On Markov Krein characterization of the mean waiting time in M/G/K and other queueing systems

Transcription

1 Queueing Syst (211) 68: DOI 1.17/s On Markov Krein characterization of the mean waiting time in M/G/K and other queueing systems Varun Gupta Takayuki Osogami Received: 1 July 21 / Revised: 7 April 211 / Published online: 29 July 211 Springer Science+Business Media, LLC 211 Abstract We propose a new research direction to reinvigorate research into better understanding of the M/G/K and other queueing systems via obtaining tight bounds on the mean waiting time as functions of the moments of the service distribution. Analogous to the classical Markov Krein theorem, we conjecture that the bounds on the mean waiting time are achieved by service distributions corresponding to the upper/lower principal representations of the moment sequence. We present analytical, numerical, and simulation evidence in support of our conjectures. Keywords M/G/K Moments Bounds Multi-server systems Markov-Krein theorem Tchebycheff system Mathematics Subject Classification (2) 6K25 1 Introduction No book on queueing theory is complete without a discussion of the M/G/K queueing system one of the classical models of multiserver systems. However even fundamental performance metrics, such as the sojourn time in the M/G/K system, are well-understood only in certain asymptotic regimes (tail of the waiting time distribution [11]; heavy [21] and light traffic [4]; Halfin Whitt regime [12]). In particular, research on obtaining the exact mean sojourn time of the M/G/K system has almost V. Gupta ( ) Carnegie Mellon University, Pittsburgh, PA, USA varun@cs.cmu.edu T. Osogami IBM Research Tokyo, Kanagawa, Japan osogami@jp.ibm.com

2 34 Queueing Syst (211) 68: come to a standstill due to analytical obstacles. While there are numerous approximations, recent work [14] has shown that these approximations can be highly inaccurate from a worst-case perspective. In this paper, we propose a different approach obtaining tight bounds on the mean sojourn time as functions of the moments of the service distribution. We feel there are at least two reasons to justify this approach: (1) explicit exact analysis of M/G/K sojourn time is unlikely to be analytically tractable, and (2) often the entire service distribution is not known, while summary statistics such as the first few moments are easy to obtain. Analogous to the classical Markov Krein theorem, we conjecture that the distributions which achieve extremal values of the mean sojourn time given the first n moments are given by the upper/lower principal representations of the moment sequence (Sect. 3). We present analytical, numerical, and simulation evidence in support of our conjectures for the cases n = 2, 3. Finally, we generalize the problem of tight moment-based bounds on the mean sojourn time in an M/G/K to the problem of characterizing the class of stochastic recursive equations whose fixed points obey a Markov Krein type characterization (Sect. 4). Progress on this problem would allow us to obtain tight moments-based bounds for many queueing systems where the stationary distribution of the performance metric of interest is given by the fixed point of a stochastic recursive equation. Please see [15] for two further examples of system for which we can provide analytical justification of similar Markov Krein characterization under suitable asymptotic regimes, and for some more recent progress. Notation An M/G/K system consists of K identical servers and a First-Come- First-Serve (FCFS) queue. The jobs (or customers) arrive according to a Poisson process with rate λ and their service requirements are assumed to be independent, identically distributed random variables having a general distribution. We use S to denote such a generic random variable, and F S (x) = Pr[S x] to denote the distribution function of S. If an arriving job finds an idle server, it immediately enters service; otherwise it waits in the FCFS queue. When a server becomes idle, it chooses the next job to process from the head of the FCFS queue. Define ρ = λ E[S] <K.We will focus on the metric of mean waiting time in this work, denoted as E[W M/G/K ], and defined to be the expected time from the arrival of a customer to the time it enters service. Overview of related literature While there is a large body of work on approximating the E[W M/G/K ], all the closed-form approximations only involve at most the first two moments of the service distribution. To the best of our knowledge, the first approximation for E[W M/G/K ] was given by Lee and Longton [22]: E [ W M/G/K] ( E[S 2 ) ] E [ W M/M/K] (1) 2E[S] where E[W M/M/K ] is the mean waiting time with exponentially distributed service requirements with mean E[S]. This approximation is very simple, is exact for K = 1 and was shown to be asymptotically exact in heavy traffic by Köllerström [21]. The

