Heavy Tails in Multi-Server Queue

Transcription

1 Queueing Systems 26) 52: 3 48 DOI.7/s z Heavy Tails in Multi-Server Queue Serguei Foss Dmitry Korshunov Received: 3 February 24 / Revised: 2 April 25 C Science + Business Media, Inc. 26 Abstract In this paper, the asymptotic behaviour of the distribution tail of the stationary waiting time W in the GI/GI/2 FCFS queue is studied. Under subexponential-type assumptions on the service time distribution, bounds and sharp asymptotics are given for the probability PW > x. We also get asymptotics for the distribution tail of a stationary two-dimensional workload vector and of a stationary queue length. These asymptotics depend heavily on the traffic load. Keywords FCFS multi-server queue Stationary waiting time Large deviations Long tailed distribution Subexponential distribution AMS subject classification: 6K25. Introduction It is well known see, for example, [,5,8]) that in the stable single server first-come-first-served queue GI/GI/ with typical interarrival time τ and typical service time σ the tail of stationary waiting time W is related to the service time distribution tail Bx) Pσ >x via the equivalence PW > x Eτ Eσ x By) dy as x, ) Supported by EPSRC grant No. R58765/, INTAS Project No and RFBR grant No S. Foss Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH4 4AS, Scotland foss@ma.hw.ac.uk D. Korshunov Sobolev Institute of Mathematics, Novosibirsk 639, Russia korshunov@math.nsc.ru provided the subexponentiality of the integrated tail distribution B I defined by its tail B I x) min, x ) By) dy, x >. As usual we say that a distribution G on R + is subexponential belongs to the class )ifg Gx) 2Ḡx) asx. The converse assertion is also true, that is, the equivalence ) implies the subexponentiality of B I, see [5, Theorem ] for the case of Poisson arrival stream and [4, Theorem ] for the general case. In this paper we consider the GI/GI/s FCFS queue which goes back to Kiefer and Wolfowitz [3]. We have s identical servers, i.i.d. interarrival times τ n with finite mean a Eτ, and i.i.d. service times σ n with finite mean b Eσ. The sequences τ n and σ n are mutually independent. The system is assumed to be stable, i.e., ρ b/a, s). We are interested in the asymptotic tail behaviour of the stationary waiting time distribution PW > x as x. It was realized recently see, for example, existence results for moments in [6,7]; an asymptotic hypothesis in [9]; asymptotic results for fluid queues fed by heavy-tailed on-off flows in [5]) that the heaviness of the stationary waiting time tail depends substantially on the load ρ in the system. More precisely, it depends on ρ via the value of k,,...,s for which k ρ<k +. In particular, Whitt conjectured that PW > x γ ηx By) dy) s k as x,

2 32 Queueing Systems 26) 52: 3 48 where γ and η are positive constants as functions of x) [sic, [9]]. In the present paper we show that, in general, the tail behaviour of W is more complicated. Let Rw) R w),...,r s w)) be the operator on R s which orders the coordinates of w R s in ascending order, i.e., R w) R s w). Then the residual work vector W n W n,...,w ns ) which the nth customer observes just upon its arrival satisfies the celebrated Kiefer-Wolfowitz recursion: W i, W n+ RW n + σ n τ n+ ) +, W n2 τ n+ ) +,...,W ns τ n+ ) + ) RW n + e σ n iτ n+ ) +, concerned, we are extremely sceptical on the possibility to get any sharp tail asymptotics in explicit form. For the two-server queue, in the maximal stability case, we prove the following: Theorem. Let s 2 and b < a. When the integrated tail distribution B I is subexponential, the tail of the stationary waiting time satisfies the asymptotic relation, as x, PW > x [ B I x)) 2 a2a b) + b ] B I x + ya) Bx + ya b)) dy. here e,,...,), i,...,) and w + max,w ),...,max,w s )). The value of W n is the delay which customer n experiences. In particular, the stationary waiting time W is a weak limit for W n. The process W n is a Markov chain in R s.itiswell known that, for general multi-dimensional Markov chains, large deviation problems are very difficult to solve even for stationary distributions. Usually they can be solved in low dimensions only, 2 or 3 at most, see [4,2]. Almost all known results are derived for so-called Cramér case which corresponds to light-tailed distributions of jumps. In the heavy-tailed case almost nothing is known for general multi-dimensional Markov chains. The process W n presents a particular but very important example of a Markov chain in R s,even ifweare interested in the first component W n.asfollows from our analysis, the case s 2 can be treated in detail. The stability condition for this particular case is b < 2a. One of the following cases can occur: i) the maximal stability case when b < a; ii) the intermediate case when b a; iii) the minimal stability case when b a, 2a). We find the exact asymptotics for PW > x in the maximal and minimal cases. We also describe the most probable way for the occurrence of large deviations. In the intermediate case, we only provide upper and lower bounds. Then we study the asymptotics for the tail of the distribution of a stationary two-dimensional workload vector and give comments on the tail asymptotics of the stationary queue length. For s > 2, the stability condition is b < sa.wehope that, for s > 2, direct modifications of our arguments may lead to exact asymptotics in two particular cases when either b < a the maximal stability) or b s ) a, sa) the minimal stability). However, one has to overcome many extra technicalities for that. Insofar as the case b [a, s ) a] is The proof follows by combining the lower bound given in Theorem 3 Section 3) and the upper bound given in Theorem 4 Section 4). Simpler lower and upper bounds for PW > x are given in the following Corollary. Under the conditions of Theorem, 2a + b 2a 2 2a b) lim PW > x inf lim x B I x)) 2 2aa b). x sup PW > x B I x)) 2 In our opinion, in Theorem it is possible to obtain a compact expression for the tail asymptotics of PW > x) only in the regularly varying case. A distribution G or its tail Ḡ)is regularly varying at infinity with index γ> belongs to the class RV ), if Ḡx) > for all x and, for any fixed c >, Ḡcx)/Ḡx) c γ as x. Corollary 2. Let b < a and the tail distribution B of service time be regularly varying with index γ>. Then PW > x c B I x)) 2, where [ c + b a2a b) γ ] dz. + za) γ + za b)) γ Recall definitions of a number of classes of heavy-tailed distributions. A distribution G is long-tailed belongs to the

