ASYMPTOTICS OF THE AREA UNDER THE GRAPH OF A LÉVY-DRIVEN
WORKLOAD PROCESS
J. BLANCHET AND M. MANDJES
A BSTRACT. Let (Qt )t∈R be the stationary workload process of a Lévy-driven queue, where
the driving Lévy process is light-tailed. For various functions T (u), we analyze
!
Z T (u)
Qs ds > u
P
0
√
for u large. For T (u) = o( u) the asymptotics resemble those of the steady-state workload
√
√
being larger than u/T (u). If T (u) is proportional to u they look like e−α u for some
√
α > 0. Interestingly, the asymptotics are still valid when u = o(T (u)), T (u) = o(u), and
T (u) = βu for β suitably small.
1. I NTRODUCTION
Let (Xt )t∈R be a two sided (one-dimensional) Lévy process with X0 = 0. It is commonly
known that under the stability condition EX1 < 0 the stationary workload process, given
by Qt := sups∈(−∞,t] (Xt − Xs ), is well-defined. It is clear that
Z
1 T
(1)
Qs ds
T 0
converges to the steady-state mean workload EQ0 as T → ∞, as a direct consequence of
the ergodic theorem.
However, so far hardly any explicit results are available on the random variable (1). The
objective of the present paper is to shed light on this, by studying the large-deviations
probabilities
!
Z
T (u)
πT (u) (u) := P
Qs ds > u
0
for u large, and various types of functions T (u); the workload is assumed to be in stationarity at time zero.
The relevance of this study is that it exhibits an important example—one could even
argue that this is actually the most fundamental example that would come to the mind to
a queueing theorist—for which the standard Donsker-Varadhan large-deviations theory
is not directly applicable, even under natural assumptions. For instance, as we shall see,
if T (u) = uβ, with β < 1/EQ0 , then the asymptotics for πT (u) (u) are subexponential in u
Date: February 9, 2013.
Key words and phrases. Queues, workload process, area, large deviations, Lévy processes.
1
2
J. BLANCHET AND M. MANDJES
despite assuming that (Xt )t∈R is a well-behaved, light-tailed Lévy process. In this paper
we study the impact of the function T (·) on the tail asymptotics of (1).
More specifically, under natural large deviations conditions, we obtain the following results:
√
• For T (u) = o( u) the asymptotics resemble those of the steady-state workload
being larger than u/T (u):
u
.
log πT (u) (u) ∼ log P Q0 >
T (u)
(2)
Intuitively this result means that, in order to ensure that the area is larger than u,
essentially it is just required that Q0 is larger than u/T (u).
√
• For T (u) of the form T u, it is shown that
1
lim √ log πT (u) (u) = −α,
u→∞
u
for a constant α > 0 that is identified explicitly. For T small (that is, T below an
explicitly given threshold), the most likely way in which the rare event happens is
√
such that, with overwhelming probability, the buffer never idles in [0, T u]; for T
above the threshold the most likely path consists essentially of a single ‘big’ busy
period.
√
• Finally it is shown that (2) remains valid in case u = o(T (u)) and T (u) = o(u),
and also if T (u) = βu for β < 1/EQ0 .
The special case of (Xt )t∈R corresponding to Brownian motion was already covered in [1].
Many steps in the analysis presented in [1] used explicit properties of Brownian motion
that are not available for light-tailed Lévy processes. Indeed, one of the challenges of
the present paper was to find their counterparts for our more general light-tailed Lévy
setting. In addition, the case of T (u) = βu with β < 1/EQ0 was not covered in [1].
Also the paper [5] is strongly related to ours. There πT (u) (u) is analyzed for the number of
customers in the M/M/1 queue, with T (u) = βu with β < 1/EQ0 . At the methodological
level, the analysis presented in this paper borrows elements from both [1] and [5].
Our paper is organized as follows. In Section 2 the main objective is to find the asymptotics for the auxiliary object
Z t
1
%t := P 2
Xs ds > a ;
t 0
this result, which is is extensively used later in the paper, relies precisely on Assumption 1.
√
Then Section 3 covers the case that T (u) is small relative to u; the analysis relies on
straightforward bounds in combination with the classical bound P(Q0 > u) ≤ e−κu [3].
√
In Section 4, T (u) is taken proportional to u. Under the assumption that a sample-path
√
large deviations principle is valid, (2) is established. Section 5 covers the cases (i) u =
o(T (u)) and T (u) = o(u), and (ii) T (u) = βu with β < 1/EQ0 . In [5] it was shown that
the probability that the sum of γu Weibullian random variables, each of them behaving
AREA SWEPT UNDER THE WORKLOAD GRAPH
3
√
as e−α u , exceeding u, essentially behaves as a single of those Weibullians exceeding u (in
terms of logarithmic asymptotics). Relying on this property it is shown that (2) is valid
in this case as well. Finally, Section 6 we discuss how to relax some of the simplifying
assumptions that we imposed, and how to obtain results assuming non-stationary initial
conditions. Section 7 provides some technical large deviations results.