3 Queueing Syst (211) 68: same expression is obtained by Nozaki and Ross [27] and by Hokstad [16] by making approximating assumptions during different phases of the solution. There is also a large literature on numerical methods for approximating E[W M/G/K ] by making weaker assumptions and solving for state probabilities. See, for example, [16, 17, 23, 24, 31, 33]. Miyazawa [24] uses basic equations to provide a unified view of approximating assumptions made in [27], [16] and [33], and to derive new approximation formulas. Other numerical algorithms include [3, 7 9, 26]. While these numerical methods are accurate and usually give an approximation for the entire waiting time distribution, the final expressions do not give any structural insight into the behavior of the queueing system and the effect of M/G/K parameters on waiting time. Heavy-traffic, light-traffic and diffusion approximations for the M/G/K system have been studied in [4, 19, 21, 37, 38, 4]. Motivated by call center applications, there is now a huge body of literature for multiserver systems with a large number of exponential servers; see the survey paper [13] and references therein. Bounds on the mean waiting time for M/G/K queues (and more generally, for GI/G/K queues) have mainly been obtained via two approaches (e.g., Sect from Wolff [39]). The first approach is by assuming various orderings (stochastic ordering, increasing convex ordering) on the service distributions (see [5, 25, 32, 34, 35]), but these tend to be very loose as approximations. Moreover, one does not always have the required strong orderings on the service distribution. The second, and more practical, approach that started with the work of Kingman [2] is obtaining bounds on mean waiting time in terms of the first two moments of the inter-arrival and service distributions. The best known bounds of this type for E[W GI/G/K ] are presented by Daley [6]. Scheller-Wolf and Sigman [29] derive bounds on for the case ρ< k 2 by reducing the GI/G/k waiting time recursion into an equivalent single-server recursion with dependent service times. Foss and Korshunov [11] and Scheller-Wolf and Vesilo [3] use dependent D/GI/1 queues to bound a GI/G/k system, and obtain necessary and sufficient conditions under which higher (even fractional) moments of delay are finite. Daley [6] also conjectures tight upper and lower bounds on E[W GI/G/K ] in terms of the first two moments of the service and the inter-arrival time distributions. Another approach with potential is to establish bounds by formulating a semidefinite program (SDP) with joint moments of service and inter-arrival time distribution forming the constraint set, and moments of waiting time as the objective function [1, 2]. Although most of the prior work obtains numerical bounds, Osogami and Raymond [28] use SDPs to establish closed-form bounds on the waiting time in a transient GI/G/1 queue. However, there are insufficient experimental results on the tightness of the resulting bounds for multiserver systems (primarily carried out for Markovian service distribution). 2 Principal representations and the Markov Krein theorem The classic Tchebycheff inequality concerns with bounds on the tail probability of a random variable X, givene[x] and E[X 2 ]. In other words, given the expecta-

4 342 Queueing Syst (211) 68: tions of functions f 1 (x) = x and f 2 (x) = x 2, one asks for bounds on the expectation of g(x) = 1 x E[X] >a. The theory of Tchebycheff systems [18] generalizes this question by asking for bounds on the expectation of some given function g( ) of a random variable, given a partial characterization of the random variable in terms of generalized moment constraints expressed as expectations of some functions f 1 ( ),...,f n ( ). In this section we will be concerned with the case f i (x) = x i. Below we present a special case of the results from this area. We will begin with the case where random variables are restricted to bounded support [,B] and where the results are easy to state. We then present results for the case where the support is [, ) and details are a little delicate. For a detailed treatment, we refer the reader to [18]. 2.1 Random variables with support on [,B] We first introduce the notion of upper and lower principal representations as presented in [1]. Define the function f (x) = 1, x B, and denote the moment space associated with {f,f 1,...,f n } as { M n+1 B = m R n+1 μ D,m i = B } f i (u) dμ(u), i n where D is the set of all non-decreasing right continuous functions for which the indicated integrals exist. For a point m in the interior of M n+1 B, we define the lower and upper principal representation (pr) to be distributions with a particular number of mass probabilities, some of which are restricted to be at or B, in such a way that the first n moments of these distributions agree with m. The precise constraints are n even n odd Upper pr ( μ) n 2 mass points in (,B), one at B n 1 2 mass points in (,B), one at, one at B Lower pr (μ) n 2 mass points in (,B), one at n+1 2 mass points in (,B) The upper and lower principal representations are uniquely determined when the functions {f,...,f n } satisfy certain linear independence constraints mentioned later. To see this, consider the case of upper pr for n even. We have n + 1 constraints, one each for m i, i n. If we are allowed n 2 +1 probability masses, then we have n+2 degrees of freedom n for the locations of the probability masses, and n 2 + 1for the actual probability values. Since one of the probability mass is constrained to be at B, we lose one degree of freedom, and thus the number of constraints match the degrees of freedom, where these constraints are linearly independent in a sense made precise next.

