Heavy-Traffic Analysis of Sojourn Time under the Foreground-Background Scheduling Policy

We consider the steady-state distribution of the sojourn time of a job entering an M/GI/1 queue with the foreground-background scheduling policy in heavy traffic. The growth rate of its mean, as well as the limiting distribution, are derived under broad conditions. Assumptions commonly used in extreme value theory play a key role in both the analysis and the results.


Introduction
One of the main insights from queueing theory is that the queue length and sojourn time are of the order 1/(1 − ρ), as the traffic intensity of the system ρ approaches 100 percent utilization.This insight dates back to Kingman [13] and Prokhorov [19] and, appropriately reformulated, remains valid for queueing networks and multiple server queues [7,11,25].This picture can change dramatically when the scheduling policy is no longer First-In-First-Out (FIFO).Bansal [1] was the first to point out that the mean sojourn time (a.k.a.response time, flow time) of a user is of o(1/(1 − ρ)) in the M/M/1 queue when the scheduling policy is Shortest Remaining Processing Time (SRPT).This result was later generalized to several non-exponential service time distributions in Lin et al. [16].More recently, Puha et al. [20] derived a process limit theorem for the SRPT queue length, under an assumption that implies that all moments of the service time are finite.
As SRPT requires information on service times in advance, the question was raised if the same growth rate in heavy traffic can be reached with a blind scheduling policy, a question that was answered negatively in Bansal et al. [3].Specifically, the authors showed that for every blind scheduling policy, there exists a service time distribution under which the growth rate in heavy traffic of the mean sojourn time is at least a factor log(1/(1 − ρ)) larger than the growth rate of SRPT.Bansal et al. also construct a scheduling policy that achieves this growth rate, but this policy rather complicated as it involves randomization.All of the results mentioned thus far only concern the mean sojourn time, and it is of interest to obtain information about the distribution of the sojourn time as well.
Motivated by these developments, we consider the Foreground-Background (FB) scheduling policy in this paper.More precisely, we investigate the invariant distribution of the sojourn time of a customer in an M/GI/1/FB queue.The FB policy operates as follows: priority is given to the customer with the least-attained service, and when multiple customers satisfy this property, they are served at an equal rate.The only heavy-traffic results for FB we are aware of are of "big-O" type and are known in case of deterministic, exponential, Pareto and specific finite-support service times [2,17].For deterministic service times, it is easy to see that all customers under FB depart in one batch at the end of every busy period, and as a result the growth rate in heavy traffic is very poor in this case O((1 − ρ) −2 ).The behaviour of FB is much better for service-time distributions with a decreasing failure rate, as FB then optimizes the mean sojourn time among all blind policies [22].For more background on the FB policy we refer to the survey by Nuyens and Wierman [17].
The main results of this paper are of three types: 1. We characterize the exact growth rate (up to a constant independent of ρ) of the sojourn time in heavy traffic under very general assumptions on the service time distribution.As in Bansal and Gamarnik [2] and Lin et al. [16], we find a dichotomy: when the service time distribution has finite variance, the mean sojourn time . Here is the tail of the service time distribution and G ← is the right-inverse of the distribution function of a residual service time; a detailed overview of notation can be found in Section 2. In the infinite variance case, we find that E[T ρ FB ] = Θ log 1 1−ρ .This result is formally stated in Theorem 3.1.The precise conditions for these results to hold involve Matuszewska indices, a concept that will be reviewed in Section 2. The behaviour of F (G ← (ρ)) is quite rich, as will be illustrated by several examples.
2. Contrary to the results in Bansal and Gamarnik [2] and Lin et al. [16], we have been able to obtain a more precise estimate of the growth rate of E[T ρ FB ].It turns out that extreme value theory plays an essential role in our analysis, and the limiting constant factor in front of the growth rate F (G ← (ρ)) (1−ρ) 2 crucially depends on in which domain of attraction the service-time distribution is.This result is summarized in Theorem 3.2 and appended in Theorem 3.4.When the service-time distribution tail is regularly varying, it is shown that the growth rate of the sojourn time under FB is equal to that of SRPT up to a finite constant.A comparison of the sojourn times under FB and SRPT is given in Corollary 3.5.