3 Queueing Systems 26) 52: class L )ifḡx) > for all x and, for any fixed t, Ḡx + t) Ḡx) as x. times σ n and deterministic interarrival times τ n a :W and W n+ RW n + e σ n ia ) +. A distribution G belongs to the class IRV of intermediate regularly varying distributions if Ḡx) > for all x and Let W system. be a stationary waiting time in this auxiliary Ḡcx) lim lim inf c x Ḡx). Clearly, RV IRV. In the minimal stability case, we prove the following Theorem 2. Let s 2 and a < b < 2a, B and B I IRV. Then PW > x ) 2a b B b I b a x as x. The proof is given in Section 7 and is based on the lower and upper bounds stated in Theorems 5 and 6 respectively. One can provide simple sufficient conditions for B and B I IRV. Let D be the class of all distributions G on R + such that Ḡx) > for all x and lim inf x Ḡ2x)/ Ḡx) >. Then the following are known: i) RV IRV L D ) ; ii) if B D has a finite first moment, then B I IRV see e.g. [6]). Therefore, if B L D and has a finite first moment, then B satisfies the conditions of Theorem 2. Note that the converse is not true, in general: there exists a distribution B with a finite first moment such that B I IRV,butB / L D see Example 2 in [9, Section 6]). The paper is organized as follows. Section 2 contains some auxiliary results. In Section 3, we formulate and prove a result concerning a lower bound for PW > x in the maximal stability case. The corresponding upper bound is given in Section 4. Sections 5, 6, and 7 deal, respectively, with lower bounds, upper bounds, and asymptotics for PW > x in the minimal stability case. In Section 8, we prove further results related to the joint distribution of a stationary workload vector. Comments on the asymptotics for a stationary queue length distribution may be found in Section 9. A number of upper and lower bounds for PW > x in s-server queue are proposed in Remarks 2, 3, 4, and Preliminaries 2.. Reduction to deterministic input stream case in assertions associated with upper bounds Consider a general GI/GI/s queue. Take any a b/s, a). Consider an auxiliary D/GI/s system with the same service Lemma. If PW > x Ḡx) for some long-tailed distribution G, then lim sup x PW > x Ḡx). Proof. Denote ξ n a τ n. Put M and, for n, M n max,ξ n,ξ n + ξ n,...,ξ n + +ξ max,ξ n + M n ) ξ n + M n ) +. First, we use induction to prove the inequality W n W n + im n a.s. 2) Indeed, for n wehave + im. Assume the inequality is proved for some n; weprove it for n +. Indeed, W n+ RW n + e σ n iτ n+ ) + RW n + im n + e σ n iτ n+ ) + RW n + e σ n ia + im n + ξ n+ )) +. Since u + v) + u + + v +, W n+ RW n + e σ n ia ) + + im n + ξ n+ ) + W n+ + im n+, and the proof of 2) is complete. Let M be the weak limit for M n which exists due to Eξ a a < and Strong Law of Large Numbers. The following stochastic equality holds: M st max,ξ,ξ + ξ 2,...,ξ + +ξ n,... Since the random variable ξ is bounded from above by a ), there exists β>such that Ee βξ. Then by Cramér estimate see, for example, [8, Section 5]), for any x, PM > x e βx. 3) The inequality 2) implies that W st W + M, where W and M are independent. Let a random variable η have distribution

4 34 Queueing Systems 26) 52: 3 48 G and be independent of M. Since η st W,wehaveW st η + M. Therefore, for any h >, PW > x x h PM > x ypη dy + Pη >x h x h by 3). Integrating by parts yields x h e βx y) Gdy) + Ḡx h), e βx y) Gdy) e βx y) Ḡy) x h + β e βx + β x h Ḡy)e βx y) dy x h Ḡx y)e βy dy. The distribution G is long-tailed, thus, for any ε> there exists xε) such that Ḡx y) Ḡx)e ε for any x xε). Hence, there exists cε) < such that Ḡx y) cε)ḡx)e εy for any x xε). Take ε β/2. Then x h Hence, x Ḡx y)e βy dy cε)ḡx) e βy/2 dy h cε) β/2 Ḡx)e βh/2. PW > x e βx + 2cε)Ḡx)e βh/2 + Ḡx h). Taking into account also that Ḡx h) Ḡx) for any fixed h >, we obtain lim sup x PW > x Ḡx) 2cε)e βh/2 +. Letting h yields the conclusion of the Lemma Reduction to deterministic input stream case in assertions associated with lower bounds Take any a > a. Asinthe previous subsection, consider an auxiliary D/GI/s system with the same service times σ n and deterministic interarrival time τ n a. Let W be a stationary waiting time in this auxiliary system. Lemma 2. If PW > x Ḡx) for some long-tailed distribution G, then lim inf x Proof. PW > x Ḡx). Put ξ n τ n a, M and M n max,ξ n,ξ n, +ξ n,...,ξ n + +ξ M n + ξ n ) +. For any n, the following inequality holds: W n W n im n. 4) Indeed, by induction arguments, W n+ RW n + e σ n iτ n+ ) + RW n im n + e σ n iτ n+ ) + RW n + e σ n ia im n + ξ n+ )) +. Since u v) + u + v +, W n+ RW n + e σ n ia ) + im n + ξ n+ ) + W n+ im n+, and the proof of 4) is complete. Let M be the weak limit for M n which exists due to Eξ a a < and the Strong Law of Large Numbers. The inequality 4) implies that W st W M where W and M are independent. Therefore, for any h >, PW > x PW > x + hpm h Ḡx + h) PM h. The distribution G is long-tailed, thus Ḡx + h) Ḡx) for any fixed h > and lim inf x PW > x Ḡx) PM h. Letting h,we obtain the desired estimate from below.

