Dominant poles and tail asymptotics in the critical Gaussian many-sources regime
DDominant poles and tail asymptotics in thecritical Gaussian many-sources regime ∗ A.J.E.M. Janssen J.S.H. van LeeuwaardenOctober 14, 2018
Abstract
The dominant pole approximation (DPA) is a classical analytic method to obtain from agenerating function asymptotic estimates for its underlying coefficients. We apply DPA to adiscrete queue in a critical many-sources regime, in order to obtain tail asymptotics for thestationary queue length. As it turns out, this regime leads to a clustering of the poles ofthe generating function, which renders the classical DPA useless, since the dominant pole isnot sufficiently dominant. To resolve this, we design a new DPA method, which might alsofind application in other areas of mathematics, like combinatorics, particularly when Gaussianscalings related to the central limit theorem are involved.
Probability generating functions (PGFs) encode the distributions of discrete random variables.When PGFs are considered analytic objects, their singularities or poles contain crucial informationabout the underlying distributions. Asymptotic expressions for the tail distributions, related tolarge-deviations events, can typically be obtained in terms of the so-called dominant singularities,or dominant poles. The dominant pole approximation (DPA) for the tail distribution is then derivedfrom the partial fraction expansion of the PGF and maintaining of this expansion the dominantfraction related to the dominant pole. Dominant pole approximations have been applied in manybranches of mathematics, including analytic combinatorics [7] and queueing theory [19]. We applyDPA to a discrete queue that has an explicit expression for the PGF of the stationary queue length.Additionally, this queue is considered in a many-sources regime, a heavy-traffic regime in whichboth the demand on and the capacity of the systems grow large, while their ratio approaches one.This many-sources regime combines high system utilization and short delays, due to economiesof scale. The regime is similar in flavor as the QED (quality and efficiency driven) regime formany-server systems [8], although an important difference is that our discrete queue fed by manysources falls in the class of single-server systems and therefore leads to a manageable closed formexpression for the PGF of the stationary queue length Q . Denote this PGF by Q ( z ) = E ( z Q ). ∗ This work was financially supported by The Netherlands Organization for Scientific Research (NWO) and by anERC Starting Grant. ∗ Eindhoven University of Technology, Department of Mathematics and Computer Science, P.O. Box 513, 5600MB Eindhoven, The Netherlands. { a.j.e.m.janssen,j.s.h.v.leeuwaarden } @tue.nl. a r X i v : . [ m a t h . P R ] D ec GFs can be represented as power series around z = 0 with nonnegative coefficients (related tothe probabilities). We assume that the radius of convergence of Q ( z ) is larger than one (in whichcase all moments of Q exist). This radius of convergence is in fact determined by the dominant sin-gularity Z , the singularity in | z | > Z is always a positive real number larger than one. Then DPA leads to the approximation P ( Q > N ) ≈ c − Z (cid:16) Z (cid:17) N +1 for large N (1.1)with c = lim z → Z ( z − Z ) Q ( z ). In many cases the approximation (1.1) can be turned into a morerigorous asymptotic expansion (for N large) for the tail probabilities P ( Q > N ). We shall nowexplain in more detail the many-sources regime, the discrete queue, and when combining both, themathematical challenges that arise when applying DPA.
Many sources and a discrete queue.
Consider a stochastic system in which demand perperiod is given by some random variable A , with mean µ A and variance σ A . For systems facinglarge demand one can set the capacity according to the rule s = µ A + βσ A , which consists of aminimally required part µ A and a variability hedge βσ A . Such a rule can lead to economies of scale,as we will now describe in terms of a setting in which the demand per period is generated by manysources. Consider a system serving n independent sources and let X denote the generic randomvariable that describes the demand per source per period, with mean µ and variance σ . Denotethe service capacity by s n , so that the system utilization is given by ρ n = nµ/s n , where the index n expresses the dependence on the scale at which the system operates. The traditional capacitysizing rule would then be s n = nµ + βσ √ n (1.2)with β some positive constant. The standard heavy-traffic paradigm [11, 15, 16], which builds onthe Central Limit Theorem, then prescribes to consider a sequence of systems indexed by n withassociated loads ρ n such that (also using that s = s n ∼ nµ ) ρ n = nµs n ∼ − βσµ √ n = 1 − γ √ s n , as n → ∞ , (1.3)where γ = βσ/ √ µ . We shall apply the many-sources regime given by (1.2) and (1.3) to a discretequeue, in which we divide time into periods of equal length, and model the net input in consecutiveperiods as i.d.d. samples from the distribution of A , with mean nµ and variance nσ . The capacityper period s n is fixed and integer valued. The scaling rule in (1.3) thus specifies how the mean andvariance of the demand per period, and simultaneously s n , will all grow to infinity as functions of n . Many-sources scaling became popular through the Anick-Mitra-Sondhi model [1], as one of thecanonical models for modern telecommunications networks, in which a switch may have hundredsof different input flows. But apart from communication networks, the concept of many sourcescan apply to any service system in which demand can be regarded as coming from many differentinputs (see e.g. [4, 17, 6, 13, 18] for specific applications. How to adapt classical DPA?