When analysing the distribution, we first show that T ρ
FB /E[T ρ FB ] converges to zero in probability as ρ ↑ 1.To still get a heavy traffic approximation for P(T ρ FB > y), we state a sample path representation for the sojourn time distribution for a job that requires a known amount of service.We then use fluctuation theory for spectrally negative Lévy processes to rewrite this representation into an expression that is amenable to analysis; in particular, we obtain a representation for the Laplace transform of the residual sojourn time distribution is of independent interest, from which a heavy-traffic limit theorem follows.Finally, the Laplace transform implies an estimate for the tail of T FB .
More specifically, our results show that P((1 − ρ) 2 T FB > y)/F (G ← (ρ)) converges to a nontrivial function g * (y), for which we give an integral expression in terms of error functions.Along the way, we derive a heavy traffic limit for the total workload in an M/GI/1 queue, with truncated service times that also seems to be of independent interest (see Proposition 7.1).As in the analysis for the mean sojourn time, ideas from extreme value theory play an important role in the analysis, and the limit function g * depends on which domain of attraction the service-time distribution falls into.A precise description of this result can be found in Theorem 3.7.
Despite the fact that extreme theory appears both in our analysis and our end results, the precise role of extreme value theory is not entirely clear from the analysis in this paper.A challenging topic for future research is to get a completely probabilistic proof of our result; e.g. a proof that does not use explicit integral expressions for the mean sojourn time.To this end, it makes sense to consider a more general class of scheduling policies, for example the class of SMART scheduling policies considered in Wierman et al. [26] and Nuyens et al. [18].This is beyond the scope of the present paper.
The rest of the paper is organized as follows.Section 2 formally introduces the model that is considered.Section 3 presents all our main results on the asymptotic behaviour of the mean and the tail of the sojourn time distribution under FB.The results on the mean are then proven in Sections 4 and 5, whereas the results on the tail distribution are supported in Sections 6 and 7.

Preliminaries
In this paper we consider a sequence of M/GI/1 queues, indexed by n, where the i-th job requires B i units of service for all n.For convenience, we say that a job that requires x units of service is a job of size x.All B i are independently and identically distributed (i.i.d) random variables with cumulative distribution function (c.d.f.) F (x) = P(B i ≤ x) and finite mean E[B].We assume that F (0) = 0, and denote x R := sup{x ≥ 0 : F (x) < 1} ≤ ∞.Jobs in the n-th queue arrive with rate λ (n) , where λ For notational convenience, we let B denote a random variable with c.d.f.F .
Let F (x) := 1 − F (x) and F ← (y) := inf{x ≥ 0 : F (x) ≥ y} denote the complementary c.d.f.(c.c.d.f.) and the right-inverse of F respectively.The random variable B * is defined by its c.d.f.G(x) as the failure rate of B * .One may deduce that h * (x) equals the reciprocal of the mean residual time;

Foreground-Background scheduling policy
Jobs are served according to the Foreground-Background (FB) policy, meaning that at any moment in time, the server equally shares its capacity over all available jobs that have received the least amount of service thus far.First, we are interested in characteristics of the sojourn time T (n) FB , defined as the duration of time that a generic job spends in the system.In order to analyse this, we consider an expression for the mean sojourn time of a generic job of size x, E[T (n) FB (x)], for which Schrage [23] states that where ρ (n) x 0 tF (t) dt are functions of the first and second moments of B ∧ x := min{B, x}, and W (n) (x) is the steady-state waiting time in a M/GI/1/FIFO queue with arrival rate λ (n) and jobs of size B i ∧x.The intuition behind this result, is that a job J 1 of size x experiences a system where all job sizes are truncated.Indeed, if another job J 2 of size x + y, y > 0, has received at least x service then FB will never dedicate its resources to job J 2 while job J 1 is incomplete.The mean sojourn time of job J 1 can now be salvaged from its own service requirement x, the truncated work already in the system upon arrival W (n) (x), and the rate 1 − ρ x at which it is expected to be served, yielding (2.1).As a consequence, the mean sojourn time E[T (n) FB ] of a generic job is given by (2.2) Second, we focus attention on the tail behaviour of for all x ∈ R and let L x (y) denote the time required by the server to empty the system given that job sizes are truncated to B i ∧ x and the current amount of work is y.The analysis of the tail behaviour is then facilitated by relation (4.28) in Kleinrock [14], stating For both the mean and tail behaviour of T (n) FB , we take specific interest in systems that experience heavy traffic, that is, systems where ρ (n) ↑ 1 as n → ∞.In the current setting, this is equivalent to sequences λ (n) that converge to 1/E [B].Most results in this paper make no assumptions on sequence λ (n) , in which case we drop the superscript n for notational convenience and just state ρ ↑ 1.
The remainder of this section introduces some notation related to Matuszewska indices and extreme value theory.
• The upper Matuszewska index α(f ) is the infimum of those α for which there exists a constant C = C(α) such that for each µ * > 1, • The lower Matuszewska index β(f ) is the supremum of those β for which there exists a constant D = D(β) > 0 such that for each µ * > 1, One may note from the above definitions that β(f ) = −α(1/f ) holds for any positive f .Intuitively, a function f with upper and lower Matuszewska indices α(f ) and β(f ) is bounded between functions Dx β(f ) and Cx α(f ) for appropriate constants C, D > 0.More accurately, however, C and D could be unbounded or vanishing functions of x.Of special interest is the class of functions that satisfy (2.6)

