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Reduced-load equivalence for Gaussian processes
Operations Research Letters 33 (2005) 502 – 510
Operations Research Letters www.elsevier.com/locate/orl
Reduced-load equivalence for Gaussian processes Bert Zwarta, b,∗ , Sem Borsta, b, c , Krzysztof D¸ebickib, d,1 a Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven,
The Netherlands b CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands c Bell Laboratories, Lucent Technologies, P.O. Box 636, Murray Hill, NJ 07974, USA d Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Received 18 February 2004; accepted 11 September 2004
Abstract We consider a fluid model fed by two Gaussian processes. We obtain necessary and sufficient conditions for the workload asymptotics to be completely determined by one of the two processes, and apply these results to the case of two fractional Brownian motions. © 2004 Elsevier B.V. All rights reserved. MSC: (primary) 60G15; (secondary) 60F10; 60G70 Keywords: Extremes; Fractional Brownian motion; Gaussian processes; Perturbed risk models; Regular variation; Ruin probabilities
1. Introduction Consider two independent stochastic processes {X(t), t 0} and {Y (t), t 0}. An important problem in applied probability is to determine the behavior of
P sup [X(t) + Y (t) − ct] > u , t 0
u → ∞.
(1)
Applications arise in queueing theory, where (1) represents an overflow probability of a fluid queue fed by ∗ Corresponding author. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail address: [email protected] (B. Zwart). 1 Supported by University of Wrocław under Grant 2529/W/IM.
0167-6377/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2004.09.004
two input sources X and Y, and insurance, where (1) can be interpreted as a ruin probability of an insurance company with total premium rate c, initial capital u and claim processes X and Y. In the latter context, an important class of risk models are ‘perturbed’ risk models. In this case the principal stream of claims (say Y) is perturbed by a second stream of smaller claims (say X). Perturbed risk models have recently received quite some attention in the literature and are reviewed by Schmidli [17]. A key issue is the influence of the noise process X on the ruin probability. This is the subject of the present paper. In particular, we investigate whether the process X has influence at all when the initial capital u is large. In mathematical terms, this question can be
B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
rephrased as follows. Assume that X is centered such that it has mean 0. Then the question we investigate is under what conditions
≈ P sup [Y (t) − ct] + sup X(t) > u 0 u − O(X (u)) . t 0
P sup [X(t) + Y (t) − ct] > u t 0
This heuristic computation indicates that (2) holds if
∼ P sup [Y (t) − ct] > u , t 0
u → ∞.
(2)
P sup [X(t) + Y (t) − ct] > u t 0
≈P
sup [X(t) + Y (t) − ct] > u 0
P sup [Y (t) − ct] > u
This type of question has been investigated extensively for fluid queues with on–off input, see e.g. [1,12,13], and has also received some attention in the case of classical risk models (see [17]). The property (2) is commonly referred to as a reduced-load equivalence (RLE). The goal of the present paper is to complement the above-mentioned results with a treatment of the case where X and Y are Gaussian processes. Such processes have become important tools in the performance analysis of communication systems because of their parsimonious nature and the fact that they serve as approximations for the traffic from the superposition of a large number of on–off sources, as is demonstrated in e.g. [6,14]. Recent papers revealing the engineering relevance of Gaussian processes are Choe and Shroff [3] and Wischik [18]. These papers also contain further useful references. The main result of this paper provides necessary and sufficient conditions for (2) to hold in a Gaussian setting. These conditions are stated in Theorems 2 and 3, respectively. When both X and Y are fractional Brownian motions (FBMs) with Hurst parameters HX and HY , respectively, these theorems imply that (2) is valid when HY > 21 + 21 HX , and (2) is not valid when the reverse inequality holds. At a heuristic level, our results can be explained as follows. The process X(t) + Y (t) − ct is well-known to reach a large value u most likely before time lu, with l a sufficiently large constant. At that time, the deviation of sups
503
t 0
∼ P sup [Y (t) − ct] > u − X (u) . t 0
(3)
As is shown in Theorem 2, this condition is indeed sufficient for (2) to hold. A similar condition has been identified earlier by Jelenkovi´c et al. [13] in the case where Y (t) is an on–off source and X(t) is a regenerative process satisfying √ some regularity conditions, implying X (u) ∼ const u. The proof of our main result is similar to the one in [13]. In particular, we almost fully rely on simple sample-path bounds, avoiding more complicated techniques like the double-sum method in Piterbarg [16]. Checking the condition (3) requires information about the exact tail asymptotics of P{supt 0 [Y (t)−ct] > u} as u → ∞. These asymptotics are available when Y is a fractional Brownian motion, see Hüsler and Piterbarg [11]. Further work in this direction has recently been pursued by Dieker [9]. Both papers also contain results for some nonstationary Gaussian processes. The remainder of the paper is organized as follows. In Section 2 we introduce some notation and state a few preliminary results. In Section 3 we present our main results and discuss some of their implications. Proof details are provided in Section 4.