As it turns out, the many-sources regime changes drastically thenature of the DPA. While the queue is pushed into the many-sources regime for letting n → ∞ ,the dominant pole becomes barely dominant, in the sense that all the other poles (the dominatedones) of the PGF are approaching the dominant pole. For the partial fraction expansion of the2GF this means that it becomes hard, or impossible even, to simply discard the contributions ofthe fractions corresponding to what we call dominated poles: all poles other than the dominantpole. Moreover, the dominant pole itself approaches 1 according to Z ∼ β √ nσ , as n → ∞ . (1.4)This implies that in (1.1) the factor c / (1 − Z ) potentially explodes, while without imposingfurther conditions on N , the factor Z − N − goes to the degenerate value 1. The many-sourcesregime thus has a fascinating effect on the location of the poles that renders a standard DPAuseless for multiple reasons. We shall therefore adapt the DPA in order to make it suitable to dealwith the complications that arise in the many-sources regime, with the goal to again obtain anasymptotic expansion for the tail distribution. First observe that the term Z − N − in (1.1) becomesnon-degenerate when we impose that N ∼ K √ nσ , with K some positive constant, in which case (cid:16) Z (cid:17) N +1 ∼ (cid:16) β √ nσ (cid:17) − K √ nσ → e − βK ∈ (0 ,
1) as n → ∞ . (1.5)The condition N ∼ K √ nσ is natural, because the fluctuations of our stochastic system are of theorder √ nσ . Of course, there are many ways in which N and n can be coupled, but due to (1.4),only couplings for which N is proportional to √ n lead to a nondegenerate limit for Z − N − . Nowlet us turn to the other two remaining issues: The fact that c / (1 − Z ) potentially explodes andthat the dominated poles converge to the dominant pole.To resolve these two issues we present in this paper an approach that relies on approximationsof the type (1.4) for all the poles (which are defined implicitly as the solutions to some equation).The approximations are accurate in the many-sources regime, and can then be substituted intothe partial fraction expansion that describes the tail distribution. We replace the partial fractionexpansion by a contour integral representation, and subsequently apply a dedicated saddle pointmethod recently introduced in [11], with again a prominent role for the dominant pole (this time inrelation to the saddle point). The key challenge is to bound the contributions of the contour integralwhen shifted beyond the dominant pole, a contribution which is substantial due to the relative largeimpact of the dominated poles. This saddle point method then provides a fully rigorous derivationof the asymptotic expression for P ( Q > N ) and is of the form P ( Q > K √ nσ ) ∼ h ( β ) · e − βK , as n → ∞ . (1.6)The function h ( β ) in this asymptotic expression involves infinite series and Riemann zeta functionsthat are reminiscent of the reflected Gaussian random walk [5, 9, 10]. Indeed, it follows from [16,Theorem 3] that our rescaled discrete queue converges under (1.3) to a reflected Gaussian randomwalk. Hence, the tail distribution of our system in the regime (1.3) should for large n be wellapproximated by the tail distribution of the reflected Gaussian random walk. We return to thisconnection in Subsection 5.Our approach thus relies on detailed knowledge about the distribution of all the poles of thePGF of Q , and in particular how this distribution scales with the asymptotic regime (1.2)–(1.3).As it turns out, in contrast with classical DPA, this many-sources regime makes that all polescontribute to the asymptotic characterization of the tail behavior. Our saddle point method leadsto an asymptotic expansion for the tail probabilities, of which the limiting form corresponds to3he heavy-traffic limit, and pre-limit forms present refined approximations for pre-limit systems( n < ∞ ) in heavy traffic. Such refinements to heavy-traffic limits are commonly referred to as corrected diffusion approximations [14, 3, 2]. Compared with the studies that directly analyzed theGaussian random walk [5, 9, 10], which is the scaling limit of our queue in the many-sources regime,we start from the pre-limit process description, and establish an asymptotic result which is valuablefor a queue with a finite yet large number of sources. Starting this asymptotic analysis from theactual pre-limit process description is mathematically more challenging than directly analyzing theprocess limit, but in return gives valuable insights into the manner and speed at which the systemstarts displaying its limiting behavior. Outline of the paper.
In Section 2 we describe the discrete queue in more detail and presentsome preliminary results for its stationary queue length distribution. In Section 3 we give anoverview of the results and the contour integration representation for the tail distribution. InSection 4, we give a rigorous proof of the main result of the leading-order term using the dedicatedsaddle point method (Subsection 4.1), and of bounding the contour integral with integration pathsshifted beyond the dominant pole (Subsection 4.2). In Section 5 we elaborate on the connectionbetween the discrete queue and the Gaussian random walk, and we present an asymptotic seriesfor P ( Q > N ) comprising not only the dominant poles but also the dominated poles.
We consider a discrete stochastic model in which time is divided into periods of equal length. Atthe beginning of each period k = 1 , , , ... new demand A k arrives to the system. The demandsper period A , A , ... are assumed independent and equal in distribution to some non-negativeinteger-valued random variable A . The system has a service capacity s ∈ N per period, so that therecursion Q k +1 = max { Q k + A k − s, } , k = 1 , , ..., (2.1)assuming Q = 0, gives rise to a Markov chain ( Q k ) k ≥ that describes the congestion in the systemover time. The PGF A ( z ) = ∞ (cid:88) j =0 P ( A = j ) z j (2.2)is assumed analytic in a disk | z | < r with r >
1, which implies that all moments of A exist.We assume that A k is in distribution equal to the sum of work generated by n sources, X ,k + ... + X n,k , where the X i,k are for all i and k i.i.d. copies of a random variable X , of which the PGF X ( z ) = (cid:80) ∞ j =0 P ( X = j ) z j has radius of convergence r >
1, and0 < µ A = nµ = nX (cid:48) (1) < s. (2.3)Under the assumption (2.3) the stationary distribution lim k →∞ P ( Q k = j ) = P ( Q = j ), j =0 , , . . . exists, with the random variable Q defined as having this stationary distribution. We let Q ( z ) = ∞ (cid:88) j =0 P ( Q = j ) z j (2.4)be the PGF of the stationary distribution. 4t is a well-known consequence of Rouch´e’s theorem that under (2.3) z s − A ( z ) has precisely s zeros in | z | ≤
1, one of them being z = 1. We proceed in this paper under the same assumptionsas in [11]. We assume that | X ( z ) | < X ( r ), | z | = r , z (cid:54) = r , for any r ∈ (0 , r ). Finally, we assumethat the degree of X ( z ) is larger than s/n . Under these conditions, z is the only zero of z s − A ( z )on | z | = 1, and all others in | z | ≤
1, denoted as z , z , ..., z s − , lie in | z | <
1. Furthermore, there areat most countably many zeros Z k of z s − A ( z ) in 1 < | z | < r , and there is precisely one, denotedby Z , with minimum modulus.There is the product form representation [4, 17] Q ( z ) = ( s − µ A )( z − z s − A ( z ) s − (cid:89) j =1 z − z j − z j , (2.5)where the right-hand side of (2.5) is analytic in | z | < Z and has a first-order pole at z = Z . Wehave for the tail probability (using that Q (1) = 1) for N = 0 , , ...P ( Q > N ) = ∞ (cid:88) i = N +1 P ( Q = i ) = C z N (cid:104) − Q ( z )1 − z (cid:105) , (2.6)where C z N [ f ( z )] denotes the coefficient of z N of the function f ( z ). By contour integration, Cauchy’stheorem and Q (1) = 1, we then get for 0 < ε < Z − P ( Q > N ) = 12 πi (cid:90) | z | =1+ ε z N +1 − Q ( z )1 − z dz = 12 πi (cid:90) | z | = R z N +1 − Q ( z )1 − z dz + c Z N +10 (1 − Z ) , (2.7)where c = Res z = Z [ Q ( z )] and R is any number between Z and min k (cid:54) =0 | Z k | . When n and s arefixed, we have that the integral on the second line of (2.7) is O ( R − N ), and so there is the DPA P ( Q > N ) = c Z N +10 (1 − Z ) (1 + exponentially small) , N → ∞ . (2.8)In this paper we crucially rely on Pollaczek’s integral representation for the PGF of Q Q ( z ) = exp (cid:16) πi (cid:90) | v | =1+ ε ln (cid:16) z − v − v (cid:17) ( v s − A ( v )) (cid:48) v s − A ( v ) dv (cid:17) (2.9)that holds when | z | < ε < Z (principal value of ln on | v | = 1 + ε ). In order to force the discrete queue to operate in the critical many-sources regimes, we shall assumethroughout the paper the following relation between the number of sources n and the capacity s : nµs = 1 − γ √ s (3.1)5ith γ > ∞ as s → ∞ . In this scaling regime, the zeros z j and Z k of z s − A ( z ) = 0 start clustering near z = 1, as described in the next lemma (proved in the appendix).Let z ∗ j and Z ∗ k denote the complex conjugates of z j and Z k , respectively. Lemma 3.1.