Extreme value theory
The following paragraphs introduce some notions and results from extreme value theory.The field of extreme value theory generally aims to assess the probability of an extreme event; however, for our purposes we restrict attention to the limiting distribution of max{B 1 , . . ., B n }.
A key result on this functional is the Fisher-Tippett theorem: then H belongs to the type of one of the following three c.d.f.s: Weibull: Gumbel: The three distributions above are referred to as the extreme value distributions.
A c.d.f.F is said to be in the maximum domain of attraction of H if there exist norming sequences c n and d n such that (2.7) holds.In this case, we write F ∈ MDA(H).A large body of literature has identified conditions on F such that F ∈ MDA(H).Excellent collections of such and related results can be found in Embrechts et al. [9] and Resnick [21].For reasons of convenience, we only mention a characterisation theorem for MDA(Λ): Theorem 2.4 (Embrechts et al. [9], Theorem 3.3.26).The c.d.f.F with right endpoint x R ≤ ∞ belongs to the maximum domain of attraction of Λ if and only if there exists some z < x R such that F has representation where c and g are measurable functions satisfying c(x) → c > 0, g(t) → 1 as x ↑ x R , and f (•) is a positive, absolutely continuous function (with respect to the Lebesgue measure) with density If F ∈ MDA(Λ), then the norming constants can be chosen as The function f (•) in the above definition is unique up to asymptotic equivalence.We refer to f as the auxiliary function of F .Also, we note the following property of f (•): Lemma 2.5 (Resnick [21], Lemma 1.2).Suppose that f (•) is an absolutely continuous auxiliary function with f ′ (x) → 0 as x ↑ x R .
Proof.According to Theorem 3.3.27 in Embrechts et al. [9], G ∈ MDA(Λ) with auxiliary function f (•) if and only if lim x↑x R G(x + tf (x))/G(x) = e −t for all t ∈ R. It is straightforward to check that the above relation holds for any auxiliary function f (•) of F by using l'Hôpital and lim x↑x R f ′ (x) = 0.

Asymptotic relations
Let f (•) and g(•) denote two positive functions and A and B two random variables.We write f ∼ g if lim z↑z * f (z)/g(z) = 1, where the appropriate limit z ↑ z * depends on and should be clear from the context; it usually equals x ↑ x R or ρ ↑ 1.Similarly, we adopt the conventions Finally, the complementary error function is defined as Erfc(x) := 2π −1/2 ∞ x e −u 2 du.

Main results and discussion
This section presents and discusses our main results.Theorems 3.1 and 3.2 consider the asymptotic behaviour of the mean sojourn time E[T FB ] for various classes of distributions.Theorem 3.4 connects the asymptotic behaviour of F (G ← (ρ)) to the literature on extreme value theory.As a consequence, the expressions obtained in Theorem 3.2 can be specified for many distributions in MDA(Λ).Theorem 3.6 shifts focus to the distribution of T FB and states that the scaled sojourn time T FB /E[T FB ] tends to zero in probability.Instead, Theorem 3.7 shows that a certain fraction of jobs experiences a sojourn time of order (1− ρ) −2 .This result is achieved through the Laplace transform of the remaining sojourn time T * FB , for which we give an integral presentation.The proofs of the theorems are postponed to later sections.
Theorem 3.1 shows that the behaviour of E[T FB ] is fundamentally different for α(F ) < −2 and β(F (x)) > −2.In the first case, the variance of F is bounded and therefore the expected remaining busy period duration is of order Θ((1 − ρ) −2 ).Our analysis roughly shows that all jobs of size G ← (ρ) and larger will remain in the system until the end of the busy period, and hence experience a sojourn time of order Θ((1 − ρ) −2 ).The theorem shows that, as the work intensity increases to unity, the contribution of these jobs to the average sojourn time determines the overall average sojourn time.
The above argumentation does not apply in case β(F (x)) > −2, since then the expected remaining busy period duration is infinite.It turns out that in this case the mean sojourn time of a large job of size x is of the same order as the time that the job is in service, which has expectation x/(1 − ρ x ).The result follows after integrating over the job size distribution.
Additionally, it can be shown that the statements in Theorem 3.1 also hold if In this case, as well as in case F (•) or F (x R − (•) −1 ) is regularly varying, one can show that (1 − ρ) 2 E[T FB ]/F (G ← (ρ)) converges.Theorem 3.2 specifies Theorem 3.1 for the aforementioned cases, as well as for distributions with an atom in their endpoint.
Theorem 3.2.The following relations hold as ρ ↑ 1: where if H = Λ, and π/(α+1) sin(π/(α+1)) The expressions in Theorems 3.1 and 3.2 give insight into the asymptotic behaviour of E[T FB ].The following corollary shows that the asymptotic expressions above may be specified further if the job sizes are Pareto distributed.This extends the result by Bansal and Gamarnik [2], who derived the growth factor of E[T FB ] but not the exact asymptotics.
Proof.One may derive that G(x) The result then follows from Theorem 3.2.