5 Queueing Systems 26) 52: Adapted versions of the Law of Large Numbers Corollary 4. Let Eη < Eξ < and ε>. Then It is well known that obtaining lower bounds for systems under assumptions of heavy tails usually requires some variant of the Law of Large Numbers. Here we provide such a tool for the two-server queue. Lemma 3. Let ξ n,η n ), n, 2,...,beindependent identically distributed pairs of random variables. Let the twodimensional Markov chain V n V n, V n2 ), n, 2,...,be defined in the following way: V has an arbitrary distribution and Vn + ξ n,η n ), if V n V n2, V n+ V n + η n,ξ n ), if V n > V n2. If E η < Eξ, then the following convergence in probability holds: V n n Eξ + Eη, Eξ ) + Eη as n. 2 2 Proof. Since V n+, + V n+,2 V n + V n2 + ξ n + η n,bythe Law of Large Numbers V n + V n2 n Eξ + Eη as n. 5) Define a Markov chain U n V n2 V n.ifu n, then U n+ U n η n ξ n, while if U n <, then U n+ U n ξ n η n η n ξ n ), so, U n is the oscillating random walk. Since Eξ > Eη, the mean drift of the chain U n is negative on the positive half-line and is positive on the negative halfline. Therefore, for any sufficiently large A, the set [ A, A] is positive recurrent for this Markov chain. In particular, the distributions of U n are tight. Hence, U n /n inprobability as n. Together with 5), it implies the desired assertion of Lemma. The classical Law of Large Numbers and Lemma 3 imply the following Corollary 3. Let Eη < Eξ < and ε>. Then PV n >, V n2 > V v,v 2 ) as N uniformly in n N and in v,v 2 )onthe set v,v 2 > n Eξ +ε),v + v 2 > n Eξ + Eη +ε). PV n >, V n2 > V v,v 2 ) as N uniformly in n N and in v,v 2 ) on the complementary set v > n Eξ ε),v 2 > n Eξ ε),v + v 2 > n Eξ + Eη ε). Corollary 5. ε>. Then Let Eη <, Eξ >, Eη + Eξ < and PV n > x, V n2 > x V v,v 2 ) as x, N uniformly in n N and in v,v 2 )onthe set v > x neξ ε),v 2 > 2x + n Eξ + Eη +ε). 3. The maximal stability case: A lower bound Theorem 3. Assume b, a). Let the integrated service time distribution B I be long-tailed. Then the tail of the stationary waiting time W admits the following estimate from below: as x, [ PW > x + o) B I x)) 2 + b B I x + ya) a2a b) ] Bx + ya b)) dy. 6) Remark. From 6), one can get the lower bound in Corollary. Namely, replace Bx + y a b)) by a smaller term Bx + ya)inthe integral in the RHS of 6). Then the new integral is equal to b B I x)) 2 /2a, and the lower bound follows since + b ) 2a + b a2a b) 2a 2a 2 2a b). Remark 2. By use of Strong Law of Large Numbers, one can get the following result for s-server queue, s 2. If b < a, then there exists a constant K K a, b, s) such that PW > x K + o)) B I x)) s. We start with some auxiliary results. The proof of the theorem is given in Section 3.4.

6 36 Queueing Systems 26) 52: An integral equality Lemma 4. Let f y)beanintegrable function. Put f I y) y f z)dz. Then, for any positive α and β,α > β, J f αy + βz) f βy + αz) dydz f I )) 2 α 2 β 2β f 2 α 2 β 2 I αu) f βu) du. Proof. Put u αy + βz and v βy + αz. Then J α 2 β 2 α 2 β 2 α 2 β 2 α α 2 β 2 β α 2 β 2 Integration by parts yields f I βu) f αu) du f I )) 2 α β α αu/β f u) du βu/α f u) f I βu/α) du f v) dv f u) f I αu/β) du f αu) f I βu) du f βu) f I αu) du. f I αu) f βu) du. By substituting this equality into the previous one, we arrive at the conclusion of the Lemma Some calculations with two big service times Fix ε> and put b b ε.fork and l, k < l n, define the events A nkl and C nkl by the equalities A nkl σ K > x + l k)a + n l)a b ), and C nkl σ l > x + n l)a b ), σ k + σ l > 2x + l k)a + n l)2a b ) n σ j x + n j)a b ). j k,l Note that the events A nkl C nkl are disjoint for different pairs k, l). Due to the existence of Eσ, uniformly in n and k < l n, P C nkl Pσ > x + ja b ) j asx. Lemma 5. Assume b, a). Let the integrated tail distribution B I be long-tailed. Then, for any fixed N and for any ε>, as x, lim k<l PA nkl a2a b ) 7) [ ] B I x)) 2 + b B I x + ya) Bx + ya b )) dy. Proof. Put A kl σ > x + ka + la b ),σ 2 > x + la b ),σ + σ 2 > 2x + ka + l2a b ), so that PA nkl PA l k,n l and lim PA nkl lim k<l ln ln Consider also the events n l PA kl PA kl. 8) Ay, z) σ > x + ya + za b ),σ 2 > x + za b ), σ + σ 2 > 2x + ya + z2a b ), which satisfy Ak, l) A kl. Since the probability PAy, z) is non-increasing in y and z, wehave the inequalities I N PAy, z) dydz ln PA kl PAy, z) dydz I +. 9)