5 Queueing Syst (211) 68: Definition 1 Functions {h,h 1,...,h n } form a Tchebycheff system over [a,b] provided the determinants h (x ) h (x 1 ) h (x n ) ( ), 1,...,n h 1 (x ) h 1 (x 1 ) h 1 (x n ) U = x,x 1,...,x n... h n (x ) h n (x 1 ) h n (x n ) are strictly positive whenever a x <x 1 < <x n b. In other words, any non-trivial linear combination of h,...,h n must have at most n zeros in the interval [,B]. Systems of polynomials: h i (x) = x α i ( α <α 1 < <α n ) indeed form Tchebycheff systems. The proof of the following theorem can be found in [18, Chap. V, Sect. 5]: Theorem 1 (Markov Krein) If {f,...,f n } and {f,...,f n,g} are Tchebycheff systems on [,B], then { [ ] [ ] [ β l inf E g(x) Pr X [,B] = 1; E fi (X) ] = m i, i n } μ X D B = g(u)dμ(u), { [ ] [ ] [ β u sup E g(x) Pr X [,B] = 1; E fi (X) ] = m i, i n } μ X D = B g(u)d μ(u), where μ and μ are the unique lower and upper pr s, respectively, of m = {1,m 1,...,m n }, and μ X denotes the measure induced by X on R. 2.2 Random variables with support on [, ) As before, denote the moment space associated with {f,f 1,...,f n } as } M n+1 {m = R n+1 μ D,m i = f i (u) dμ(u), i n where D is the set of non-negative regular measures of bounded variation for which the indicated integrals exist. The definition of lower pr, μ, remains unchanged when we extend the support to [, ) as there are no atoms placed at the upper bound of the support. Hence, for a large enough B, thelowerprofm on [,B] will coincide with the lower pr on [, ). In particular, for n even, the lower pr will constitute of n 2 mass points in (, ) and one mass point at ; for n odd, there will be n+1 2 mass points in (, ). To define the upper pr, μ, we first need another definition.

