x δ eventually. We then get K p /a 1 (p which implies r(x r(k p /a p for l 1 (x K p /a. This gives pl 1 (x h(x. Note that pk p /a (1/a 1 2δ e (1 δ g( 2δlog p+(1 δ g( log K p, so pl 1 (x log x for x K p /a. Hence dividing the second part of (2.17 by (1/p F(x gives exp{ pl 1 (x + h(x µ + 2log x + log p} exp{ pl 1 (x(1 + o(1} { ( } K exp pl 1 p a (1 + o(1 o(1 This concludes our proof of Lemma (2.5. Proof of Proposition (1.7. The case for x < is obvious, since we have P(S n > x, exactly one X i > C p,n n F( and hence P(S M > x, exactly one X i > C p,m pq n n F( q p F( o(1 e θ x, uniformly over x <. Thus we shall focus on x. Note that n1 P(S M > x, exactly one X i > C p,m K p ( pq n n F(x(F( n 1 + P(S n 1 > x y,max X i df(y n1 ( + pq n n F(x(F(µ + n n 1 + µ+ n P(S n 1 > x y,max X i µ + ndf(yi(x µ + n q p F(x K p + pq n n P(S n 1 > x y,max X i df(y n2 + pq n n P(S n 1 > x y,max X i µ + ndf(yi(x µ + n q p F(x + + n2 µ+ n pq n n P(S n 1 > x y,max X i df(y pq n n P(S n 1 > x y,max X i df(y pq n n µ+ P(S n 1 > x y,max X i µ + ndf(yi(x µ + n. n We will finish the proof by invoking the following lemmas:

15 UNIFORM LARGE DEVIATIONS FOR LEMMA (2.18. n2 pq n n P(S n 1 > x y,max X i df(y uniformly over x. ( 1 p + x y e θ (x y df(y(1 + o(1 µ LEMMA (2.19. We have µ + K p for p small enough, and uniformly over x. pq n n pq n n ( 1 p + x y µ Proof of Lemma (2.18. We write n2 pq n np(s n 1 > x,max X i q p P(S n 1 > x y,max X i df(y P(S n 1 > x y,max X i µ + ndf(y e θ (x y df(y + 1 p F(x p 2 q n (n + 1P(S n > x,max X i n1 q p P(S N > x,max X i where N is a negative binomial variable with parameter 2 and p. Let {X i } i1,2,..., M and S M be independent and identical copies of {X i} i1,2,..., M and S M, and let F M (x P(S M x,max X i and F M (x be its complement defined by P(S M > x,max X i. Note that by Lemma (2.4 we have P(S M > x,max X i e θ x (1 + u(x, p where sup x>bp u(x, p. We have P(S N > x,max X i P(S M + S M > x, max 1iM X i, max 1 jm X j F M (x ydf M (y + F M (x F M ( e θ (x y (1 + u(x y, pdf M (y + F M (x F M ( e θ (x y df M (y + F M (x F M ( e θ x F M ( F M (x + e θ x F M ( + F M (yθ e θ (x y d y + F M (x F M ( θ e θ x (1 + u(y, pd y F M (x(1 F M ( e θ x + θ xe θ x uniformly over x >. The fourth equality is obtained using integration by parts, and the last equality follows from the observation that F M ( P(max 1iM X i n pq n F( n 1 as p. Noting that θ p/µ, the conclusion of the lemma is then an easy consequence.