7 Queueing Systems 26) 52: The values of integrals I and I + are close to each other in the following sense: I + I N + PAy, z) dydz NPσ 2 > x + Pσ > x PAy, z) dydz Pσ > x + yady Pσ > x + za b ) dz. Recall that the distribution B I x) islong tailed, which is equivalent to Bx) o B I x)). Therefore, as x, I + I N + a b Bx) B I x) o B I x)) 2 ). Now it follows from 9) that, as x, ln Further, PA kl PAy, z) PAy, z)dydz + o B I x)) 2 ). ) Bx + ya + za) Bx + za b )) + Px + ya + za b ) <σ x + ya + za, σ 2 > x + za b ),σ + σ 2 > 2x +ya + z2a b ) Bx + ya + za) Bx + za b )) + Px + ya + za b ) <σ x + ya + za, σ + σ 2 > 2x + ya + z2a b ) Bx + ya + za) Bx + za b )) + Qy, z), since the event σ x + ya + za,σ + σ 2 > 2x + ya + z2a b ) implies the event σ 2 > x + za b). Consequently integrating over y and z,weobtain a Bx + ya + za) Bx + za b )) dydz B I x + za) Bx + za b )) dz. By the total probability formula, Qy, z) zb Pσ x + ya + za b ) + dt Pσ 2 > x + za t zb Bx + za t)bx + ya + za b ) + dt). The integration against y leads to the equalities zb Qy, z) dy Bx + za t) a B I x + za b ) + dt) zb Bx + za t) a Bx + za b ) + t) dt z b Bx + za tb ) a Bx + za b ) + tb ) dt. Integrating against z, weobtain: Qy, z)dydz b a z Bx + za tb ) Bx + za b ) + tb ) dt dz b Bx + za tb ) a t Bx + za b ) + tb ) dz dt b Bx + za + ta b )) a Bx + za b ) + ta) dz dt. By Lemma 4 with f y) Bx + y),α a, and β a b, the latter integral is equal to a2a b ) B I x)) 2 2a b ) a2a b ) B I x + ya) Bx + ya b )) dy. Putting everything together into ), we obtain the following equivalence, as x : l PA kl a2a b ) B I x)) 2 b + a2a b ) B I x + ya) Bx + ya b )) dy, which due to 8) completes the proof of the Lemma.

8 38 Queueing Systems 26) 52: Proof of Theorem 3 If B I x) islong-tailed, then the function in x B I x)) 2 + b B I x + ya) Bx + ya b)) dy is long-tailed as well. Indeed, for any fixed t, wehave,as x, B I x + t + ya) Bx + t + ya b)) dy B I x + ya) Bx + t + ya b)) dy. Integrating by parts we get the equality for RHS integral a b B I x + ya) B I x + t + ya b)) a b B I x)) 2 Bx + ya) B I x + t + ya b)) dy B I x + ya) Bx + ya b)) dy. Bx + ya) B I x + ya b)) dy So, we can apply Lemma 2, and it is sufficient to prove the lower bound of Theorem 3 for the queueing system D/GI/2 with deterministic input stream. Let the interarrival times τ n be deterministic, i.e., τ n a. Then the event A nkl implies the event W k+,2 > x + l k)a + n l)a b ) a, W l+, > x + n l)a b ) a, W l+, + W k+,2 > 2x + l k)a + n l)2a b ) 2a, which implies W l+,2, W l+, > x + n l)a b ) a, W l+, + W l+,2 > 2x + n l)2a b ) 2a. Thus, by Corollary 3 with ξ σ τ, and η τ) there exists N such that PW n > x A nkl ε ) for any n > N and k < l < n N. Taking into account that the events A nkl C nkl are disjoint for distinct pairs k, l), we obtain the following estimates: PW n > x lk+ lk+ lk+ PW n > x, A nkl, C nkl PW n > x, A nkl PA nkl, C nkl. Since the events A nkl and C nkl are independent, PW n > x lk+ sup kl lk+ PW n > x, A nkl PC nkl o) lk+ PA nkl PW n > x A nkl PA nkl lk+ PA nkl as x uniformly in n, by7). Together with ) it implies that, for sufficiently large x and n > N, PW n > x 2ε) lk+ PA nkl. Letting now n,wederive from Lemma 5 the following lower bound, for all sufficiently large x: PW > x [ 3ε B a2a b I x)) 2 + b ) ] Bx + ya b )) dy. Note that, for any b < b < a, B I x + ya) Bx + ya b )) dy a b a b B I x + ya) Bx + ya b)) dy. B I x + ya) We complete the proof of the Theorem by letting ε.

9 Queueing Systems 26) 52: The maximal stability case: An upper bound Theorem 4. Assume b, a). Suppose that the distribution B I is subexponential. Then, as x, [ PW > x + o) B I x)) 2 + b a2a b) ] Bx + ya b)) dy. B I x + ya) By Lemma, it is sufficient to prove this upper bound for the queueing system D/GI/2 with deterministic input stream. So, let the interarrival times τ n be deterministic, i.e., τ n a. Let σ n ) and σ n 2), n, be independent random variables with common distribution B. Inthis Section, define the service times σ n recursively. For that, we have to associate workloads with servers. Put U U,, U,2 ), ) and introduce the recursion U n+ U n + e αn σ n ia ) + 2) where α n ifu n, < U n,2 and α n 2ifU n, > U n,2.if U n, U n,2, then α n takes values and 2 with equal probabilities independently of everything else. Note that W n RU n ) a.s. for any n, 2,... Now we can define σ n by induction. Indeed, α is chosen at random from the set, 2. Put σ σ α ). Then U is defined by recursion 2) with n. Assume that U n is defined for some n >. Then α n is defined, too. Put σ n σ α n) n and determine U n+ by 2). Due to the symmetry, for any n, Pα n Pα n 2 /2. 3) Consider two auxiliary D/GI/ queueing systems which work in parallel: at any time instant T n na, n, 2,..., one customer arrives in the first queue and one in the second. Service times in queue i, 2 are equal to σ n i). Denote by W n i), i, 2, the waiting times in the ith queue and put W n ) W n 2). Since b < a, both queues are stable. Let W i) be a stationary waiting time in the ith queue. By monotonicity, with probability, W n min W n ), W n 2) ) 4) for any n. Hence, W min W ), W 2). 5) Lemma 6. The waiting times W n ) 2) and W n are independent. Proof follows from the observation that the input deterministic) stream and service times in the first queue do not depend on the input also deterministic) stream and service times in the second one. Provided B I is a subexponential distribution, P W i) > x a b B I x) asx. 6) Then Lemma 6 together with 5) implies the following simple upper bound: lim sup x PW > x B I x)) 2 a b). 7) 2 Remark 3. For a GI/GI/s queue with a < b and subexponential distribution B I, similar arguments lead to lim sup x PW > x B I x)) s a b). s Introduce the events, for k < n, A ) nk σ ) k > x + n k)a b), A 2) nk σ 2) k > x + n k)a b). Lemma 7. See also [3, Theorem 5]). Provided the distribution B I is subexponential, for any fixed N, lim sup P W ) n > x, n N A ) nk o B I x)) as x. Proof. For any δ>, consider the disjoint events C ) nk σ ) k > x + n k)a b + δ) n σ ) j x + n j)a b). j k Due to the Law of Large Numbers, there exists M > N such that P W n ) > x C ) nk δ