2. A SYMPTOTICS OF THE INTEGRAL OF A L ÉVY PROCESS
The main objective of this section is to study large deviations asymptotics for the auxiliary
quantity
Z t
1
Xs ds > a .
%t := P 2
t 0
In our future analysis of πT (u) (u), which is the main goal of the paper, only logarithmic
results for %t might suffice; however, because we believe that this quantity is of independent interest and because a sharp analysis is not difficult to perform, we provide exact
asymptotics.
Rt
First, define φ(ϑ) := log EeϑX1 and let Z (t) := t−1 0 Xs ds. Introducing the notation
ηt (ϑ) := E exp (ϑZ (t)) and applying integration by parts, we obtain
Z t
Z t
Z t
Xs ds = tXt −
sdXs =
(t − s) dXs ,
0
0
0
to conclude that
Z
ηt (ϑ) = exp
0
t
Z 1
t−s
ds = exp t
φ (ϑu) du .
φ ϑ
t
0
Consequently, we have that
(3)
1
χt (ϑ) := log ηt (ϑ) =
t
Z
1
φ (ϑu) du.
0
The quantities χt (ϑ) and ηt (ϑ) play an important role in the characterization of the exact
asymptotics of %t . In order to develop such asymptotics, we shall impose throughout the
rest of the paper the following assumption, which is fairly standard when developing
large deviation estimates.
Assumption 1. (Steepness to the right): If φ (θ) = log(EeθX1 ), then for every a > EX1 there
exists θ∗ > 0 such that φ0 (θ∗ ) = a.
Using expression (3) we can obtain logarithmic asymptotics (via an application, for instance, of the Gärtner-Ellis theorem, see [8]) leading to
Z 1
1
(4)
lim log %t = − sup aϑ −
φ(ϑx)dx =: −J(a).
t→∞ t
ϑ≥0
0
R1
The convexity of φ (·) implies the convexity of ϑ 7→ 0 φ (ϑx) dx, which in turn implies
(together with the fact that φ(0) = 0, and that a > φ0 (0)/2) that the supremum in the
previous display is the same if one optimizes over ϑ ∈ R. Moreover, it also follows
that conditions for local optimality imply global optimality in the optimization problem
4
J. BLANCHET AND M. MANDJES
underlying the definition of J(a). The next lemma, therefore, shows that there is a unique
optimizer ϑ? to the previous optimization problem.
Lemma 1. For every a > EY /2, there exists ϑ? > 0 such that
Z 1
a=
φ0 (ϑ? x)x dx.
0
Proof: Because of the monotone convergence theorem, it follows that for ϑ ∈ (0, θ∗ ),
Z 1
Z 1
d
φ0 (ϑx)x dx,
φ(ϑx)dx =
dϑ 0
0
where φ0 (·) is the derivative of φ (·). In turn, we have that
Z 1
Z ϑ
1
0
φ (ϑx)dx = 2
φ0 (y)y dy.
ϑ
0
0
We must show that there exists a unique solution to the equation
Z ϑ
2
0
(5)
ϑ a − φ (0)/2 =
(φ0 (y) − φ0 (0))y dy.
0
By the strict convexity of φ, which follows because Y is a non-degenerate random variable
in view of Assumption 1, we have that φ0 (y) − φ0 (0) > 0 if y > 0. Moreover, both the
right-hand side and the left-hand side of Eqn. (5) are convex functions of ϑ. The derivative
of the left-hand side is larger than the derivative of the right-hand side for values of ϑ that
are sufficiently close to zero, but because of Assumption 1 eventually the derivative of the
right-hand side increases superlinearly in ϑ. So, eventually the right hand side overtakes
the left-hand side. By continuity, thus, there exists a solution to Eqn. (5). The solution
must be unique because the optimization problem underlying the definition of J (a) is a
strictly concave program.
2
Now we are ready to sharpen the large deviations result (4).
Theorem 1. Suppose, in addition to Assumption 1, that φ (θ) < ∞ for θ in a neighborhood of the
origin. Define
Z 1
2
σ :=
φ00 (ϑ? x)x2 dx.
0
Then, as t → ∞,
1
1
%t := P (Z (t) > at) ∼ √ √
e−tJ(a) .
t 2πϑ? σ
Proof of Thm. 1: According to [6, Thm. 3.3] we need to verify the following properties.
