Using Robust Queueing to Expose the Impact of Dependence in Single-Server Queues

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Ward Whitt, Wei You

To cite this article: Ward Whitt, Wei You (2018) Using Robust Queueing to Expose the Impact of Dependence in Single-Server Queues. Operations Research 66(1):184-199. https://doi.org/10.1287/opre.2017.1649

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INFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org OPERATIONS RESEARCH Vol. 66, No. 1, January–February 2018, pp. 184–199 http://pubsonline.informs.org/journal/opre/ ISSN 0030-364X (print), ISSN 1526-5463 (online)

Using Robust Queueing to Expose the Impact of Dependence in Single-Server Queues

Ward Whitt,a Wei Youa a Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027 Contact: [email protected], http://orcid.org/0000-0003-4298-9964 (WW); [email protected], http://orcid.org/0000-0003-0844-4194 (WY)

Received: March 4, 2016 Abstract. Queueing applications are often complicated by dependence among interar- Revised: October 10, 2016; March 11, 2017 rival times and service times. Such dependence is common in networks of queues, where Accepted: May 3, 2017 arrivals are departures from other queues or superpositions of such complicated pro- Published Online in Articles in Advance: cesses, especially when there are multiple customer classes with class-dependent service- July 31, 2017 time distributions. We show that the robust queueing approach for single-server queues Subject Classifications: queues: proposed in the literature can be extended to yield improved steady-state performance approximations, networks, algorithms approximations in the standard stochastic setting that includes dependence among inter- Area of Review: Stochastic Models arrival times and service times. We propose a new functional robust queueing formulation for the steady-state workload that is exact for the steady-state mean in the M GI 1 model https://doi.org/10.1287/opre.2017.1649 and is asymptotically correct in both heavy traffic and light traffic. Simulation/ experiments/ Copyright: © 2017 INFORMS show that it is effective more generally. Funding: Support was received from the National Science Foundation [Grants CMMI 1265070 and 1634133]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2017.1649.

Keywords: robust queueing • queueing approximations • dependence among interarrival times and service times • indices of dispersion • heavy traﬃc • queueing network analyzer

1. Introduction The RQ in Bandi et al. (2015) emphasizes the high Robust optimization is proving to be a useful approach dimension arising when we consider a network of to complex optimization problems involving signifi- queues instead of a single queue. Instead, in this paper cant uncertainty; e.g., see Bandi and Bertsimas (2012), we focus on the high dimension that occurs in a sin- Bertsimas et al. (2011), and references therein. In that gle queue when there is complex stochastic depen- context, the primary goal is to create an efficient algo- dence over time in the arrival and service processes. rithm to produce useful, practical solutions that appro- In a sequel, Whitt and You (2016), we focus on the priately capture the essential features of the uncer- high dimension that occurs in a single queue when the tainty. Bandi et al. (2015) have applied this approach deterministic arrival-rate function is time varying. For to create a robust queueing (RQ) theory, which can both problems, we find that the robust optimization be used to generate performance predictions in com- approach is remarkably effective. Here, we show that, plex queueing systems, including networks of queues with an appropriate choice of parameters, all our new as well as single queues. Indeed, they construct a full RQ solutions are asymptotically correct in the heavy- robust queueing analyzer (RQNA) to develop rela- traffic limit. Our most promising new RQ solutions tively simple performance descriptions such as those in in (18) and (28) are asymptotically correct in both light the queueing network analyzer (QNA) in Whitt (1983). traffic and heavy traffic. Our simulation experiments Our goal in this paper is to make further progress show that the new RQ solutions provide useful approx- in the same direction. We do so by introducing new imations more generally. RQ formulations and evaluating their performance. We too want to obtain useful performance descriptions for 1.1. Dependence Among Interarrival Times and complex queueing networks, but here we only con- Service Times sider a single queue. We judge our RQ formulations by Even though we only focus on one single-server queue, their ability to efficiently generate useful performance ultimately we also want to develop methods that apply approximations for the given stochastic model, which to complex networks of queues. We view the present so far has been mostly intractable. paper as an important step in that direction, because As emphasized in Bandi and Bertsimas (2012), the experience from applications of QNA has shown that a intractability is usually due to high dimension, but major shortcoming is its inability to adequately capture high dimensionality can occur in many different ways. the dependence among interarrival times and service

Figure 1. Common Queueing Network Structure That Can Induce Dependence Among Interarrival Times: Superpositions of Arrival Processes (Top) and Flow Through a Series of Queues (Bottom)

Queue

… Server

Queue 1 Queue 2 … Queue n times at the individual queues in the network. That process is again Poisson. In other words, there are no was dramatically illustrated by comparisons of QNA to nondeterministic non-Poisson renewal departure pro- model simulations in Sriram and Whitt (1986), Fendick cesses from an M GI 1 queue; e.g., see Disney and et al. (1989), and Suresh and Whitt (1990). Konig (1985). / / Dependence among successive interarrival times at The dependence among interarrival times and ser- a queue is a common phenomenon, usually because vice times has long been recognized as a major that queue is actually part of a network of queues. difficulty in developing effective approximations for For example, arrival processes in queueing networks open queueing networks, such as in QNA in Whitt are often superpositions of other arrival processes or (1983); e.g., see Whitt (1995) and references therein. departure processes from other queues, as depicted in Refined performance approximations have been pro- Figure1. posed using second-order partial characterizations of In most manufacturing production lines, an exter- dependence, using indices of dispersion (variance-time nal (or initial) arrival process is often far less variable functions), which involve correlations among interar- than a Poisson process by design, while complicated rival times as well as means and variances; e.g., see Cox processing operations, such as those involving batch- and Lewis (1966), Heffes (1980), Heffes and Luantoni