10 4 Queueing Systems 26) 52: 3 48 for any n M and k n M and, by the limit at 7), P C ) nk δ)p σ ) k > x + n k)a b + δ). The events C ) nk, k n M, are disjoint, hence, P W ) n > x, n M C ) nk n M P W n ) > x, C ) nk δ) 2 km + ka b + δ)). P σ ) k The latter implies the following lower bound: lim inf P δ) 2 W ) n > x, km n M C ) nk Bx + k)a b + δ)) > x δ)2 a b + δ B I x) as x. Since A ) nk C) nk and since M > N and δ> can be chosen arbitrarily, lim inf P W ) n > x, n N A ) nk + o) a b B I x) as x. Together with 6), it implies the assertion of the Lemma. Proof of Theorem 4 continued. Estimate 4) and Lemma 6 imply P W n > x, P P n N A ) n N nk l A 2) nl W ) n > x, W 2) n > x, W ) n > x, + P W ) n n N > x P A ) nk n N W 2) n > x, A ) n N nk l P W 2) n > x n N l A 2) nl. A 2) nl Applying now Lemma 7 and relation 6), we conclude that, as x, lim inf P Since n N W n > x, A ) n N nk l n N A ) n N nk l A 2) nl A 2) n N nl A ) nk ) A2) nl n N A ) nk ) A2) nl, we obtain the equivalent relation, as x, lim inf P W n > x, n N o B I x)) 2 ). A ) nk ) A2) nl o B I x)) 2 ). 8) Fix ε> and put b b + ε. Forany n and k l n, define D ) nk σ ) k > x + l k)a + n l)a b ), D 2) nl σ 2) l > x + n l)a b ), D nkl σ ) k + σ 2) l > 2x + l k)a + n l)2a b ). For any n and l k n, define D ) nk σ ) k > x + n k)a b ), D 2) nl σ 2) l > x + k l)a + n k)a b ), D nkl σ ) k + σ 2) l > 2x + k l)a + n k)2a b ). Denote F nkl D ) nk D2) nl D nkl. We can derive an upper bound on the probability of the event W n > x as follows: PW n > x P W n > x, n N + P W n > x, + P W n > x, n N n N F nkl F nkl, n N A ) nk ) A2) nl A ) nk ) A2) nl P n + P n2 + P n3. 9)

11 Queueing Systems 26) 52: Here the first term is not greater than n P n P W n > x, F nkl + P W n > x, k<l n + P W n > x, F nkk n k>l F nkl P n + P n2 + P n3. 2) The third probability is negligible in the sense that P n3 P n Bx) D ) nk ) D2) nk P D ) nk P D 2) nk Bx + ka b ε)) Bx) B I x)/a b ε) o B I x)) 2 ) 2) as x, since B x) o B I x)). The first probability in 2) admits the following upper bound: P n P W n > x, D ) n nk,α k, D 2) ) nl D nkl + P + 2. W n > x, D ) nk,α k 2, lk+ n lk+ D 2) nl D nkl ) For,wehave the following inequality and equalities: 2 k<l k<l k<l P D ) nk,α k, D 2) nl, D nkl Pα k P D ) nk, D2) nl, D nkl PF nkl, 22) by independence of the event α k from D ) nk, D2) nl and D nkl and by the symmetry 3). The sum 2 is not greater than 2 P W N > x, D ) nk,α k 2 P D ) nk PWn > x,α k 2 PW n > x P D ) nk. Hence, 2 opw n > x)asx uniformly in n. Combining the latter fact with estimate 22) for,weget P n 2 k<l PF nkl +opw n > x). 23) Taking into account the equality P n P n2,weobtain from 2), 2) and 23) the following estimate: P n k<l PF nkl +o B I x)) 2 ) as x uniformly in n. Now applying the calculations of Section 3.3 we can write down the following estimate, as x : limsup P n [ + o) B a2a b I x)) 2 + b B I x + ya ) ) ] Bx + ya b )) dy. 24) It is proved in 8) that, uniformly in n, P n3 o B I x)) 2 ) as x. 25) n N Thus, We have P n2 F nkl n N n N A ) nk ) A2) nl A ) nk ) A2) nl F nkl. P W n > x A ) nk, A2) nl, F nkl P A ) nk A2) nl. Conditioning on W nk and W nl yields, for any w>; P W n > x A ) nk, A2) nl, F nkl P W n > x W k w, W l2 w, A ) nk, A2) nl, F nkl + PW k >w+pw l2 >w. 26) Since b < 2a, the two-server queue is stable and, in particular, the sequence of distributions of random variables W n, W n2 )istight. It means that, for any fixed ε>, there exists w such that, for any k and l, PW k >w ε and PW l2 >w ε.