Corollary 3.3 exemplifies that the asymptotic growth of E[T FB
] may be specified in some cases.However, it is often non-trivial to analyse the behaviour of F (G ← (ρ)) or equivalently h * (G ← (ρ)).Theorem 3.4 aims to overcome this problem if F ∈ MDA(Λ) by presenting a relation between h * (G ← (ρ)) and norming constants c n of F , which can often be found in the large body of literature on extreme value theory.
The same results hold if x R < ∞, provided that the F (•) and h * (•) in (i) and (ii) are replaced by , respectively.Remark 1. Condition (3.5) in part (i) of Theorem 3.4 is a Tauberian condition, and origins from Theorem 1.7.5 in Bingham et al. [6].A Tauberian theorem makes assumptions on a transformed function (here h * ), and uses these assumptions to deduce the asymptotic behaviour of that transform.It is non-restrictive in the sense the result in the theorem holds, i.e. if h * (x) ∼ αl(x)x α−1 , then obviously condition (3.5) is met and therefore the Tauberian condition does not restrict the class of functions F to which the theorem applies.However, condition (3.5) is necessary for the result to hold.The interested reader is referred to Section XIII.5 in Feller [10] and Section 1.7 in Bingham et al. [6].
Theorem 2.4 implies that ) exists, and exploits this limit to write E[T ).As to illustrate the implications of Theorem 3.4, the exact asymptotic behaviour of several well-known distributions is presented in Table 1.
We take a brief moment to compare the asymptotic mean sojourn time under FB to that under SRPT in M/GI/1 models.Clearly, FB can perform no better than SRPT due to SRPT's optimality [24].The ratio of their respective mean sojourn time is shown to be unbounded if the job sizes are Exponentially distributed or if the job size distribution has finite support [1,14,16,17], but bounded if the job sizes are Pareto distributed [2,16].To the best of the authors' knowledge, no results of this nature are known if job sizes are Weibull distributed.
The following corollary specifies the asymptotic advantage of SRPT over FB if the job sizes are Pareto distributed, and presents the first such results for Weibull distributed job sizes.Its statements follow directly from Corollaries 1 and 2 in Lin et al. [16] and the earlier results in this section.
Corollary 3.5.The following relations hold as ρ ↑ 1: Now that the asymptotic behaviour of the mean sojourn time under FB has been quantified, it is natural to investigate more complex characteristics.One such characteristic is the behaviour of the tail of the sojourn time distribution, where one usually starts by analysing the distribution of the sojourn time normalized by its mean, T FB /E[T FB ].The following theorem indicates that this random variable converges to zero in probability, meaning that almost every job experiences a sojourn time that is significantly shorter than the mean sojourn time as ρ ↑ 1: Benktander-II Table 1: Asymptotic expressions for the mean sojourn time for several well-known distributions in MDA(Λ), characterized by either their tail distribution or their probability density function (p.d.f.).These expressions follow from Table 3.4.4 in Embrechts et al. [9] through Theorem 3.4, where it is assumed that relation (3.5) holds.
Theorem 3.6 indicates that a decreasing fraction of jobs experiences a sojourn time of at least duration E[T FB ].Our final main result aims to specify both the size of this fraction, and the growth factor of the associated jobs' sojourn time.
The intuition from the proof of Theorem 3.1 suggests that T FB scales as (1 − ρ) −2 , but only for jobs of size at least G ← (ρ).This makes it conceivable that the scaled probability P((1 − ρ) 2 T FB > y)/F (G ← (ρ)) may be of Θ(1) as ρ ↑ 1. Theorem 3.7 confirms this hypothesis, and additionally shows that the residual sojourn time T * FB with density P( Theorem 3.7.Assume F ∈ MDA(H), where H is an extreme value distributions with finite (2 + ε)-th moment for some ε > 0. Let r(H) be as in relation (3.3).Then (1 − ρ) 2 T * FB converges to a non-degenerate random variable with monotone density g * as ρ ↑ 1, and almost everywhere.Here, ) All theorems presented in this section are now proven in order.First, Theorems 3.1 and 3.2 are proven in Section 4.Then, Theorem 3.4 is justified in Section 5. Finally, Sections 6 and 7 respectively validate Theorems 3.6 and 3.7.