6 344 Queueing Syst (211) 68: Definition 2 Functions {h,h 1,...,h n } form a Tchebycheff system of Type II over [, ) provided: (i) {h,...,h n 1 } and {h,...,h n } are Tchebycheff systems on [, ); and (ii) there exists A> such that h n (x) > forx A, and h i (x) lim x h n (x) = for i<n. If {f,...,f n } is a Tchebycheff system of Type II, then for m in the interior of M n+1, the upper pr puts one mass at, n 2 mass points in (,B), and additionally one at if n is odd. The following example might help readers uncomfortable with the idea of mass at infinity: Consider the case f i (x) = x i and n = 2. In this case, the upper pr can be seen as a limit ɛ of distributions with support [, 1 ɛ ] which put (ɛ2 ) mass on 1 ɛ. Thus, this mass at is needed to satisfy the constraint corresponding to f n, but does not contribute to constraints for f,...,f n 1 when {f,...,f n } is a Tchebycheff system of Type II. Theorem 2 (Markov Krein) (18, Theorem V5.1) If {f,...,f n } and {f,...,f n,g} are Tchebycheff systems on [, ), and m lies in the interior of M n+1, then there exists { [ ] [ ] [ β l inf E g(x) Pr X [,B] = 1; E fi (X) ] = m i, i n } μ X D which is achieved uniquely for μ X = μ, the lower pr of m. The upper bound { [ ] [ ] [ β u sup E g(x) Pr X [,B] = 1; E fi (X) ] = m i, i n } μ X D in general may not be attained, or may be infinite. However, if {f,...,f n } is a Tchebycheff system of Type II, and lim x g(x) f n (x) = γ<, then β u exists and is achieved by the upper pr of m. In the last sentence of the theorem, we say achieved to emphasize the fact that the upper pr has a mass point at and thus it is not a finite measure. However, β u exists and is achieved as a limit. The Markov Krein Theorem has been successfully applied in the context of queueing systems [1, 36]. In particular, for a G/M/1 system, Theorem 1 proves that given the first n moments of the inter-arrival time distribution, the mean number of jobs in the system is extremized by inter-arrival time distributions which correspond to the upper and lower pr s. The proof follows by noting that the mean number of jobs in a G/M/1 is an increasing function of the Laplace-Stieltjes transform of the inter-arrival time distribution, and the functions g s (x) = e sx form Tchebycheff

7 Queueing Syst (211) 68: system with f i (x) = x i.infact,whitt[36] goes further to show that a similar result holds if one restricts the inter-arrival time distribution to lie in the class of hyperexponential distributions (mixture of exponential distributions). This is achieved by expressing the Laplace transform of the inter-arrival time distribution in terms of the Laplace transform of its spectrum. 3 Tight bounds for M/G/K mean waiting time We begin by presenting conjectures on tight moment-based bounds for E[W M/G/K ]. 3.1 Conjectures Let m = (m = 1,m 1,m 2,...,m n ) R n + be such that there exists a positive random variable X with E[X i ]=m i,i=,...,n.forn odd, let D(m) denote the unique lower pr with moments m and support [, ) (and therefore has mass at ), and let DB (m) denote the unique upper pr with moments m and support [,B]. Forn even, let D (m) denote the unique lower principal representation with moments m and support (, ], and let D B (m) denote the unique upper pr with moments m and support [,B]. (The star in the superscript is to emphasize a point mass at, and the B in the subscript emphasizes the point mass at the upper bound, B, of the support.) Let { [ W h (m) = sup E W M/G/K ] E [ X i] = m i,i=,...,n }, μ X D { [ W l (m) = inf E W M/G/K ] E [ X i] = m i,i=,...,n }. μ X D Conjecture 1 Let m = (m = 1,m 1,...,m n ), n 1, be a valid moment sequence for positive distributions. Let m = (m,m 1,...,m n 1 ). Then, Case 1: n odd (i) W h (m) = lim B E[W M/D B (m)/k ]=E[W M/D (m )/K ]. (ii) W l (m) = E[W M/D(m)/K ]. (iii) W l (m 1,...,m n 1,x)is strictly decreasing in x when K>1. Case 2: n even (i) W h (m) = E[W M/D (m)/k ]. (ii) W l (m) = lim B E[W M/DB(m)/K ], and if n>2, or n = 2 and ρ<k 1, W l (m) = E[W M/D(m )/K ]. (iii) W h (m 1,...,m n 1,x)is strictly increasing in x when K>1. Further, for n odd, W h (m) = W h (m ); and for n even (and ρ<k 1ifn = 2), W l (m) = W l (m ). Plainly stated, the conjectures say the following: If we are given only the mean of the service distribution, we only have enough information to fix a lower bound on E[W M/G/K ]. This lower bound is given by E[W M/D/K ]. If we are additionally