12 42 Queueing Systems 26) 52: 3 48 Also, from the stability and from Corollary 4, for any fixed ε>, and w>, there exists N such that, for any n N and k, l n N, P W n > x W k w, W l2 w, A ) nk, A2) nl, F nkl ε. Combining these estimates we obtain from 26), P n2 3ε Hence, P A ) nk A2) nl nn 3ε P A ) ) 2 nk. 2 P n2 3ε Bx + ka b ))) 3ε a b ) B 2 I x)) 2. 27) Since the choice of ε> is arbitrary, relations 9) 25) and 27) imply the conclusion of Theorem The minimal stability case: Lower bounds Theorem 5. Let b a, 2a) and the integrated tail distribution B I be long tailed. Then the tail of the stationary waiting time satisfies the following inequality, for any fixed δ>: PW > x + o) ) 2a b B b + δ I b a x as x. Notice that if b a, 2a) then b b a > 2. Remark 4. By use of similar arguments, one can get the following result for an s-server queue, s 2: if the integrated distribution B I is long tailed and b s )a, sa) then, for any δ>, PW > x + o) ) sa b B s )b ss 2)a + δ I b s )a as x. Theorem 5 implies the following Corollary 6. Assume that B I IRV. Then, as x, PW > x + o) ) 2a b B b I b a x. In the case b [a, 2a) one can also derive a lower bound which is similar to 6). More precisely, assume b [a, 2a). Then introduce another two-server queue with the same service times and with inter-arrival times τ n cτ n, where c > b/a. For this queue, denote by W a stationary waiting time of a typical customer. Due to monotonicity, PW > x PW > x for all x. Applying Theorem 3 and Remark, we get the following lower bound for the case b [a, 2a): if the integrated tail distribution B I is long-tailed, then, for any c > b/a, 2ca + b PW > x + o)) 2c 2 a 2 2ca b) B I x)) 2. 28) Proof of Theorem 5. By Lemma 2, it is sufficient to prove the lower bound for the queueing system D/GI/2 with deterministic input stream. Let the inter-arrival times τ n be deterministic, i.e., τ n a. Forany δ>, set ε δb a) b b ε and N the events x b a A nk σ k > 2x + 2a b )n k), n C nk σ l 2x + 2a b )n l). l l k Since Eσ is finite, P C nk a+δ. Put.Forany k [, n N] consider Pσ > 2x + 2a b )l O B I 2x)) l 29) as x uniformly in n and k n. Since the events A nk C nk, k [, n], are disjoint, we obtain PW n > x PW n > x, A nk, C nk PW n > x, A nk PA nk, C nk. Since the events A nk and C nk are independent, we get PW n > x PW n > x, A nk sup k n P C nk PA nk PW n > x A nk PA nk o) PA nk 3)

13 Queueing Systems 26) 52: as x uniformly in n, by 29). The event A nk implies the event W k+,2 > 2x + 2a b )n k) a. Thus, it follows from Corollary 5 that PW n > x A nk Proof of Theorem 6. From Lemma 3, it is sufficient to consider the case of constant interarrival times τ n a only. Put M n, and M n,i+ M n,i + σ n+i a) +. Since b > a, M,n a.s. as n and, due to the Law of Large Numbers, as x uniformly in n and k n N. Therefore, we can derive from 3) the estimate M,n n b a a.s. 3) n N PW > x lim PW n > x ε) lim PA nk ε) kn B2x + 2a b )k), which is valid for all sufficiently large x. Since the tail B I v) is long-tailed, B2x + 2a b )k) B 2a b I 2x + 2a b )k)n) b ) ) B 2a b I b a x b + δ B 2a b I b a x kn as x. The proof is complete. 6. The minimal stability case: An upper bound Theorem 6. Assume b [a, 2a). Let both B and B I be subexponential distributions. Then the tail of the stationary waiting time satisfies the following inequality, as x : PW x + o) 2a b B I 2x). Remark 5. By use of the same arguments, one can get the following result for any s-server queue, s 2: if B I and b < sa, then PW x + o) sa b B I sx) as x provided that either i) σ s )a a.s., or ii) B. Remark 6. For an s-server queue, Foss and Chernova [] have proposed another way of obtaining upper bounds; it is based on comparison with a queue with the so-called cyclic service discipline. and in mean. Note that EM,n nb a), since M,n σ + +σ n na. Forany given ε>, choose an integer L > such that EM,L L [b a, b a + ε). 32) Consider any initial workload vector W W,, W,2 ). Put Z n W n, + W n,2 Since the increments of the minimal coordinate of the waiting time vector is not greater than the increments of M o,n, W,n W, M,n for any n. Hence, provided W n,2 a, wehave the inequality Z n+ Z n M,n+ M,n a. If Z 2aL, then W,2 al and, for n,...,l, W n,2 al n) a. Therefore, if Z 2aL, then Z L Z + M,L al. Monotonicity implies, for any initial vector W, Z L max2al, Z +M,L al. Thus, the following inequalities are valid for any n: Z n+)l max2al, Z nl +M nl,l al. 33) Consider a single-server queue with i.i.d. service times ˆσ n M nl,l and constant inter-arrival times ˆτ n La and denote by Ŵ n awaiting time of nth customer. This queue is stable since ˆb E ˆσ < al â. Put Ŵ. Assuming that Z, we can derive from 33) the following bounds: for all n,,..., Z nl 2aL + Ŵ n a.s. 34)

