Queueing Syst (2013) 73:425–446DOI 10.1007/s11134-013-9349-7

Large deviations for the empirical mean of an M/M/1queue

Jose Blanchet · Peter Glynn · Sean Meyn

Received: 12 March 2011 / Revised: 14 June 2012 / Published online: 14 March 2013© Springer Science+Business Media New York 2013

Abstract Let (Q(k) : k ≥ 0) be an M/M/1 queue with traffic intensity ρ ∈ (0, 1).

Consider the quantity

Sn(p) = 1

n

n∑

j=1

Q ( j)p

for any p > 0. The ergodic theorem yields that Sn(p) → μ(p) := E[Q(∞)p], whereQ(∞) is geometrically distributed with mean ρ/(1 − ρ). It is known that one canexplicitly characterize I (ε) > 0 such that

limn→∞

1

nlog P

(Sn(p) < μ (p) − ε

) = −I (ε) , ε > 0.

In this paper, we show that the approximation of the right tail asymptotics requires adifferent logarithm scaling, giving

limn→∞

1

n1/(1+p)log P

(Sn(p) > μ

(p) + ε

) = −C(

p)ε1/(1+p),

J. Blanchet (B)Columbia University, New York, NY 10027, USAe-mail: jose.blanchet@columbia.edu

P. GlynnStanford University, Stanford, CA 94305, USA

S. MeynUniversity of Florida, Gainesville, FL 32611, USA

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426 Queueing Syst (2013) 73:425–446

where C(p) > 0 is obtained as the solution of a variational problem.We discuss why this phenomenon—Weibullian right tail asymptotics rather than expo-nential asymptotics—can be expected to occur in more general queueing systems.

Keywords Large deviations · Markov random walks · M/M/1 queue

Mathematics Subject Classification 60K25 · 60F10

1 Introduction

The theory of large deviations for random walks is important both in its own right andas a starting point for establishing large deviations for more complex models (suchas “small noise” diffusions and “slow Markov random walks”); see, for example,[7,10,15]. Furthermore, because of the close connection between queues and randomwalks, a good understanding of the large deviations for random walks is an essentialingredient in building a successful large deviations theory for queues.

A natural extension to the existing theory of large deviations for random walksis that of Markov random walks (i.e., sums of increments that are a function of aMarkov chain). As in the conventional independent and identically distributed (i.i.d.)setting, such large deviations computations are of both intrinsic interest, and as a keyingredient in the theory of queues that are fed by exogenous Markov-dependent inputsequences (e.g., Markov modulated arrival streams). Our goal in this paper is to studya qualitatively new phenomenon that can arise in the context of such Markov randomwalks, in particular those with unbounded increments. Our results provide an extensionof the analysis considered in [8]; see also [1] for additional asymptotic regimes in thesetting of reflected Brownian motion. As in [8], we focus our attention on the randomwalk generated by the number in system of the M/M/1 queue.

Let Q = (Q(k) : k ≥ 0) be the embedded discrete time Markov chain correspond-ing to the number in system of an M/M/1 queue with arrival rate λ and service rateμ; we assume that ρ = λ/μ ∈ (0, 1) so that the stability of the system is ensured.Define

Sn(p) :=n∑

j=1

Q ( j)p

for any p > 0. This class of integrated processes are, of course, specials case ofMarkov random walks of the form

Sn :=n∑

j=1

f(X j

),

where f ( · ) is real-valued and unbounded and X = (X j : j ≥ 0) is a suitably reg-ular Markov process (e.g., geometrically ergodic). We will now provide a heuristicexplanation as to why the large deviations theory for Markov random walks in this

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Queueing Syst (2013) 73:425–446 427

unbounded setting will often look fundamentally different both from the large devia-tions for conventional i.i.d. random walks and for Markov random walks with boundedincrements. Recall that in the i.i.d. light-tailed setting, the most likely way in whicha large deviation of order n will occur for a random walk involving n increments isthrough “conspiratorial” behavior among the increments that persists over essentiallythe entire time interval of length n, and its corresponding probability is roughly expo-nential in n. In fact, and more precisely, the Gibbs conditioning principle asserts thatthe random walk continues to have i.i.d. increments under the conditioning with amarginal distribution that is a suitably exponentially twisted version of the originalmarginal distribution; see [5], Sect. 3.3. A similar picture typically asserts itself in theMarkov random walk setting when f ( · ) is bounded. In particular, when Sn exhibits alarge deviation from its equilibrium mean of order n, the conditional behavior of theprocess over essentially the entire interval of length n follows that of a new Markovprocess for which the transition function is an exponentially twisted version of theoriginal underlying transition function of the process, and its corresponding proba-bility is roughly exponential in n. This exponential twist is determined by solving asuitable eigenvalue problem; see [14] for the details in the finite state space case and,for example, [11] for a discussion of general state spaces. At an intuitive level, thefact that the probability is exponential in n follows from the observation that becausef ( · ) is bounded, the probability distribution of roughly n increments (that are essen-tially independent due to mixing) must be modified, each increment providing a O(1)

contribution to the probability.On the other hand, when f ( · ) is unbounded from above, a large deviation of order

n from the steady-state mean can be achieved in o(n) time steps. Because of thecorrelation that is present in Markov processes, this suggests that the probability ofsuch a large deviation will be exponential with an exponent that is o(n). The intuitionmight be explained as follows. The conditional dynamics of X that achieve the largedeviation might not typically involve modifying the dynamics of the process over theentire time interval [0, n] and will take advantage of the fact that the “cost” of pushingX into a region having high f -values can be amortized over the entire time scaleover which X then relaxes back to equilibrium, during which further contributionsto the large deviation event will be accumulated. Furthermore, if f ( · ) is unboundedfrom above, but bounded from below, the intuition explained at the end of the pre-vious paragraph suggests that the large deviations that are smaller than normal willbe qualitatively identical to those in the setting in which f ( · ) is bounded (e.g., largedeviations probabilities that are exponential in n), while only the large deviations thatare larger than typical will exhibit the qualitatively new features that can arise in theunbounded (from above) setting (which is strikingly different from what is manifestedin the theory of large deviations for random walk in the conventional context). Thus, anumber of qualitatively new features manifest themselves in the setting of unboundedf ( · ).