8 346 Queueing Syst (211) 68: m 2 m 2 1 given the second moment of the service distribution, we can fix an upper bound on E[W M/G/K ] (it can be shown that this conjectured upper bound is given by E[W M/D/K ] [14]). But if ρ K 1 we also refine our lower bound. By determining the third moment of service distribution, we can refine (tighten) our lower bound but this lower bound decreases as the third moment increases. Further, the upper bound remains unchanged (Case 1, (i)). The upper pr has probability mass at, and hence the service distribution is not a finite measure, but W h (m) can be obtained via a limit of systems with bounded service times. Similarly, knowledge of the fourth moment will refine the upper bound on the mean waiting time (bring it down), but leave the lower bound unchanged (Case 2, (ii)), and so forth for alternating higher even and odd moments. An interesting questions is whether the upper and lower bounds converge to a single value as n. We believe that this should be true under certain regularity conditions on the job size distribution, e.g., the moment sequence uniquely determines the distribution. For example, we have seen numerically that a moment sequences from lognormal distributions (whose moments grow very quickly) yield a slow rate of convergence. We present numerical evidence in support of Conjecture 1 in Fig. 1 by restricting attention to the class of hyperexponential distributions and using Matrix analytic methods to evaluate the performance. Notice that a hyperexponential distribution is a probabilistic mixture of exponential distributions with varying means, which can be interpreted as an exponential distribution having a random mean. We can define the lower (respectively, upper) pr of a moment sequence, m, within the class of hyperexponential distributions to be the exponential distribution whose random mean has the lower (respective, upper) pr and has the moment sequence determined by m (see [15] for details). In Fig. 1(a), we plot the mean waiting time for two-phase hyperexponential distributions which allows us to vary E[S 3 ] while holding the first two moments fixed. We see that indeed the curve lies between the mean waiting time of the H 2 distributions representing upper and lower pr for the given first two moments, the curve is decreasing in E[S 3 ], and since n = 2 and ρ>k 1, the upper pr Fig. 1 Numerical results to support Conjecture 1: (a) and(c) plot the mean waiting time with H 2 service distribution while keeping E[S]=1, E[S 2 ]=2 and varying E[S 3 ].(b)and(d) plot the mean waiting time with H3 service distribution while varying E[S4 ] and maintaining E[S]=1, E[S 2 ]=2, E[S 3 ]=8

9 Queueing Syst (211) 68: Fig. 2 Bounding mean delay in an M/G/4 queue when the job size has a Weibull distribution refines the lower bound obtained from exponential service distribution. In Fig. 1(c) we see the same trend, except that since ρ<k 1, the performance under upper pr (i.e. lower bound on mean waiting time) coincides with the lower bound obtained from exponential service distribution. In Figs. 1(b) and (d), we plot the mean waiting time for H3 service distribution (two exponential phases with non-zero mean, 1 phase with mean ), which allows us to vary E[S 4 ] while holding fixed the first three moments. Here again we see that the numerical experiments agree with Conjecture 1. Figure 2 shows simulation results E[W M/G/K ] and its bounds obtained with principal representations, when the job size distribution is a Weibull distribution. We fix the parameters of the Weibull distribution such that its probability density function is f(x)= 1 2 x 1/2 exp( x 1/2 ) for x. We also fix the number of servers, k = 4, and vary the load, ρ λe[s], as indicated below each figure. The dashed line shows the exact value of E[W M/G/K ], and the dots show bounds on E[W M/G/K ] obtained with principal representations. The horizontal axis indicates the number of moments used to determine the principal representations. The moments of the Weibull distribution under consideration are E[S n ]=(2n)! for n 1. Notice that the lower (respectively, upper) bound obtained with an even (respectively, odd) number n of moments in general does not improve the corresponding lower (respectively, upper) bound obtained with n 1 moments. An exception is that the lower bound obtained with n = 2 moments improves upon that with n = 1forρ = 3.6, but not for ρ = 2.4, in agreement with Conjecture 1. The lower bound with n = 2 moments is given by a limiting distribution where one of mass points, B, approaches infinity. The lower bound shown corresponds to the principal representation with B = 1 6. (It appears that the mean delay with the principal representation hardly changes in the interval between B = 1 4 and B = 1 6.ForaB>1 6, the analysis of the mean delay suffers from numerical errors.) Also observe that, as the number of moments used to determine principal representations grows, the upper and lower bounds approach each other quickly particularly at high load (Fig. 2(b)). This makes intuitive sense, because E[W M/G/K ] becomes insensitive to third and higher moments in heavy traffic. A Markov Krein characterization for E[W M/G/K ] by expressing it as an increasing function of the expectation of a function g( ) which forms a Tchebycheff system with f i (x) = x i, and then directly applying Theorem 1, eludes us. We believe that