First, it should be checked that χt (·) is analytic, for every t > 0, in a neighborhood of the
R1
origin. This property is immediate from the fact that χt (·) = 0 φ (·u) du is independent
of t, and φ (·) is clearly analytic around the origin since we are assuming the existence of
exponential moments of X1 in a neighborhood of the origin. Second, we need to show
AREA SWEPT UNDER THE WORKLOAD GRAPH
5
that χ00t (ϑ? ) > 0. Applying ‘dominated convergence’ (which is justified due to Assumption 1) we obtain that
Z 1
φ00 (ϑ? x)x2 dx > 0;
σ 2 = χ00t (ϑ? ) =
0
the inequality follows because X1 cannot be a degenerate random variable (i.e., a deterministic quantity), once again, due to Assumption 1. Finally, the third and last property
that we need to verify is that that there exists δ1 such that for any δ0 , λ > 0, with the
property that 0 < δ0 < δ1 < λ,
|ηt (ϑ? + iθ)|
=o
lim sup
t→∞ δ ≤|θ|≤λ
|ηt (ϑ? )|
0
1
√
t
as t → ∞, where |z| denotes the modulus of a complex number z (so if z is a real number
the definition corresponds to the absolute value). To this end, first note that ηt (ϑ? +
i ·)/ηt (ϑ? ) is simply the characteristic function of a member of the natural exponential
family of Z(t), corresponding to the natural parameter ϑ? . So, it suffices to show that for
any random variable such as Z(t) one can find δ1 > 0 such that for any δ0 , λ > 0 satisfying
0 < δ0 < δ1 < λ,
(6)
sup
δ0 ≤|θ|≤λ
Z
exp
1
0
φ(iθx)dx < 1.
It is now noted that this property holds, as can be seen as follows. Suppose first that X1
has a lattice distribution with span h. If 0 < δ0 < δ1 < min (π/h, 1) then we have that
ξ (δ0 , δ1 ) :=
sup
| exp (φ(iθ)) | < 1.
δ02 ≤|θ|≤δ12
Rw
Now, observe that for any v < w, the quantity exp( v φ(iθx)dx) can be interpreted as
Rw
the characteristic function of the random variable v sdXs , and hence such quantity has
modulus less than or equal to unity. Consequently, if 0 < v < w < 1, then
Z
exp
(7)
0
1
Z
φ(iθx)dx ≤ exp
v
w
φ(iθx)dx .
Now, given δ0 , λ > 0, select w < δ1 /λ and v = δ0 , to conclude that if |θ| ∈ [δ0 , λ], then
Z
exp
v
(8)
w
n Y
w − v exp φ iθ v + (w − v) k
φ(iθx)dx = lim
n→∞
n
n
k=0
≤ξ
w−v
< 1.
Inequalities (7) and (8) together imply (6) and thus the result. The case in which X1 is
even easier as in this case one can select δ1 = 1.
2
6
J. BLANCHET AND M. MANDJES
3. S HORT TIMESCALE
From now on we consider the queueing model described in the introduction; in this sec√
tion we concentrate on the short timescale regime in which T (u) = o( u). Throughout
the rest of the paper, in addition to Assumption 1, we shall impose the following standard
large deviations assumptions.
Assumption 2. (Cramér condition) There is κ > 0 such that E exp (κX1 ) = 1.
Under Assumption 2, the celebrated Cramér-Lundberg asymptotics hold, meaning that the
tail probability P(Q0 > u) decays exponentially in u. More precisely, as follows from [3],
with the κ introduced in Assumption 2, the asymptotics take the form, as u → ∞,
log P(Q0 > u) = −κu (1 + o (1)) .
In addition, we impose the following simplifying assumption.
Assumption 3. There exists δ < 0 such that E exp (δ |X1 |) < ∞.
As we shall discuss in Section 6, the results in this paper can be derived only under Assumptions 1 and 2. Assumption 3 allows us to take advantage of sample-path large deviations, a concept that we shall review now.
With the local rate function given by I(x) := supϑ (ϑx − φ(ϑ)), we define the sample-path
rate function I(·), for T > 0, by
Z T
I(f ) =
I(f 0 (t))dt
0
for any absolutely continuous f , and ∞ else. A sample-path large deviations principle
then says that for any closed A
1
X· u
lim sup log P
∈ A ≤ − inf I(f ),
f ∈A
u
u→∞ u
and for any open B
1
X· u
lim inf log P
∈ B ≥ − inf I(f ).
u→∞ u
f ∈B
u
To determine whether a given set is open or closed we use the norm
||f || :=
|f (s)|
.
1
s∈(−∞,∞) + |s|
sup
We state the following technical result whose proof is given in the appendix at the end of
this paper.
Lemma 2. The sample-path large deviations principle is valid under the norm ||f ||.
It will also be useful to keep in mind the representation
(9)
Qt = max Q0 + Xt , max (Xt − Xs ) .