The M/M/1 queue and the associated Markov random walks S(p) = (Sn(p) : n ≥0) form an excellent vehicle for illustrating these qualitatively new behaviors. Since wehave assumed that the M/M/1 queue is stable, we have Q(n) �⇒ Q(∞) as n → ∞,where Q(∞) is geometric with mean ρ/(1 − ρ), so that μ(p) := E Q(∞)p < ∞ foreach p > 0. Note first that f p(x) := |x |p is bounded below, so our above discussion

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428 Queueing Syst (2013) 73:425–446

suggests that large deviations for Sn(p) that are order n smaller than nμ(p) will followthe same pattern as in the setting of bounded f ( · ). This has previously been verified(see [12], following [11]), in which it was established that for ε > 0,

limn→∞

1

nlog P0 (Sn(p)/n < μ(p) − ε) = −I (ε) , (1)

where I (ε) > 0 can be explicitly computed. On the other hand, for large deviationsthat are larger than nμ(p), observe that the geometric stationary distribution for Qimplies that the probability of exceeding nr within a busy cycle is exponential in aterm of order nr , thereby contributing an f -value to the Markov random walk Sp oforder n pr . Once Q has achieved a level of order nr , it takes a time of order nr forthe system to relax back to equilibrium, during which the typical value of f p(Q(·))continues to be of order n pr , thereby yielding a total contribution to S(p) of ordernr+pr . Thus, to produce a large deviation in Sn(p) of order n above nμ(p), we shouldchoose r = 1/(p + 1), leading to an associated large probability that is exponentialin n1/(p+1). Rigorously verifying this intuition, by means of the following theorem,is the main contribution of this paper.

Theorem 1 For each p > 0, there exists C(p) > 0 such that the following limit holdsfor any ε > 0,

limn→∞

1

n1/(p+1)log P0

(Sn(p)/n > μ(p) + ε

) = −C (p) ε1/(p+1). (2)

The function C(p) is characterized in Sect. 2—see (12).The limit in (2) for the case p = 1 might be conjectured from the results of [8]. It

follows from the main results there that there is a function K : (0,∞) → (0,∞] suchthat for positive δ,

limn→∞

1

nlog P0

(Sn(1)/n > μ(1) + nδ

) = −K (δ). (3)

See also Proposition 11.3.4 of [13]. For the M/M/1 queue, one can show that K (δ) ≈C(1)

√δ for δ sufficiently close to zero (as illustration, see Fig. 7 of [8]). If one formally

sets δn = ε/n in (3), we would obtain, for all δ > 0,

limn→∞

1√n

log P0(Sn(1)/n > μ(1) + ε

) = −C (1)√

ε (4)

This coincides with (2) in the case p = 1.

The rigorous verification of (4) was posed as an open problem in the Workshopon Simulation of Networks organized at the Newton Institute in the Summer of 2010.The motivation from a simulation perspective relates to the speed of convergence ofsteady-state estimation estimators.

Note that a key property of the M/M/1 queue is that it relaxes to equilibrium inlinear time (as a function of the initial condition). It is this property that allows us to

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Queueing Syst (2013) 73:425–446 429

“amortize” the effort of pushing the process Q to level nr over a time scale of sufficientduration so as to produce the desired large deviation. Furthermore, the conditionaldynamics can now involve non-time homogeneous dynamics (over a time scale oforder o(n)), which in turn are essentially randomly and uniformly initiated at sometime within the interval [0, n]. This is consistent with heavy-tailed large deviationsphenomena which will allow us to explain the subexponential large deviations rate. Inthe end, if we decompose the path in regenerative cycles, the contribution of each cycleinvolves a heavy-tailed (Weibullian-type) random variable. Therefore, the principle ofthe big jump applies leading to the subexponential asymptotics and the conditionaldynamics explained that we have explained intuitively.

We end our discussion here by pointing out that in the context of more generalMarkov random walks the amortization strategy described earlier would fail if theprocess were one that relaxes to equilibrium so rapidly that only a small contributionto the area under f (X ( · )) ensued. As a consequence, when f ( · ) is unbounded, itis still possible that the interaction of f ( · ) with the dynamics of X can be such theconventional large deviations explanation (i.e., modifying the dynamics of X over aninterval of length n, thereby producing a large deviation that is exponential in n) holds.An interesting such example (in the context of the natural continuous time analog)is that of a mean-reverting Ornstein-Uhlenbeck process X with f (x) = x . However,even in this setting, the conventional explanation can break down with f (x) = x p

with p large enough.The rest of the paper is devoted to the proof of Theorem 1.

2 Proof of Theorem 1

We will describe the strategy of the proof and along the way introduce necessarynotation. We will complete the asymptotic lower bound leading to (2) in Sect. 2.2,the upper bound in Sect. 2.3, and auxiliary results will be proved in the appendix inSect. 2.4.

2.1 Strategy and notation

Our strategy is to relate the limit in (2) to a large deviations problem involving heavy-tailed random variables. For simplicity, we concentrate on the case p = 1. GivenQ(0) = 0, we define T0 = 0, Tj = inf{k > Tj−1 : Q(k) = 0} for j = 1, 2, . . . , andτ j = Tj − Tj−1 for j ≥ 1. Then Q( · ) is regenerative with respect to the sequence{Tj : j ≥ 0}, which in turn induces a renewal process defined via N (n) = max{k ≥0 : Tk ≤ n}. We write

Sn(p) =N (n)∑

j=1

Tj −1∑

k=Tj−1

Q ( j)p +n∑

k=TN (n)

Q ( j)p .