ing, often produce complicated dependence among the (1986), Sriram and Whitt (1986), Fendick et al. (1989, interarrival times at subsequent queues; e.g., see the 1991), and Fendick and Whitt (1989). Our new RQ for- example in section 3 of Segal and Whitt (1989). In both mulations will exploit these same partial characteriza- manufacturing and communication systems, depen- tions of the dependence among interarrival times and dence among successive interarrival times and among service times; see Sections 3.3 and4. Even though we successive interdeparture times at a queue often occurs only consider a single queue here, in Section6 we intro- because there are multiple classes of customers with duce a new framework in which we hope to develop a different characteristics (e.g., Bitran and Tirupati 1988). full RQNA based on the results in this paper. Multiple classes can even cause significant dependence (i) among interarrival times, (ii) among service times, 1.2. Main Contributions and (iii) between interarrival times and service times, 1. In this paper, we introduce several new RQ for- which all can contribute to a major impact on perfor- mulations for the steady-state waiting time and work- mance, as shown by Fendick et al. (1989) and reviewed load in a single-server queue, and we make useful in section 9.6 of Whitt (2002). connections to the general stationary G G 1 stochas- / / In service systems, an external customer arrival tic model and the GI GI 1 special case. In particular, / / process often is well modeled by a Poisson process, we show how to choose the RQ parameters so that because it is generated by many separate people mak- these RQ solutions all are asymptotically exact for the ing decisions independently, at least approximately, steady-state mean in the heavy-traffic limit. but dependence may be induced by overdispersion; 2. In addition to new parametric versions of RQ as in e.g., see Kim and Whitt (2014) and references there. By Bandi et al. (2015), we introduce new functional formu- contrast, internal arrivals within a network of queues lations that capture the impact of dependence among are less likely to be well approximated by a Poisson the interarrival times and service times over time upon process, because the flow through queues disrupts the the steady-state performance of the queue as a function statistical regularity of a Poisson process. In particular, of the traffic intensity ρ. (See the uncertainty sets in (9) service-time distributions are often not nearly expo- and (15).) nential, while the interdeparture times in steady state 3. We evidently introduce the first RQ formulations from an M GI 1 queue, with GI meaning that the ser- for the continuous-time workload process and show vice times are/ independent/ and identically distributed that it is advantageous to do so. We show how to (i.i.d.), are themselves i.i.d. only if the service-time dis- choose the RQ parameters so that the solution of the tribution is exponential, in which case the departure functional RQ for the workload coincides with the Whitt and You: Dependence in Single-Server Queues 186 Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS steady-state mean in the M GI 1 model for all traffic These experiments demonstrate (i) the strong impact intensities and is simultaneously/ / asymptotically cor- of dependence upon performance and (ii) the value rect in both heavy traffic and light traffic for the general of the new RQ in capturing the impact of that depen- G G 1 model, including the dependence. dence. Finally, in Section6 we introduce a new frame- / / 4. We conduct simulation experiments showing that work for applying the results in this paper to develop a the new functional RQ for the workload is effective new RQNA that better captures the dependence. Addi- in exposing the impact of the dependence among the tional supporting material appears in the e-companion interarrival times and service times over time upon the (EC)—in particular, (i) a short summary of the main mean steady-state workload as a function of the traffic paper; (ii) additional discussion; (iii) additional theo- intensity. retical support, including central limit theorems and 5. We provide a road map for the application to net- heavy-traffic limit theorems; (iv) more results for the works of queues by introducing a new framework for discrete-time waiting time and indices of dispersion; an RQNA based on indices of dispersion. We show that and (v) more simulation examples. such an RQNA is feasible and provide support with a simulation comparison for a series queue network. 2. Robust Queueing for the Steady-State 1.3. More Related Literature Waiting Time Mamani et al. (2016) also incorporated dependence We start by reviewing the RQ developed in sections 2 within a robust optimization formulation for a problem and 3.1 of Bandi et al. (2015), which involves sepa- in inventory management (which we might call RI), rate uncertainty sets for the arrival times and service but otherwise, there is relatively little overlap with times. We then construct an alternative formulation this paper; we discuss the connection in Remark4. with a single uncertainty set and show, for the GI GI 1 Mamani et al. (2016) point to early RI work by Scarf queue, that a natural version of the RQ solution/ coin-/ (1958) and then Moon and Gallego (1994). The new RQ cides with the Kingman (1962) bound and so is asymp- work is also related to Whitt (1984a), which used opti- totically correct in the heavy-traffic limit. We show that mization to study performance approximations used both formulations provide insight into the relaxation in QNA. In particular, Whitt (1984a) studied the range time for the GI GI 1 queue, the approximate time of possible values for the mean steady-state num- required to reach/ steady/ state. ber in a GI M 1 queue subject to specified first and / / We use the representation of the waiting time (before second moments of the interarrival-time distribution. receiving service) in a general single-server queue Klincewicz and Whitt (1984) and Whitt (1984b) con- with unlimited waiting space and the first-come first- struct tighter bounds based on additional constraints to served (FCFS) service discipline, without imposing any enforce a realistic shape on the underlying interarrival- stochastic assumptions. The waiting time of arrival n time distribution. This work showed that we can hope satisfies the Lindley (1952) recursion to obtain useful accuracy such as 20% relative error, but that we cannot hope to obtain extraordinarily high + + Wn Wn 1 Vn 1 Un 1 ( − − − − ) accuracy, such as an only 5% error, given the usual par- + max Wn 1 Vn 1 Un 1, 0 , (1) tial information based on the first two moments. And ≡ { − − − − } that is not yet considering the dependence. Ignoring where Vn 1 is the service time of arrival n 1, Un 1 the dependence can lead to much bigger errors, as in − − is the interarrival time between arrivals n −1 and n, Fendick et al. (1989) and section 9.6 of Whitt (2002). and denotes equality by definition. If we− initialize ≡ 1.4. Organization of the Paper the system by having an arrival 0 finding an empty In Section2, after reviewing RQ for the steady-state system, then Wn can be represented as the maximum waiting time in the single-server queue from sections 2 of a sequence of partial sums, using the Loynes (1962) and 3.1 of Bandi et al. (2015), we develop an alter- reverse-time construction; i.e., native formulation whose solution coincides with the Wn Mn max Sk , n > 1, (2) Kingman (1962) bound and is asymptotically correct in ≡ 06k6n { } heavy traffic. In Section3 we introduce new parametric + + and functional RQ formulations for the continuous- using reverse-time indexing with Sk X Xk and ≡ 1 ··· time workload process and characterize their solutions. Xk Vn k Un k , 1 6 k 6 n, and S0 0. (Bandi et al. ≡ − − − ≡ In Section4 we introduce the index of dispersion for 2015 actually look at the system time, which is the sum work (IDW) and incorporate it in the RQ. We develop of an arrival’s waiting time and service time. These closed-form RQ solutions and show that the func- representations are essentially equivalent.) tional RQ is asymptotically correct in both heavy and If we extend the reverse-time construction indefi- light traffic. In Section5 we conduct simulation exper- nitely into the past from a fixed present state, then Wn iments for the two network structures in Figure1. W sup S with probability 1 as n , allowing↑ ≡ k>0 { k } → ∞ Whitt and You: Dependence in Single-Server Queues Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS 187

W ˜ x + + for the possibility that might be infinite. For the sta- where X Xk : k > 1 ∞, Sk X1 Xk , k > 1, ble stationary G G 1 stochastic model with E U < , and S 0≡, while{ m and} ∈b are constant≡ ··· parameters to / / [ k ] ∞ 0 x E V < and ρ E V E U < 1, P W < 1; e.g., be specified.≡ To avoid excessively strong constraints for [ k ] ∞ ≡ [ k ]/ [ k ] ( ∞) see Loynes (1962) or section 6.2 of Sigman (1995). small values of k, not justified by the CLT, we could Bandi et al. (2015) propose an RQ approximation replace k in the constraint bounds on the right in (5) by for the steady-state waiting time W by performing a max k, kL , but that lower bound kL has no impact if deterministic optimization in (2) subject to determinis- chosen{ appropriately.} Combining (2) and (5), we obtain tic constraints, where we can ignore the time reversal. the alternative RQ optimization a Treating the partial sums Sk of the interarrival times s x Uk and the partial sums Sk of the service times Vk sep- W ∗ sup sup Sk , (6) ≡ X˜ Ux k>0 { } arately leads to the two uncertainty sets (for W): ∈