0≤s≤t
AREA SWEPT UNDER THE WORKLOAD GRAPH
7
Recall that κ > 0 solves EeκX1 = 1, and, as earlier,
Z 1
J(a) := sup aϑ −
φ(ϑx)dx .
ϑ>0
0
√
The main result of this section states that if T (u) = o( u), the probabilities πT (u) (u) and
P(Q0 > u/T (u)) are equivalent on a logarithmic scale.
√
Theorem 2. If T (u) = o( u), then
T (u)
log πT (u) (u) = −κ.
u
Lower bound: Using the first term in the right-hand side of (9),
!
Z T (u)
Xs ds > u .
πT (u) (u) ≥ P Q0 T (u) +
lim
u→∞
0
Under the light-tailed conditions, we have
T (u)
log P (Q0 T (u) > u) = −κ,
u
where κ > 0 solves φ(κ) = 0. Also, using Chernoff’s bound and our expression for J (·)
derived in the previous section we obtain
!
Z T (u)
u
.
P
Xs ds > u ≤ exp −T (u)J
(T (u))2
0
lim
u→∞
Hence, to prove the lower bound, it suffices to prove that for u sufficiently large
u
u
κ
.
<J
(T (u))2
(T (u))2
This follows from the strict convexity of J(·) in conjunction with u/(T (u))2 → ∞ as u →
∞. This completes the proof of the lower bound.
Upper bound: Note that, due to the union bound and the stationarity of the workload
process,
!
Z 2
Z T (u)
Z 1
Qs ds +
Qs ds + ... +
Qs ds > u
πT (u) (u) ≤ P
0
bT (u)c
1
≤ (T (u) + 1) P ∃t ∈ [0, 1] : Qt ≥
u
T (u)
,
so that it suffices to prove that
T (u)
u
lim sup
log P ∃t ∈ [0, 1] : Qt ≥
≤ −κ.
u
T (u)
u→∞
Now, from (9) we obtain that
u
u
P ∃t ∈ [0, 1] : Qt ≥
≤ P ∃t ∈ [0, 1] : Q0 + Xt ≥
T (u)
T (u)
+ : P ∃t ∈ [0, 1] : max (Xt − Xs ) ≥
0≤s≤t
u
T (u)
.
8
J. BLANCHET AND M. MANDJES
We now analyze both probabilities in the right-hand side of the previous display in detail.
• Note that for each ε > 0 we have that
u
P ∃t ∈ [0, 1] : Q0 + Xt ≥
T (u)
u
u
≤ P Q0 ≥ (1 − ε)
+ P max Xt ≥ ε
.
0≤t≤1
T (u)
T (u)
(10)
We know that, by Assumption 2,
T (u)
u
lim
= −κ (1 − ε) .
log P Q0 ≥ (1 − ε)
u→∞ u
T (u)
In addition, with n(u) := u/T (u) and δ > 1/n(u),
Xn(u)·t
u
P max Xt ≥ ε
≤ P max
≥ε .
0≤t≤1
0≤t≤δ n(u)
T (u)
Because of Lemma 2,
1
log P
lim sup
u→∞ n(u)
Xn(u)·t
max
≥ε
0≤t≤δ n(u)
= −δI
ε
δ
.
By the strict convexity of I(·), we have that I(x/δ) ≥ κx/δ for δ sufficiently small.
Combining this with (10), we conclude that
T (u)
u
lim sup
log P ∃t ∈ [0, 1] : Q0 + Xt ≥
≤ −κ.
u
T (u)
u→∞
• On the other hand, with n(u) = u/T (u) for any δ > 1/n(u), we note that
u
P ∃t ∈ [0, 1] : max (Xt − Xs ) ≥
0≤s≤t
T (u)
u
= P max max (Xt − Xs ) ≥
T (u)
t∈[0,1] 0≤s≤t
Xn(u)·t − Xn(u)·s
≤ P max max
≥1 .
0≤t≤δ 0≤s≤t
n(u)
By Lemma 2,
1
lim sup
log P
u→∞ n(u)
Xn(u)·t − Xn(u)·s
1
max max
≥ 1 = −δI
.
0≤s≤t
0≤t≤δ
n(u)
δ
As δ > 0 can be chosen arbitrarily small, the convexity of I(·) entails that δI(1/δ)
can be made arbitrarily large and we conclude the proof.
2
4. I NTERMEDIATE TIMESCALE
In this section we analyze
!