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430 Queueing Syst (2013) 73:425–446

Now, define

Y j (p) =Tj −1∑

k=Tj−1

Q ( j)p =Tj∫

Tj−1

Q (�s�)p ds, (5)

and note that the Y j (p)’s are i.i.d. Moreover, observe that μ(p) = E Q(∞)p =EY1(p)/Eτ1 and therefore

P0(Sn (p) > nμ (p) + nε) ≈ P0(

�n/Eτ1�∑

j=1

(Y j (p) − μ (p)) +n∑

k=TN (n)

Q ( j)p > nε).

(6)

In our intuitive discussion, here, we use “≈” non-rigorously, but all the statements areproved in the sense of logarithmic asymptotics as n → ∞ (which is ultimately thestatement of Theorem 1). In approximation (6), we have replaced N (n) by n/Eτ1 andwe shall provide rigorous support for this substitution in the arguments following (16)and (21) below.

Now, since n is large, then the distribution of Q(n) is close to that of Q(∞) this willbe discussed in (19). In turn, since the M/M/1 queue is reversible, then the currentcycle in progress can be interpreted as the area enclosed starting from steady-stateuntil hitting zero. In other words, we can write the approximation

P0(Sn (p) > nμ (p)) ≈ Pπ (Y0 (p) +�n/Eτ1�∑

j=1

(Y j (p) − μ) > nμ + nε), (7)

where

Y0 (p) =T0−1∑

j=0

Q ( j)p =T0∫

0

Q (�s�)p ds, and T0 = inf{k ≥ 0 : Q (k) = 0

}.

The rigorous justification behind the approximation in (7) starting from the steady-state distribution will eventually be given in the argument leading to Eq. (19) belowand it is partly based on the following result (see [3]); we provide a proof at the endof this section in order to make our exposition self-contained.

Lemma 1 For any selection of measurable sets A1, . . . , An

P0(Q (k) ∈ Ak : 1 ≤ k ≤ n

)

= 1

π (0)Pπ

(Q (0) ∈ An, Q (1) ∈ An−1, . . . , Q (n − 1) ∈ A1, Q (n) = 0

).

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Queueing Syst (2013) 73:425–446 431

Returning to (7). Intuitively,

Y0 (p) ≈ Θ

⎛

⎝Q(∞)∑

j=1

j p

⎞

⎠ = Θ(Q (∞)p+1 ). (8)

In turn, it is straightforward to see that Q(∞)p+1 has Weibullian-type tails withindex 1/(p + 1), and therefore Y0(p) is expected to have Weibullian tails with index(or shape parameter) 1/(p + 1); the precise statement of this assertion correspondsto Proposition 2 given below. A similar argument follows for the random variablesY1(p), Y2(p), . . . , which actually have lighter tails than Y0(p) (this follows easilyby coupling because Y j (p) with j ≥ 1 constitutes an accumulated area starting fromthe origin to the first time the process returns to the origin, whereas Y0(p) is thearea starting from steady state). We now can take advantage of the way in which largedeviations occur for the sum of mean-zero heavy-tailed random variables, which statesthat the large deviations behavior arises due to the contribution to a single large jump;this in particular explains the scaling n1/(p+1) that appears in (2). In order to make theprinciple of the single large jump rigorous, we take advantage of the following resultwhich is proved in the appendix at the end of this section.

Proposition 1 Consider a sequence {Z1, Z2, . . . }of mean zero i.i.d. random variablesand assume that Z0 is independent of the Z j ’s for j ≥ 1. Moreover, suppose thatE(Z2

1) + E(Z20) < ∞ and that

P (Z0 > t) = exp(−ctβ + o

(tβ))

(9)

as t → ∞ for some c ∈ (0,∞) and β ∈ (0, 1). Finally, suppose that there existsκ > 0 such that

P(Z j > t

) ≤ κ P (Z0 > t) , (10)

then for any a > 0

log P (Z0 + Z1 + · · · + Zn > an) ∼ log P (Z0 > an)

as n → ∞.

Although the large deviations theory for heavy-tailed (more precisely, subexponen-tial random variables) is well developed; see, for instance, [4,6,16] and the textbooksof [2] and [9]. The subexponential or semiexponential property in our current settingis difficult to verify. If we could verify some of these properties, we might be ableto obtain more precise results than just logarithmic asymptotics. Nevertheless, theassumption on the tail of Z0 as indicated in (9) is enough for the issue that is of mainconcern for us, namely, logarithmic asymptotics as indicated by (2). This is why wecannot apply the existing theory directly and we need to develop the correspondingasymptotics in Proposition 1.

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432 Queueing Syst (2013) 73:425–446

Our strategy so far, combining the Weibullian tail behavior expected from (8) andProposition 1, suggests a path to establishing the subexponential scaling in (2). Letus now discuss the nature of the constant C(p). Let us consider for simplicity p = 1in this discussion. Applying then the principle of the largest jump from heavy-tailedanalysis (i.e., Proposition 1) yields

P0(Sn > nμ

) ≈ Pπ

(T0−1∑

k=0

Q ( j) > nε

)

= Pπ

⎛

⎝1

n

T0∫

0

Q (�s�) ds > ε

⎞

⎠

= Pπ

⎛

⎜⎝T0/n1/2∫

0

Q(⌊

un1/2⌋)

n1/2 du > ε

⎞

⎟⎠

The right hand side is easy to evaluate by Laplace’s principle and standard largedeviations analysis given that now the initial condition can be chosen to be Q(0) =�ny1/2� with probability π(�ny1/2�) = ρ�ny1/2�(1 − ρ). Large deviations analysis isnow directly applicable because the random variable of interest has been expressedas a functional of a properly scaled (by n1/2 in time and space) queue length processthat satisfies a large deviations principle with rate n1/2. In particular, evaluating theconstant C(1) in (2) is equivalent to obtaining logarithmic asymptotics for the tailof Y0(p) assuming that Q(0) follows the steady-state distribution. Now, suppose thatQ(0) = y > 0, then the distribution of (Q(k) : k ≤ T0) coincides with that of(R(k) : k ≤ T ), where R( · ) is the random walk