2. Notation and preliminaries In this section we introduce some notation and state a few preliminary results. We consider a fluid queue with infinite buffer size and constant drain rate c fed by two independent traffic processes X and Y having stationary increments. Denote by X(t) and Y (t) the amount of traffic generated by the two processes during the time interval [−t, 0]. If E{X(1) + Y (1)} < c,
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B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
then the random variable c VX+Y = sup [X(t) + Y (t) − ct] t 0
is finite a.s. and can be identified with the buffer content in steady state. We will frequently make comparisons with the buffer content for each of the two processes in isolation. For c > E{X(1)} and D ⊆ [0, ∞), define VXc (D) := supt∈D [X(t) − ct], and let VXc := VXc ([0, ∞)) be a random variable representing the stationary workload in a buffer with drain rate c fed by the process X(t) only. Similarly, for c > E{Y (1)} and D ⊆ [0, ∞), define VYc (D) := supt∈D [Y (t) − ct], and let VYc := VYc ([0, ∞)) be a random variable representing the stationary workload in a buffer with drain rate c fed by the process Y (t) only. For any two real functions f (·) and g(·), we use the notational convention f (u) ∼ g(u) to denote that f (u) = g(u)(1 + o(1)) as u → ∞, i.e., limu→∞ f (u)/g(u) = 1. We assume without loss of generality that X and Y are centered, i.e., E{X(t)} = E{Y (t)} = 0. Moreover, X(t) and Y (t) are assumed to be independent Gaussian processes with stationary increments and continuous sample paths. Thus, the distributional properties of X and Y are fully characterized by their variance functions, which are denoted by 2X (t) and 2Y (t), respectively. Finally, we assume that the following conditions are satisfied for X (and an analogous set of conditions for Y): C1 2X (t), ∈ C([0, ∞)) is increasing; C2 2X (t) is regularly varying at 0 with index X ∈ (0, 2] and 2X (t) is regularly varying at ∞ with index X ∈ (0, 2). Two important classes of processes which satisfy these conditions are FBM [11] and Gaussian integrated input [5]. The logarithmic asymptotics for VYc are due to D¸ebicki [4] and Dieker [8]. Proposition 1. Let Y satisfy conditions C1 and C2. Then there exists a constant d = d(Y , c) such that − log P{VYc > u} ∼ d
u2 . 2Y (u)
Exact asymptotics are currently known only for some special cases. The case of FBM is treated in [11]. A large class of short-range dependent processes has been treated by D¸ebicki [5]. We refer to Dieker [9] for more recent work in this area. The logarithmic asymptotics given above also apply to X + Y , which is Gaussian with variance function X (t) + Y (t). From this, it is not difficult to prove that, if X, Y satisfy conditions C1 and C2, then c log P{VX+Y > u} ∼ log P{VYc > u},
(4)
if and only if Y (u)/X (u) → ∞. Indeed, this condition determines when X (t) + Y (t) ∼ Y (t), and from Proposition 1 we know that the constant d is completely determined by Y . Using Proposition 1 once more, we see that an RLE holds on a logarithmic scale when log P{VYc > u} = o(log P{VXc > u}). Thus, on a logarithmic scale, an RLE holds whenever VYc has a heavier tail than VXc . As will be shown in the next section, the situation is more subtle when exact asymptotics are considered. Before we turn to this topic, we provide an extension of the logarithmic asymptotics in Proposition 1 to a sharper asymptotic bound for P{VYc > u}. Such a bound has been considered before by Choe and Shroff [3]. In fact, the result presented here is an extension of the main result in [3]. In order to formulate this result, we need a few additional definitions. Let mc,Y (u) := mint 0
ct + u , Y (t)
and tu := arg mint 0 (ct + u/Y (t)). In the case that (ct + u/Y (t)) attains its minimum in several points, we take tu to be the minimal such point. Using the fact that Y (t) is regularly varying at ∞ with index Y /2, we have that mc,Y (u) and tu are regularly varying at ∞ with indices 1 − Y /2 and √ 1, respectively. ∞ Moreover, let (u) = (1/ 2) u exp(−x 2 /2) dx be the tail of the standard normal random distribution. We denote by HY /2 the classical Pickand’s constant, see [16], p. 16. Theorem 1. Let Y (t) be a centered Gaussian process with stationary increments satisfying conditions C1
B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
finding simple, explicit conditions in terms of X and Y . When it comes to determining necessary conditions, this is indeed possible.
and C2. Then
P{VYc
> u} HY /2 tu (mc,Y (u))
4/Y
× (mc,Y (u))(1 + o(1)),
Proposition 2. Let X(t), Y (t) be centered Gaussian processes satisfying conditions C1 and C2. If
as u → ∞. The proof of Theorem 1 is given in Section 4.1.
lim sup u→∞
P{VYc > u − X (u)} < ∞, P{VYc > u}
3. Main results
then
The following theorem is the main result of the paper.
Y 1 + 21 X .
Theorem 2 (Sufficient conditions for RLE). Let X(t) and Y (t) be centered Gaussian processes with stationary increments satisfying conditions C1 and C2 and let c > 0. Assume that
P{VYc > u − X (u)} ∼ P{VYc > u}.
(5)
Then, c P{VX+Y > u} ∼ P{VYc > u}.
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(6)
The heuristic interpretation of (5) has already been given in the introduction. The assumption Y (u)/X (u) → ∞ which emerged from the analysis in the previous section follows from Condition (5), as is shown in Corollary 2 below. The next result shows that Condition (5) is close to necessary.
A proof of this result can be found in Section 4.4. Proposition 2 shows that if Y < 1 + 21 X , the RLE property (5) does not hold. It also implies that Y > X , in particular that X (t) = o(Y (t)) as t → ∞. We now turn to sufficient conditions for (5). Unfortunately, in this case it seems difficult to obtain conditions which are as simple as the one in Proposition 2. One must invoke additional regularity assumptions, which reveal the connections with extreme-value theory. All relevant notions in the next proposition can be found in the excellent textbook of Embrechts et al. [10]. Proposition 3. Let Y be a Gaussian process with stationary increments satisfying conditions C1 and C2.
(7)
(i) In addition, assume that P{VYc > u} is in the maximum domain of attraction of the Gumbel distribution with auxiliary function a(x). Then (6) holds if X (u) = o(a(u)). If a(u) = o(X (u)), then (6) does not hold. (ii) Assume in addition that P{VYc > u} ∼ e−Q(u) , with Q (u) ultimately monotone. Then (6) holds if Y > 1 + X /2.
We conjecture that (6) also implies (5), i.e., that (5) is necessary and sufficient for (6). This seems hard to prove, although our proof technique would allow us to strengthen (7) to “ 2”. Further details can be found in Section 4, which contains proofs of Theorems 2 and 3. The sufficient condition (5) and the necessary condition (6) still involve the distribution of VYc . In the remainder of the section, we explore the prospects of
Proof. Assertion (i) follows immediately from standard results in extreme-value theory, see e.g. Embrechts et al. [10]. To establish the second assertion, note that, according to Proposition 1, Q(u) is regularly varying of index 2 − Y . In view of the monotone density theorem (see [2]), the derivative of Q is regularly varying of index 1− Y . Now, (5) can be rewritten into u Q (y) dy → 0.
Theorem 3 (Necessary conditions for RLE). Let X(t) and Y (t) be centered Gaussian processes with stationary increments satisfying conditions C1 and C2 and let c > 0. Furthermore, assume that the RLE property (6) holds. Then, lim sup u→∞
P{VYc > u − X (u)} < ∞. P{VYc > u}
u−X (u)
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B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
Since Q (u) is both ultimately monotone and regularly varying, this integral behaves like Q (u)X (u), which is regularly varying of index X /2+1− Y . This easily yields Assertion (ii). The above regularity conditions can be easily shown to hold when Y is an FBM, the reason being that the exact (rather than only the logarithmic) asymptotics for P{VYc > u} are available. This gives the following corollary. Corollary 1. If Y is an FBM with Y > 1 and X(t) satisfies conditions C1 and C2, then (6) holds if Y > 1 + X /2. If Y < 1 + X /2, then (6) does not hold. This corollary covers in particular the case where both X and Y are FBM with Hurst parameters HX and HY , respectively. In that case (noting that · =2H· ), (6) holds if HY > 21 + 21 HX , and does not hold otherwise. Proof. By Theorem 1 in [11], we can write
P{VYc > u} ∼ p(u)e−u
2−Y
,
P
sup X(s) > x X (t) P
0s t
→P
sup Xt (s) > x
0s 1
sup BH (s) > x ,
0s 1
as t → ∞. Hence there exists q > 0 such that for all sufficiently large t 1 P sup X(s) > q X (t) . (8) 2 0s t Now using Theorem 12.2 in [15], we have that for sufficiently large t and u > q X (t) u P sup X(s) > u −q , X (t) 0s t with q chosen such that (8) holds. This immediately implies (1). We also need several characterizations of (5).
with some constant and p(u) a power function. Hence, Q (y) is ultimately monotone.