Z T u
Z T √u
1
1
2
lim √ log P
Qs ds ≥ M u = lim log P
Qs ds ≥ M u .
u→∞
u→∞ u
u
0
0
AREA SWEPT UNDER THE WORKLOAD GRAPH
9
The probability in the right-hand side of the previous display can be rewritten immediately in terms of an expressions that can be analyzed using a sample-path large deviations
principle for Lévy processes. Indeed, observe that
Z T u
Z T u
2
2
P
Qs ds ≥ M u = P
sup(Xs − Xr )ds ≥ M u
0
r≤s
0
T
Z
=P
0
sup (Xtu − Xr )dt ≥ M u
r≤tu
Z T
1
sup (Xtu − Xr )dt ≥ M
=P
u 0 r≤tu
Z T
1
X· u
=P
Qtu dt ≥ M = P
∈A ,
u 0
u
with
(11)
A :=
Z
T
sup (f (t) − f (s))dt ≥ M
f:
0
.
0≤s≤t
We can now state the main result of this section.
Theorem 3.
1
log P
u→∞ u
Z
lim
Tu
Qs ds ≥ M u2
M − as
κa + sJ
.
a≥0 s∈[0,T ]
s2
= − inf inf
0
To prove this theorem, we first need to show an auxiliary result. To this end, define
Z Tu
pu (T, M, a) := P
Qs ds ≥ M u Q0 = au .
0
Lemma 3.
1
log pu (T, M, a) ≤ − inf sJ
u→∞ u
s∈[0,T ]
lim
M − as
s2
.
Proof: The proof of this lemma is very similar to the proof of the Brownian case in [1], and
we therefore just sketch the most important steps. First it is observed that A (as defined
in (11)) is closed; see the proof of the second part in the appendix of [1]. As a result,
1
log pu (T, M, a) ≤ − inf I(f ).
u→∞ u
f ∈A
lim
Now take an arbitrary path f in A (where f is absolutely continuous), we construct f¯ as
follows. Define q[f ](t) := sup0≤s≤t (f (t) − f (s)); f represents a trajectory of the input to
the queue in fluid scale. We have subintervals of [0, T ] in which the queue is empty (idle
periods), and subintervals in which it is non-empty. Clearly there are at most countably
many idle periods because the total length of time the queue is empty is less than T . Now,
permute these such that all busy periods are moved to the front, and the idle periods to
the back. For example, if f (s) = min(s, 2 − s)+ + min(s − 4, 6 − s)+ , then f¯ (s) = min(s, 2 −
s)+ + min(s − 2, 4 − s)+ . The corresponding transformed input path, which we call f¯,
has the property that I(f ) = I(f¯); to see this, the essential property here is that (Xt )t∈R
10
J. BLANCHET AND M. MANDJES
has stationary and independent increments, which yields that the large deviations rate
function of a trajectory f just depends on the derivative of the path.
As a result,
f (r)dr ≥ M − as .
f : ∃s ∈ [0, T ] :
inf I(f ) = inf I(f ), with M̄ :=
f ∈A
s
Z
f ∈M̄
0
Evidently,
inf I(f ), with M̄s :=
inf I(f ) = inf
f ∈M̄
Z
s
f (r)dr ≥ M − as .
f:
s∈[0,T ] f ∈M̄s
0
But, due to the sample-path large deviations principle,
1
inf I(f ) = lim log P
u→∞
u
f ∈M̄s
Z
0
s
M − as
,
Xtu dt ≥ (M − as)u = −sJ
s2
2
where the last equality is due to Thm. 1.
Proof of Thm. 3. The proof consists of two bounds. The lower bound is straightforward,
whereas the upper bound relies on Lemma 3.
Lower bound: The starting point is, for any s ∈ [0, T ],
Z T
Z s
Z s
1
1
P
Qtu dt ≥ M ≥ P
Qtu dt ≥ M ≥ P Q0 s +
Xtu dt ≥ M u .
u 0
u 0
0
Evidently, this majorizes, for any a ≥ 0,
Z
P (Q0 ∈ [a, a + ε)u) P
s
Xtu dt ≥ (M − as)u .
0
The exact asymptotics [3] of the first factor look like Ce−κau (1 − e−κεu ), for some C > 0,
which has logarithmic asymptotics −κa. The second factor obeys, due to Thm. 1,
1
lim log P
u→∞ u
Z
0
s
M − as
Xtu dt ≥ (M − as)u = −sJ
.
s2
As the lower bounds hold for any s ∈ [0, T ] and a ≥ 0, we obtain
1
lim inf log P
u→∞ u
Z
Tu
2
Qs ds ≥ M u
0
≥ − inf inf
a≥0 s∈[0,T ]
κa + sJ
M − as
s2
.