R (k + 1) = R (k) + W (k + 1) ,

where P(W (k) = 1) := r(λ) := λ/(λ + μ) = 1 − P(W (k) = −1), R(0) = y andT = inf{k ≥ 0 : R(k) = 0}. Note that E[W (1)] = −(μ − λ)/(μ + λ) < 0. Recallthat the local large deviations rate function of R( · ) is given by

I (z) = supθ

[θ z − log Eexp (θW (k))

]

= supθ

[θ z − log(exp(θ)p (λ) + exp (−θ) q (λ)

].

Mogulskii’s theorem (see [5]) establishes that the sequence {n−1 R(�n · �)} satisfiesa large deviations principle in the space of D[0,∞) with the standard Skorokhodtopology with a rate function equal to

J (x ( · )) =∞∫

0

I (x (s)) ds (11)

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Queueing Syst (2013) 73:425–446 433

where x(s) denotes the derivative of the function x( · ) (the rate function J (x( · ))evaluated at a function x( · ) that is not absolutely continuous is set equal to infinity).Our characterization of the tail of Y1(p) given that Q(0) follows the steady-statedistribution takes advantage of the associated sample path large deviations for therandom walk. This is precisely the content of the next result, which is proved in theappendix in Sect. 2.4.

Proposition 2 Let C(p) be the solution of the variational problem of minimizing

y log (1/ρ) +υ∫

0

I (φ (s))ds (12)

subject to φ(0) = y, φ(s) > 0 for all s ∈ (0, υ), φ(υ) = 0 and∫ v

0 φ(s)pds ≥ 1. Theminimization is performed over y ∈ [0, (p + 1)1/(p+1)] and over functions φ( · ) thatare absolutely continuous and that satisfy the constraints we just indicated. Then, wehave

limt→∞

1

t1/(p+1)log Pπ

(Y0 (p) > t

) = −C (p) .

Remark It might be possible to solve for the optimal φ( · ) more explicitly in somecases, for instance, when p = 1, in which case the constraints can be expressedlinearly in φ( · ). We do not pursue these simplifications here as our objective is simplyto establish the logarithmic asymptotics with the characterization of the constant C(p)

in (2).

We now are ready to provide the rigorous details behind our strategy. We prove thelower and upper bounds leading to the statement of Theorem 1 separately.

2.2 Proof of Theorem 1: The lower bound

First define Q( j) = Q( j) − μ(p) and put

Sn (p) = Sn (p) − nμ (p) . (13)

We have, owing to Lemma 1,

P0(Sn(p) ≥ n(μ (p) + ε)

)

= P0(Sn(p) ≥ nε

) = 1

π (0)Pπ

(Sn(p) ≥ nε, Q (n) = 0

). (14)

Note that the initial position of the process given in the last equality of the previousdisplay is not the origin any longer. So, the induced regenerative process is nowdelayed. We set T0 = inf{k ≥ 0 : Q(k) = 0} (note that if Q(0) = 0, T0 = 0, so we

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434 Queueing Syst (2013) 73:425–446

are consistent with our notation introduced for the non-delayed process given in theIntroduction). As before, we put Tj = inf{k > Tj−1 : Q(k) = 0} and

N (n) = max{k : Tk ≤ n

}.

Observe that (14) induces the representation

Sn(p) =T0−1∑

k=0

Q (k) +N (n)∑

j=1

Tj −1∑

i=Tj−1

Q (k) .

We let

Z0 =T0−1∑

k=0

Q (k) , and Z j =Tj −1∑

i=Tj−1

Q (k) , j = 1, 2, . . . ,

so that the equality in (14) becomes equivalent to

P0(Sn(p) ≥ nμ (p) + nε

) = 1

π (0)Pπ

⎛

⎝Z0 +N (n)∑

j=1

Z j ≥ nε, Q (n) = 0

⎞

⎠ . (15)

Now, let ε0 > 0 be arbitrary and note that

Pπ

⎛

⎝Z0 +N (n)∑

j=1

Z j ≥ nε, Q (n) = 0

⎞

⎠

≥ Pπ

⎛

⎝Z0 +N (n)∑

j=1

Z j ≥ nε,

∣∣∣N (n)∑

j=1

Z j

∣∣∣ ≤ ε0n, Q (n) = 0

⎞

⎠

≥ Pπ

⎛

⎝Z0 ≥ n(ε + ε0),

∣∣∣N (n)∑

j=1

Z j

∣∣∣ ≤ nε0, Q (n) = 0

⎞

⎠ . (16)

To see that we actually have

limn→∞

1

n p/(1+p)log Pπ

⎧⎨

⎩Z0 ≥ nε,

∣∣∣N (n)∑

j=1

Z j

∣∣∣ ≤ nε0, Q (n) = 0

⎫⎬

⎭

= −C (p) (ε + ε0)p/(1+p) , (17)

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Queueing Syst (2013) 73:425–446 435

we take advantage of the large deviations analysis behind the proof of Proposition 2.Note that

Pπ

(Z0 ≥ n (ε + ε0)

) = Pπ

(Y0(p) − T0μ (p) ≥ n (ε + ε0)

)

≤ Pπ

(Y0(p) ≥ n (ε + ε0)

)

and also note that for each ε1 > 0 we have

Pπ

(Y0(p) − T0μ (p) ≥ n (ε + ε0)

)

≥ Pπ

(Y0(p) ≥ n (ε + ε0 + ε1) , T0μ (p) ≤ nε1

)

= Pπ

(Y0(p) ≥ n (ε + ε0 + ε1)

)− Pπ

(T0μ (p) > nε1

).