14 44 Queueing Systems 26) 52: 3 48 Denote Ḡx) P ˆσ > x. Weshow that the integrated tail distribution G I is a subexponential one. We need to consider only the case L >. Note first that σ + +σ L La ˆσ σ + +σ L a.s. 35) Since the distribution of σ,isassumed to be subexponential, the asymptotics for the lower and upper bounds in the latter inequalities are the same: as x, P L σ i La > x P L σ i > x L Bx). 36) Therefore, the tail Ḡx) has the same asymptotics and G is a subexponential distribution. Thus, Ḡ I x) x Ḡy) dy L B I x), 37) and G I is a subexponential distribution, too. Thus, by classic result ) for the single server queue, the steady state distribution of the waiting time Ŵ n satisfies the following relations, as x : lim PŴ n > x Ḡ I x) â ˆb 2a b ε)l Ḡ I x) 2a b ε B I x), 38) by 32) and 37). Since Z n W n, + W n,2 2W n,, PW > x P2W > 2x lim PZ nl > 2x. Now it follows from 34) and 38) that PW > x lim PŴ n > 2x 2aL + o) 2a b ε B I 2x 2aL) 2a b ε B I 2x), since B I is long-tailed. Letting ε concludes the proof. 7. The minimal stability case: Exact asymptotics In this Section, we prove Theorem 2. First note that, as follows from 28), the tail PW > x may be heavier than that in Theorem 2, in general. For instance, this happens if ) b B I b a x o B I 2 x)) as x. 39) Assume b a, 2a) and consider, for example, a service time distribution with the Weibull integrated tail B I x) e xβ,β, ). Then 39) holds if b b a )β > 2. Proof of Theorem 2. Since B I IRV, both the lower bound in Theorem 5 and the upper bound in Theorem 6 are of the same order, B I 2x) O b B I b a x )). 4) We use the notation from the previous Section. In particular, we fix ε> and choose L satisfying 32). For any constant c, 35) implies L σ kl+i > x + La + L i)c i L σ kl+i La > x ˆσ k > x. i Therefore, from 35) and 36), L P ˆσ k > x\ σ kl+i > x + La + L i)c i o Bx)). 4) Take c â ˆb)/L. By34) PW > x lim PW nl, > x lim PW nl, > x, Ŵ n > 2x 2aL. Standard arguments concerning how large deviations in the single server queue Ŵ n occur imply the relation PW > x lim PW nl, > x, k ˆσ k > 2x + n k)â ˆb)+o B I 2x)) nl lim PW nl, > x,σ i > 2x + n i)c i + o B I 2x)), by 4). Now it follows from 32) that nl PW > x lim PW nl, > x, k σ i > 2x + n i)2a b ε)+o B I 2x))

15 Queueing Systems 26) 52: lim nl PW nl, > x, σ nl j > 2x + j2a b + ε)+εo B I 2x)) ) N ε) nl + o B I 2x)) lim + + εo B I 2x)) lim + 2 ) + εo B I 2x)), jn ε) where N x/b a). The second term admits the following estimate 2 jn ε) Pσ >2x + j2a b) 2a b B I 2x + N ε)2a b)) 2a b B b I b a x ε 2a b ) b a x. It follows from B I RV ε>such that 2 ) 2a b B b I b a x + δ B I 2x), that, for any δ>, there exists which coincides with the lower bound in Theorem 5. Now consider the first term. Since the queue is stable, one can choose K > such that PW n,2 K ε for all K. Then N ε) + N ε) PW nl j,2 > K,σ nl j > 2x + 2a b + ε) j PW nl, > x, W nl j,2 K, σ nl j > 2x + 2a b + ε) j, +,2. We have, ε N ε) PW nl j,2 > K Pσ > 2x + 2a b + ε) j ε Pσ > 2x + 2a b) j 2a b B I 2x). Note that if W nl j,2 K, then W nl, K + M nl j+, j. Therefore,,2 N ε) Pσ nl j > 2x + 2a b) j, K + M nl j+, j > x N ε) Pσ nl j > 2x + 2a b) j PK + M, j > x. Since the sequence M, j stochastically increases,,2 P K + M,N ε) > x Pσ > 2x + 2a b) j Since P M,N ε) > x K 2a b B I 2x). x K N ε) b a > b a as x, ε we have by 3) P M,N ε) > x K MN ε) P N ε) > x K. N ε) Thus, we have shown that the upper bound for PW > x is not bigger than the lower bound in Theorem 5 plus a term of order )) bx ε + δ)ob I 2x)) ε + δ)o B I b a due to 4). Since ε> and δ> may be chosen as small as we please, the proof of Theorem 2 is complete. 8. Tail asymptotics for the two-dimensional workload vector Denote by W W, W 2 )aweak limit for the vectors W n as n. Clearly, W W. 8.. Maximal stability case First, we obtain simple lower and upper bounds which are equivalent up to some constant. Second, we give without a proof) a result related to the exact asymptotics.

16 46 Queueing Systems 26) 52: 3 48 Theorem 7., x y, Let b < a and B I L. Then, as x, y P W > x, W 2 > y + o) a 2 If, in addition, B I, then B I x) B I y). P W > x, W 2 > y 2 + o) a b) 2 B I x) B I y). Proof. Fix ε> and put a a + ε. Fork, l n, k l, define the events A nkl and C nkl by the equalities A nkl σ k > x + n k)a,σ l > y + n l)a and C nkl n σ j x + n j)a. j k,l Note that the events A nkl C nkl are disjoint for different pairs k, l) and PW n > x, W n2 > y n n lk+ PW n > x, W n2 > y, A nkl, C nkl. Then the same calculations as in Section 3.3 imply the estimate, as x, y, PW n > x, W n2 > y n n + o)) + o)) lk+ n k l Bx + n k)a ) By + n l)a ) Bx + ka ) By + la ). Proceed to the upper bound. Due to construction of the majorant W n ), W n 2) )insection 4, we have the inequality P W > x, W 2 > y [ lim P W ) n > x, W n 2) > y + P W n ) > y, W n 2) > x ] 2 lim P W ) n > x P W n 2) > y. Together with 6) it implies the desired upper bound. Theorem 7 is proved. Turn now to the exact asymptotics. Below is the result. The proof is rather complicated and will be presented in another paper. Denote Rx, y) B I x) B I y) + b B I y + za) Bx + xa b)) dz. Recall that Theorem states that PW > x Rx, x)/a2a b) givenb I. Theorem 8. Assume b < a and B I. Let x, y, x y. Then P W > x, W 2 > y a2a b) Ry, y) + Rx, y) Ry, y)). a Minimal stability case We prove the following Theorem 9. Assume a < b < 2a, B, and B I IRV. Let x, y, in such a way that y/x c [, ]. Then P W > x, W 2 > y )) a B a I y + + b a a2a b) B I cb a) b y b a ). Hence, P W > x, W 2 > y + o)) By + la ) B I x) B I y)/a 2 and the lower bound is proved. l Bx + ka ) Proof. Start with the case c. From Theorem in [3], one can get the following: Corollary 7. Assume b a, 2a). If B and B I, then, as y, P W 2 > y a B I y) + b a a2a b) B I y b b a ).