AREA SWEPT UNDER THE WORKLOAD GRAPH
11
Upper bound: Clearly,
Z T u
2
P
Qs ds ≥ Mu
0
≤
∞
X
P(Q0 ∈ [kε, (k + 1)ε)u)pu (T, M, (k + 1)ε)
k=0
≤
N
−1
X
P(Q0 ≥ kεu)pu (T, M, (k + 1)ε) +
k=0
≤
N
−1
X
=
P(Q0 ≥ kεu)
k=N
−κ·kεu
e
pu (T, M, (k + 1)ε) +
k=0
N
−1
X
∞
X
∞
X
e−κ·kεu
k=N
e−κ·kεu pu (T, M, (k + 1)ε) +
k=0
e−κ·N εu
.
1 − e−κ·εu
Applying [8, Lemma 1.2.15], it follows that the decay rate of interest is bounded from
above by
1
max
max
lim log pu (T, M, (k + 1)ε) − κ · kε, −κ · N ε .
k=0,...,N −1 u→∞ u
As a result of Lemma 3, we can bound the above decay further by
M − (k + 1)εs
max
max
− inf sJ
− κ · kε, −κ · N ε
k=0,...,N −1
s2
s∈[0,T ]
M − as
≤ − min inf inf sJ
+ κa − κε, κ · N ε .
a≥0 s∈[0,T ]
s2
The upper bound then follows after letting N ↑ ∞, and then ε ↓ 0.
Remark 1. The decay rate
inf inf
a≥0 s∈[0,T ]
κa + sJ
M − as
s2
2
can be analyzed in more detail. Define T ? as the solution to
M
0
,
κ=J
T2
which exists because J 0 (·) is increasing (following from the fact that J(·) is strictly convex
by Assumption 2), and continuous
Z 1
0
J (0) := arg inf
φ(ϑx)dx < κ.
ϑ≥0 0
Elementary analysis yields that for T ∈ [0, T ? ), the infimum over s is attained at s? = T .
Then the optimal a follows from optimizing
M − aT
,
inf κa + T J
a≥0
T2
12
J. BLANCHET AND M. MANDJES
which is a convex function in a. It is seen that, due to the inequality T < T ? , the maximum
is attained for a positive a? ; this follows from the definition of T ? . The intuition behind
this solution is that the most likely path is such that there is, most likely, already a positive
buffer content at time 0, and the buffer remains non-idle during the entire interval [0, T ].
For T ≥ T ? , the optimum is achieved at a? = 0 and T = T ? . In this case, with overwhelming probability, the buffer content starts off at a value near to 0 at time 0, to become idle
again at roughly T ? .
2
5. L ONG TIME - SCALE
In this section we prove the following result.
√
Theorem 4. If either (i) u = o(T (u)) and T (u) = o(u), or (ii) T (u) = βu for β < 1/EQ, then
!
Z T (u)
1
1
?
.
lim √ log P
Qr dr > u = −T J
u→∞
(T ? )2
u
0
Proof of Thm. 4. We argue the lower and upper bound separately.
Lower bound: The lower bound is trivial, as we have that
!
!
Z T (u)
Z T √u
Qr dr > u ;
P
Qr dr > u ≥ P
0
0
applying the results of Section 4 yields the claim.
Upper bound: The upper bound is more involved, but can be established by applying
√
techniques developed in [1] and [5]. Let us first consider case (i), that is, u = o(T (u))
and T (u) = o(u). Replace the process Qr by a process Q̄r , as follows. Start in stationarity,
and let Q̄r equal Qr until the buffer idles, say at time τ. Then let Q̄τ be sampled from
the stationary distribution, and let it follow the workload process from then on; again, as
soon as it hits zero, it is replaced by a sample from the stationary distribution, etc. Clearly,
Q̄r ≥ Qr , and with N (u) the number of these i.i.d. cycles in T (u), we have that
Z
N (u)
T (u)
Qr dr ≤
0
X
Hi ,
i=1
with the Hi distributed as a random variable H such that
Z τ
d
H=
Qr dr.
0
With the techniques used in the proof of Lemma 3, we can prove that
1
M
?
lim √ log P (H > M u) ≤ −T J
.
u→∞
(T ? )2
u
Similar to the upper bound in the proof of [1, Thm. 3], we have, with K := 2/Eτ ,




N (u)
dKT (u)e
X
X
P
Hi > u ≤ P 
Hi > u + P (N (u) ≥ bKT (u)c) .
i=1
i=1
AREA SWEPT UNDER THE WORKLOAD GRAPH
13
Due to [5, Prop. 1],


dKT (u)e
1
lim √ log P 
u→∞
u
X
Hi > u ≤ −T ? J
i=1
1
(T ? )2
.
Also, P (N (u) ≥ bKT (u)c) decays exponentially. This implies the upper bound.