For finite j, k = 1 , , ... and s → ∞ , z = 1 , Z = 1 + 2 a b √ s + O ( s − ) , (3.2) z j = z ∗ s − j = 1 + a √ s ( b − (cid:113) b − πij ) + O ( s − ) , (3.3) Z k = Z ∗− k = 1 + a √ s ( b + (cid:113) b − πik ) + O ( s − ) (3.4) with a = √ µσ , b = γ √ µσ √ and principal roots in (3.3) - (3.5) . Due to this clustering phenomenon, the main reasoning that underpin classical DPA cannot becarried over. Starting from the expression (2.8) we need to investigate what becomes of the term c / (1 − Z ), and moreover, the validity of the exponentially-small phrase in (2.8) and the actual N -range both become delicate matters that need detailed information about the distribution of thezeros as in Lemma 3.1.Let us first present a result that identifies the relevant N -range: Proposition 3.2. Z N +10 = exp (cid:16) − Lγµσ (cid:17) (1 + O ( s − / )) (3.6) when N + 1 = L √ s with L > bounded away from 0 and ∞ .Proof. We have from (3.2) and (3.5) that Z = 1 + γµσ √ s + O ( s ). Hence1 Z N +10 = exp (cid:16) − L √ s ln (cid:16) γµσ √ s + O ( s − ) (cid:17)(cid:17) = exp (cid:16) − Lγµσ + O ( s − / ) (cid:17) (3.7)when L is bounded away from 0 and ∞ , and this gives the result.From (2.5) we obtain the representation c − Z = − s − µ A sZ s − − A (cid:48) ( Z ) s − (cid:89) j =1 Z − z j − z j . (3.8)The next result will be proved in Section 4. Lemma 3.3. − s − µ A sZ s − − A (cid:48) ( Z ) = (cid:16) Z (cid:17) s − (cid:16) O ( s − / ) (cid:17) . (3.9)6e thus get from (3.8) and Lemma 3.3 c − Z = P ( Z ) P (1) (cid:16) O ( s − / ) (cid:17) , (3.10)where P ( Z ) = s − (cid:89) j =1 (1 − z j /Z ) = exp (cid:16) s − (cid:88) j =1 ln(1 − z j /Z ) (cid:17) (3.11)for Z ∈ C , | Z | ≥ P ( z ), in Lemma 3.4 below, weevaluate ln P ( Z ) for | Z | ≥ I ( Z ) = 12 πi (cid:90) | z | =1+ ε ln(1 − z − s A ( z )) Z − z dz, (3.12)where ε > < ε < Z . Lemma 3.4.
Let ε > , < ε < Z and | Z | ≥ . Then ln P ( Z ) = − ln(1 − Z − ) + I ( Z ) , ε < | Z | < r, − ln(1 − Z − ) + ln(1 − Z − s A ( Z )) + I ( Z ) , < | Z | < ε, ln( γ √ s ) + I (1) , Z = 1 . (3.13)The dedicated saddle point method, as considered in [11], applied to I ( Z ), with saddle point z sp = 1 + ε of the function g ( z ) = − ln z + ns ln( X ( Z )), yields I (1) = − ln[ Q (0)] + O ( s − / ) , I ( Z ) = ln[ Q (0)] + O ( s − / ) . (3.14)Combining (3.2), (3.5), (3.8), (3.9) and (3.14), then gives one of our main results: Proposition 3.5. ln (cid:16) c − Z (cid:17) = − ln(4 b ) + 2 ln[ Q (0)] + O ( s − / ) . (3.15)The next step consists of bounding the integral on the second line of (2.7), that can be writtenas − πi (cid:90) | z | = R Q ( z ) z N +1 (1 − z ) dz, (3.16)by choosing R appropriately. To do this, we consider the product representation (2.5) of Q ( z ), andwe want to choose R such that | z s − A ( z ) | ≥ C | z | s , | z | = R , for some C > s .It will be shown in Section 4 that this is achieved by taking R such that the curve | z s | = | A ( z ) | ,on which Z and Z ± lie, is crossed near a point z (also referred to as Z ± / ), where z s and A ( z )have opposite sign. A further analysis, using again the dedicated saddle point method to boundthe product (cid:81) s − j − in (2.5), then yields that the integral in (3.16) decays as R − N . Finally using theasymptotic information in (3.2)-(3.4) for Z and Z ± , with Z ± / lying midway between Z and Z ± , the integral on the second line of (2.7) can be shown to have relative order exp( − DN/ √ s ),for some D > s , compared to the dominant-pole term in (2.8).7o summarize, we have now that P ( Q > N ) = c − Z (cid:16) Z (cid:17) N +1 (cid:16) O (e − DN/ √ s ) (cid:17) , (3.17)for some D > s , N = 1 , , . . . . The DPA c / (1 − Z ) − Z − N − of P ( Q > N ) thushas a relative error that decays exponentially fast.In Subsection 5.1 the stationary queue length Q , considered in the many-sources regime, isshown to be connected to the Gaussian random walk. This connection will imply that the frontfactor of the DPA in (3.17) satisfies c − Z = H ( b ) (cid:16) O ( s − / ) (cid:17) , (3.18)where ln H ( b ) has a power series in b with coefficients that can be expressed in terms of theRiemann zeta function. Combining this with Proposition 3.2 and (3.17) yields P ( Q > N ) = H ( b ) exp (cid:16) − Lγµσ (cid:17)(cid:16) O ( s − / ) (cid:17) (3.19)when N + 1 = L √ s with L bounded away from 0 and ∞ . The leading