4. Additional proofs In this section we prove Theorems 1–3, as well as Proposition 2. The proofs of these results can be found in Section 4.1–4.4, respectively. First, we state and prove two important lemmas. Lemma 1. If X(t) has stationary increments and satisfies conditions C1 and C2, then there exist constants K > 0 and q > 0 such that for t large enough and every u > q X (t),
P
each x > 0
sup X(s) > u Ke−1/2((u/X (t))−q) . 2
s∈[0,t]
Proof. Define Xt (s) = X(st)/X (s). From Lemma 4.2 in [7], we have that the process {Xt (s), 0 s 1} converges weakly to {BH (s), 0 s 1} in C([0, 1]) as t → ∞, with H = X /2. Since the sup operator is continuous in C([0, 1]), we conclude that for
Lemma 2. Assume that conditions C1 and C2 are satisfied for X and Y. The following statements are equivalent:
P{VYc > u − X (u)} ∼ P{VYc > u}; P{VYc > u − X (lu)} ∼ P{VYc > u}, l > 0; P{VYc > u − k X (lu)} ∼ P{VYc > u}, k, l > 0; P{VYc > u − Z X (lu) | Z > k} ∼ P{VYc > u}, k, l > 0; with Z a random variable which is independent of Y 2 and has density 1 ze−2 z .
(i) (ii) (iii) (iv)
Proof. The equivalence between (ii) and (iii) is trivial. The equivalence between (i) and (ii) follows from the equivalence between (ii) and (iii), combined with the bounds l 1 (u) X (lu) l 2 X (u), (for suitable choices of 1 , 2 ), which follow from Potter’s theorem. To prove that (i) implies (iv), we use a similar argument as in the proof of Theorem 1 in [13] and write,
B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
for some large M,
4.1. Proof of Theorem 1
P{VYc > x − Z X (lx) | Z > k} P{VYc > x − Mk Y (lx)} x/(2X (lx)) 2 2 ze−z /2 ek +
Following Theorem 1 in Dieker [8] we have
P{VYc
Mk
t 0
=P
The last term can be neglected in view of (5) and Proposition 2. To bound the integral, we use the following bound: for every > 0, there exists an x such that > x − zX (lx)} e
(z+1)
P{VYc
> x},
P{VYc > x} e P{VYc > x + X (lx)}, for x x . Hence, for any integer k, we obtain using the monotonicity of (lx) in x,
i=1
k i=1 k x
e
[Y (t) − ct] > u (10)
as u → ∞. Note that
P
sup
t∈[1/2tu ,3/2tu ]
[Y (t) − ct] > u
Y (t) =P sup >1 t∈[1/2tu ,3/2tu ] ct + u Y (t) Y (t) sup =P t∈[1/2tu ,3/2tu ] Y (t) ct + u × mc,Y (u) > mc,Y (u) Y (t) sup P > mc,Y (u) t∈[1/2tu ,3/2tu ] Y (t)
P{VYc > x + iX (lx)} P{VYc > x + iX (lx) + X (lx)} P{VYc > x + iX (lx)} c P{VY > x + iX (lx) + (l(x + i X (lx)))} .
[tu ]
k=0
Y (t) sup P > mc,Y (u) . t∈[1/2tu +k,1/2tu +k+1] Y (t) (11)
Let {Z(s) : s 0} be a centered stationary Gaussian process with covariance function R(t) = 1 − t −Y /2 + o(|t|−Y /2 ) as t → 0. Note that, for s, t ∈ [1/2tu , 3/2tu ], |t − s| 1 and sufficiently large u,
We conclude that for any z 0 and x x :
P{V > x} e(z+1)x P{V > x + z(lx)}.
Finally, note that
P{VYc
> x − zX (lx)} P{VYc > x} P{VYc > x − zX (lx)} . P{VYc > x − zX (lx) + zX (l(x − zX (lx)))}
Combining the last two inequalities yields (9).
sup
t∈[1/2tu ,3/2tu ]
P{VYc > x} P{VYc > x + k X (lx)} k
× (1 + o(1))
(9)
for all z 0 and x − zX (lx) x . Substituting this bound in the integral and invoking (i) then easily yields (iv) after letting M → ∞. The reverse implication is trivial. Thus, it remains to prove the bound (9). For this, we note that Property (ii) implies that for every > 0 there exists an x such that
=
> u} = P sup [Y (t) − ct] > u
× P{VYc > x − zX (lx)} dz + P{Z > x/(2X (lx)) | Z > k}.
P{VYc
507
Cov
Y (s) Y (t) , Y (s) Y (t)
=
2Y (s)+2Y (t)−2Y (|s−t|) 2Y (s)Y (t)
1 −
2Y (|s − t|) 2Y (s)Y (t)
1 − |t − s|Y /2 .
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B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
Thus from the Slepian inequality (see Theorem C.1 in [16]), for sufficiently large u and k = 0, 1, ..., [tu ],
P
Y (t) > mc,Y (u) t∈[1/2tu +k,1/2tu +k+1] Y (t) sup
P
sup Z(t) > mc,Y (u) .
Lemma 1 holds. Moreover let k q. Write
P{VYc u − k X (lu); VYc + VX0 [0, lu] > u} u−k X (lu) P{VX0 [0, lu] > u − y} dP{VYc y} = 0 u−k X (lu) 2 K e−1/2(((u−y)/X (lu))−q) 0
t∈[0,1]
dP{VYc y}.