We now consider case (ii), that is, T (u) = βu for β < 1/EQ. Replace the process Qr by a
(ε)
(ε)
process Q̄r , as follows; fix an ε > 0. Start in stationarity, and let Q̄r equal Qr until the
(ε)
buffer idles, say at time τ. Then let Q̄τ be set to ε, and let it follow the workload process
from then on with the same driving Lévy process as Qr ; again, as soon as it hits zero, it is
(ε)
(ε)
set to ε, etc. Clearly, Q̄r ≥ Qr , but, in addition, 0 ≤ Qr − Qr ≤ ε for all r, which also
entails that 0 ≤ EQ(ε) − EQ ≤ ε.
Now the proof of case (i) can be mimicked, the only term to be still taken care of is, with
K := (1 + δ)/Eτ (ε) ,


dKT (u)e
P H +
X
(ε)
Hi
> u ,
i=1
(ε)
with H distributed as the Hi in case (i), and the Hi corresponds to the area under the
(ε)
graph of the workload process between two subsequent epochs in which Qr is empty.
(ε)
(ε)
(ε)
Define H̄i := Hi − EHi . Then the above probability reads


dKT (u)e
(ε)
X
EH
(ε)
.
P H +
H̄i > u − βu(1 + δ)
(ε)
Eτ
i=1
Choosing δ, ε sufficiently small, the requirement β < 1/EQ entails that
1 − β(1 + δ)
EH (ε)
> 0,
Eτ (ε)
realizing that EH (ε) /Eτ (ε) → EQ. Then the upper bound follows as in the proof of Theorem 1 in [5].
2
6. F INAL R EMARKS
We note that we can dispense from Assumption 3 as follows. Suppose that X· only satisfies Assumptions 1 and 2. Let us consider the standard decomposition
Xt = µt + σBt + Yt ,
where (Yt )t∈R is a pure jump process independent of the Brownian motion (Bt )t∈R , µ is a
drift parameter and σ ≥ 0 is a volatility parameter. Note that by truncating the negative
jumps of (Yt )t∈R (i.e., keeping only those whose size is smaller c) we obtain a process
(Qct )t∈R such that Qct ≥ Qt . The whole large deviations analysis can be performed under
(Qct )t∈R , which now satisfies Assumptions 1, 2, and 3. Then finally one can let c → ∞.
It is natural to consider, particularly in the regime studied in Section 5, the asymptotics
assuming that Q0 = 0. It is easily shown that under Assumptions 1 and 2 the asymptotic
14
J. BLANCHET AND M. MANDJES
results derived in Section 5 remain unchanged. To see this note that the analysis obtained provides a correct upper bound for the asymptotics (by monotonicity, bearing in
mind that starting in stationarity dominates the zero initial condition). To show the lower
bound one simply isolates a path that falls within the event under consideration. Note
√
that since a? = 0, as discussed in Remark 1, one can consider a path for which Q0 ≤ u
and the area reaches level βu in the first busy period. Since is arbitrary one thus obtain
that the large deviations asymptotics hold also for the case Q0 = 0 as claimed.
7. A PPENDIX : P ROOF OF L EMMA 2
We start by quoting the following result due to [7] which is immediately applicable under
Assumptions 2 and 3.
Lemma 4. If there exists δ > 0 such that E exp (δ |X1 |) < ∞, then the process Xu· /u satisfies a
large deviations principle under the topology of convergence on compact sets.
We need to extend the previous result to accommodate the topology that we use, namely
that generated by the norm ||f || = sup0≤t<∞ |f (t)| / (1 + t). Let us define L1 [0, ∞) to be
the space of continuous functions endowed with the topology generated by the norm ||·||.
A sequence of probability measures (Pn )n∈N is exponentially tight if for every λ > 0 there
exists a compact set Kλ such that
1
log Pn (Kλ ) ≤ −λ.
n→∞ n
lim sup
Exponential tightness is a crucial concept in order to lift the upper bound in the samplepath large deviations principle from compact sets to all closed sets. The next result characterizes exponential tightness in L1 [0, ∞).
Lemma 5. Consider a sequence of probability measures (Pn )n∈N on L1 [0, ∞) (which are such
that Pn (x : x (0) = 0) = 1) and acting on the Borel sigma-field corresponding to the topology
generated by the norm k·k. Then, (Pn )n∈N is exponentially tight if and only if: a) (Pn )n∈N is
exponentially tight under the relative topology generated by uniform convergence on compact sets
(called Stone’s topology), and b) for each δ > 0,
1
|x (t)|
(12)
limn→∞ log Pn x : sup
> δ → −∞ as t0 → ∞.
n
t
t≥t0
Proof. [10, Lemma 3.3] establishes that relatively compact sets in (L1 [0, ∞), k·k1 ) are those
sets B with compact closure under the relative Stone topology, and satisfying
lim sup
t→∞ x∈B
|x (t)|
= 0.
t
Also, recall that (if x (0) = 0 a.s. with respect to each Pn ) for exponential tightness under
Stone’s topology, it is necessary and sufficient (see [9, p. 30]) that, for each ε, T > 0,
(13)
limn→∞
1
log Pn (x : ω (x, δ, T ) > ε) → −∞ as δ ↓ 0,
n
AREA SWEPT UNDER THE WORKLOAD GRAPH
15
where ω (x, δ, T ) is the modulus of continuity of x, on the interval [0, T ], evaluated at δ.