16 198 JOSE BLANCHET AND HENRY LAM Proof of Lemma (2.19. The inequality is obvious. We will thus focus on the order relation. We first prove that pq n np(s n 1 > x,max X i µ + n e θ x + 1 p F(x uniformly over x. The proof is very similar to that of Lemma (2.5. Adopting the notation there, we can write pq n np(s n 1 > x,max X i µ + n R 2 p l 1 (x pq n np( S n 1 > x (n 1µ,max X i ni( R 2 p l 1 (x > K p + + l 1 (x n R 2 p +1 n K p l 1 (x +1 pq n np( S n 1 > x (n 1µ,max X i ni( l 1 (x > R 2 p > K p pq n np( S n 1 > x (n 1µ,max X i n Using the same analysis, the first term will be less than or equal to l 1 (x(l 1 (x+1 2 pq l 1 (x e βαh2 ( l 1 (x for l 1 (x R 2 p R 2 p (R2 p +1 2 pe (x/µlog q for l 1 (x > R 2 p the second term will be less than or equal to l 1 (x(l 1 (x + 1 and the third term will be less than or equal to 2 pe βxr( l 1 (x (K p 1 p + 1 e pk p/a for l 1 (x K p /a (l 1 (x 1 p + 1 e pl 1 (x for l 1 (x K p /a For the first two terms the same analysis carries over while for the last one we only have to observe again that px log x for x K p /a, which will show our claim. Hence we have pq n n P(S n 1 > x y,max X i µ + ndf(y df(y ( e θ (x y + 1 p F(x y ( 1 p + x y e θ (x y df(y + 1 µ p F(x where the last order relation is obtained by using property of class S that F(x y/ F(xdF(y 2 as x. We conclude our proof of Lemma (2.19.

17 UNIFORM LARGE DEVIATIONS FOR Acknowledgements We thank the Referee for the careful review and the useful recommendations. Blanchet acknowledges support from the NSF through the grants CMMI and Received June 1, 213 Final version received October 17, 213 JOSE BLANCHET ASSOCIATE PROFESSOR OF INDUSTRIAL ENGINEERING AND OPERATIONS RESEARCH, ASSOCIATE PROFESSOR OF STATITICS. COLUMBIA UNIVERSITY HENRY K. LAM ASSISTANT PROFESSOR OF MATHEMATICS AND STATISTICS BOSTON UNIVERSITY REFERENCES [1] S. ASMUSSEN, Ruin Probabilities, World Scientific, Singapore, 21. [2] S. ASMUSSEN, Applied Probability and Queues, Springer, New York, 23. [3] J. BLANCHET AND P. GLYNN, Complete Corrected Diffusion Approximations for the Maximum of a Random Walk, Annals of Applied Probability 16 (2, (26, [4] J. BLANCHET AND P. GLYNN, Uniform renewal theory with applications to expansions of geometric sums, Adv. Appl. Prob., 39(4, (27, [5] J. BLANCHET AND P. GLYNN, Efficient rare-event simulation for the maximum of heavy-tailed random walks, Annals of applied Probability 18 (4, (28, [6] J. BLANCHET AND H. LAM, State-dependent importance sampling for rare-event simulation: An overview and recent advances, Surveys in Operations Research and Management Science17 (1, (212, [7] A. A. BOROVKOV AND K. A. BOROVKOV, On probabilities of large deviations for random walks. I. Regularly varying distribution tails, Theory Prob. Appl. 49 (2, (21, [8] P. EMBRECHTS, C. KLUPPELBERG, AND T. MIKOSCH, Modelling Extremal Events for Insurance and Finance, Springer, 23. [9] P. EMBRECHTS AND N. VERAVERBEKE, Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance: Mathematics and Economics 1 (1, (1982, [1] J. KINGMAN, The single server queue in heavy traffic, Proc. Camb. Phil. Soc. 57, (1961, [11] H. LAM, J. BLANCHET, M. BAZANT, AND D. BURCH, Corrections to the Central Limit Theorem for heavy-tailed probability densities, Journal of Theoretical Probability 24 (4, (211, [12] A. V. NAGAEV, Integral limit theorems taking large deviations into account when Cramer s condition does not hold II, Theory Prob. Appl. 14 (2, (1969, [13] M. OLVERA-CRAVIOTO, J. BLANCHET, AND P. GLYNN, On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case, Annals of Applied Probability 21 (2, (211, [14] M. OLVERA-CRAVIOTO AND P. GLYNN, Uniform Approximations for the M/G/1 Queue with Subexponential Processing Times, Queueing Systems 68 (1, (211, 1 5. [15] L. V. ROZOVSKII, Probabilities of large deviations of sums of independent random variables with common distribution function in the domain of attraction of the normal law, Theory Prob. Appl. 34 (4, (1989, [16] L. V. ROZOVSKII, Probabilities of large deviations on the whole axis, Theory Prob. Appl. 38 (1, (1993,

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