Hence [tu ]
P
k=1
In the last step, we applied Lemma 1. Next, use integration by parts to get the upper bound
Y (t) > mc,Y (u) t∈[1/2tu +k,1/2tu +k+1] Y (t) sup
[tu + 1]P
Ke
sup Z(t) > mc,Y (u)
t∈[0,1]
×
= HY /2 tu (mc,Y (u))4/Y × (mc,Y (u))(1 + o(1))
(12)
as u → ∞, where (12) follows from Theorem D.2 in [16]. Combination of (10) and (11) with (12) completes the proof.
The proof consists of a lower and an upper bound. We will repeatedly use the equivalence between (i), (ii), and (iii) in Lemma 2 without mention. We start with the upper bound. Note that the class of Gaussian processes satisfying conditions C1 and C2 is closed under superposition. Thus, by Theorem 1 in Dieker [8] we have for some constant l, c c P{VX+Y > u} ∼ P{VX+Y [0, lu] > u}.
Using sample-path arguments, we obtain
=P
sup [X(t) + Y (t) − ct] > u
0 t lu P{VYc > u − k Y (lu)} + P{VYc u − k X (lu); VYc
+ VX0 [0, lu] > u}.
We need to show that the second term can be asymptotically neglected. Let K, q > 0 be constants such that
1 + 2K 2
u−k X (lu)
0
P{VYc > y}
u − y −1/2((u−y/X (lu))−q)2 dy. e 2X (lu)
The first term can be neglected in view of the fact that X (u) = o(Y (u)), combined with Proposition 1. Substituting z = (u − y)/X (lu), the second term can be rewritten as K
4.2. Proof of Theorem 2
c P{VX+Y [0, lu] > u}
−1/2(u2 /2X (lu))
u X (lu)
k
ze−1/2z P{VYc > u − X (lu)z} dz 2
K1 P{VYc > u − X (lu)Z; Z > k}, where Z is a random variable with density proportional 2 to ze−(1/2)z and K1 is some constant independent of k. Using Lemma 2 (iv), we conclude that
P{VYc u − k X (u); VYc + VX0 [0, lu] > u} P{VYc > u} u→∞ K1 P{Z > k}.
lim sup
The proof of the upper bound now follows by letting k → ∞. We now turn to the lower bound. Using properties of the sup operator, we obtain, with l as defined above, c 0 P{VX+Y > u} P{VYc [0, lu] − V−X [0, lu] > u}.
Hence, for some k > l, c > u} P{VYc [0, lu] > u + k X (u)} P{VX+Y 0 × P{V−X [0, lu] k X (u)}.
B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
Now, write
509
4.4. Proof of Proposition 2
c P{VX+Y > u} P{VYc [0, lu] > u + k X (u)} c P{VY > u} P{VYc > u + k X (u)}
×
Note that
P{VYc > u + k X (u)} P{VYc > u}
0 × P{V−X [0, lu] k X (u)},
and take the lim inf of each of the three terms as u → ∞. The first term converges to 1 by Theorem 1 of [8]. The second term tends to 1 in view of Assumption (5). The third term converges to a limit U2 (k, l), which has the property that U2 (k, l) → 1, as k → ∞ for every l, in view of Lemma 1 applied to −X. 4.3. Proof of Theorem 3 We may write for some constant g which is suitably small, c c P{VX+Y > u} P{VX+Y > u | VYc
> u − X (gu)}P{VYc > u − X (gu)}.
L(u) := P{VYc > u − X (u)} sup P{X(t) − ct > u − X (u)} t 0 ct u + u − X (u) Y (tu ) 1 (ct u + u)2 Y (tu ) =√ exp − 22 (tu ) 2 ct u + u − X (u) Y 2X (u) (ct u + u)X (u) × exp exp − 2 2Y (tu ) 2Y (tu ) × (1 + o(1)). On the other hand, from Theorem 1, R(u) := P{VYc > u} ct u + u f (u) (1 + o(1)) Y (tu ) 1 Y (tu ) (ct u + u)2 = f (u) √ exp − 22Y (tu ) 2 ct u + u × (1 + o(1))
(13)
Denote the right-hand side as I(u)II(u). Define (u) := inf{t : Y (t) − ct u − X (gu)}. Then c I(u) P{VX+Y > u; gu < (u) < ∞ | (u) < ∞}
P{X( (u)) > X (gu); gu < (u) < ∞ | (u) < ∞}
P{X(gu) > X (gu); gu < (u) < ∞ | (u) < ∞} = P{X(1) > 1}P{gu < (u) < ∞}, where the third inequality follows from the fact that 1(X(t) > X (gu)) is stochastically monotone in t. Since P{gu < (u) < ∞} ∼ P{ (u) < ∞} if g is sufficiently small (see Theorem 1 in Dieker [8]), we obtain lim inf u→∞ I (u) > 0. Combining this with the fact that reduced-load equivalence is assumed to hold (i.e. (6)) and (13), the desired statement follows.
as u → ∞, where f (u) is regularly varying at infinity with index 1 + 4(1 − Y /2)/Y . Hence, using that L(u)/R(u) is bounded and X (u) = o(u) as u → ∞, the index of the regularly varying function (ct u + u)X (u) 2Y (tu ) has to be non-positive. This implies X 2Y − 2, which completes the proof. Acknowledgements The authors are indebted to Ton Dieker for useful discussions and for showing us his preprint [8], which enabled us to shorten several proofs. References [1] R. Agrawal, A.M. Makowski, Ph. Nain, On a reduced load equivalence for fluid queues under subexponentiality, Queueing Syst. 33 (1999) 5–41.
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[2] N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. [3] J. Choe, N.B. Shroff, Use of the supremum distribution of Gaussian processes in queueing analysis with long-range dependence and self-similarity, Stochastic Models 16 (2000) 209–231. [4] K. D¸ebicki, A note on LDP for supremum of Gaussian processes over infinite horizon, Stat. Probab. Lett. 44 (1999) 211–219. [5] K. D¸ebicki, Ruin probability for Gaussian integrated processes, Stoch. Proc. Appl. 98 (2002) 151–174. [6] K. D¸ebicki, Z. Palmowski, On-off fluid models in heavy traffic environment, Queueing Syst. 33 (1999) 327–338. [7] K. D¸ebicki, A.P. Zwart, S.C. Borst, The supremum of a Gaussian process over a random interval, Stat. Probab. Lett. 68 (2004) 221–234. [8] A.B. Dieker, Conditional limit theorems for queues with Gaussian input, a weak convergence approach, Report PNA R0305, CWI, Amsterdam, 2003 Available at http://www.cwi.nl/∼ton. [9] A.B. Dieker, Extremes of Gaussian processes with stationary increments, 2004, to appear in Stoch. Proc. Appl. [10] P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling Extremal Events, Springer, Berlin, 1997.