We now show that if conditions (12) and (13) are satisfied, then the sequence (Pn )n∈N is
exponentially tight. Pick λ > 0, choose δk so that
Pn (x : ω (x, δk , T ) > 1/k) ≤ e−nλ /2k+1 ,
and let Bk = {x : ω (x, δk , T ) ≤ 1/k}. Also, pick tk so that
|x (t)|
Pn x : sup
> 1/k ≤ e−nλ /2k+1 ,
t
t≥tk
and let Ck = {x : supt>tk |x (t)| /t ≤ 1/k}. Consider the closure, Āλ , of Aλ := ∩k (Bk ∩ Ck ).
Note that
1 − P Āλ ≤ 1 − P (Aλ ) = P (∪k (Bkc ∩ Ckc )) ≤ e−nλ
We now claim that Aλ is relatively compact (i.e., that Āλ is compact). To see this, choose
ε > 0 and let k0 > 1/ε. Then, for all δ < δk0 we have that
sup ω (x, δ, T ) < ε.
x∈A
Similarly, for every T > tk0 we have that
ε > sup sup
x∈A t>T
|x (t)|
,
t
which implies that
|x (t)|
≤ε
t
x∈A
limt→∞ sup
for all ε > 0. Thus, by virtue of the Arzelà-Ascoli theorem (see [4, p. 81] and [10, Lemma
3.3]), we conclude the argument for sufficiency. The necessity part is easier and it follows
just as in [9, p. 30]. Therefore, it is omitted.
We now are ready to provide the proof of Lemma 2.
Proof of Lemma 2. Using a discrete polygonal approximation similar to the one used in
the proof of [7, Thm. 1.2], we can introduce a continuous exponential approximation to
Xu· /u. So, we might assume that the underlying space is L1 [0, ∞). In view of Lemma 4,
in order to lift the large deviations result to the topology generated by k·k, it suffices to
verify, according to [8, Corollary 4.2.6], that X·u /u is exponentially tight. Because the rate
function I (·) is known to have compact level sets, Lemma 4 implies exponential tightness
of X·u /u under the uniform topology on compact sets. Because of Lemma 5 it then suffices
to verify that, for any δ > 0,
Xut /u − tEX1 lim sup log P sup ≥ δ → −∞ as t0 → ∞.
t
u→∞
t>t0
Note that for any 0 < a < b < ∞, the mapping x 7→ supt∈[a,b] |x (t) /t| is continuous under
the topology generated by k·k. This implies that the family Vu = supt∈[a,b] |Xtu /u − tEX1 |
16
J. BLANCHET AND M. MANDJES
satisfies an LDP with rate function H (·), say. Hence, we can write
!
X
∞
Xtu /u − tEX1 Xtu /u − tEX1 ≥δ ≤
≥δ
P sup P
sup t
t
t>t0
t∈t
[k,k+1]
0
k=1


∞
∞
X
X
Xrkt0 u /u − rkt0 EX1 ≥ δ =
≤
P  sup exp (− (H (δ) + okt0 u (1)) kt0 u) ,
rkt0
t
k=1
r= kt ∈[1,2]
k=1
0
where the subindex in okt0 /α (1) has been used just to emphasize that okt0 u (1) → 0 as
kt0 u → ∞. So we can choose k0 big enough so that for every k > k0 we have H (δ) +
okt0 /α (1) > H (δ) /2 > 0. From these estimates the statement of the lemma follows.
Acknowledgement: We thank the Editors and the Referees for their careful reviews and
their useful recommendations. Blanchet acknowledges support from the NSF through
the grants CMMI-0846816 and 1069064.
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C OLUMBIA U NIVERSITY, D EPARTMENT OF I NDUSTRIAL E NGINEERING & O PERATIONS R ESEARCH , 340 S.
W. M UDD B UILDING , 500 W. 120 S TREET, N EW Y ORK , NY 10027, U NITED S TATES .
E-mail address: [email protected]
K ORTEWEG - DE V RIES I NSTITUTE FOR M ATHEMATICS , U NIVERSITY OF A MSTERDAM , THE N ETHERLANDS ;
E URANDOM , E INDHOVEN U NIVERSITY OF T ECHNOLOGY, THE N ETHERLANDS ; CWI, A MSTERDAM , THE
N ETHERLANDS
E-mail address: [email protected]