Asymptotic behaviour of the mean sojourn time
In this section, we prove Theorems 3.1 and 3.2 in order.The intuition behind the theorems is that jobs of size x can only be completed once the server has finished processing of all jobs of size at most x.Additionally, jobs of size x experience a system with job sizes B i ∧ x since no job will receive more than x units of processing as long as there are size x jobs in the system.One thus expects all jobs of size x to stay in the system for the duration of a remaining busy period in the truncated system, which is expected to last for Θ It turns out that the asymptotic behaviour of (1 − ρ) 2 E[T FB ] is now determined by the fraction of jobs for which ν takes values away from zero.
If instead E[B 2 ] = ∞, then it will be shown that the growth rate of the second term in (2.1) is bounded by the growth rate of xG(x).It turns out that the sojourn time is of the same order as the time that a job receives service, which is of order Θ(x/(1 − ρ x )).
Both theorems follow after integrating E[T FB (x)] over all possible values of x, as shown in (2.2).By integrating by parts, we find that the first integral in (2.2) can be rewritten as Similarly, the second integral can be rewritten as and therefore We will now derive Theorems 3.1 and 3.2 from this relation.

General Matuszewska indices
This section proves Theorem 3.1.Relation (4.2) will be analysed separately for the cases −∞ < β(F ) ≤ α(F ) < −2 and −2 < β(F ) ≤ α(F ) < 1, which will be referred to as the finite and the infinite variance case, respectively.The finite variance case also considers Prior to further analysis, however, we introduce several results that will facilitate the analysis.

Finite variance
In this section, we assume either Noting that G ← is a continuous, strictly increasing function, it follows that the function . For this choice of x ν ρ , we have Dividing both sides by F (G ← (ρ)) yields We will show that I(ρ) + II(ρ) = o(1) and III(ρ) = Θ(1).Assume x R = ∞.Then, by Lemma 4.1 and Corollary 4.7 we find that and consequently I(ρ) = o(1) as ρ ↑ 1 by Lemma 4.2.
Next, fix 0 Let q(w) denote the integrand in the last line.A similar analysis to (4.4) indicates that the term in front of the integral vanishes as ρ ↑ 1, so we only need to show that the integral is bounded.This is implied by Lemma 4.4(i) after noting that where the inequalities follow from Lemmas 4.1 and 4.6 and Corollary 4.7.

Infinite variance
Assume β(F ) > −2 and recall that m . By Lemmas 4.1 and 4.5, one sees that β((•)G(•)) > 0 and therefore it follows from Lemma 4.3 that Consequently, it follows from relation (4.2) that, for some C, D > 0 and all ρ sufficiently close to one, we have and therefore

Special cases
This section proves Theorem 3.2.The maximum domain of attraction of each of the extreme value distributions are considered in order, followed by a distribution with an atom in its right endpoint.The Fréchet and Weibull cases follow rapidly from Theorem 3.1 and the Dominated Convergence Theorem.The same approach works for the Gumbel case, although Theorem 3.1 is not directly applicable.Finally, the atom case follows readily by analysing the sojourn time of maximum-sized jobs.  1) uniformly for all 0 < c < x, y < ∞.Therefore, we substitute w = ν−(1−ρ) ρ and exploit the Dominated Convergence Theorem to obtain Similarly, Theorem 3.3.12 in Embrechts et al. [9] states that F ∈ MDA(Ψ α ), α > 0, if and only if x R < ∞ and F (x R − x −1 ) = L(x)x −α is regularly varying with index −α.The corresponding result then follows after noting that α+1) uniformly for all 0 < c < x, y < ∞.

Gumbel
If F ∈ MDA(Λ), then so is G by Lemma 2.6 and we may choose h * as the auxiliary function of G. Propositions 0.9(a), 0.10 and 0.12 in Resnick [21] together state that Following the analysis in Section 4.1.1,we obtain α(I) = −1 < 0 as before.Consider term II(ρ).By Markov's inequality, we have The term in front of the integral and the integrand both have upper Matuszewska index −1/2, and therefore II(ρ) → 0. Lastly, consider term III(ρ).The relation lim sup ρ↑1 III(ρ) < ∞ follows analogously to the analysis in Section 4.1.1.Then, along the lines of Section 4.2.1, one may apply the Uniform Convergence Theorem and the Dominated Convergence Theorem to derive the theorem statement.