10 348 Queueing Syst (211) 68: a more fruitful approach might be to start a new research direction Markov Krein characterization of fixed points of Stochastic Recursive equations (Sect. 4). 3.2 Analytical justification via light traffic limit The major obstacle in the analysis of the M/G/K model is that the Kiefer Wolfowitz workload vector can only be obtained in an implicit form through the Lindley recursion. However, in certain asymptotic regimes the waiting time distribution has explicit limits. As we mentioned earlier, in heavy traffic [21] (K fixed, ρ K), W M/G/K depends only on E[S] and E[S 2 ], and hence is insensitive to higher moments of S. However, in light traffic (K fixed, ρ ), W M/G/K depends on the entire distribution, and has a rather explicit characterization [4]. We now support Conjecture 1 by verifying it for n = 2, 3 in light traffic. Theorem 3 Given the first n (n = 2 or 3) moments of the service distribution S, E[W M/G/K ] under light traffic is extremized by service distributions given by the lower and upper principal representations of the moment sequence. Proof Due to lack of space we sketch the proof for the case n = 3 (for n odd, we have conjectured that the lower pr minimizes, and upper pr maximizes E[W M/G/K ]). Recall that under light traffic, the probability of waiting is asymptotically given by ρk K!, and conditioning on the event that an arrival waits, W M/G/K is distributed as the minimum of K independent copies of the stationary excess of S, denoted by S e. u=x The survival function of S e is given by F Se (x) = Pr[S e >x]= F S(u) du. Denote x Pr[S>u] du F Se = h for succinctness. Since 1 h(x) = E[S] is the integral of a bounded, non-negative, decreasing function, (1 h(x)) is a continuous, non-decreasing, concave function. The problem of extremizing E[W M/G/K ] in the light-traffic asymptote can thus be equivalently formulated as min / max h(u) K du E[S] subject to h( ) continuous, non-negative, non-increasing, convex; h() = 1; h ( +) Pr[S >] = 1 E[S] E[S] ; h(u) du = E[S2 ] 2E[S] ; u h(u) du = E[S3 ] 12E[S]. Note that a solution to the above problem exists because h(u) K h(u), and h(u) du is finite. Let h represent the survival function of Se corresponding to the lowerprofs for the given moment sequence. Now, suppose that h is not the solution to the minimization problem above, and the solution is instead given by h min.for n = 3, we see that the lower pr has two point masses, say at <x 1 <x 2 <, as shown on the right.

11 Queueing Syst (211) 68: The absolute value of the slope of h min at + must be at most that of h since the lower pr has no mass at for n = 3, and because h min is convex, it follows that δ = h min h satisfies (i) δ(u)du = (i.e., areas under h and h min are equal), (ii) u δ(u)du = (from moment conditions), and (iii) δ( ) changes sign exactly twice, and the sequence of signs is + + (see the figure above). We obtain a contradiction: h min (u) K du h(u) K du = δ(u) [ h min (u) K 1 + h min (u) K 2 h(u) + +h(u) K 1] du >. To see the last inequality, denote the function in the square brackets by l( ), and note that l is convex. Now δ(u)l(u) du =[δ(u)l (u)] u= l (u) u v= δ(v)dvdu. The first term is zero because δ() = δ( ) =. Now assuming derivatives exist (by approximating by smoothed versions) l (u) is an increasing function, and δ(v)dv is a function that changes sign only once, from + to and integrates to. Thus u= l (u) u v= δ(v)dvdu <. The proof for upper pr is identical: the sequence of signs of δ in this case is +. For n = 2, δ changes sign once. Note that following the above proof, one can prove a stronger result: consider a random variable Y with a continuous and convex survival function. Then given the first n (n = 2, 3) moments of a positive random variable X (independent of Y ), E[min{X e,y}] is extremized by X given by upper/lower pr s. 4 Stochastic recursive equations and Markov Krein characterization The proof of Theorem 3 is far from elegant in that it would be quite tedious to extend it to n 4. Even if one were to prove Theorem 3 for arbitrary n, this would only be a partial justification for Conjecture 1 unless one could prove that the ordering of E[W M/G/K ] for different service distributions is invariant to ρ. We believe that a more promising approach would be to start with the Lindley recursion for the Kiefer Wolfowitz workload vector, and then try to prove an analog of Theorem 1. We believe that a similar result is likely to hold in other application areas as well, and hence we propose a general framework by posing the following moment problem.