a a x ˜ U U˜ ∞: S > km b √k, k > 0 and where Sk is the function of X specified above. The RQ ≡ { ∈ k a − a } (3) s s formulations in (4) and (6) are attractive because the U V˜ ∞: S 6 km + b √k, k > 0 , { ∈ k s s } optimization problems have simple solutions in which ˜ ˜ all constraints are satisfied as equalities. That follows where U Uk : k > 1 and V Vk : k > 1 are arbitrary ≡ { } ≡ { a }+ + easily from the fact that Wn is a nondecreasing (nonin- sequences of real numbers in ∞; Sk U1 Uk and s + + ≡ ··· creasing) function of Vk (of Uk ) for all k and n. The sim- Sk V1 Vk , k > 1; S0 0; and ma, ms , ba, and bs are≡ parameters··· to be specified.≡ The constraints in (3) ple closed-form solution follows from the triangular are one-sided because that is what is required to bound structure of the equations; see section 3.1 of Bandi et al. the waiting times above, as we can see from (1) and (2). (2015). The following is a direct extension of theorem 2 Thus, the RQ optimization can be expressed as of Bandi et al. (2015) to include the new RQ formulation in (6). The final statement involves an interchange s a W ∗ sup sup sup Sk Sk , (4) of suprema, which is justified by Lemma EC.1. ≡ U˜ Ua V˜ Us k>0 { − } ∈ ∈ Theorem 1 (RQ Solutions for the Steady-State Waiting a s ˜ ˜ where Sk (Sk ) is a function of U (V) specified above. Time). The RQ optimizations (4) with ma > ms > 0 and (6) Versions of this formulation in (4) and others in this with m > 0 have the solution paper also apply to the transient waiting time Wn, but we will focus on the steady-state waiting time. W ∗ sup mk + b√k 6 sup mx + b√x {− } {− } Thinking of the general stationary G G 1 stochas- k>0 x>0 tic model, where the distributions of U/ and/ V are b2 b2 k k mx∗ + b√x for x∗ , (7) independent of k (but stochastic independence is not − ∗ 4m 4m2 assumed), Bandi et al. (2015) assume that m E U a k where m m m > 0. For (4), b b + b ; for (6), b b . and m E V and assume that m > m , so that≡ [ρ ] a s s a x s k a s In (7), W is maximized− at one of≡ the integers immediately≡ m m <≡1.[ The] square-root terms in the constraints≡ ∗ s a above or below x . in (/3) are motivated by the central limit theorem (CLT). ∗ Thinking of the GI GI 1 model in which the interar- We now establish implications for the GI GI 1 and / / / / rival times Uk and service times Vk come from indepen- general stationary G G 1 models. To discuss heavy- dent sequences of i.i.d. random variables with finite traffic limits, it is convenient/ / to introduce the traffic 2 2 variances σa and σs , the CLT suggests that ba βa σa and intensity ρ as a scaling factor applied to the interar- bs βs σs for some positive constants βa and βs , perhaps rival times. Hence, we start with a sequence Uk , Vk {( )} with β βa βs . With this choice, these new parameters where E Uk E Vk 1 for all k. Then in model ρ we [ ] [ ] 1 measure the number of standard deviations away from let the interarrival times be ρ− Uk , where 0 < ρ < 1. 1 the mean in a Gaussian approximation. Bandi et al. Thus, ms 1 and ma ρ− , so that m 1 ρ ρ and 2 ≡ ( − )/ (2015) also provide an extension to cover the heavy- W ∗ b ρ 4 1 ρ in (7). tailed case, where finite variances might not exist; then Since the/ ( CLT− underlies) the heavy-traffic limit theory 1 α √k in (3) is replaced by k / for 0 < α 6 2, as we would as well as the RQ formulation, it should not be sur- expect from sections 4.5, 8.5, and 9.7 of Whitt (2002), prising that we can make strong connections to heavy- but we will not discuss that extension here. traffic approximations. The new formulation in (6) is From (1), it is evident that the waiting times depend attractive because, with a natural choice of the con- on the service times and interarrival times only stant bx there, it matches the Kingman (1962) bound through their difference Xn. Thus, instead of the two for the mean steady-state wait E W in the GI GI 1 uncertainty sets in (3), we propose the single uncer- stochastic model and so is asymptotically[ ] correct/ / in tainty set (for each n) heavy traffic, whereas that is not the case for (4) with a natural choice of b. To quantify the variability indepen- x ˜ x + √ U X ∞: Sk 6 mk bx k, k > 0 , (5) dent of the scale, let c2 Var V E V 2 Var V and ≡ { ∈ − } s ≡ ( 1)/( [ 1]) ( 1) Whitt and You: Dependence in Single-Server Queues 188 Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS

2 2 2 ca Var U1 E U1 ρ Var U1 be the squared coeffi- in (4) and (6) can be considered instances of a para- cients≡ of( variation)/( [ (scvs).]) Let ( denote) approximately metric RQ, because they depend on the stochastic equal, without any precise asymptotic≈ meaning. model only through a few parameters—in particular, m , m , b , b in (4) and m, b in (6). We can expose Corollary 1 (RQ Yields the Kingman Bound for GI GI 1). ( a s a s ) ( x) p / / the impact of dependence among the interarrival times In the setting of (6), if we let b β Var X and β √2, p x ≡ ( 1) ≡ and service times on the steady-state waiting time in then b 2 c2 + ρ 2c2 for the GI GI 1 model with traffic x ( s − a ) / / the general stationary G G 1 model as a function of the intensity ρ, so that traffic intensity ρ by introducing/ / a new functional RQ 2 + 2 2 formulation. (With the G G 1 model, we assume sta- Var X1 ρ cs ρ− ca / / W ∗ W ∗ ρ ( ) ( ) , (8) tionarity, so that there is a well-defined steady state, but ≡ ( ) 2 E X 2 1 ρ | [ 1]| ( − ) we allow dependence among the interarrival times and service times.) To treat the G G 1 model, we replace which is the upper bound for E W in Theorem 2 of King- / / [ ] 2 2 the uncertainty set in (6) by man (1962), so that 1 ρ W ∗ ρ c + c 2 as ρ 1, ( − ) ( ) → ( a s )/ ↑ which supports the heavy-traffic approximation W ∗ ρ x x x x ( ) ≈ U X˜ : S 6 E S + b0 √Var S , k > 0 . (9) ρ c2 + c2 2 1 ρ , just as for E W in the stochastic model. f ≡ { k [ k ] x ( k ) } ( a s )/ ( − ) [ ] On the other hand, in the setting of (4), if we let bs and similarly for the two constraints in (4). For the p p +≡ β Var V1 and ba β Var U1 , then we obtain b bs GI GI 1 model, the new uncertainty set (9) is essen- ( ) 1 ≡ ( )p 2 2 p 2 2 2 b β c + ρ− c instead of b b + b β c + ρ c , / / a ( s a) s a s − a tially equivalent to the previous one in (5), but they as needed. can be very different with dependence. It is significant Remark 1 (The Significance for Approximations). The dif- that there are CLTs to motivate the form of the con- ference between the RQ solutions for (4) and (6) men- straints in (9), just as there are in the i.i.d. case under- tioned at the end of Corollary1 can have serious impli- lying (5). These supporting CLTs are reviewed here in 2 2 Section EC.5. The CLT supports the spatial scaling by cations for approximations; e.g., if ca cs x, then 2 + 2 + 2 √Var Sk instead of √k, as we show in Section EC.5.3. ca cs 2 x, while ca cs 2 2x, a factor of 2 ( ) ( )/ ( ) / Of course, the functional RQ produces a more compli- larger. Hence, if we apply (4) with ba bs to the simple M M 1 queue, one is forced to have a 100% error in cated optimization problem, but it is potentially more heavy/ / traffic. These two coincide only when at least one useful, in part because it too can be analyzed. For brevity, we discuss this functional RQ for the wait- of ba and bs is 0 (i.e., in D GI 1 or GI D 1 models), and the percentage error is/ the/ largest/ when/ service ing time in the EC because we will next develop such times and arrival times have the same variability. For- a functional RQ formulation for the continuous-time tunately, robust optimization has flexibility that makes workload. As discovered in Fendick and Whitt (1989), it possible to circumvent the difficulties in the form of it is convenient to focus on the steady-state workload the optimization in (4). For example, Bandi et al. (2015) when we want to expose the performance impact of use statistical regression in their section 7 to refine their the dependence among interarrival times and service solution to (4). Of course, such refinements complicate times. algorithms. Remark 3 (Asymptotically Correct in Heavy Traffic for the G G 1 Model). In Section EC.6.2 we observe that Corol- These RQ formulations provide insight into the rate / / of approach to steady state for the GI GI 1 model, as lary1 can be extended, with the aid of Sections EC.5 captured by the relaxation time; see section/ / III.7.3 of and EC.6, to show that both the new parametric RQ Cohen (1982) and section XIII.2 of Asmussen (2003). in (6) and the new functional RQ with uncertainty set For RQ, steady state is achieved at a fixed time, whereas in (9) are asymptotically correct in heavy traffic for the more general stationary G G 1 model, where we in the stochastic model, steady state is approached / / E W E W regard Uk , Vk as a stationary sequence with the gradually, with the error n typically being {( )} 3 2 n r | [ ]− [ ]| same mean values, including E Vk 1 and E Uk of order O n− / e− / as n , where r r ρ is called 1 [ ] [ ] the relaxation( time. As) usual,→ ∞ we say f t≡ is( O) g t as ρ− > 1 for all k. Now we must choose the param- t , where f and g are positive real-valued( ) ( ( func-)) eter bx appropriately to account for the dependence → ∞ tions, if f t g t c as t , where 0 < c < . among the interarrival times and service times. Just ( )/ ( ) → → ∞ ∞ as before, that can be motivated by the CLT, but now Corollary 2 (Relaxation Time for the GI GI 1 Queue). we need a CLT that accounts for the dependence, as / / With both (4) and (6), the place where the RQ supremum is in theorem 4.4.1 and section 9.6 of Whitt (2002); see 2 attained is x∗ ρ O 1 ρ − as ρ 1, which is the same Section EC.5. order as the relaxation( ) (( time− in) the) GI ↑GI 1 model. / / Remark 4 (Connection to Mamani et al. 2016). At first Remark 2 (A Functional RQ to Expose the Impact of glance, the connection to Mamani et al. (2016) may not Dependence in the G G 1 Model). The RQ problems be obvious, because we have introduced no explicit / / Whitt and You: Dependence in Single-Server Queues Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS 189 covariances, like what appears in uncertainty set (6) in while the workload (the remaining workload) at time t, their section 2.5. The Lindley recursion in (1) here leads starting empty at time 0, is the reflection map Ψ directly to the expression for the steady-state waiting applied to N; i.e., time in terms of the partial sums Sk in (2), so it is natural for us to focus on the variances Var Sk . How- Z t Ψ N t N t inf N s , t > 0. (12) ( ) ( )( ) ≡ ( ) − 06s6t { ( )} ever, the variances Var S in our uncertainty( ) set (9) ( k ) are variances of sums of random variables, which in- As in section 6.3 of Sigman (1995), we again use a cludes covariances of the summands X j when these reverse-time construction to represent the workload in summands are not required to be independent. As a single-server queue as a supremum, so that the RQ indicated above, our uncertainty sets are motivated optimization problem becomes a maximization over by CLTs, but CLTs without the usual independence constraints expressed in an uncertainty set, just as assumption. The second paragraph of section 2.5 in before, but now it is a continuous optimization prob- Mamani et al. (2016) also mentions CLTs for depen- lem. Using the same notation, but with a new mean- dent random variables but seems to be suggesting that ing, let Z t be the workload at time 0 of a system the conditions are too restrictive to be useful. Unlike that started( ) empty at time t. Then Z t can be repre- Mamani et al. (2016), the CLT and the heavy-traffic the- sented as − ( ) ory play a big role here to expose what properties of the model have the greatest impact upon the queue Z t sup N s , t > 0, (13) performance; see Section EC.5. ( ) ≡ 06s6t { ( )} where N is defined in terms of Y as before, but Y is 3. Robust Queueing for the interpreted as the total work in service time to enter Continuous-Time Workload over the interval s, 0 . That is achieved by letting We now develop RQ formulations for the continuous- [− ] Vk be the kth service time indexed going backwards time workload in the single-server queue. We develop from time 0 and A s counting the number of arrivals both a parametric RQ paralleling (6) and a functional in the interval s(, 0). Paralleling the waiting time in RQ with an uncertainty set paralleling (9) in Remark2. Section2, Z t increases[− ] monotonically to Z as t . t The workload at time is the amount of unfinished For the stable( ) stationary G G 1 stochastic queue,→ ∞Z work in the system at time t; it is also called the virtual corresponds to the steady-state/ / workload and satisfies waiting time because it represents the waiting time a P Z < 1; see section 6.3 of Sigman (1995). hypothetical arrival would experience at time t. The ( ∞) workload is more general than the virtual waiting time 3.2. Parametric and Functional RQ for the because it applies to any work-conserving service dis- Steady-State Workload cipline. We consider the workload primarily because it Just as in Section2, to create appropriate RQ formu- can serve as a convenient, more tractable alternative to lations for the steady-state workload, it is helpful to the waiting time. have a reference stochastic model, which can be the sta- We start by developing a reverse-time representa- ble stationary G G 1 model, where such a steady-state tion of the workload process paralleling (2). Then we workload is well/ defined./ In discrete time, our formula- develop both parametric and functional RQ formulation can be developed by scaling the interarrival times, tions and give their solutions, which closely parallels assuming that E V E U 1 for all k for a base Theorem1. We then show that natural versions of both k k stationary sequence[ ] U [, V ] and introducing ρ by RQ formulations for the workload are exact for the k k {( )} 1U M GI 1 model and are asymptotically correct in both letting the interarrival times be ρ− k when the traffic light/ traffic/ and heavy traffic for the general stationary intensity is ρ, 0 < ρ < 1. (That was done in Section2, G G 1 model. right after Theorem1.) Now, in continuous time, we do / / essentially the same, but now we need to work with 3.1. The Workload Process and Its Reverse-Time continuous-time stationarity instead of discrete-time Representation stationarity; e.g., see Sigman (1995). Hence, we assume As before, we start with a sequence Uk , Vk of inter- that there is a base stationary process A t , Y t : arrival times and service times. The{( arrival)} counting t > 0 with E A t E Y t t for all t >{(0 and( ) intro-( )) process can be defined by duce}ρ by simple[ ( )] scaling[ via( )] A t max k 1: U + + U t for t U (10) > 1 k 6 > 1 A t A ρt and Y t Y ρt , ( ) ≡ { ··· } ρ( ) ≡ ( ) ρ( ) ≡ ( ) and A t 0 for 0 6 t < U1, while the total input of work t > 0 and 0 < ρ < 1, (14) over 0(,)t ≡and the net-input process are, respectively, [ ] A t which implies that E Aρ t E Yρ t ρt for all t > 0. X( ) [ ( )] [ ( )] Y t V and N t Y t t, t > 0, (11) Then N t Y t t and Z t Ψ Y t , t > 0, just k ρ( ) ≡ ρ( ) − ρ( ) ( ρ)( ) ( ) ≡ k1 ( ) ≡ ( ) − as in (11) and (12). With the reverse-time construction, Whitt and You: Dependence in Single-Server Queues 190 Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS

Z t can be expressed as a supremum over the interval We immediately obtain the following corollary, ρ( ) 0, t , just as in (13). which states that the RQ formulation in (16) yields [ Within] that scaling framework, the natural paramet- the exact mean steady-state workload for the M GI 1 / / ric and functional (see Remark2) uncertainty sets for model. the steady-state workload are, respectively, Corollary 3 (Exact for M GI 1). For the M GI 1 model, / / / / p ˜ + ˜ + the total input process Yρ t : t > 0 in (14) is a com- Uρ Nρ: : Nρ s 6 1 ρ s bp√s, s >0 and { ( ) } ≡{ → ( ) −( − ) } pound Poisson process with E Y t ρt and Var Y t f + ρ ρ ˜ ˜ 2 [ ( )]2 2 2 (2 ( )) Uρ Uρ Nρ: : Nρ s 6E Nρ s ρt c + 1 , so that Z∗ Z∗ if b b ρ c + c . If, in ≡ ≡ → ( ) [ ( )] s f , ρ p, ρ p f s a ( ) ( ) +b √Var N s , s >0 , addition, b f √2, then f ( ρ( )) ≡ ˜ + ˜ Nρ: : Nρ s 6 1 ρ s 2 + 2 → ( ) −( − ) ρ cs ca Z∗ Z∗ ( ) E Z , (19) +b √Var N s , s >0 , p, ρ f , ρ 2 1 ρ [ ρ] f ( ρ( )) (15) ( − ) where E Z is the mean steady-state workload in the [ ρ] ˜ ˜ M GI 1 model with traﬃc intensity ρ. where we regard Nρ Nρ s : 0 6 s 6 t as an arbi- ≡ { ( ) + } / / trary real-valued function on 0, , while we This corollary suggests a natural choice of b f in (15). N s s ≡ [ ∞) regard ρ : > 0 as the underlying stochastic pro- From now on, we assume that b √2 unless otherwise { ( ) N s } s Y s s f cess and Var ρ : > 0 Var ρ : > 0 as its stated. variance-time{ ( function,( )) which} { can be( either( )) calculated} for a stochastic model or estimated from simulation 3.3. The Variance-Time Function for the or system data; see Section 4.3. In (15), bp and b f are Total Input Process parameters to be speciﬁed. For further progress, we focus on the variance-time

function Var Yρ t in (18). As regularity conditions for Remark 5 (Choosing the Parameters bp and b f ). The pa- ( ( )) Y t , we assume that V t Vρ t Var Yρ t is dif- rameters bp and b f in (15) add a degree of freedom ( ) ( ) ≡ ( ) ≡ ( ( )) V t in the algorithm, but some choices lead to asymptoti- ferentiable with derivative Û having ﬁnite positive limits as t and t 0; i.e.,( ) cally correct values of the steady-state mean workload, → ∞ → while others do not. On the basis of Corollary3 below, V t σ2 as t and we will let b √2 after this section. Û ( ) → Y → ∞ (20) V t V t Û Û 0 > 0 as 0 Paralleling Section2, the associated parametric and ( ) → ( ) → 2 functional RQ formulations are , for an appropriate constant σY. These assumptions are known to be reasonable; see section 4.5 of Cox and ˜ Zp∗ , ρ sup sup Nρ t , Lewis (1966), Fendick and Whitt (1989), and Section 4.3. ≡ ˜ p s>0 { ( )} Nρ Uρ A common case in models for applications is to have ∈ (16) ˜ positive dependence in the input process Y, which Zρ∗ Z∗f , ρ sup sup Nρ t . ≡ ≡ f s { ( )} N˜ U >0 holds if ρ ∈ ρ