[11] J. Hüsler, V.I. Piterbarg, Extremes of a certain class of Gaussian processes, Stochastic Process. Appl. 83 (1999) 257–271. [12] P.R. Jelenkovi´c, A.A. Lazar, Asymptotic results for multiplexing subexponential on-off processes, Adv. Appl. Probab. 31 (1999) 394–421. [13] P.R. Jelenkovi´c, P. Momˇcilovi´c, A.P. Zwart, Reduced-load equivalence and subexponentiality, Queueing Syst. 46 (2004) 97–112. [14] T.G. Kurtz, Limit theorems for workload input models, in: F.P. Kelly, S. Zachary, I. Ziedins (Eds.), Stochastic Networks, Springer, Berlin, 1996. [15] M. Lifshits, Gaussian Random Functions, Kluwer Academic Publishers, Boston, 1995. [16] V.I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, Translations of Mathematical Monographs, vol. 148, AMS, Providence, RI, 1996. [17] H.P. Schmidli, Perturbed risk processes: a review, Theory Stochastic Process. 5 (1999) 145–165. [18] D.J. Wischik, Moderate deviations in queueing theory, submitted for publication, 2001. Available at http:// www.statslab.cam.ac.uk/∼djw1005/Stats/ Research/moddev.html.
Operations Research Letters www.elsevier.com/locate/orl
Reduced-load equivalence for Gaussian processes Bert Zwarta, b,∗ , Sem Borsta, b, c , Krzysztof D¸ebickib, d,1 a Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven,
The Netherlands b CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands c Bell Laboratories, Lucent Technologies, P.O. Box 636, Murray Hill, NJ 07974, USA d Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Received 18 February 2004; accepted 11 September 2004
Abstract We consider a fluid model fed by two Gaussian processes. We obtain necessary and sufficient conditions for the workload asymptotics to be completely determined by one of the two processes, and apply these results to the case of two fractional Brownian motions. © 2004 Elsevier B.V. All rights reserved. MSC: (primary) 60G15; (secondary) 60F10; 60G70 Keywords: Extremes; Fractional Brownian motion; Gaussian processes; Perturbed risk models; Regular variation; Ruin probabilities
1. Introduction Consider two independent stochastic processes {X(t), t 0} and {Y (t), t 0}. An important problem in applied probability is to determine the behavior of
P sup [X(t) + Y (t) − ct] > u , t 0
u → ∞.
(1)
Applications arise in queueing theory, where (1) represents an overflow probability of a fluid queue fed by ∗ Corresponding author. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail address: [email protected] (B. Zwart). 1 Supported by University of Wrocław under Grant 2529/W/IM.
0167-6377/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2004.09.004
two input sources X and Y, and insurance, where (1) can be interpreted as a ruin probability of an insurance company with total premium rate c, initial capital u and claim processes X and Y. In the latter context, an important class of risk models are ‘perturbed’ risk models. In this case the principal stream of claims (say Y) is perturbed by a second stream of smaller claims (say X). Perturbed risk models have recently received quite some attention in the literature and are reviewed by Schmidli [17]. A key issue is the influence of the noise process X on the ruin probability. This is the subject of the present paper. In particular, we investigate whether the process X has influence at all when the initial capital u is large. In mathematical terms, this question can be
B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
rephrased as follows. Assume that X is centered such that it has mean 0. Then the question we investigate is under what conditions
≈ P sup [Y (t) − ct] + sup X(t) > u 0
P sup [X(t) + Y (t) − ct] > u t 0
This heuristic computation indicates that (2) holds if
∼ P sup [Y (t) − ct] > u , t 0
u → ∞.
(2)
P sup [X(t) + Y (t) − ct] > u t 0
≈P
sup [X(t) + Y (t) − ct] > u 0
P sup [Y (t) − ct] > u
This type of question has been investigated extensively for fluid queues with on–off input, see e.g. [1,12,13], and has also received some attention in the case of classical risk models (see [17]). The property (2) is commonly referred to as a reduced-load equivalence (RLE). The goal of the present paper is to complement the above-mentioned results with a treatment of the case where X and Y are Gaussian processes. Such processes have become important tools in the performance analysis of communication systems because of their parsimonious nature and the fact that they serve as approximations for the traffic from the superposition of a large number of on–off sources, as is demonstrated in e.g. [6,14]. Recent papers revealing the engineering relevance of Gaussian processes are Choe and Shroff [3] and Wischik [18]. These papers also contain further useful references. The main result of this paper provides necessary and sufficient conditions for (2) to hold in a Gaussian setting. These conditions are stated in Theorems 2 and 3, respectively. When both X and Y are fractional Brownian motions (FBMs) with Hurst parameters HX and HY , respectively, these theorems imply that (2) is valid when HY > 21 + 21 HX , and (2) is not valid when the reverse inequality holds. At a heuristic level, our results can be explained as follows. The process X(t) + Y (t) − ct is well-known to reach a large value u most likely before time lu, with l a sufficiently large constant. At that time, the deviation of sups
503
t 0
∼ P sup [Y (t) − ct] > u − X (u) . t 0
(3)
As is shown in Theorem 2, this condition is indeed sufficient for (2) to hold. A similar condition has been identified earlier by Jelenkovi´c et al. [13] in the case where Y (t) is an on–off source and X(t) is a regenerative process satisfying √ some regularity conditions, implying X (u) ∼ const u. The proof of our main result is similar to the one in [13]. In particular, we almost fully rely on simple sample-path bounds, avoiding more complicated techniques like the double-sum method in Piterbarg [16]. Checking the condition (3) requires information about the exact tail asymptotics of P{supt 0 [Y (t)−ct] > u} as u → ∞. These asymptotics are available when Y is a fractional Brownian motion, see Hüsler and Piterbarg [11]. Further work in this direction has recently been pursued by Dieker [9]. Both papers also contain results for some nonstationary Gaussian processes. The remainder of the paper is organized as follows. In Section 2 we introduce some notation and state a few preliminary results. In Section 3 we present our main results and discuss some of their implications. Proof details are provided in Section 4.