term in (3.19) agrees with(1.6) when we identify L = σK/ √ µ, γ = βσ/ √ µ, s = nµ + βσ √ n ≈ nµ, b = β/ √ H ( b ) = h ( β ).In Subsection 5.2 we extend for a fixed M = 1 , , . . . the approach in Section 4 by increasingthe radius R of the integration contour in (3.16) to R M such that the poles Z , Z ± , . . . , Z ± M areinside | z | = R M . this lead to P ( Q > N ) = Re (cid:104) c (1 − Z ) Z N +10 + 2 M (cid:88) k =1 c k (1 − Z k ) Z N +1 k (cid:105) + O (cid:16) | Z M +1 | − N (cid:17) . (3.21)The front factors c k / (1 − Z k ) in the series in (3.21) satisfy c k − Z k = H k ( b ) (cid:16) O ( s − / ) (cid:17) , (3.22)with H k ( b ) some explicitly defined integral. When N + 1 = L √ s with L bounded away from 0and ∞ , we find from (3.22) and Proposition 3.2 that c k − Z k = H ( b k ) exp (cid:16) − La ( (cid:113) b − πik + b ) (cid:17)(cid:16) O ( s − / ) (cid:17) , (3.23)compare with (3.18), and it can be shown that this gives rise to an exp( − DL/ √ k ) decay of theright-hand side of (3.23). The results in (3.19) and (3.23) together give precise information as tohow the DPA arises, with leading behavior from the dominant pole, and lower order refinementscoming from the dominated poles. 8 DPA through contour integration
In this section we present the details of getting approximations of the tail probabilities usinga contour integration approach as outlined in Section 3. In Subsection 4.1, we concentrate onapproximation of the front factor c / (1 − Z ) and the dominant pole Z , and combine these toobtain an approximation of the leading-order term in (2.8). This gives Lemma 3.3, Lemma 3.4 andProposition 3.5.In Subsection 4.2 we assess and bound the integral on the second line of (2.7) and thereby makeprecise what exponentially small in (2.8) means in the present setting. From Z s = A ( Z ) = X n ( Z ) , µ A = nµ = s (cid:16) − γ √ s (cid:17) , A (cid:48) ( Z ) = nX (cid:48) ( Z ) X n − ( Z ) , (4.1)we compute s − µ A sZ s − − A (cid:48) ( Z ) = γ/ √ s − (1 − γ/ √ s ) X (cid:48) ( Z ) Z X (cid:48) (1) X ( Z ) 1 Z s − . (4.2)With the approximation (3.2), written as Z = 1 + d √ s + O ( s − ) , d = 2 γµσ , (4.3)we get X (cid:48) ( Z ) = X (cid:48) (1) + X (cid:48)(cid:48) (1)( Z −
1) + O ( s − ) = µ + X (cid:48)(cid:48) (1) d √ s + O ( s − ) (4.4)and X ( Z ) = X (1) + X (cid:48) (1)( Z −
1) + O ( s − ) = 1 + µd √ s + O ( s − ) . (4.5)Hence, by (4.3–4.5) and (A.6),1 − (cid:16) − γ √ s (cid:17) X (cid:48) ( Z ) Z X (cid:48) (1) X ( Z )= 1 − (cid:16) − γ √ s (cid:17) (cid:16) µ + X (cid:48)(cid:48) (1) d √ s + O ( s − ) (cid:17)(cid:16) d √ s + O ( s − ) (cid:17) µ (cid:16) µd √ s + O ( s − ) (cid:17) = 1 − (cid:0) − γ √ s (cid:17)(cid:16) d √ s (cid:16) X (cid:48)(cid:48) (1) µ + 1 − µ (cid:17) + O ( s − ) (cid:17) = 1 − (cid:0) − γ √ s (cid:17)(cid:16) d √ s σ µ + O ( s − ) (cid:17) = − γ √ s + O ( s − ) , (4.6)where we have used d of (4.3) in the last step. This gives (3.9).9 .1.2 Proof of Lemma 3.4 We have | A ( z ) | < | z s | when z (cid:54) = 1, 1 < | z | < Z , and so ln(1 − A ( z ) /z s ) is analytic in z (cid:54) = 1,1 < | z | < Z . When | Z | > ε , we have by partial integration and Cauchy’s theorem I ( Z ) = 12 πi (cid:90) | z | =1+ ε ln (cid:16) − zZ (cid:17) (1 − A ( z ) /z s ) (cid:48) − A ( z ) /z s dz = s − (cid:88) j =0 ln (cid:16) − z j Z (cid:17) = ln (cid:0) − Z (cid:17) + ln P ( Z ) . (4.7)This gives the upper-case formula in (3.13), and the middle case follows in a similar manner bytaking the residue at z = Z inside | z | = 1 + ε into account. For the lower case in (3.13), we use theresult of the middle case, in which we take 1 < Z < ε , Z ↓
1. We have I ( Z ) → I (1) as Z ↓ Z ↓ (cid:104) − ln (cid:16) − Z (cid:17) + ln (cid:16) − X n ( Z ) Z s (cid:17)(cid:105) = ln[( Z s − X n ( Z )) (cid:48) | Z =1 ] = ln( s − nµ ) = ln( γ √ s ) , (4.8)and this completes the proof. By (3.10) and Lemma 3.4 we haveln (cid:16) c − Z (cid:17) = ln P ( Z ) − ln P (1) + O ( s − / )= − ln (cid:16) − Z (cid:17) − ln( γ √ s ) + I ( Z ) − I (1) + O ( s − / ) . (4.9)From (3.2) it follows that − ln (cid:16) − Z (cid:17) − ln( γ √ s ) = − ln (cid:0) γ µσ (cid:17) + O ( s − / ) = − ln(4 b ) + O ( s − / ) , (4.10)and so ln (cid:16) c − Z (cid:17) = I ( Z ) − I (1) − ln(4 b ) + O ( s − / ) . (4.11)Next, we consider the integral representation (3.12) of I ( Z ), where we take ε such that1 + ε = z sp = 1 + γµσ √ s + O ( s − ) , (4.12)with z sp , see [11, Section 3], the unique point z ∈ (1 , Z ) such that ddz (cid:104) − ln z + ns ln( X ( z )) (cid:105) = 0 . (4.13)10bserve that z sp = (1 + Z ) + O ( s − ) , Z − z sp = z sp − O ( s − ) , (4.14)and this suggests that I ( Z ) ≈ − I (1), a statement made precise below, since the main contributionto I ( Z ) comes from the z ’s in (3.12) close to z sp .We have, see [11], ln[ P ( Q = 0)] = 12 πi (cid:90) | z | = z sp ln (cid:16) − z − s A ( z ) (cid:17) z ( z − dz. (4.15)Now 1 z − z ( z −