It is easy to see that T0 has a finite moment generating function and therefore

Pπ

(T0μ (p) > nε1

)= O

(exp (−cnε1)

)

for some c > 0. Since ε1 > 0 is arbitrary, by Proposition 2 we conclude that

limn→∞

log Pπ

(Z0 ≥ n (ε + ε0)

)

log Pπ

(Y0(p) ≥ n (ε + ε0)

) = 1. (18)

Now, the large deviations analysis behind Proposition 2, in particular the associatedcalculus of variations problem, indicates that if δ ∈ (0, 1/(1 + p)) then

limn→∞ P

(T0 ≤ n1−δ|Z0 ≥ n (ε + ε0)

)= 1.

Consequently, it follows that

limn→∞ Pπ

⎛

⎝∣∣∣

N (n)∑

j=1

Z j

∣∣∣ ≤ nε0, Q (n) = 0|Z0 ≥ n (ε + ε0)

⎞

⎠ (19)

= limn→∞ P0

⎛

⎝∣∣∣

N (n−o(n))∑

j=0

Z j

∣∣∣ ≤ nε0, Q (n − o (n)) = 0

⎞

⎠ = π (0) , (20)

which, together with (18), yields (17).

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436 Queueing Syst (2013) 73:425–446

2.3 Proof of Theorem 1: The Upper Bound

Our departing point is equation (15), which combined with (14) implies that

P0(Sn(p) ≥ nε

) ≤ 1

π (0)Pπ

⎛

⎝Z0 +N (n)∑

j=1

Z j ≥ nε

⎞

⎠ .

In turn, we have that for some c > 0, and each ε0 > 0,

Pπ

⎛

⎝Z0 +N (n)∑

j=1

Z j ≥ nε

⎞

⎠ = Pπ

⎛

⎝Z0 +N (n)∑

j=1

Z j ≥ nε,

∣∣∣N (n) − n/Eτ1

∣∣∣ ≤ nε0

⎞

⎠

+O(

exp(−ncε0))

(21)

Moreover, we have

Pπ

⎛

⎝Z0 +N (n)∑

j=1

Z j ≥ nε,

∣∣∣N (n) − n/Eτ1

∣∣∣ ≤ nε0

⎞

⎠

≤nε0∑

k=−nε0

Pπ

⎛

⎝Z0 +n/Eτ1+k∑

j=1

Z j ≥ (n/Eτ1) εEτ1

⎞

⎠ .

Using Proposition 1 we conclude that for each k ≤ ε0n

lim supn→∞

1

n p/(1+p)log Pπ

⎛

⎝Z0 +n/Eτ1+k∑

j=1

Z j ≥ (n/Eτ1) εEτ1

⎞

⎠

≤ lim supn→∞

1

n p/(1+p)log Pπ

(Z0 ≥ n (1/Eτ1 − ε0) εEτ1

)

≤ −C (p)(ε − ε0εEτ1

)p/(1+p).

Therefore, we conclude from these observations that

lim supn→∞

1

n1/(p+1)log P0

(Sn(p) ≥ nε

) ≤ −C (p) (ε − ε0εEτ1)−1/(p+1) .

Since ε0 > 0 is arbitrary, we arrive at the result.

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Queueing Syst (2013) 73:425–446 437

2.4 Appendix: Proof of auxiliary results

Proof of Lemma 1 Let K (i, j) be the Markov transition matrix of the process Q; inparticular,

K (i, j) =⎧⎨

⎩

λ/(λ + μ) i > 0, j = i + 1μ/ (λ + μ) i > 0, j = i − 11 i = 0

,

and K (i, j) = 0 otherwise. It is well known that Q is reversible, so

π (i) K (i, j) = π ( j) K ( j, i)

and then

π (0) P0(Q (k) ∈ Ak : 1 ≤ k ≤ n

)

=∑

q1,q2,...,qn

π (0) K (0, q1) . . . K (qn−1, qn) I (qk ∈ Ak : 1 ≤ k ≤ n)

=∑

q1,q2,...,qn

K (q1, 0) π (q1) . . . K (qn−1, qn) I (qk ∈ Ak : 1 ≤ k ≤ n)

. . .

=∑

q1,q2,...,qn

π (qn) K (qn, qn−1) . . . K (q2, q1) K (q1, 0) I (qk ∈ Ak : 1 ≤ k ≤ n).

If we write ik = qn−k for k = 0, 1, . . . , n we obtain

π (0) P0(Q (k) ∈ Ak : 1 ≤ k ≤ n

)

=∑

i0,i1,...,in=0

π (i0) K (i0, i1) . . . K (in−1, in) I(in−k ∈ Ak : 1 ≤ k ≤ n

).

The right hand side is just

Pπ (Q (0) ∈ An, Q (1) ∈ An−1, . . . , Q (n − 1) ∈ A1, Q (n) = 0) .

Dividing by π (0) we complete the lemma. ��Proof of Proposition 1 Let us introduce ε0 > 0 to be selected later and let us writeδn = ε0/ log(n). Note that

P(Z0 + Z1 + · · · + Zn > an

)

= P(

Z0 + Z1 + · · · + Zn > an, ∪nj=0

{Z j > an1−δn

})

+P(

Z0 + Z1 + · · · + Zn > an, ∩nj=0

{Z j ≤ an1−δn

}). (22)

123

438 Queueing Syst (2013) 73:425–446

Now, we have that for any ε1 > 0,

P(

Z0 > (a + ε1)n)

P(∣∣Z1 + · · · + Zn

∣∣ ≤ nε1

)

= P(

Z0 > (a + ε1)n,∣∣Z1 + · · · + Zn

∣∣ ≤ nε1

)

≤ P(

Z0 + Z1 + · · · + Zn > an, ∪nj=0

{Z j > an1−δn

})

≤ P(

∪nj=0

{Z j > an1−δn

}) ≤ (n + 1)κ P(

Z0 > an1−δn).