2. Notation and preliminaries In this section we introduce some notation and state a few preliminary results. We consider a fluid queue with infinite buffer size and constant drain rate c fed by two independent traffic processes X and Y having stationary increments. Denote by X(t) and Y (t) the amount of traffic generated by the two processes during the time interval [−t, 0]. If E{X(1) + Y (1)} < c,
504
B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
then the random variable c VX+Y = sup [X(t) + Y (t) − ct] t 0
is finite a.s. and can be identified with the buffer content in steady state. We will frequently make comparisons with the buffer content for each of the two processes in isolation. For c > E{X(1)} and D ⊆ [0, ∞), define VXc (D) := supt∈D [X(t) − ct], and let VXc := VXc ([0, ∞)) be a random variable representing the stationary workload in a buffer with drain rate c fed by the process X(t) only. Similarly, for c > E{Y (1)} and D ⊆ [0, ∞), define VYc (D) := supt∈D [Y (t) − ct], and let VYc := VYc ([0, ∞)) be a random variable representing the stationary workload in a buffer with drain rate c fed by the process Y (t) only. For any two real functions f (·) and g(·), we use the notational convention f (u) ∼ g(u) to denote that f (u) = g(u)(1 + o(1)) as u → ∞, i.e., limu→∞ f (u)/g(u) = 1. We assume without loss of generality that X and Y are centered, i.e., E{X(t)} = E{Y (t)} = 0. Moreover, X(t) and Y (t) are assumed to be independent Gaussian processes with stationary increments and continuous sample paths. Thus, the distributional properties of X and Y are fully characterized by their variance functions, which are denoted by 2X (t) and 2Y (t), respectively. Finally, we assume that the following conditions are satisfied for X (and an analogous set of conditions for Y): C1 2X (t), ∈ C([0, ∞)) is increasing; C2 2X (t) is regularly varying at 0 with index X ∈ (0, 2] and 2X (t) is regularly varying at ∞ with index X ∈ (0, 2). Two important classes of processes which satisfy these conditions are FBM [11] and Gaussian integrated input [5]. The logarithmic asymptotics for VYc are due to D¸ebicki [4] and Dieker [8]. Proposition 1. Let Y satisfy conditions C1 and C2. Then there exists a constant d = d(Y , c) such that − log P{VYc > u} ∼ d
u2 . 2Y (u)
Exact asymptotics are currently known only for some special cases. The case of FBM is treated in [11]. A large class of short-range dependent processes has been treated by D¸ebicki [5]. We refer to Dieker [9] for more recent work in this area. The logarithmic asymptotics given above also apply to X + Y , which is Gaussian with variance function X (t) + Y (t). From this, it is not difficult to prove that, if X, Y satisfy conditions C1 and C2, then c log P{VX+Y > u} ∼ log P{VYc > u},
(4)
if and only if Y (u)/X (u) → ∞. Indeed, this condition determines when X (t) + Y (t) ∼ Y (t), and from Proposition 1 we know that the constant d is completely determined by Y . Using Proposition 1 once more, we see that an RLE holds on a logarithmic scale when log P{VYc > u} = o(log P{VXc > u}). Thus, on a logarithmic scale, an RLE holds whenever VYc has a heavier tail than VXc . As will be shown in the next section, the situation is more subtle when exact asymptotics are considered. Before we turn to this topic, we provide an extension of the logarithmic asymptotics in Proposition 1 to a sharper asymptotic bound for P{VYc > u}. Such a bound has been considered before by Choe and Shroff [3]. In fact, the result presented here is an extension of the main result in [3]. In order to formulate this result, we need a few additional definitions. Let mc,Y (u) := mint 0
ct + u , Y (t)
and tu := arg mint 0 (ct + u/Y (t)). In the case that (ct + u/Y (t)) attains its minimum in several points, we take tu to be the minimal such point. Using the fact that Y (t) is regularly varying at ∞ with index Y /2, we have that mc,Y (u) and tu are regularly varying at ∞ with indices 1 − Y /2 and √ 1, respectively. ∞ Moreover, let (u) = (1/ 2) u exp(−x 2 /2) dx be the tail of the standard normal random distribution. We denote by HY /2 the classical Pickand’s constant, see [16], p. 16. Theorem 1. Let Y (t) be a centered Gaussian process with stationary increments satisfying conditions C1
B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
finding simple, explicit conditions in terms of X and Y . When it comes to determining necessary conditions, this is indeed possible.
and C2. Then
P{VYc
> u} HY /2 tu (mc,Y (u))
4/Y
× (mc,Y (u))(1 + o(1)),
Proposition 2. Let X(t), Y (t) be centered Gaussian processes satisfying conditions C1 and C2. If
as u → ∞. The proof of Theorem 1 is given in Section 4.1.
lim sup u→∞
P{VYc > u − X (u)} < ∞, P{VYc > u}
3. Main results
then
The following theorem is the main result of the paper.
Y 1 + 21 X .
Theorem 2 (Sufficient conditions for RLE). Let X(t) and Y (t) be centered Gaussian processes with stationary increments satisfying conditions C1 and C2 and let c > 0. Assume that
P{VYc > u − X (u)} ∼ P{VYc > u}.
(5)
Then, c P{VX+Y > u} ∼ P{VYc > u}.
505
(6)
The heuristic interpretation of (5) has already been given in the introduction. The assumption Y (u)/X (u) → ∞ which emerged from the analysis in the previous section follows from Condition (5), as is shown in Corollary 2 below. The next result shows that Condition (5) is close to necessary.
A proof of this result can be found in Section 4.4. Proposition 2 shows that if Y < 1 + 21 X , the RLE property (5) does not hold. It also implies that Y > X , in particular that X (t) = o(Y (t)) as t → ∞. We now turn to sufficient conditions for (5). Unfortunately, in this case it seems difficult to obtain conditions which are as simple as the one in Proposition 2. One must invoke additional regularity assumptions, which reveal the connections with extreme-value theory. All relevant notions in the next proposition can be found in the excellent textbook of Embrechts et al. [10]. Proposition 3. Let Y be a Gaussian process with stationary increments satisfying conditions C1 and C2.
(7)
(i) In addition, assume that P{VYc > u} is in the maximum domain of attraction of the Gumbel distribution with auxiliary function a(x). Then (6) holds if X (u) = o(a(u)). If a(u) = o(X (u)), then (6) does not hold. (ii) Assume in addition that P{VYc > u} ∼ e−Q(u) , with Q (u) ultimately monotone. Then (6) holds if Y > 1 + X /2.
We conjecture that (6) also implies (5), i.e., that (5) is necessary and sufficient for (6). This seems hard to prove, although our proof technique would allow us to strengthen (7) to “ 2”. Further details can be found in Section 4, which contains proofs of Theorems 2 and 3. The sufficient condition (5) and the necessary condition (6) still involve the distribution of VYc . In the remainder of the section, we explore the prospects of
Proof. Assertion (i) follows immediately from standard results in extreme-value theory, see e.g. Embrechts et al. [10]. To establish the second assertion, note that, according to Proposition 1, Q(u) is regularly varying of index 2 − Y . In view of the monotone density theorem (see [2]), the derivative of Q is regularly varying of index 1− Y . Now, (5) can be rewritten into u Q (y) dy → 0.