12 35 Queueing Syst (211) 68: Consider a queueing system which can be represented by a stochastic recursive equation. That is, there exists a function φ such that W d = φ(w,s) (2) where W is the unknown random vector capturing the performance of the system, and d = denotes equality in distribution. For example, for the M/G/K queueing system, let W = (W 1,W 2,...,W k ) where W 1 W 2 W k denote the Kiefer Wolfowitz workload vector seen by arriving customer (equivalently, the ordered vector of times at which the K servers will idle, assuming the customer arrives at time t = ). The distribution of W is then given by W d = R ( (W + S e 1 A e) +) where S is the random variable for the service requirement, A is the random variable for inter-arrival time, e 1 is a K-vector whose first element is 1 and the rest are, e is a K-vector all of whose elements are 1, and R is a function that reorders the elements of its argument in ascending order. The final performance metric of interest is then E[g(W)] for some function g (e.g. E[W 1 ] for E[W M/G/K ]). We now pose the following problem: Our goal is to seek bounds on E[g(W)], given the first n moments of S. For what class of probability flows φ( ) and functions g( ) can these bounds be characterized along the Markov Krein Theorem? 5 Conclusions Through this paper we hope to have provided sufficient justification to warrant exploration of Markov Krein characterization of solutions of Stochastic recursive equations. More than just being a theoretical curiosity, such a characterization would permit numerical computation of provable and tight bounds based on partial characterization of the random variables driving the system. One might then extend the question we posed in Sect. 4 to the case where S is a vector, and we are given the first n (joint) moments of S. This problem is relevant to multi-class systems, systems with general inter-arrival time distributions, and a large class of systems which have multiple general input distributions. Acknowledgements The authors express sincere gratitude to the referees for numerous suggestions, which have helped improve the exposition of this paper. References 1. Bertsimas, D., Natarajan, K.: A semidefinite optimization approach to the steady-state analysis of queueing systems. Queueing Syst. 56(1), (27) 2. Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: A convex optimization approach. SIAM J. Optim. 15, (25) 3. Breuer, L.: Transient and stationary distributions for the GI/G/k queue with Lebesgue-dominated inter-arrival time distribution. Queueing Syst. 45(1), (23)