As in Section2, our RQ formulations in (16) are moti- Cov Y t2 Y t1 , Y t4 Y t3 > 0 ( ( ) − ( ) ( ) − ( )) vated by a CLT but here for Y t (which implies an for all 0 6 t < t 6 t < t . (21) ρ( ) 1 2 3 4 associated CLT for Nρ t ), which we review in Sec- tion EC.5; in particular,( see) (EC.14) and (EC.16). The Negative dependence holds if the inequality is re- same reasoning as before yields the following analog versed. These are strict if the inequality is a strict of Theorem1. The proof can be found in Section EC.7. inequality. From (17) and (18) of section 4.5 in Cox and Lewis (1966), which is restated in (48) and (49) Theorem 2 (RQ Solutions for the Workload). The solutions of Fendick and Whitt (1989), with positive (negative) of the RQ optimization problems in (16) are dependence, under appropriate regularity conditions,

2 V t > 0 and V t > 6 0. bp Û ( ) Ü ( ) ( ) Z∗ 1 ρ x∗ + b √x p, ρ −( − ) p ∗ 4 1 ρ Remark 6 (Example of Negative Dependence). Negative | − | dependence in Y occurs if greater input in one inter- b2 p val tends to imply less input in another interval. Such for x∗ x∗ ρ (17) ≡ ( ) 4 1 ρ 2 negative dependence occurs when there is a specified ( − ) number of arrivals in a long time interval, as in the and ∆ i GI 1 model, where the arrival times (not interar- ( )/ / + √ rival times) are i.i.d. over an interval; see Honnappa Zρ∗ Z∗f , ρ sup 1 ρ s b f Var Yρ s . (18) ≡ s>0 − ( − ) ( ( )) et al. (2015). This phenomenon can also occur in Whitt and You: Dependence in Single-Server Queues Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS 191 queues with arrivals by appointment, where there 4. The Indices of Dispersion for are i.i.d. deviations about deterministic appointment Counts and Work times; e.g., see Kim et al. (2017). The workload process is convenient not only because it leads to the continuous RQ optimization problem Theorem 3 (RQ Exposing the Impact of the Depen- in (16) with a solution in (18) but also because the work- dence). Consider the functional RQ optimization for the load process scales with ρ in a more elementary way steady-state workload in the general stationary G G 1 queue / / than the waiting times, as indicated in (14). By con- with ρ < 1 formulated in (16) and solved in (18). Assume trast, the scaling of the waiting times (specified in the that (20) holds for the variance-time function V t V t ρ first paragraph after Theorem1) is more complicated Var Y t . ( ) ≡ ( ) ≡ ( ρ( )) because the interarrival times are scaled with ρ but the (a) For each ρ, 0 < ρ < 1, there exists (possibly not service times are not. unique) x∗ x∗ ρ , such that a finite maximum is attained ≡ ( ) It is also convenient to relate the variances of the at x∗ for all t > x∗. In addition, 0 < x∗ < and x∗ satisfies ∞ arrival counting process A s and the cumulative the equation work input process Y s to associated( ) continuous-time ( ) p indices of dispersion, studied in Fendick and Whitt 1 ρ h x , where h x b0 V x . (22) ( − ) Û( ) ( ) ≡ z ( ) (1989) and Fendick et al. (1991). We define the index of dispersion for counts (IDC) associated with the rate-1 The time x∗ is unique if h x is strictly concave or ( ) arrival process A as in section 4.5 of Cox and Lewis strictly convex—i.e., if hÛ x is strictly increasing or strictly (1966) by decreasing. ( ) (b) If there is positive negative dependence, as in (21) Var A t Var A t Ia t ( ( )) ( ( )) , t > 0 (23) (with sign reversed), the variance( function) V x is convex ( ) ≡ E A t t p ( ) [ ( )] (concave), so that the function h x V x is concave. and the index of dispersion for work (IDW) associated ( ) ≡ ( ) Moreover, a strict inequality is inherited. Thus, there exists with the rate 1 cumulative input process Y by a unique solution to the RQ if there is strict positive depen- Var Y t V t dence or strict negative dependence. Moreover, the optimal Iw t ( ( )) ( ) , t > 0. (24) ( ) ≡ E V1 E Y t t time x∗ ρ is strictly increasing in ρ, approaching 1 as ρ 1, [ ] [ ( )]

( ) 2 ↑ so that Z∗ V I σ as ρ 1. Clearly, these indices of dispersion are just scaled ver- ρ Û w Y → (∞) (∞) ↑ sions of the associated variance functions, but they are Proof. The inequalities can be satisfied as equalities important for understanding because they expose the just as before. There are finite values s such that p p 0 variability over time, independent of the scale. The rea- V s 6 2σ2 s for all s > s by virtue of the limit in (20). ( ) Y 0 son for using these indices of dispersion is just like (Also see (EC.1) and (EC.12).) That shows that the opti- the reason for using the scvs (introduced before Corol- mization can be regarded as being over closed bounded lary1) instead of the variances. More generally, this intervals. The assumed differentiability of V implies is consistent with the well-established practice of care- that it is continuous, which implies that the supremum fully focusing on units in physics and engineering. is attained over the compact interval. Because V x Fendick and Whitt (1989) show that the IDW Iw is Û ( ) → 2 V 0 > 0, we see that there exists a small s0 such that intimately related to a scaled mean workload c ρ , Û ( ) Z which can be defined by comparing to what it would( ) p p 1 ρ s + b0 V s > 1 ρ s + b0 sV 0 2 > 0 be in the associated M D 1 model; i.e., −( − ) z ( ) −( − ) z Û ( )/ / / for all s 6 s0. E Zρ 2 1 ρ E Zρ c2 ρ [ ] ( − ) [ ] Z( ) ≡ E Z ; M D 1 E V ρ As a consequence, the maximum in (18) must be strictly [ ρ / / ] [ 1] positive and must be attained at a strictly positive time. 2 1 ρ E Zρ p ( − ) [ ] . (25) The results for V x with positive dependence fol- ρ low from convexity properties( ) of compositions. First, p The normalization in (25) exposes the impact of vari- with positive dependence, V x is a convex func- − ( ) ability separately from the traffic intensity. Hence, it tion of an increasing convex function and thus convex should not be surprising that c2 ρ should be related p Z so that V x is concave. Second, with negative depen- to the IDW. Indeed, under regularity( ) conditions (see ( ) dence, we have V > 0, V t > 0 and V t 6 6 0. Thus, Section EC.5.5), the following finite positive limits exist Û ( ) Ü ( ) ( ) by direct differentiation, and are equal: 2 2 2 lim Iw t Iw σY cZ 1 lim cZ ρ , and 1 V x V x t { ( )} ≡ (∞) ( ) ≡ ρ 1 { ( )} h x Ü ( ) Û ( ) 6 0, →∞ → Ü( ) p 2 − 4V x + 2 2 2 V x lim Iw t Iw 0 1 cs cZ 0 lim cZ ρ ( ) t 0 ρ 0 ( ) → { ( )} ≡ ( ) ( ) ≡ → { ( )} with strictness implying a strict inequality. (26) Whitt and You: Dependence in Single-Server Queues 192 Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS for c2 Var V E V 2 and c2 in (20) and (EC.7). The for b in (18). The associated RQ optimal workload in (28) s ≡ ( 1)/ [ 1] Y f limits for Iw above and the differentiability of Iw follow can be expressed as from the assumed differentiability for V t and limits ( ) 2 2 b f ρIw x∗ x I x in (20). For t 0 and ρ 0, see section IV.A of Fendick ( ) ∗ Ûw ∗ → → Zρ∗ 1 ( ) , (31) and Whitt (1989). 4 1 ρ − Iw x∗ The challenge is to relate c2 ρ to the IDW I t for ( − ) ( ) Z( ) w( ) 0 < ρ < 1 and t > 0. As observed by Fendick and Whitt which is a valid nonnegative solution provided that 2 (1989), a simple connection would be c ρ I t x∗Iw x∗ 6 Iw x∗ . If b f √2, then the associated scaled RQ Z( ) ≈ w( ρ) Û ( ) ( ) for some increasing function tρ, reflecting that the im- workload satisfies pact of the dependence among the interarrival times x I x 2 and service times has impact on the performance of a 2 ∗ Ûw ∗ c ρ Iw x∗ 1 ( ) . (32) queue over some time interval 0, t , where t should Z∗ ( ) ( ) − I x [ ρ] ρ w ∗ increase as ρ increases. The extreme cases are sup- ( ) Proof. Note that xI x V x . Because we have ported by (26), but we want more information about w( ) ( ) the cases in between. assumed that V x is differentiable, so too is Iw. We obtain (30) by differentiating( ) with respect to x in (28) 4.1. Robust Queueing with the IDW and setting the derivative equal to 0. After substitut- To obtain more information, RQ can help. As a first ing (30) into (28), algebra yields (31). The limits in (20) step, we express the solution in (18) as imply that x∗Iw x∗ 0 and Iw x∗ Iw as ρ 1. Û ( )→ ( )→ (∞) → xI x x b √ + p Given that Ûw 0 as , if f 2, then it is Zρ∗ sup 1 ρ s b f Var Yρ s ( ) → → ∞ s>0 − ( − ) ( ( )) natural to consider the approximation q + 2 sup 1 ρ s b f ρsIw ρs , (27) ρ − ( − ) ( ) x∗ ρ I x∗ ρ so that s>0 ( ) ≈ 2 1 ρ 2 w( ( )) ( − ) (33) ρI x∗ ρ using (24). Making the change of variables x ρs, we w 2 ≡ Z∗ ( ( )) and c ρ Iw x∗ ρ . can write ρ ≈ 2 1 ρ Z∗ ( ) ( ( ))