1) + 1 z , (4.16)and (cid:90) | z | = z sp (cid:12)(cid:12)(cid:12) ln (cid:16) − z − s A ( z ) (cid:17)(cid:12)(cid:12)(cid:12) | dz | = O ( s − / ) , (4.17)see [11, Subsection 5.3]. Hence, I (1) = − πi (cid:90) | z | = z sp ln (cid:16) − z − s A ( z ) (cid:17) z − dz = ln[ P ( Q = 0)] + O ( s − / ) . (4.18)As to I ( Z ), we observe that, see (4.14),1 Z − z = 1 z − z − (1 + Z )( Z − z )( z −
1) = 1 z − z − z sp ( Z − z )( z −
1) + O (1) . (4.19)Thus, I ( Z ) = − I (1) + 2 (cid:90) | z | = z sp z − z sp ( Z − z )( z −
1) ln (cid:16) − z − s A ( z ) (cid:17) dz + O ( s − / ) . (4.20)We next estimate the remaining integral in (4.20). With the substitution z = z ( v ), − δ ≤ v ≤ δ ,we have A ( z ( v )) /z s ( v ) = B exp( − sηv ) with 0 < B < η > z ( v ) = z sp + iv + ∞ (cid:88) n =2 c n ( iv ) n , − δ ≤ v ≤ δ, (4.21)where c n are real. Then we get with exponentially small error (cid:90) | z | = z sp z − z sp ( Z − z )( z −
1) ln (cid:16) − z − s A ( z ) (cid:17) dz = δ (cid:90) − δ ( z ( v ) − z sp ) z (cid:48) ( v )( Z − z ( v ))( z ( v ) −
1) ln(1 − Be − sηv ) dv. (4.22)Now we get from (4.14) and (4.21) that( Z − z ( v ))( z ( v ) −
1) = ( z sp − − iv + O ( v ))( z sp − iv + O ( v ))+ O (cid:16) s − ( z sp − iv + O ( v )) (cid:17) = | z sp − | + v + O (cid:16)(cid:16) s − + v (cid:17) / (cid:17) . (4.23)11urthermore, ( z ( v ) − z sp ) z (cid:48) ( v ) = − v + O ( v ) . (4.24)Thus ( z ( v ) − z sp ) z (cid:48) ( v )( Z − z ( v ))( z ( v ) −
1) = − v + O ( v ) | z sp − | + v + O (cid:16)(cid:16) s − + v (cid:17) / (cid:17) . (4.25)Inserting this into the integral on the second line of (4.22), we see that the − v in (4.25) cancelsupon integration. Also δ (cid:90) − δ v | z sp − | + v ln(1 − Be − sηv ) dv = O ( s − / ) , (4.26)and this finally shows that the integral in (4.22) is O ( s − / ). Then combining (4.11), (4.18), (4.20),we get the result. We have from (2.7) P ( Q > N ) = c Z N +10 (1 − Z ) − πi (cid:90) | z | = R Q ( z ) z N +1 (1 − z ) dz, (4.27)where R ∈ ( Z , | Z ± | ), and we intend to bound the integral at the right-hand side of (4.27). Weuse in (4.27) the Q ( z ) as represented by the right-hand side of (2.5) which is defined and analyticin z , | z | < r , z (cid:54) = Z k . We write for | z | < r , z (cid:54) = Z k Q ( z )(1 − z ) z N +1 = − z N +2 − s s − µ A (cid:81) s − j =1 (1 − z j ) 1 z s − A ( z ) s − (cid:89) j =1 (1 − z j /z ) . (4.28)Now s − µ A = γ √ s , and by Lemma 3.4 and (4.18), we have (cid:81) s − j =1 (1 − z j ) = P (1) ≥ Cγ √ s for some C > s . Hence ( s − µ A ) / (cid:81) s − j =1 (1 − z j ) is bounded in s . Next, for | z | ≥ Z , we haveby Lemma 3.4 s − (cid:89) j =1 (1 − z j /z ) = zz − I ( z )) , (4.29)with I ( z ) given by (3.12) and admitting an estimate | I ( z ) | = O (cid:16) | z − z sp | − ∞ (cid:90) −∞ ln(1 − Be − sηt ) dt (cid:17) = O (1) (4.30)since √ s | z − z sp | , B ∈ (0 ,
1) and η > z s − A ( z )) − . We show below that there is a C >
0, independent of s , such that | z s − A ( z ) | ≥ C | z | s (4.31)12hen z is on a contour K as in Figure , consisting of a straight line segment z = ξ + iη, ξ = Re[ ˆ Z ( ± )] , − √ s y ≤ η ≤ √ s y , (4.32)and a portion of the circle | z | = R = (cid:114) Re [ ˆ Z ( ± )] + 1 s y (4.33)that are joined at the points (Re[ ˆ Z ( ± )] , ± √ s y ). Hereˆ Z ( t ) = 1 + a √ s (( b − πit ) / + b ) , (4.34)with a , b > s , approximates the solution z = Z ( t ), for real t small compared to s , of the equation ns ln X ( z ) − ln z = 2 πits (4.35)outside the unit disk, according to Z ( t ) = ˆ Z ( t ) + O (cid:16) ts (cid:17) . (4.36)Thus on K we have from (4.31) (cid:12)(cid:12)(cid:12) Q ( z )(1 − z ) z N +1 (cid:12)(cid:12)(cid:12) = O (cid:16) z − z N +1 (cid:17) , (4.37)and we estimate (cid:12)(cid:12)(cid:12) πi (cid:90) | z | = R Q ( z ) z N +1 (1 − z ) dz (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) πi (cid:90) z ∈ K Q ( z ) z N +1 (1 − z ) dz (cid:12)(cid:12)(cid:12) = O (cid:16)(cid:16) Z ( ± )] (cid:17) N +1 (cid:90) z ∈ K | dz || z − | (cid:17) = O (cid:16) ln s (cid:16) Z ( ± )] (cid:17) N +1 (cid:17) . (4.38)Here we have used that | z − | ≥ Re[ ˆ Z ( ± )] − ≥ E/ √ s , z ∈ K , for some E > s . Observing that ˆ Z (0) = 1 + 2 a b √ s , (4.39)Re[ ˆ Z ( ± )] = 1 + a √ s [( b + ( b + π ) / ) + b ] , (4.40)we see that (cid:18) ˆ Z (0)Re[ ˆ Z ( ± )] (cid:19) N = (cid:32) − a √ s [( b + ( b + π ) / ) / − b ]1 + a √ s [( b + ( b + π ) / ) / + b ] (cid:33) N = O (exp( − ˆ DN/ √ s )) (4.41)13or some ˆ D > s . Hence, by (4.36), we see that the relative error in (4.27) due toignoring the integral at the right-hand side is of order exp( − DN/ √ s ) with some D >
0, independentof s .We show the inequality (4.31) for z ∈ K using the following property of X : there is a δ > ϑ ∈ (0 , π/
2) such that for any R ∈ [1 , δ ] the function | X ( Re iϑ ) | is decreasing in | ϑ | ∈ [0 , ϑ ]while ϑ ≤ | ϑ | ≤ π ⇒ | X ( Re iϑ ) | ≤ | X ( Re iϑ ) | . (4.42)This property follows from strict maximality of | X ( e iϑ ) | in ϑ ∈ [ − π, π ] at ϑ = 0 and analyticity of X ( z ) in the disk | z | < r (with r > K in (4.31–4.32), we consider the quantity n ln[ X (1 + v )] − ln(1 + v ) , (4.43)where v is of the form v = 2 γµσ √ s + x + iy √ s = ˆ Z (0) − x + iy √ s (4.44)with x > y ∈ R . We choose x such that the outer curve Z ( t ) is crossed by z = 1 + v near the points Z ( ± ), where z s − A ( z ) equals 2 z s . Thus, we choose x = Re[ √ s ( ˆ Z ( ± ) − ˆ Z )] = Re[ a (( b + πi ) / − b )] . (4.45)We have, as in the analysis in the appendix, that n ln[ X (1 + v )] − s ln(1 + v ) = 2 γµσ x + ( x − y ) + 2 i (cid:16) γµσ − x (cid:17) y + O ( sv ) + O ( v √ s ) . (4.46)With x > s , see (4.45), the leading part of the right-hand side in (4.46)is independent of s and describes, as a function of the real variable y , a parabola in the complexplane with real part bounded from above by its real value at y = 0 and that passes the imaginaryaxis at the points ± πi . Therefore, this leading part has a positive distance to all points 2 πik ,integer k . Now take y such that2 γµσ + ( x − y ) = − (cid:16) γµσ + x (cid:17) . (4.47)In Figure 1 we show the curve K (heavy), the approximation ˆ Z ( t ) of the outer curve, and thechoice y = η √ s for the case that γ = 1, µ/σ = 2, s = 100.It follows from the above analysis, with v as in (4.44), that1 − X n (1 + v )(1 + v ) s , − y ≤ y ≤ y , (4.48)is bounded away from 0 and has a value 1 − c, − c ∗ at y = ± y , where c is bounded away from 1and | c | <
1. Now write Re iϑ = ˆ Z (0) + x + iy √ s = 1 + v . (4.49)When s is large enough, we have that R ∈ [1 , δ ] and 0 ≤ ϑ ≤ ϑ , where δ and ϑ are as abovein (4.42). We have | X n (1 + v ) | ≤ | c | | v ) | s , (4.50)14 |z| = R z= + i ξ ηξ δ η
00 1
RR 1O ^ z sp ^ Z(0) ^ Z(-1/2) ^ Z(-1) ^ Z(-2) ^ Z(1/2) ^ Z(1) ^ Z(2)
Figure 1: Integration curve K consisting of line segment z = ξ + iη , − η ≤ η ≤ η , where ξ = Re[ ˆ Z ( ± )] and η = y / √ s , and portion of the circle | z | = R with R = ( ξ + η ) / . Choice ofparameters: γ = 1, µ/σ = 2, s = 100.and by (4.42) and monotonicity of | X ( Re iϑ ) | , ϑ ≤ | ϑ | ≤ ϑ , | X n ( Re iϑ ) | ≤ | X n ( Re iϑ ) | ≤ | c | R s , ϑ ≤ | ϑ | ≤ π. (4.51)Therefore, (4.31) holds on K with C = min (cid:110) − | c | , min | y |≤ y (cid:12)(cid:12)(cid:12) − X n (1 + v )(1 + v ) s (cid:12)(cid:12)(cid:12)(cid:111) (4.52)positive and bounded away from 0 as s gets large. In this section we give a series expansion for the leading term in (3.15) involving the Riemann zetafunction. We also show how to find an asymptotic series for P ( Q > N ) as N → ∞ of which theterm involving the dominant pole is the leading term. Before we do so, we first discuss how thisleading term is related to the Gaussian random walk and a result of Chang and Peres [5]. We know from [16, Theorem 3] that under the critical many-sources scaling, the rescaled queueingprocess converges to a reflected Gaussian random walk. The latter is defined as ( S β ( k )) k ≥ with S β (0) = 0 and S β ( k ) = Y + . . . + Y k (5.1)15ith Y , Y , . . . i.i.d. copies of a normal random variable with mean − β and variance 1. Assume β > M β .Denote by Q ( s ) the stationary congestion level for a fixed s (that arises from taking k → ∞ in(2.1)). Then, using ρ s = 1 − γ/ √ s , with γ = βσ √ sµ √ n , (5.2)the spatially-scaled stationary queue length reaches the limit Q ( s ) / ( σ √ n ) d → M β as s, n → ∞ (see[12, 15, 16]).The random variable M β was studied in [5, 9, 10]. In particular, [9, Thm. 1] yields, for β < √ π , P ( M β = 0) = √ β exp (cid:110) β √ π ∞ (cid:88) r =0 ζ ( − r )( − β ) r r !