Therefore, since E Z1 = 0 we conclude that

lim supn→∞

1

nβlog P

(Z0 + Z1 + · · · + Zn > an, ∪n

j=0

{Z j > an1−δn

})

≤ −caβexp (−βε0) (23)

and the corresponding lower bound,

lim infn→∞

1

nβlog P

(Z0 + Z1 + · · · + Zn > an, ∪n

j=0

{Z j > an1−δn

})

≥ −c (a + ε1)β . (24)

Later on, we will argue that ε0, ε1 > 0 can be chosen arbitrarily small, but before wedo that first let us provide an upper bound for the second term in the right hand sideof (22).

Without loss of generality, we can assume that E Z0 = 0; if not, we redefineZ0 − E Z0 and analyze

P(

Z0 − E Z0 + Z1 + · · · + Zn > an − E Z0, ∩nj=0

{Z j ≤ an1−δn − E Z0

}).

The whole analysis to be presented would be identical, just keeping track of E Z0 whichasymptotically does not play any role; note that given the form of (9), assumption (10)would still hold after possibly redefining the constant κ.

Now, define θn = �nγn such that γn < 1 − δn and

P(

Z0 + Z1 + · · · + Zn > an, ∩nj=0

{Z j ≤ an1−δn

})

= P((

Z0 + Z1 + · · · + Zn)/n1−δn > anδn , ∩n

j=0

{Z j ≤ an1−δn

})

≤(

Eexp(θn Z0/n1−δn

)I (Z0 ≤ an1−δn )

)

×(

Eexp(θn Z1/n1−δn

)I (Z1 ≤ an1−δn )

)nexp

(−θnanδn).

123

Queueing Syst (2013) 73:425–446 439

Our strategy is to show that if � and γn are selected appropriately then the contributionof the term

n∏

j=0

(Eexp

(θn Z j/n1−δn

)I (Z j ≤ an1−δn )

)= O (1) (25)

and that we can find a constant κ ′ > 0 such that

exp(−θnanδn

) ≤ κ ′exp(−aβcnβ(1 − ε3)

)

for an arbitrary ε3 > 0. To execute this strategy, we first provide a Taylor developmentfor the expectations in the product (25). Note that

E exp(θn Z j/n1−δn

)I(Z j ≤ an1−δn

)

= P(Z j ≤ an1−δn

) + θn

n1−δnE Z j I

(Z j ≤ an1−δn

)

+ θ2n

2n2(1−δn)E{

exp(θn Z j/n1−δn

)Z2

j I(Z j ≤ an1−δn

)}.

Note that

E{

exp(θn Z j/n1−δn

)Z2

j I(Z j ≤ an1−δn

)}

≤ E{

exp(θn Z j/n1−δn

)Z2

j I(0 ≤ Z j ≤ an1−δn

)}

+ E{

Z2j I(0 ≤ Z j < 0

)}.

Now we have

E exp(θn Z j/n1−δn

)Z2

j I(0 ≤ Z j ≤ an1−δn

)

= E

∞∫

−Z j

θn

n1−δnexp

(−θnu/n1−δn

)Z2

j I(0 ≤ Z j ≤ an1−δn

)du

= E

∞∫

−∞

θn

n1−δnexp

(−θnu/n1−δn

)Z2

j I(0 ≤ Z j ≤ an1−δn , −u ≤ Z j

)du

=∞∫

−∞

θn

n1−δnexp

(−θnu/n1−δn

)E Z2

j I( − u ∨ 0 ≤ Z j ≤ an1−δn

)du

= E Z2j I(0 ≤ Z j ≤ an1−δn

)

123

440 Queueing Syst (2013) 73:425–446

+0∫

−an1−δn

θn

n1−δnexp

(−θnu/n1−δn

)E Z2

j I( − u ≤ Z j ≤ an1−δn

)du .

Observe that

0∫

−an1−δn

θn

n1−δnexp

(−θnu/n1−δn

)E Z2

j I( − u ≤ Z j ≤ an1−δn

)du

=an1−δn∫

0

θn

n1−δnexp

(θnt/n1−δn

)E Z2

j I(t ≤ Z j ≤ an1−δn

)dt

=an1−δn∫

0

θn

n1−δnexp

(θnt/n1−δn

)E Z2

j I(Z j > t

)dt

− E Z2j I(Z j > an1−δn

)(exp (θna) − 1)

≤an1−δn∫

0

θn

n1−δnexp

(θnt/n1−δn

)E Z2

j I(Z j > t

)dt .

Now, let us write

an1−δn∫

0

θn

n1−δnexp

(θnt/n1−δn

)E Z2

j I(Z j > t

)dt

=an1−δn−γn∫

0

θn

n1−δnexp

(θnt/n1−δn

)E Z2

j I(Z j > t

)dt

+an1−δn∫

an1−δn−γn

θn

n1−δnexp

(θnt/n1−δn

)E Z2

j I(Z j > t

)dt .

We analyze the previous two integrals, first we note that

an1−δn−γn∫

0

θn

n1−δnexp

(θnt/n1−δn

)E Z2

j I(Z j > t

)dt

≤ θn

n1−δnexp (a�)

∞∫

0

E Z2j I(Z j > t

)dt.

123

Queueing Syst (2013) 73:425–446 441

For the second integral, we use assumption (10), which implies that for any ε2 > 0there exists K (ε2) > 0 such that for t ≥ 0

E Z2j I(Z j > t

) ≤ K (ε2) exp(−c(1 − ε2)t

β).