( − ) + p Zρ∗ sup 1 ρ x ρ b f xIw x . (28) The first equation in (33) is a variability fixed-point x>0 − ( − ) / ( ) equation of the form in suggested in (15) of Fendick and Whitt (1989). Clearly, from an algorithmic perspective, (28) is essentially the same as (18) and (27), but (28) is helpful 4.2. Heavy-Traffic and Light-Traffic Limits for developing approximations and insights, includ- The following result shows the great advantage of ing supporting theory. Our algorithm will exploit the doing RQ with (i) the continuous-time workload and one-dimensional optimization problem in (28), which (ii) the functional version of the RQ in (28). A proof is is easy to solve given the IDW I x . We will discuss w given in Section EC.7. methods of estimating and calculating( ) IDW in Sec- tions 4.3 and6. Theorem 5 (Heavy-Traffic and Light-Traffic Limits). Under To further relate the RQ solution in (28) to the steady- the regularity conditions assumed for the IDW I t , if b w( ) f ≡ state workload in the G G 1 queue, we define an RQ √2, then the functional RQ solution in (28) is an asymptot- analog of the normalized/ mean/ workload in (25)—in ically correct characterization of steady-state mean workload particular, both in heavy traffic (as ρ 1) and light traffic (as ρ 0). Specifically, we have the following↑ supplement to (26): ↓ 2 1 ρ Z∗ 2 ( − ) ρ cZ ρ . (29) 2 2 ∗ lim cZ ρ Iw lim cZ ρ and ( ) ≡ ρ ρ 1 ∗ ( ) (∞) ρ 1 ( ) ↑ ↑ (34) 2 2 The RQ approach allows us to establish versions of the lim cZ ρ Iw 0 lim cZ ρ . ρ 0 ∗ ( ) ( ) ρ 0 ( ) variability fixed-point equation suggested in (9), (15), ↓ ↓ and (127) of Fendick and Whitt (1989). Remark 7. Theorem5 greatly generalizes results in Theorem3(b) with both light and heavy traffic ad- Theorem 4 (Restatement of Theorem2 in Terms of the dressed in the general case beyond positive or negative IDW). Any optimal solution of the RQ in (28) is attained at correlations. We also note that the parametric RQ solu- s∗ ρ x∗ ρ, where x∗ x∗ ρ satisfies the equation ( ) ≡ / ≡ ( ) tion can be made correct in heavy traffic or in light 2 2 2 traffic, as above, by choosing the parameter b appro- b ρ I x∗ p f w x∗Iw x∗ x∗ ( ) + Û priately, but both cannot be achieved simultaneously 2 1 ( ) (30) 4 1 ρ Iw x∗ unless I I 0 . ( − ) ( ) w(∞) w( ) Whitt and You: Dependence in Single-Server Queues Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS 193

P 4.3. Estimating and Calculating the IDW 4.3.4. The Balanced i Gi GI 1 Model. An important / / P For applications, it is significant that the IDW Iw t special case of Section 4.3.3 is the balanced i Gi GI 1 used in Section4 can readily be estimated from data( ) model in which the arrival process is the superposi-/ / from system measurements or simulation and calcu- tion of n i.i.d. non-Poisson processes each with rate lated in a wide class of stochastic models. The time- ρ n, so that the overall arrival rate is ρ, and asymptotic / 2 dependent variance functions can be estimated from variability parameter is ca . From the results above, we the time-dependent first and second moment func- obtain tions, as discussed in section III.B of Fendick et al. I ρt I ρt n and I ρt I ρt n , (1991). Calculation depends on the specific model a, n( ) a, 1( / ) w, n( ) w, 1( / ) structure. t > 0, (39) G GI 1 If the service times are i.i.d. 4.3.1. The Model. so that the superposition IDI and IDW differ from those with a general/ distribution/ having mean τ and scv c2 s of a single-component process only by the time scaling, and are independent of a general stationary arrival pro- but that time scaling involves both n and ρ. cess, then as indicated in (58) and (59) in section III.E As discussed in section 9.8 of Whitt (2002), this of Fendick and Whitt (1989), model is known to have complex behavior as a func- 2 + tion of n and ρ, so that RQ may be helpful. In partic- Iw t c Ia t , t > 0, (35) ( ) s ( ) ular, under regularity conditions, (i) the superposition 2 where cs is the scv of a service time and Ia is the IDC arrival process is known to be non-Poisson and nonre- of the general arrival process. newal, unless the component arrival streams are Pois- son. (ii) If we let n but keep the total rate fixed, 4.3.2. The Multiclass P G G 1 Model. As indicated i i i then the superposition→ ∞ process approaches a Poisson in (56) and (57) in section( / III.E)/ of Fendick and Whitt process. (iii) However, for any n, no matter how large, (1989), if the input comes from independent sources, if we let t , then the superposition process obeys each with their own arrival process and service times, the same CLT→ ∞ as a single component arrival process then the overall IDC and IDW are revealing functions and so has asymptotic variability parameter c2. Thus, of the component ones. Let λ be the arrival rate, let a i we have I 0 1 and I c2, but I t depends on n τ be the mean service time of class i, and let ρ a a a a i i and ρ in a( complicated) (∞) way for 0 < t <( ). λ τ be the traffic intensity for class i with λ P λ≡, i i i i As shown in Whitt (1985), important∞ insight can be τ P λ λ τ 1 so that ρ λ. With our≡ scaling i i i gained by considering the joint limit as n and ρ 1. conventions,≡ ( / ) It turns out the asymptotic behavior depends↑∞ on the↑ P 2 Var A t i Var Ai t X λi limit of n 1 ρ . The separate limits occur if that limit I λt ( ( )) I λ t (36) a ( ( )) a, i i is either infinite( − ) or zero. A complex interaction occurs ( )≡ E A t λt i λ ( ) [ ( )] at finite limits. We will show that RQ provides impor- and tant insight when we conduct simulation experiments P for this model in Section 5.1. Var X t i Vi t X ρi τi I λt ( ( )) ( ) I λ t w( ) ≡ τE X t ρt ρτ w, i( i ) 4.3.5. The IDCs for Common Arrival Processes. The [ ( )] i for all t > 0. (37) two previous subsections show that for a large class of models the main complicating feature is the IDC of the