(2 r + 1) (cid:111) , (5.3)and from [5] we have P ( M β > K ) = h ( β, K ) e − βK with h ( β, K ) → h ( β ) = exp (cid:110) β √ √ π ∞ (cid:88) r =0 ζ ( − r )( − β ) r r !(2 r + 1) (cid:111) , (5.4)exponentially fast as K → ∞ .Hence, there are the approximations P ( Q > K √ nσ ) ≈ P ( M β > K ) ≈ h ( β ) · e − βK , as n → ∞ , (5.5)where the second approximation holds for small values of β . We will now show how this secondapproximation in (5.5) follows from our leading term in the expansion. Proposition 5.1. ln (cid:16) c − Z (cid:17) = 2 b √ π ∞ (cid:88) r =0 ζ ( − r )( − b ) r r !(2 r + 1) + O ( s − / ) , < b < √ π. (5.6) Proof.
It is shown in [11], Subsection 5.3 thatln[ Q (0)] = ln[ P ( M β = 0)] + O ( s − / ) , (5.7)in which we take the drift parameter β according to β = b √ γ (cid:114) µσ . (5.8)From [10] we haveln[ P ( M β = 0)] = ln(2 b ) + b √ π ∞ (cid:88) r =0 ζ ( − r )( − b ) r r !(2 r + 1) , < b < √ π. (5.9)Then from Proposition 3.5, (5.7) and (5.9), we get the results in (5.6).16 .2 Asymptotic series for P ( Q > N ) as N → ∞ When inspecting the argument that leads to (2.7), it is obvious that one can increase the radius R of the integration contour to values R M between | Z ( ± M ) | and | Z ( ± ( M + 1)) | when M = 1 , , ... is fixed. Here it must be assumed that s is so large that Z k increases in k = 0 , , ..., M + 1. Then,the poles of Q ( z ) at z = Z ± k , k = 0 , , ..., M , are inside | z | = R M , and we get P ( Q > N ) = c (1 − Z ) Z N +10 + 2 M (cid:88) k =1 Re (cid:104) c k (1 − Z k ) Z N +1 k (cid:105) − πi (cid:90) | z | = R M Q ( z )(1 − z ) z N +1 dz. (5.10)As in Subsection 4.2, one can argue that the integral on the second line of (5.10) is relatively smallcompared to | Z M | − N − when R M is chosen between but away from | Z M | and | Z M +1 | .We now need the following result. Lemma 5.2.
There holds − s − µ A sZ s − k − A (cid:48) ( Z k ) = b (cid:112) b − πik Z s − k (cid:16) O (cid:16) | k |√ s (cid:17)(cid:17) (5.11) when k = o ( s ) .Proof. This follows from the appendix with a similar argument as in the proof of Lemma 3.3.As to the terms in the series in (5.10), we have for bounded k , see Lemma 5.2, c k − Z k = − s − µ A sZ s − k − A (cid:48) ( Z k ) s − (cid:89) j =1 Z k − z j − z j = b (cid:112) b − πik · s − (cid:89) j =1 − z j /Z k − z j · (cid:16) O ( s − / ) (cid:17) . (5.12)Furthermore, according to Lemma 3.4,ln (cid:104) s − (cid:89) j =1 − z j /Z k − z j (cid:105) = I ( Z k ) − I (1) − ln (cid:16) − Z k (cid:17) − ln( γ √ s )= I ( Z k ) − I (1) − ln[2 b ( b + (cid:113) b − πik )] + O ( s − / ) . (5.13)Thus, we get the following result. Proposition 5.3.
For bounded k ∈ Z , c k − Z k = exp( I ( Z k ) − I (1))2 (cid:112) b − πik ( b + (cid:112) b − πik ) (cid:16) O ( s − / ) (cid:17) . (5.14)17e aim at approximating I ( Z k ), showing, in particular, that c k / (1 − Z k ) (cid:54) = 0 is bounded awayfrom 0 for bounded k and large s . To that end, we conduct the dedicated saddle point analysis for I ( Z k ). We have for | Z | ≥ Z , Re( Z ) > z sp , I ( Z ) = 12 πi (cid:90) | z | = z sp ln (cid:16) − z − s A ( z ) (cid:17) Z − z dz = 12 πi δ (cid:90) − δ z ( v ) Z − z ( v ) ln(1 − Be − sηv ) dv, (5.15)with exponentially small error in the last identity as s → ∞ . With g ( z ) = − ln z + ns ln X ( z ), welet B = exp( sg ( z sp )) = e − b (cid:16) O ( s − / ) (cid:17) , η = g (cid:48)(cid:48) ( z sp ) = σ µ + O ( s − / ) , (5.16)and z ( v ) is as in (4.21) and defined implicitly by g ( z ( v )) = g ( z sp ) − v g (cid:48)(cid:48) ( z sp ). We then find, byusing z ( v ) = z sp + iv + O ( v ) and z (cid:48) ( v ) = i + O ( v ), that I ( Z ) = − πi ∞ (cid:90) −∞ ln(1 − Be − sηv ) v + i ( Z − z sp ) dv + O ( s − / )= − πi ∞ (cid:90) −∞ ln(1 − Be − t ) t + i ( Z − z sp ) (cid:112) sη/ dt + O ( s − / ) , (5.17)where in the last step the substitution t = v (cid:112) sη/ t and − t for t ≥
0, we get the following result.