Theorem 3 (Necessary conditions for RLE). Let X(t) and Y (t) be centered Gaussian processes with stationary increments satisfying conditions C1 and C2 and let c > 0. Furthermore, assume that the RLE property (6) holds. Then, lim sup u→∞
P{VYc > u − X (u)} < ∞. P{VYc > u}
u−X (u)
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B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
Since Q (u) is both ultimately monotone and regularly varying, this integral behaves like Q (u)X (u), which is regularly varying of index X /2+1− Y . This easily yields Assertion (ii). The above regularity conditions can be easily shown to hold when Y is an FBM, the reason being that the exact (rather than only the logarithmic) asymptotics for P{VYc > u} are available. This gives the following corollary. Corollary 1. If Y is an FBM with Y > 1 and X(t) satisfies conditions C1 and C2, then (6) holds if Y > 1 + X /2. If Y < 1 + X /2, then (6) does not hold. This corollary covers in particular the case where both X and Y are FBM with Hurst parameters HX and HY , respectively. In that case (noting that · =2H· ), (6) holds if HY > 21 + 21 HX , and does not hold otherwise. Proof. By Theorem 1 in [11], we can write
P{VYc > u} ∼ p(u)e−u
2−Y
,
P
sup X(s) > x X (t) P
0s t
→P
sup Xt (s) > x
0s 1
sup BH (s) > x ,
0s 1
as t → ∞. Hence there exists q > 0 such that for all sufficiently large t 1 P sup X(s) > q X (t) . (8) 2 0s t Now using Theorem 12.2 in [15], we have that for sufficiently large t and u > q X (t) u P sup X(s) > u −q , X (t) 0s t with q chosen such that (8) holds. This immediately implies (1). We also need several characterizations of (5).
with some constant and p(u) a power function. Hence, Q (y) is ultimately monotone.
4. Additional proofs In this section we prove Theorems 1–3, as well as Proposition 2. The proofs of these results can be found in Section 4.1–4.4, respectively. First, we state and prove two important lemmas. Lemma 1. If X(t) has stationary increments and satisfies conditions C1 and C2, then there exist constants K > 0 and q > 0 such that for t large enough and every u > q X (t),
P
each x > 0
sup X(s) > u Ke−1/2((u/X (t))−q) . 2
s∈[0,t]
Proof. Define Xt (s) = X(st)/X (s). From Lemma 4.2 in [7], we have that the process {Xt (s), 0 s 1} converges weakly to {BH (s), 0 s 1} in C([0, 1]) as t → ∞, with H = X /2. Since the sup operator is continuous in C([0, 1]), we conclude that for
Lemma 2. Assume that conditions C1 and C2 are satisfied for X and Y. The following statements are equivalent:
P{VYc > u − X (u)} ∼ P{VYc > u}; P{VYc > u − X (lu)} ∼ P{VYc > u}, l > 0; P{VYc > u − k X (lu)} ∼ P{VYc > u}, k, l > 0; P{VYc > u − Z X (lu) | Z > k} ∼ P{VYc > u}, k, l > 0; with Z a random variable which is independent of Y 2 and has density 1 ze−2 z .
(i) (ii) (iii) (iv)
Proof. The equivalence between (ii) and (iii) is trivial. The equivalence between (i) and (ii) follows from the equivalence between (ii) and (iii), combined with the bounds l 1 (u) X (lu) l 2 X (u), (for suitable choices of 1 , 2 ), which follow from Potter’s theorem. To prove that (i) implies (iv), we use a similar argument as in the proof of Theorem 1 in [13] and write,
B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
for some large M,
4.1. Proof of Theorem 1
P{VYc > x − Z X (lx) | Z > k} P{VYc > x − Mk Y (lx)} x/(2X (lx)) 2 2 ze−z /2 ek +
Following Theorem 1 in Dieker [8] we have
P{VYc
Mk
t 0
=P
The last term can be neglected in view of (5) and Proposition 2. To bound the integral, we use the following bound: for every > 0, there exists an x such that > x − zX (lx)} e
(z+1)
P{VYc
> x},
P{VYc > x} e P{VYc > x + X (lx)}, for x x . Hence, for any integer k, we obtain using the monotonicity of (lx) in x,
i=1
k i=1 k x
e
[Y (t) − ct] > u (10)
as u → ∞. Note that
P
sup
t∈[1/2tu ,3/2tu ]
[Y (t) − ct] > u
Y (t) =P sup >1 t∈[1/2tu ,3/2tu ] ct + u Y (t) Y (t) sup =P t∈[1/2tu ,3/2tu ] Y (t) ct + u × mc,Y (u) > mc,Y (u) Y (t) sup P > mc,Y (u) t∈[1/2tu ,3/2tu ] Y (t)
P{VYc > x + iX (lx)} P{VYc > x + iX (lx) + X (lx)} P{VYc > x + iX (lx)} c P{VY > x + iX (lx) + (l(x + i X (lx)))} .
[tu ]
k=0
Y (t) sup P > mc,Y (u) . t∈[1/2tu +k,1/2tu +k+1] Y (t) (11)
Let {Z(s) : s 0} be a centered stationary Gaussian process with covariance function R(t) = 1 − t −Y /2 + o(|t|−Y /2 ) as t → 0. Note that, for s, t ∈ [1/2tu , 3/2tu ], |t − s| 1 and sufficiently large u,
We conclude that for any z 0 and x x :
P{V > x} e(z+1)x P{V > x + z(lx)}.
Finally, note that
P{VYc
> x − zX (lx)} P{VYc > x} P{VYc > x − zX (lx)} . P{VYc > x − zX (lx) + zX (l(x − zX (lx)))}
Combining the last two inequalities yields (9).
sup
t∈[1/2tu ,3/2tu ]
P{VYc > x} P{VYc > x + k X (lx)} k
× (1 + o(1))
(9)
for all z 0 and x − zX (lx) x . Substituting this bound in the integral and invoking (i) then easily yields (iv) after letting M → ∞. The reverse implication is trivial. Thus, it remains to prove the bound (9). For this, we note that Property (ii) implies that for every > 0 there exists an x such that
=
> u} = P sup [Y (t) − ct] > u
× P{VYc > x − zX (lx)} dz + P{Z > x/(2X (lx)) | Z > k}.