Atom in right endpoint
First, we show that I(ρ) + II(ρ) = o(1).Lemma 2.5 states that lim x↑x R h * (x) = ∞, and therefore The following lemma facilitates the analysis of this term.The proof the lemma is postponed until the end of this section.
Lemma 4.8.Let f : D → R be any function that maps D ⊆ R onto R, and assume that lim y↑x f (y) = p for some x in the closure D of D. Then, there exist z > 0 and q > 0 such that for all y ∈ (0, z] that satisfy x − y ∈ D.
Let q > 0 and δ * > 0 be such that and hence, for q = q(1 + ε)E[B]/p 2 , the relations hold for all ν > 1−ρ 1−ρ•ρ 0 , ρ > ρ 0 .Consider term III(ρ).On the one hand, we find On the other hand, we have Since ρ 0 may be chosen arbitrarily close to unity, we find Here, the last equivalence follows from (4.7).The section is concluded with the proof of Lemma 4.8.
We will obtain a contradiction by showing that (q n ) n∈N also converges as n → ∞.Assume that lim sup n→∞ q n ξ n > 0.Then, relation (4.9) implies lim sup n→∞ f (x − ξ n ) ≥ lim sup n→∞ p + q n ξ n > p which contradicts the lemma assumptions.Therefore, lim sup n→∞ q n ξ n must equal zero.It then follows that 0 ≤ lim sup n→∞ q n /q n+1 ≤ lim sup n→∞ q n ξ n = 0 and as such the ratio test states that the sequence (q n ) n∈N converges.
Note that Lemma 4.8 can be applied generally to yield lower and upper bounds for f (y) around any point x ∈ D for which either lim y↑x f (y) or lim y↓x f (y) exists.

Asymptotic behaviour of h
This section is dedicated to the proof of Theorem 3.4.Theorem 2.4 states that c n may be chosen as 1/h * (F ← (1 − n −1 )), so that the theorem follows from Theorem 3.2 after analysing the limit lim The proof heavily relies upon the work by De Haan [12] and Resnick [21], who both consider Γ-and Π-varying functions: for all t ∈ R. The function f (•) is called an auxiliary function and is unique up to asymptotic equivalence.
Definition 5.2.A function V : (x L , ∞) → R ≥0 is in the class of Π-varying functions if it is non-decreasing, and there exist functions a(x) > 0, b(x) ∈ R, such that for all t ∈ R. The function a(•) is called an auxiliary function and is unique up to asymptotic equivalence.
It turns out that Γ-and Π-varying functions are closely related to MDA(Λ).In particular, if F ∈ MDA(Λ) with auxiliary function 1/h * , then Proposition 1.9 in Resnick [21] states that U F := 1/F ∈ Γ with auxiliary function f F := 1/h * .Proposition 0.9(a) then states that ).Similarly, using Lemma 2.6, we find that ).Now, since Theorem 2.4 states that the norming constants c n may be chosen as 1/h * (F ← (1− 1/n)), we are done once we show that lim a G (x) tends to the right quantity for all cases in the theorem.
Corollary 3.4 in De Haan [12] states that   [21, p.44], this is equivalent to finding a function P (x), of the given form, that satisfies We use the following lemma, proven at the end of this section, to construct a suitable P (x): Lemma 5.3.Let F be a c.d.f.Then, there exists a strictly increasing, continuous c.d.f.F ↑ (x) As G ← (F ↑ (x)) is strictly increasing, there exists a positive function b(•) such that . Therefore, we see that (5.3) is satisfied with b 2 = 1 and b 3 = 0.The result follows once we show that , and once we show that The right-hand sides of both (5.4) and (5.5) depend on the function G ← (F (x)).The advantage of this representation is apparent from the following key relation, which connects G ← (F (x)) to h * (x): h * (t) dt .

Infinite support
First assume x R = ∞.The following theorem relates the assumptions on F (x) to properties of h * (x): (i) If there exists α > 0 and a slowly varying function l(x) such that − log then l(x) is slowly varying and h * (x) ∼ (l(x) − 1)/x as x → ∞.
The cases in Theorem 3.4 correspond to the cases in Theorem 5.4.We will consider the implications of Theorem 5.4 as to derive the results presented in Theorem 3.4.
Finally, if L = ∞ then h * (x) ↓ 0 and therefore G ← (F (x)) ≥ x by (5.6).For sake of contradiction, assume lim inf x→∞ G ← (F (x))/x < ∞.Then there exists M 0 ≥ 1 such that for all M ≥ M 0 there exists a sequence (x n ) n∈N , x n → ∞, such that G ← (F (x n ))/x n ≤ M for every n ∈ N. A similar analysis as in (i) then shows that this contradicts relation (5.9), and therefore lim x→∞ G ← (F (x))/x = ∞.