From (36) and (37), we see that Ia and Iw are convex arrival process from a single source. The only really combinations of the component Ia i and Iw i modiﬁed simple case is a Poisson arrival process with rate λ. , , by additional time scaling. Then Ia t 1 for all t > 0. A compound (batch) Poisson process( is) also elementary because the process Y has 4.3.3. The Multiclass P G GI 1 Model. An important i i independent increments; then the arrival process itself special case of Section 4.3.2/ arising/ in open queueing is equivalent to M GI source. However, for a large class networks is the P G GI 1 model in which there are i i of models, the variance/ Var A t and thus the IDC I t multiple general arrival/ / streams coming to a queue a can either be calculated directly( ( )) or be characterized via( ) where all arrivals experience common i.i.d. service their Laplace transforms and thus calculated by invert- times. We can combine (35) and (36) to get the expres- ing those transforms and approximated by performing sion for the IDW, asymptotic analysis. For all models, we assume that the + 2 processes A and Y have stationary increments. Iw λt Ia λt cs , t > 0, (38) ( ) ≡ ( ) An important case for A is the renewal process; to where Ia λt is given by (36). Of course, if all the have stationary increments, we assume that it is the component( arrival) streams are Poisson processes, then equilibrium renewal process, as in section 3.5 of Ross Ia λt 1 for all t > 0, but otherwise, the IDC Ia λt can (1996). Then Var A t can be expressed in terms of be( quite) complicated. ( ) the renewal function,( ( )) which in turn can be related Whitt and You: Dependence in Single-Server Queues 194 Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS to the interarrival-time distribution and its transform. 5.1. A Queue with a Superposition Arrival Process The explicit formulas for renewal processes appear in We start by looking at an example of a balanced P (14), (16), and (18) in section 4.5 of Cox (1962). The i Gi GI 1 model from Section 4.3.4, where (39) can be required numerical transform inversion for the renewal applied./ Let/ the rate 1 arrival process A be the superpo- function is discussed in section 13 of Abate and Whitt sition of n 10 i.i.d. renewal processes, each with rate

(1992). The hyperexponential (H2) and Erlang (E2) spe- 1 n, where the times between renewals have a lognor- / 2 cial cases are described in section III.G of Fendick and mal distribution with mean n and scv ca 10. Let the Whitt (1989). service-time distribution be hyperexponential (H2), a It is also possible to carry out similar analyses for mixture of two exponential distributions with mean 1, 2 much more complicated arrival processes. Neuts (1989) cs 2, and balanced means as on page 137 of Whitt applies matrix-analytic methods to give explicit rep- (1982). Then (39) and (26) imply that the IDW has lim- resentations of the variance Var A t for the versatile its I 0 1 + c2 3 and I c2 + c2 12, so that the w( ) s w(∞) a s Markovian point process or Neuts( ( )) process; see sec- IDW is not nearly constant. tion 5.4, especially theorem 5.4.1. Explicit formulas for The left panel of Figure2 shows a comparison be- the Markov-modulated Poisson process are given on tween the simulation estimate of the normalized work- pages 287–289. load c2 ρ in (25) and the approximation c2 ρ in (29) Z( ) Z∗ ( ) All of these explicit formulas above have the asymp- for this example. We make two important observa- 2 totic form tions: (i) the normalized mean workload cZ ρ in (25) as a function of ρ is not nearly constant, and( (ii)) there 2 γt Var A t σ t + ζ + O e− as t . ( ( )) A ( ) → ∞ is a close agreement between the RQ approximation c2 ρ in (29) and the direct simulation estimate; the Z∗ ( ) 5. Simulation Comparisons close agreement for all traffic intensities is striking. It is We illustrate how the new RQ approach can be used important to note that the parametric RQ approxima- with system data from queueing networks by apply- tions produce constant approximations and so cannot ing simulation to analyze two common but challeng- be simultaneously good for all traffic intensities. For this example, we see that c2 ρ 3 for ρ 6 0.5, ing network structures in Figure1: (i) a queue with a Z( ) ≈ superposition arrival process and (ii) several queues which is consistent with the Poisson approximation for in series. The specific examples are chosen to capture the arrival process and the associated M G 1 queue, 2 / / a known source of difficulty: there is complex depen- where cZ ρ 3 for all ρ, but the normalized workload ( ) dence in the arrival process to the queue, so that the increases steadily to 12 after ρ 0.5, as explained in relevant variability parameter of the arrival process at section 9.8 of Whitt (2002). the queue can depend strongly on the traffic inten- The estimates for Figure2 were obtained for ρ over a sity of that queue, as discussed in Whitt (1995). Our grid of 99 values, evenly spaced between 0.01 and 0.99. RQ approximations are obtained by applying (28) after Similarly, the RQ optimization was performed using estimating the IDC and applying (35). (28) with a discrete-time estimate of the IDW. By doing

2 Figure 2. (Color online) Left: A Comparison Between Simulation Estimates of the Normalized Mean Workload cZ ρ in (25) 2 Pn 2 ( ) and Its Approximation c ρ in (29) as a Function of ρ for the GIi H2 1 Model with cs 2 and a Superposition of n i.i.d. Z∗ ( ) i / / p Lognormal Renewal Arrival Processes for n 10 and c2 10; Right: Graphical RQ Solution Showing h x 2xI x and the a ( ) ≡ w ( ) Tangent Line with Slope 1 ρ ρ at x∗ 482 for ρ 0.9 and at x∗ 17 for 0.7, as Dictated by (22) ( − )/ ≈ ≈ 12 120 RQ = 0.7 Simulation = 0.9 10 100

8 80 ) x

6 ( 60 h

4 40 Normalized workload

2 20

0 0 0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 500 600 Traffic intensity Time Whitt and You: Dependence in Single-Server Queues Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS 195 multiple runs, we ensured that the statistical variation last queue should inherit the variability of the external was not an issue. For the main simulation of the arrival arrival process and behave like an H2 M 1 queue with 6 2 / / process and the queue we used 5 10 replications, dis- scv ca 5. This behavior is substantiated by Figure3, carding a large initial portion of× the workload process which compares simulation estimates of the normal- to ensure that the system is approximately in steady ized mean workload c2 ρ in (25) at the last queue of Z( ) state. (The component renewal arrival processes thus 10 queues in series as a function of the mean service can be regarded as equilibrium renewal processes, as time and traffic intensity ρ there with the correspond- in section 3.5 of Ross 1996.) We let the run length and ing values in the E M 1 queue (left panel) and with 2/ / amount discarded be increasing in ρ, as dictated by the RQ approximation c2 ρ in (29) (right panel). The Z∗ ( ) Whitt (1989). We provide additional details about our left panel of Figure3 shows that the last queue behaves simulation methodology in the appendix. like a E M 1 queue for all traffic intensities 6 0.8 but 2/ / then starts behaving more like an H2 M 1 queue as the 5.2. A Series of Ten Queues traffic intensity approaches the value/ 0./95 at the ninth This second example is a variant of examples in Suresh queue. The right panel of Figure3 shows that RQ suc- and Whitt (1990), exposing the complex impact of vari- cessfully captures this phenomenon and provides an ability on performance in a series of queues if the accurate approximation for all ρ. external arrival process and service times at a pre- To elaborate on this series-queue example, we show vious queue have very different levels of variability. the IDW for the last queue in Figure4. The plot This example has 10 single-server queues in series. shows the IDW assuming continuous-time stationar- The external arrival process is a rate 1 renewal pro- ity (which we use) together with the plot using the 2 cess with H2 interarrival times having ca 5. The first discrete-time Palm stationarity (see Sigman 1995) over 2 2 5 nine queues all have Erlang service times with c 0.5 the long interval 10− , 10 in log scale. The good per- a [ ] denoted by E2, i.e., the sum of two i.i.d. exponential formance in Figure3 for small values of ρ depends on random variables. The first eight queues have mean using the proper (continuous-time) version. service time and thus traffic intensity 0.6, while the We conclude this example by illustrating the dis- ninth queue has mean service time and thus traffic crete-time approach for approximating the expected intensity 0.95. The last (10th) queue has an exponential steady-state waiting time E W using the RQ optimiza- service-time distribution with mean and traffic inten- tion in (6) with uncertainty[ set] in (9). Figure5 is the dis- sity ρ; we explore the impact of ρ on the performance crete analog of Figure3. Figure5 compares simulation 2 of that last queue. estimates of the normalized mean waiting time cW ρ , The Erlang services act to smooth the arrival pro- defined just as in (25), at the last queue of 10 queues( in) cess at the last queue. Thus, for sufficiently low traffic series as a function of the mean service time and traf- intensities ρ at the last queue, the last queue should fic intensity ρ there with the corresponding values in behave essentially the same as a E2 M 1 queue, which the E2 M 1 queue (left) and with the RQ approxima- 2 / / 2/ / has ca 0.5, but as ρ increases, the arrival process at the tion c ρ , defined just as in (29). Figures5 and3 look W∗ ( )