Therefore,

an1−δn∫

an1−δn−γn

θn

n1−δnexp

(θnt/n1−δn

)E Z2

j I(Z j > t

)dt

≤ K (ε2)

an1−δn∫

an1−δn−γn

θn

n1−δnexp (gn (t)) dt ,

where

gn (t) = θnt

n1−δn− c(1 − ε2)t

β.

Note that

maxan1−δn−γn ≤t≤an1−δn

gn (t) = gn(an1−δn

),

thus

an1−δn∫

an1−δn−γn

θn

n1−δnexp (gn (t)) dt ≤ an1−δn exp

(aθn − c

(1 − ε2

)aβn(1−δn)β

)

= an1−δn exp(

a�nγn − c(1 − ε2

)aβn(1−δn)β

).

Now, let us pick D > 0 and note that (since β < 1), for n large enough,

γn = (1 − δn

)β − Dδn < 1 − δn .

So we have

an1−δn exp(

a�nγn − c(1 − ε2

)aβn(1−δn)β

)

≤ an1−δn exp(−n(1−δn)β

[aβc(1 − ε2) − a� exp(−Dε0)

]).

We can clearly select D := D (ε0,�) such that

exp( − Dε0

) = aβ−1c(1 − 2ε2

)/� (26)

123

442 Queueing Syst (2013) 73:425–446

and obtain

an1−δn∫

an1−δn−γn

θn

n1−δnexp

(θnt/n1−δn

)E Z2

j I(Z j > t

)dt

≤ K (ε2) an1−δn exp(−n(1−δn)β

[ε2aβc

]).

Overall, then we conclude (redefining K (ε2))

E exp(θn Z j/n1−δn

)I(Z j ≤ an1−δn

)

= 1 + K (ε2)

n2(1−δn)+ O

(exp

(−c(ε2)nβ))

for some c(ε2) > 0 independent of the selection of the ε j ’s, as long as D(�) is selectedas indicated above. We then obtain that there exists K (ε2) such that

(E exp

(θn Z0/n1−δn

)I(Z0 ≤ an1−δn

) )

×(

E exp(θn Z1/n1−δn

)I(Z1 ≤ an1−δn

))nexp

(−θnanδn).

≤ exp(K (ε2) n2δn−1 + K (ε2)

)exp

(−�anδn+γn)

≤ exp(K (ε2) n2δn−1 + K (ε2)

)exp

(−�anβ+δn(1−β−D)

)

= exp(K (ε2) n2δn−1 + K (ε2)

)exp

(−�anβexp(ε0 (1 − β))exp(−Dε0))

= exp(K (ε2) n2δn−1 + K (ε2)

)exp

( − aβnβcexp(ε0(1 − β))(1 − 2ε2)),

where in the last equality we used (26). Consequently, we conclude that

P(

Z0 + Z1 + · · · + Zn > an, ∩nj=0

{Z j ≤ an1−δn

})

≤ exp(K (ε2) n2δn−1+ K (ε2)

)exp

(−aβnβcexp(ε0(1−β))(1−2ε2)). (27)

So, in summary, we have from (24)

lim infn→∞

1

nβlog P

(Z0 + Z1 + · · · + Zn > an

)

≥ lim infn→∞

1

nβlog P

(Z0 + Z1 + · · · + Zn > an, ∪n

j=0

{Z j > an1−δn

})

≥ −c(a + ε1

)β.

123

Queueing Syst (2013) 73:425–446 443

On the other hand, combining (23) and (27) we obtain

lim infn→∞

1

nβlog P (Z0 + Z1 + · · · + Zn > an)

≤ max( − caβexp (−βε0) , −caβnβexp(ε0(1 − β))(1 − 2ε2)

).

Since ε0, ε1 and ε3 are arbitrary we conclude the result. ��Proof of Proposition 2 Note that for any ξ ∈ (0, 1)

Pπ (Y1 (p) > t) = Pπ

⎛

⎝1

t

T1∫

0

Q (�s�)p ds > 1

⎞

⎠

= Pπ

⎛

⎜⎝T1/tξ∫

0

(Q(⌊

utξ⌋)

/t (1−ξ)/p)pdu > 1

⎞

⎟⎠ .

Pick ξ = 1/(p + 1) so that ξ = (1 − ξ)/p. Let x = tξ and recall that π(k) =P(Q(∞) = k) = ρk(1 − ρ), so we can write

Pπ (Y1 (p) > t) = Pπ

⎛

⎜⎝T1/x∫

0

(Q (�ux�) /x)pdu > 1

⎞

⎟⎠

=∞∑

k=0

π (k) Pk

⎛

⎜⎝T1/x∫

0

(Q (�ux�) /x)pdu > 1

⎞

⎟⎠

= (1 − ρ)

∞∫

0

ρ�s� P�s�

⎛

⎜⎝T1/x∫

0

(Q (�ux�) /x)pdu > 1

⎞

⎟⎠ ds

= (1 − ρ)

∞∫

0

xρ�xy� P�xy�

⎛

⎜⎝T1/x∫

0

(Q (�ux�) /x

)pdu > 1

⎞

⎟⎠ dy.

Moreover, as mentioned earlier before we stated our current proposition,

∞∫

0

xρ�xy� P�xy�

⎛

⎜⎝T1/x∫

0

(Q (�ux�) /x

)pdu > 1

⎞

⎟⎠ dy

=∞∫

0

xρ�xy� P

⎛

⎜⎝T /x∫

0

(R(�ux�) + �xy� /x

)pdu > 1

⎞

⎟⎠ dy .

123

444 Queueing Syst (2013) 73:425–446

By the (functional) Law of Large Numbers, we have

T /x∫

0

(R(�ux�) + �xy� /x

)pdu

−→y/|EW (1)|∫

0

(− u |EW (1)| + y

)pdu =

y∫

0

s pds = y p+1

p + 1.