Finite support
Now assume x R < ∞.Theorem 2.4 states that F (x) can be represented as where c and g are measurable functions satisfying c(x) → c > 0, g(t) → 1 as x ↑ x R , and the auxiliary function f F (•) = 1/h * (•) is positive, absolutely continuous and has density . From this representation it is easy to obtain a finite-support equivalent of Theorem 5.4: (i) If there exists α > 0 and a slowly varying function l(x) such that − log (5.10) Again, the cases in Theorem 3.4 correspond to the cases in Corollary 5.5.The proof for the finite support case is similar to the infinite support case, yet we state it for completeness.Note that h * (x) → ∞ as x ↑ x R in both cases, and therefore x R −G ← (F (x)) x R −x ≥ 1 for all x sufficiently close to x R by (5.6).
We will show that lim x→∞ x R −G ← (F (x)) x R −x = 1 by contradiction.By our previous remark, we If this is false, then there exists ε ∈ (0, 1) and a sequence ( As before, the Uniform Convergence Theorem [6, Theorems 1.2.1 and 1.5.2]then implies (5.12) = 1 = e 0 by the analysis in (i).Alternatively, if L ∈ (0, ∞) then (5.6) and (5.12) imply where (x R − x)u(x) → 0 for all x sufficiently close to x R .One then obtains Lastly, consider L = ∞ and assume lim sup < ∞ for sake of contradiction.Then there exists M 0 ≥ 1 such that for all M ≥ M 0 there exists a sequence ( x R −xn ≤ M for every n ∈ N. A similar analysis as in (i) then shows that this contradicts relation (5.12), and therefore lim

Proof of Lemma 5.3
For any positive, non-increasing φ : [0, 1) → (0, 1) that vanishes as the argument tends to unity, we may define where s 1 := 0 and forms a strictly increasing sequence.Now, if s n ↑ s * < x R then F (s n ) ↑ p for some p ∈ (0, 1) and therefore, for any ε ∈ (0, 1) and all n sufficiently large, we have (1 , which yields a contradiction if ε < φ(p) 1−p p .We conclude that F φ is a strictly increasing, continuous c.d.f. that satisfies F φ (x) ≤ F (x) for all x.
Define n(x) := sup{n ∈ N : as x ↑ x R , so that F ↑ (x) ∼ F (x) by our earlier remark.
Let (s n ) n∈N and ( s n ) n∈N be the sequences associated with F φ and F φ and assume φ(y) ≤ φ(y) for all y ∈ [0, 1).We prove s n ≤ s n for all n ∈ N by induction.The inequality s 1 ≤ s 1 is immediate from the definition.Now, assume that s n ≤ s n and observe that (F (s) + q)/(1 + q) is non-decreasing in s for every q ≥ 0, and in q for every s ∈ R. Thus, any x that satisfies ), the proof is complete once we show that there is a version of φ such that lim To this end, we construct a suitable φ inductively.
Fix φ 1 := 1/2.Then, for n = 1, 2, . .., let r n+1 := inf x ≥ 0 : F (x) ≥ F (sn)+φn 1+φn , denote )) for notational convenience, one may now use (5.14) to deduce where the last inequality follows from the relation 6 Scaled sojourn time tends to zero in probability The current section is dedicated to the proof of Theorem 3.6.The intuition behind the proof is that the sojourn time of all jobs of size at most x ρ grows slower than E[T FB ], where x ρ is a function that depends on F .Alternatively, the fraction of jobs of size at least x ρ tends to zero, since x ρ → x R as ρ ↑ 1. Section 7 discusses the sojourn time of these jobs in more detail.
For any ε > 0 we have where the final term vanishes as ρ ↑ 1 by choice of x ρ .The proof is completed if the first probability at the right-hand side also vanishes as ρ ↑ 1.
In preparation of the analysis of P(T FB ( x ρ ) > εE[T FB ]), reconsider the busy period representation T FB (x) The relation states that the sojourn time of a job of size x is equal in distribution to a busy period with job sizes B i ∧ x, initiated by the job of size x itself and the time W x required to serve all jobs already in the system up to level x.Here, the random variable W x is equal in distribution to the steady state waiting time in an M/GI/1/FIFO queue with job sizes B ∧ x.
Let N x (t) denote a Poisson process with rate ρ x /E[B ∧ x].Then, it follows from the busy period representation of T FB that Additionally, application of Chebychev's inequality to the above relation yields At this point, similar to the approach in Section 4, we differentiate between the finite and infinite variance cases.

Finite variance
This section considers all functions F that satisfy one of the conditions in the theorem statement and have finite variance.Specifically, this excludes the case and γ ∈ ( p(F )/2, 1), and define ν(ρ) := (1 − ρ) γ and x ρ := x . Indeed x ρ → x R , and we proceed with the analysis in (6.3).