P{VYc
507
Cov
Y (s) Y (t) , Y (s) Y (t)
=
2Y (s)+2Y (t)−2Y (|s−t|) 2Y (s)Y (t)
1 −
2Y (|s − t|) 2Y (s)Y (t)
1 − |t − s|Y /2 .
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B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
Thus from the Slepian inequality (see Theorem C.1 in [16]), for sufficiently large u and k = 0, 1, ..., [tu ],
P
Y (t) > mc,Y (u) t∈[1/2tu +k,1/2tu +k+1] Y (t) sup
P
sup Z(t) > mc,Y (u) .
Lemma 1 holds. Moreover let k q. Write
P{VYc u − k X (lu); VYc + VX0 [0, lu] > u} u−k X (lu) P{VX0 [0, lu] > u − y} dP{VYc y} = 0 u−k X (lu) 2 K e−1/2(((u−y)/X (lu))−q) 0
t∈[0,1]
dP{VYc y}.
Hence [tu ]
P
k=1
In the last step, we applied Lemma 1. Next, use integration by parts to get the upper bound
Y (t) > mc,Y (u) t∈[1/2tu +k,1/2tu +k+1] Y (t) sup
[tu + 1]P
Ke
sup Z(t) > mc,Y (u)
t∈[0,1]
×
= HY /2 tu (mc,Y (u))4/Y × (mc,Y (u))(1 + o(1))
(12)
as u → ∞, where (12) follows from Theorem D.2 in [16]. Combination of (10) and (11) with (12) completes the proof.
The proof consists of a lower and an upper bound. We will repeatedly use the equivalence between (i), (ii), and (iii) in Lemma 2 without mention. We start with the upper bound. Note that the class of Gaussian processes satisfying conditions C1 and C2 is closed under superposition. Thus, by Theorem 1 in Dieker [8] we have for some constant l, c c P{VX+Y > u} ∼ P{VX+Y [0, lu] > u}.
Using sample-path arguments, we obtain
=P
sup [X(t) + Y (t) − ct] > u
0 t lu P{VYc > u − k Y (lu)} + P{VYc u − k X (lu); VYc
+ VX0 [0, lu] > u}.
We need to show that the second term can be asymptotically neglected. Let K, q > 0 be constants such that
1 + 2K 2
u−k X (lu)
0
P{VYc > y}
u − y −1/2((u−y/X (lu))−q)2 dy. e 2X (lu)
The first term can be neglected in view of the fact that X (u) = o(Y (u)), combined with Proposition 1. Substituting z = (u − y)/X (lu), the second term can be rewritten as K
4.2. Proof of Theorem 2
c P{VX+Y [0, lu] > u}
−1/2(u2 /2X (lu))
u X (lu)
k
ze−1/2z P{VYc > u − X (lu)z} dz 2
K1 P{VYc > u − X (lu)Z; Z > k}, where Z is a random variable with density proportional 2 to ze−(1/2)z and K1 is some constant independent of k. Using Lemma 2 (iv), we conclude that
P{VYc u − k X (u); VYc + VX0 [0, lu] > u} P{VYc > u} u→∞ K1 P{Z > k}.
lim sup
The proof of the upper bound now follows by letting k → ∞. We now turn to the lower bound. Using properties of the sup operator, we obtain, with l as defined above, c 0 P{VX+Y > u} P{VYc [0, lu] − V−X [0, lu] > u}.
Hence, for some k > l, c > u} P{VYc [0, lu] > u + k X (u)} P{VX+Y 0 × P{V−X [0, lu] k X (u)}.
B. Zwart et al. / Operations Research Letters 33 (2005) 502 – 510
Now, write
509
4.4. Proof of Proposition 2
c P{VX+Y > u} P{VYc [0, lu] > u + k X (u)} c P{VY > u} P{VYc > u + k X (u)}
×
Note that
P{VYc > u + k X (u)} P{VYc > u}
0 × P{V−X [0, lu] k X (u)},
and take the lim inf of each of the three terms as u → ∞. The first term converges to 1 by Theorem 1 of [8]. The second term tends to 1 in view of Assumption (5). The third term converges to a limit U2 (k, l), which has the property that U2 (k, l) → 1, as k → ∞ for every l, in view of Lemma 1 applied to −X. 4.3. Proof of Theorem 3 We may write for some constant g which is suitably small, c c P{VX+Y > u} P{VX+Y > u | VYc
> u − X (gu)}P{VYc > u − X (gu)}.
L(u) := P{VYc > u − X (u)} sup P{X(t) − ct > u − X (u)} t 0 ct u + u − X (u) Y (tu ) 1 (ct u + u)2 Y (tu ) =√ exp − 22 (tu ) 2 ct u + u − X (u) Y 2X (u) (ct u + u)X (u) × exp exp − 2 2Y (tu ) 2Y (tu ) × (1 + o(1)). On the other hand, from Theorem 1, R(u) := P{VYc > u} ct u + u f (u) (1 + o(1)) Y (tu ) 1 Y (tu ) (ct u + u)2 = f (u) √ exp − 22Y (tu ) 2 ct u + u × (1 + o(1))
(13)
Denote the right-hand side as I(u)II(u). Define (u) := inf{t : Y (t) − ct u − X (gu)}. Then c I(u) P{VX+Y > u; gu < (u) < ∞ | (u) < ∞}
P{X( (u)) > X (gu); gu < (u) < ∞ | (u) < ∞}
P{X(gu) > X (gu); gu < (u) < ∞ | (u) < ∞} = P{X(1) > 1}P{gu < (u) < ∞}, where the third inequality follows from the fact that 1(X(t) > X (gu)) is stochastically monotone in t. Since P{gu < (u) < ∞} ∼ P{ (u) < ∞} if g is sufficiently small (see Theorem 1 in Dieker [8]), we obtain lim inf u→∞ I (u) > 0. Combining this with the fact that reduced-load equivalence is assumed to hold (i.e. (6)) and (13), the desired statement follows.
as u → ∞, where f (u) is regularly varying at infinity with index 1 + 4(1 − Y /2)/Y . Hence, using that L(u)/R(u) is bounded and X (u) = o(u) as u → ∞, the index of the regularly varying function (ct u + u)X (u) 2Y (tu ) has to be non-positive. This implies X 2Y − 2, which completes the proof. Acknowledgements The authors are indebted to Ton Dieker for useful discussions and for showing us his preprint [8], which enabled us to shorten several proofs. References [1] R. Agrawal, A.M. Makowski, Ph. Nain, On a reduced load equivalence for fluid queues under subexponentiality, Queueing Syst. 33 (1999) 5–41.
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