17 Queueing Systems 26) 52: It is clear that P W > x, W 2 > y P W 2 > y. On the other hand, for any N, 2,..., P W > x, W 2 > y lim PW n, > x, W n,2 > y lim P W n N,2 > y + Na, σ j τ j ) > x jn N N lim PW n N,2 > y + NaP σ j τ j ) > x. Fix ε>. Put N Nx) x + ε)/b a). Then by LLN N P σ j τ j ) > x ε for all sufficiently large x and, as n, PW n N,2 > y + Na P W 2 > y + Na. Since B I IRV, P W 2 > y + Na P W 2 > y as y. By letting ε, we get the result. Now consider the case c <. Ifc, then the result follows from Theorem 2. Let c, ). We give here only a sketch of the proof, by making links to the proof of Theorem 2. Since P W > y P W > x, W 2 > y P W > x and P W > y P W > cx K + o))p W > x where K inf t B I ct)/ B I t) >, one can get from the proof of Theorem 2 the following equivalences: for N x x/b a), N y y/b a), and for ε, / c), P W > x, W 2 > y n N x ε) lim PW n, > x, W n,2 > y, i σ i > 2x + n i)2a b)+εo B I 2x)) n Ny ) +ε) n N x ε) lim + + εo B I 2x)) i in N y +ε) + 2 ) + εo B I 2x)). Choose K > such that PW n,2 > K ε for all n. Then 2 lim n N x ε) in N y +ε) PW i,2 K, W n, > x, W n,2 > y, σ i > 2x + n i)2a b)+εo B I 2x)). From Lemma 2 and its Corollaries, 2 + o)) + o) a N y +ε) jn x ε) B I y + + εo B I x)) + o) B I y a + + ε + δ)o B I x)), Pσ > y + ja+εo B I x)) ) )) x ε)a yb + εa) B I b a b a )) a cb a) )) yb B I b a due to B I IRV. From Lemma 3 and its Corollaries, one can also conclude that, for i < n N y + ε), if σ i < 2y + n i)2a b ε) and W i,2 K, then, with probability close to one, both coordinates of the vector W n,, W n,2 ) take values less then y for all sufficiently large n. From the other side, if σ i > 2y + n i)2a b + ε) then, with probability close to one, y < W n, W n,2. Therefore, + o)) + εo B I x)) + o) 2 b B I jn y +ε) yb b a Pσ > 2y + j2a b) ) + εo B I x)). Summing up the terms and letting ε and δ concludes the proof. 9. Comments on stationary queue length Let Q n be a queue length viewed by an arriving customer n, and Q its stationary version in discrete time i.e. Palmstationary). Due to the distributional Little s law, PQ > n PW > T n where W is the stationary waiting time, T n τ + +τ n, and W, and T n do not depend on each other. When a distribution of W is long-tailed, the asymptotics for PW > T n, n,have been found in [2] and in []. If, in addition, τ n has a non-lattice distribution, there exists a stationary

18 48 Queueing Systems 26) 52: 3 48 distribution G for a queue length in continuous time. Then, from Lemma in [], Ḡn) PQ > n as n. Acknowledgment The authors gratefully acknowledge helpful discussions with Onno Boxma and Bert Zwart, and comments from Daryl Daley. References. S. Asmussen, Applied Probability and Queues, 2nd ed., New York, 23). 2. S. Asmussen, C. Kluppelberg and K. Sigman, Sampling at subexponential times, with queueing applications, Stoch. Process. Appl ) F. Baccelli and S. Foss, Moments and tails in monotone-separable stochastic networks, Ann. Appl. Probab 4 24) A.A. Borovkov and A.A. Mogul skii, Large deviations for Markov chains in the positive quadrant, Russ. Math. Surv 56 2) S. Borst, M. Mandjes and A.P. Zwart, Exact asymptotics for fluid queues fed by heavy-tailed On-Off flows, Ann. Appl. Probab 4 24) O.J. Boxma, S.G. Foss, J.-M. Lasgouttes and R. Nunez Queija, Waiting time asymptotics in the single server queue with service in random order, Queueing Systems 46 24) O.J. Boxma, Q. Deng and A.P. Zwart, Waiting-time asymptotics for the M/G/2 queue with heterogeneous servers, Queueing Systems 4 22) H. Cramér, Collective Risk Theory Esselte, Stockholm, 955). 9. D. Denisov, S. Foss and D. Korshunov, Tail asymptotics for the supremum of a random walk when the mean is not finite, Queueing Systems 46 24) S.G. Foss and N.I. Chernova, On optimality of FCFS discipline in multi-channel queueing systems and networks, Siberian Math. J 42 2) S. Foss and D. Korshunov, Sampling at a random time with a heavytailed distribution, Markov Processes and Related Fields 6 2) I.A. Ignatyuk, V. Malyshev and V. Scherbakov, Boundary effects in large deviation problems, Russ. Math. Surv ) J. Kiefer and J. Wolfowitz, On the theory of queues with many servers, Tran. Amer. Math. Soc ) D. Korshunov, On distribution tail of the maximum of a random walk, Stochastic Process. Appl ) A.G. Pakes, On the tails of waiting-time distribution, J. Appl. Probab 2 975) A. Scheller-Wolf, Further delay moment results for FIFO multiserver queues, Queueing Systems 34 2) A. Scheller-Wolf and K. Sigman, Delay moments for FIFO GI/GI/s queues, Queueing Systems ) N. Veraverbeke, Asymptotic behavior of Wiener-Hopf factors of a random walk, Stochastic Process. Appl 5 977) W. Whitt, The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution, Queueing Systems 36 2) 7 87.