Infinite variance
This section regards all functions F that satisfy x R = ∞, β(F ) > −2.In this case, x ρ can be any function that satisfies both lim ρ↑1 x ρ = ∞ and lim ρ↑1 xρ for all ρ sufficiently close to one.Again, denote ε = εC.The analysis resumes with relation (6.3),where we substitute y by ε(1 − ρ) 2 log 1 1−ρ to obtain By relation (4.5), there exists a function b(x) that is bounded for all x sufficiently large and satisfies m Substituting this into the above relation yields , so that The result follows after noting that 1 − ρ x = 1 − ρG(x) ≥ G(x) and substituting x ρ for x.

Asymptotic behaviour of the sojourn time tail
In this section, we prove Theorem 3.7 after presenting two facilitating propositions.The proofs of the propositions are postponed to Sections 7.1 and 7.2.Throughout this section, e(q) will denote an Exponentially distributed random variable with rate q > 0. We abuse notation by writing e(0) = +∞.
Reconsider the relation T FB (x) d = L x (W x +x) to gain some intuition.A rough approximation of the duration of a busy period, given W x + x units of work at time t = 0, is (W x + x)/(1 − ρ x ).The scaled sojourn time (1 − ρ) 2 T FB (x) is then approximated by ).We will show that (1 − ρ)x ν ρ → 0 for all fixed ν ∈ (0, 1).Instead, the following proposition shows that (1 − ρ)W x ν ρ behaves as an exponentially distributed random variable as ρ ↑ 1: ), and let W ρ x denote the steady state waiting time in an M/GI/1/FIFO queue with job sizes B i ∧ x and arrival rate ρ Then, for any fixed ν ∈ (0, 1), ( 1 ∞ denotes the steady state waiting time in the non-truncated system, then Kingman [13] ). Proposition 7.1 shows how jobs can be truncated such that the exponential behaviour is preserved, and quantifies how the truncation affects the parameter of the exponential distribution.
Substituting the result in Proposition 7.1 into our approximation yields ) for every fixed ν ∈ (0, 1).We will show that the fraction of jobs for which ν is in (ε, 1 − ε) scales as F (G ← (ρ)), and that the contribution of other jobs to the tail of (1 − ρ) 2 T FB is negligible.The result is presented in Proposition 7.2, where we focus on the probability P((1 − ρ) 2 T FB > e(q)) for its connection to the Laplace transform of T * FB .
Proposition 7.2.Assume F ∈ MDA(H), where H is an extreme value distribution.Let p(H) 1) for all q ≥ 0. Here, the integral is finite for all q ≥ 0.
The Laplace transform inversion formula (12) in Bateman [4, p.234] states that f and hence relation (7.2) may be rewritten as We conclude that the limiting random variable lim ρ↑1 (1 − ρ) 2 T * FB has density g * .Furthermore, as for all q ≥ 0, we also see that lim ρ↑1 ) almost everywhere.To see that g * is monotone, it suffices to show that f (t) is monotone.To this end, we exploit the continued fraction representation (13.2.20a) in Cuyt et al. [8] and find Erfc(x) = x √ π e −x 2 1 x 2 + 1/2 1 + 1 As a consequence, one sees that which is negative for all t ≥ 0. We conclude the section with the postponed proofs of Propositions 7.1 and 7.2.
A more general version of this lemma has been stated in several other works [e.g. 6, 16]; however, these works refer to an unpublished manuscript by De Haan and Resnick for the corresponding proof.
Our final results relate the Matuszewska indices of F to those of related functions.First, Lemma 4.5 relates the Matuszewska indices of F to those of G. Its proof is similar to the proof of Lemma 6 in Lin et al. [16].Similarly, if x R < ∞ then α(1/G(x R − (•) −1 )) < ∞ and The result then follows from β )) and application of Lemma 4.5.

Lemma 4 .
1 states some closure properties of Matuszewska indices.Lemma 4.2 gives a sufficient condition for f to vanish.Lemmas 4.3 and 4.4 state helpful results on the asymptotic behaviour of the ratio between a function and certain integrals over this function, depending on its Matuszewska indices.Lemmas 4.5 and 4.6 and Corollary 4.7 specify and append the earlier lemmas by giving bounds on the Matuszewska indices of G ← and the composition of F and G ← .The proofs of Lemmas 4.1, 4.2, 4.5 and 4.6, along with several additional results, are postponed to Appendix A. Corollary 4.7 follows immediately from Lemmas 4.1 and 4.6.
Theorem 2.3(Resnick [21], Proposition 0.3).Let (B n ) n∈N be a sequence of i.i.d.random variables and define M n := max{B 1 , . . ., B n }.If there exist norming sequences c n > 0, d n ∈ R and some non-degenerate c.d.f H such that