So, if y > (p + 1)1/(p+1) + ε for any ε > 0, we have

limx→∞ P

⎛

⎜⎝T /x∫

0

(R(�ux�) + �xy� /x

)pdu > 1

⎞

⎟⎠ = 1.

This implies that

limx→∞

1

xlog

∞∫

(p+1)1/(p+1)

xρ�xy� P

⎛

⎜⎝T /x∫

0

(R(�ux�) + �xy� /x

)pdu > 1

⎞

⎟⎠ dy

= −(p + 1)1/(p+1) log (1/ρ) . (28)

Moreover, if z(p) = (p + 1)1/(p+1) then Pxz(p)(T > xk) ≤ P(R(�kx�) + xz(p) >

0) = O(exp(−kc(p)x)) for some c(p) > 0 by a simple Chernoff bound argument.So, it suffices to obtain logarithmic asymptotics for the quantity (assuming k has beenchosen sufficiently large),

(p+1)1/(p+1)∫

0

xρ�xy� P( T /x∫

0

(R(�ux�) + �xy� /x

)pdu > 1, T /x ≤ k)

dy.

On the other hand, if y < (p + 1)1/(p+1), we can apply large deviations theory andLaplace’s principle. We wish to invoke the contraction principle; but, to do this, we firstneed to define a suitable continuous function. Let Rx ( · ) be the continuous piecewiselinear approximation to R(� · x�)/x such that x Rx (k/x) = R(k) for each integerk ≥ 0. The process Rx ( · ) is an exponentially good approximation to R(� · x�)/xunder the uniform topology on compact sets (see [5]). So, the rate function (11) givenby Mogulskii’s theorem applies to Rx ( · ). Now, given a continuous function φ(·), letT−(φ) = inf{s > 0 : φ(s) < 0} and note that the mapping F( · ) defined as

F(φ) =T−(φ)∫

0

φ (s)p ds

123

Queueing Syst (2013) 73:425–446 445

is a continuous mapping under the uniform topology on compact sets. So, by thecontraction principle it follows easily that

1

xlog

(p+1)1/(p+1)∫

0

xρ�xy� P�xy�(F(Rx ) ≥ 1

)dy → −C (p) ≥ −(p + 1)1/(p+1)

log (1/ρ) , (29)

where C(p) is the solution of the variational problem stated in (12); note, as observedearlier, that it suffices to consider the variational problem only on functions φ suchthatT−(φ) ≤ k. The inequality in the right hand side of (29) follows because onecan choose y arbitrarily close to (p + 1)1/(p+1) as a test value in the variationalproblem.

Now, clearly, T−(Rx ) ≥ T /x so we have, using again exponential equivalence,that for y < (p + 1)1/(p+1)

limx→∞1

xlog P�xy�

⎛

⎜⎝T /x∫

0

(R(�ux�)/x

)pdu > 1, T /x ≤ k

⎞

⎟⎠ ≤ limx→∞1

xlog P�xy�

(F(Rx ) ≥ 1

).

This implies that

limx→∞1

xlog

(p+1)1/(p+1)∫

0

xρ�xy� P�xy�

⎛

⎜⎝T /x∫

0

(R(�ux�)/x

)pdu ≥ 1

⎞

⎟⎠ dy ≤ −C (p) .

The lower bound is obtained simply by isolating the event that Rx ( · ) tracks a piece-wise linear approximation that is arbitrarily close to any ε-optimal solution of theassociated calculus of variations problem in (12); this tracking procedure is standardin every analysis of the lower bound when proving the so-called weak contractionprinciples. ��

References

1. Arendarczyk, A., Debicki, K., Mandjes, M.: On the tail asymptotics of the area swept under theBrownian storage graph. Math. Oper. Res. (accepted)

2. Asmussen, S.: Applied Probability and Queues. Springer, New York (2003)3. Blanchet, J.: Optimal sampling of overflow paths in Jackson networks. Math. Oper. Res. http://www.

bernoulli-society.org/index.php/publications/bernoulli-journal/bernoulli-journal-papers (accepted)4. Borovkov, A.A.: Estimates for the distribution of sums and maxima of sums of random variables

without the cramer condition. Sib. Math. J. 41, 811–848 (2000)5. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York

(1998)6. Denisov, D., Dieker, A., Shneer, V.: Large deviations for random walks under subexponentiality: the

big-jump domain. Ann. Probab. 36, 1946–1991 (2008)

123

446 Queueing Syst (2013) 73:425–446

7. Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations forlarge time. I. II. Commun. Pure Appl. Math. 28, 1–47 (1975); ibid. 28, 279–301 (1975)

8. Duffy, K.R., Meyn, S.P.: Most likely paths to error when estimating the mean of a reflected randomwalk. Perform. Eval. 67(12), 1290–1303 (2010)

9. Embrechts, P., Klppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance.Springer, New York (1997)

10. Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems, Series in ComprehensiveStudies in Mathematics. Springer, New York (1984)

11. Kontoyiannis, I., Meyn, S.P.: Spectral theory and limit theorems for geometrically ergodic Markovprocesses. Ann. Appl. Probab. 13, 304–362 (2003)

12. Meyn, S.P.: Large deviation asymptotics and control variates for simulating large functions. Ann. Appl.Probab. 16(1), 310–339 (2006)

13. Meyn, S.P.: Control Techniques for Complex Networks. Cambridge University Press, Cambridge(2007)

14. Miller, H.: A convexity property in the theory of random variables defined on a finite Markov chain.Ann. Math. Stat. 32, 1260–1270 (1961)

15. Ney, P., Nummelin, E.: Markov additive processes. II. Large deviations. Ann. Probab. 15(2), 593–609(1987)

16. Rozovskii, L.V.: Probabilities of large deviations of sums of independent random variables. TheoryProbab. Appl. 34, 625–644 (1990)

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