13 Queueing Syst (211) 68: Burman, D., Smith, D.: A light-traffic theorem for multi-server queues. Math. Oper. Res. 8, (1983) 5. Daley, D., Rolski, T.: Some comparability results for waiting times in single- and many-server queues. J. Appl. Probab. 21, (1984) 6. Daley, D.J.: Some results for the mean waiting-time and workload in GI/GI/k queues. In: Dshalalow, J.H. (ed.) Frontiers in Queueing: Models and Applications in Science and Engineering, Boca Raton, FL, USA, pp (1997) 7. de Smit, J.H.A.: A numerical solution for the multiserver queue with hyper-exponential service times. Oper. Res. Lett. 2(5), (1983) 8. de Smit, J.H.A.: The queue GI/M/s with customers of different types or the queue GI/H m /s.adv. Appl. Probab. 15(2), (1983) 9. de Smit, J.H.A.: The queue GI/H m /s in continuous time. J. Appl. Probab. 22(1), (1985) 1. Eckberg, A. Jr.: Sharp bounds on Laplace Stieltjes transforms, with applications to various queueing problems. Math. Oper. Res. 2(2), (1977) 11. Foss, S., Korshunov, D.: Heavy tails in multi-server queue. Queueing Syst. 52(1), (26) 12. Gamarnik, D., Momčilović, P.: Steady-state analysis of a multiserver queue in the Halfin Whitt regime. Adv. Appl. Probab. 4(2), (28) 13. Gans, N., Koole, G., Mandelbaum, A.: Telephone call centers: tutorial, review, and research prospects. Manuf. Serv. Oper. Manag. 5, (23) 14. Gupta, V., Dai, J., Harchol-Balter, M., Zwart, B.: On the inapproximability of M/G/K: why two moments of job size distribution are not enough. Queueing Syst. 64(1), 5 48 (21) 15. Gupta, V., Osogami, T.: On Markov-Krein characterization of mean sojourn time in queueing systems. Technical Report CMU-CS-11-19, School of Computer Science, Carnegie Mellon University (211) 16. Hokstad, P.: Approximations for the M/G/m queue. Oper. Res. 26(3), (1978) 17. Hokstad, P.: The steady state solution of the M/K 2 /m queue. Adv. Appl. Probab. 12(3), (198) 18. Karlin, S., Studden, W.J.: Tchebycheff Systems: With Applications in Analysis and Statistics. Wiley, New York (1966) 19. Kimura, T.: Diffusion approximation for an M/G/m queue. Oper. Res. 31, (1983) 2. Kingman, J.: Inequalities in the theory of queues. J. R. Stat. Soc. 32(1), (197) 21. Köllerström, J.: Heavy traffic theory for queues with several servers. I. J. Appl. Probab. 11, (1974) 22. Lee, A., Longton, P.: Queueing process associated with airline passenger check-in. Oper. Res. Q. 1, (1959) 23. Ma, B., Mark, J.: Approximation of the mean queue length of an M/G/c queueing system. Oper. Res. 43(1), (1995) 24. Miyazawa, M.: Approximation of the queue-length distribution of an M/GI/s queue by the basic equations. J. Appl. Probab. 23, (1986) 25. Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley Series in Probability and Statistics. Wiley, Chichester (22) 26. Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover, New York (1995), revised edn. 27. Nozaki, S., Ross, S.: Approximations in finite-capacity multi-server queues with Poisson arrivals. J. Appl. Probab. 15(4), (1978) 28. Osogami, T., Raymond, R.: Semidefinite optimization for transient analysis of queues. ACM SIG- METRICS Perform. Eval. Rev. 38(1), (21) 29. Scheller-Wolf, A., Sigman, K.: New bounds for expected delay in FIFO GI/GI/c queues. Queueing Syst. 26(1 2), (1997) 3. Scheller-Wolf, A., Vesilo, R.: Structural interpretation and derivation of necessary and sufficient conditions for delay moments in FIFO multiserver queues. Queueing Syst. 54(3), (26) 31. Stoyan, D.: Approximations for M/G/s queues. Math. Oper.forsch. Stat., Ser. Optim. 7, (1976) 32. Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, Chichester (1983). Translation from the German edited by D.J. Daley 33. Tijms, H.C., Van Hoorn, M.H., Federgruen, A.: Approximations for the steady-state probabilities in the M/G/c queue. Adv. Appl. Probab. 13, (1981) 34. Whitt, W.: The effect of variability in the GI/G/s queue. J. Appl. Probab. 17, (198)

14 352 Queueing Syst (211) 68: Whitt, W.: Comparison conjectures about the M/G/s queue. Oper. Res. Lett. 2(5), (1983) 36. Whitt, W.: On approximations for queues, I: Extremal distributions. AT&T Bell Lab. Tech. J. 63, (1984) 37. Whitt, W.: Approximations for the GI/G/m queue. Prod. Oper. Manag. 2(2), (1993) 38. Whitt, W.: A diffusion approximation for the G/GI/n/m queue. Oper. Res. 52, (24) 39. Wolff, R.W.: Stochastic Modeling and the Theory of Queues. Prentice Hall, New York (1989) 4. Yao, D.: Refining the diffusion approximation for the M/G/m queue. Oper. Res. 33, (1985)