Proposition 5.4.
For | Z | ≥ Z , Re( Z ) > z sp , I ( Z ) = J ( d ) + O ( s − / ) , (5.18) where d = ( Z − z sp ) (cid:112) sη/ , and J ( d ) = 1 π ∞ (cid:90) dt + d ln(1 − Be − t ) dt. (5.19)In the context of Proposition 5.3, we consider d = d k = ( Z k − z sp ) (cid:112) sη/ , (5.20)with z sp = 1 + a b √ s + O ( s − ) , Z k = 1 + a √ s ( (cid:113) b − πik + b ) + O ( s − ) . (5.21)18sing the definitions of a , b in (A.10) and η in (5.16), we get d k = ˆ d k + O ( s − / ) ; ˆ d k = ( b − πik ) / . (5.22)We have that | ˆ d k | ≥ b , arg( ˆ d k ) ∈ ( − π, π ) , k ∈ Z . (5.23)Since for t ≥ d ) ∈ ( − π, π ), we haveln(1 − Be − t ) < , arg (cid:16) dt + d (cid:17) ∈ ( − π, π ) , (5.24)we see that we have complete control on the quantities J ( ˆ d k ) (also note (5.16) for this purpose).Using that − I (1) = I ( Z ) + O ( s − / ), see (4.20), we get the following result. Proposition 5.5.
For bounded k ∈ Z , c k − Z k = exp( J ( ˆ d k ) + J ( ˆ d ))2 ˆ d k ( ˆ d k + ˆ d ) + O ( s − / ) , (5.25) where the leading quantity in (5.25) (cid:54) = 0 , ∞ , ˆ d k is given in (5.22) with ˆ d = b , and J is given in(5.19). Theorem 5.6.
There is the asymptotic series P ( Q > N ) ∼ Re (cid:104) c (1 − Z ) Z N +10 + 2 ∞ (cid:88) k =1 c k (1 − Z k ) Z N +1 k (cid:105) , (5.26) where the ratio of the terms in the series with index M and M − is O ( | Z M − /Z M | N ) .Proof. This follows from (5.10), in which the integral is o ( | Z M | − N ) and the term with k = M is O ( | Z M | − N ), while the reciprocal of the term with k = M − O ( | Z M − | − N ) by Proposition 5.5.In the consideration of the terms with k = M − , M , it is tacitly accumed that s is so large that | Z k | , k = 0 , , ..., M is a strictly increasing sequence. A Proof of Lemma 3.1
We consider the zeros z j , j = 0 , , ..., s −
1, and Z k , k ∈ Z , of the function z s − A ( z ) in the unit disk | z | ≤ < | z | < r , respectively, in particular those that are relatively close to1. These zeros are elements of the set S A,s = { z ∈ C || z | < r, | z s | = | A ( z ) |} . For z ∈ S A,s , we havethat ln( z s X − n ( z )) is purely imaginary. We thus consider the equation s ln z = n ln X ( z ) + 2 πit (A.1)with z near 1 and t small compared to s . Writing u = 2 πt, z = 1 + v, (A.2)we get by Taylor expansion around z = 1 the equation s ( v − v + O ( v )) = n ln(1 + X (cid:48) (1) v + X (cid:48)(cid:48) (1) v + O ( v )) + iu. (A.3)19ividing by s and using that ns X (cid:48) (1) = 1 − γ/ √ s , yields v − v + O ( v ) = (cid:16) − γ √ s (cid:17)(cid:16) v + X (cid:48)(cid:48) (1) − ( X (cid:48) (1)) X (cid:48) (1) v + O ( v ) (cid:17) + i us , (A.4)i.e., − γ √ s v + σ µ v + i us = O (cid:16) v √ s (cid:17) + O ( v ) , (A.5)where we have used that µ = X (cid:48) (1) , σ = X (cid:48)(cid:48) (1) − ( X (cid:48) (1)) + X (cid:48) (1) > . (A.6)Dividing in (A.5) by σ / µ and completing a square, we get (cid:16) v − γµσ √ s (cid:17) = (cid:16) γµσ √ s (cid:17) (cid:16) − iuσ γ µ (cid:17) + O (cid:16) v √ s (cid:17) + O ( v ) . (A.7)Taking square roots at either side of (A.7) and using that the leading term at the right-hand sideof (A.7) has order (1 + | u | ) /s , we get v = γµσ √ s ± γµσ √ s (cid:16) − iuσ γ µ (cid:17) / + O (cid:16) | v | + | v | √ s (cid:112) | u | (cid:17) . (A.8)Irrespective of the ± -sign, the leading part of right-hand side of (A.8) has order ((1 + | u | ) /s ) / (and even ( | u | /s ) / in the case of the − -sign), and the O -term has order (1 + | u | ) /s which is o (((1 + | u | ) /s ) / ) as long as | u | /s = o (1). Thus, in that regime of u , we have z = 1 + v = 1 + γµσ √ s (cid:16) ± (cid:16) − iuσ γ µ (cid:17) / (cid:17) + O (cid:16) | u | s (cid:17) = 1 ± a √ s (( b − iu ) / ± b ) + O (cid:16) | u | s (cid:17) , (A.9)where we have inserted a = (cid:16) µσ (cid:17) / , b = (cid:16) γ µ σ (cid:17) / = γa . (A.10)In the case of the minus sign in (A.9), the O -term may be replaced by O ( | u | /s ). Choosing u = 2 πj with j = 0 , , ..., and j = o ( s ), we get from (A.9) with the minus sign, (3.3). Choosing u = 2 πk with k ∈ Z and k = o ( s ), we get from (A.9) with the plus sign, (3.4). References [1] D. Anick, D. Mitra, and M.M. Sondhi. Stochastic theory of a data-handling system withmultiple sources.
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