Figure 3. (Color online) A Comparison Between Simulation Estimates of the Normalized Mean Workload c2 ρ in (25) at the Z( ) Last Queue of the 10 Queues in Series with Highly Variable External Arrival Process, but Low-Variability Service Times, as a Function of the Mean Service Time and Traﬃc Intensity ρ There with the Corresponding Value in the E M 1 Queue (Left) 2/ / and with the RQ Approximation c2 ρ in (29) (Right) Z∗ ( ) 6 6 Queues in series RQ Simulation 5 E2/M/1 5

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0 0 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 Traffic intensity Traffic intensity Whitt and You: Dependence in Single-Server Queues 196 Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS

Figure 4. (Color online) The IDW at the Last Queue Over 6. An IDC Framework for a New RQNA 2 5 the Interval 10− , 10 in Log Scale A main contribution of Bandi et al. (2015) was to [ ] 6 develop a full RQNA. While we have established good Palm RQ results for one single-server queue, it still remains Stationary to develop a full RQNA exploiting the indices of dis- 5 persion and the results in the previous sections. To conclude this paper, we propose a candidate framework in 4 which we hope to develop an initial IDC-based RQNA. To start, we make several simplifying assumptions:

3 (i) all queues are single-server queues with unlimited IDW waiting space and the FCFS discipline; (ii) with m queues, the service times at these queues come from 2 m independent sequences of i.i.d. random variables, independent of all the external arrival processes, where 1 these service times have finite means and variances; (iii) each queue has its own external arrival process (which may be null), assuming that each is a general 0 10–2 100 102 104 stationary point process; (iv) these m external arrival Time processes are mutually independent and exogenous, each having a finite arrival rate, with the arrival pro- Note. The continuous-time stationary version used for RQ with the workload is contrasted with the discrete-time Palm version. cess satisfying a functional central limit theorem with a Brownian motion limit; (v) as in the basic form of QNA in Whitt (1983), we let departures be routed to similar, except that there is a significant difference for other queues or out of the network by Markovian rout- small values of ρ. In general, we do not expect RQ to ing, independent of the rest of the model history; and be effective for extremely low ρ because (i) the CLT is (vi) given that the traffic-rate equations are used to not appropriate for only a few summands, and (ii) the find the net arrival rate at each queue, as in section 4.1 mean waiting time is known to depend on other fac- of Whitt (1983), the resulting traffic intensities satisfy tors when ρ is small. The mean waiting time and mean ρi < 1 for all i, so that the final open network produces a workload actually are quite different in light traffic; see stable general stationary G GI 1 m stochastic network section IV.A of Fendick and Whitt (1989). As explained model, which has a proper( / steady-state/ ) distribution. there, the mean workload tends to be more robust to As discussed in section 2.3 of Whitt (1983) and Segal model detail. and Whitt (1989), practical applications require much

Figure 5. (Color online) Contrasting the Discrete-Time and Continuous-Time Views: The Analog of Figure3 for the Waiting Time

6 6 Queues in series RQ 5 E2/M/1 5 Simulation

4 4

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0 0 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 Traffic intensity Traffic intensity

2 Note. Simulation estimates of the normalized mean waiting time cW ρ , defined as in (25), at the last queue of the 10 queues in series with highly variable external arrival process, but low-variability service times,( ) as a function of the mean service time and traffic intensity ρ there 2 with the corresponding value in the E2 M 1 queue (left) and with the RQ approximation c ρ , defined as in (29) (right). / / W∗ ( ) Whitt and You: Dependence in Single-Server Queues Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS 197 more complicated models—e.g., including multiserver to show that the IDC of the split stream can be repre- queues, non-FCFS disciplines and, as in section 2.3 of sented exactly by Whitt (1983), input by classes with basic routes that + must be converted into the framework above—but here IB t pIA t 1 p, t > 0, (41) ( ) ( ) − we suggest the G GI 1 m model above as a candidate reference stochastic( / / ) model in which we want to which is analogous to (36) in Whitt (1983). Finally, it remains to treat the flow through a G GI 1 develop an initial RQNA. / / We propose going beyond QNA by letting the vari- queue. Of course, the rate out is just the rate in, so it suffices to calculate the IDC I t for the departure ability of each arrival process, external or internal, be d( ) partially characterized by its rate and IDC. Let the net process. We propose a candidate approximation that can serve as a basis for a full RQNA, but it remains arrival process at queue i have rate λi and IDC Ia, i t . We let the service-time cumulative distribution func-( ) to be more thoroughly tested and refined. In particular, a candidate approximation for the IDC I t of the tion (cdf) Gi at queue i be partially characterized by its d( ) 2 departure process from a G GI 1 queue is mean τi and scv cs , but we might use the full cdf Gi. / / By (35), the associated IDW is then I t I t + w, i a, i + 2 ( ) ( ) Id, ρ t wρ t Ia t 1 wρ t Is t , (42) cs, i, t > 0. Thus, we can approximate the mean steady- ( ) ≈ ( ) ( ) ( − ( )) ( ) state workload at queue i, E Z ρ for each i, by i i where I t is the IDC of the equilibrium renewal pro- solving the one-dimensional RQ[ optimization( )] problem s cess with( specified) service-time distribution, w t , 0 6 in (28). We consider that as the initial objective, even ρ w t 6 1, is a weight function, which depends( on) the though we want to extend the algorithm to develop a ρ( ) traffic intensity ρ. Preliminary study indicates that the full performance description. As a first cut to describe weight function might be network performance, we would follow section VI of

Whitt (1983). 2 √t w t w c 1 ρ t , t > 0, where w t 1 e− (43) For the G GI 1 m model introduced above, we spec- ρ( ) ≡ ( ( − ) ) ( ) ≡ − ify the service( / time/ ) at queue i by its mean τ and scv c2 , i s, i and c is a properly chosen scale parameter; here, c is as in QNA, but now we specify the external arrival pro- chosen to be 0.25. The component IDCs Ia t and Is t cess at queue i by its rate λo, i and IDC Ia, o, i t :t > 0 , ( ) ( ) in (42) can readily be estimated from simulations or o { ( ) } with designated from outside. Paralleling QNA, the calculated, as indicated in Section 4.3.5. The IDC of IDC-based RQNA would apply the familiar traffic-rate the equilibrium renewal process I t can be obtained i s equations to determine the net arrival rate λi at queue from the associated variance function( ) via I t V t t, for each i, just as in section 4.1 of Whitt (1983), and asso- s assuming that it has rate 1. In turn, the variance( ) func-( )/ ciated traffic variability equations, based on a network tion of the rate 1 equilibrium renewal process is calculus for the three operations—(i) superposition or merging, (ii) splitting, and (iii) flow through a queue ¹ t or departure—to determine the final net IDC I t at V t 1 + 2m u 2u du, (44) a, i ( ) ( ( ) − ) queue i for each i. ( ) 0 With the framework above, it suffices to specify and where m t is the renewal function (mean function of apply a network calculus to determine the IDC of the the standard( ) renewal process), which can be calculated net arrival process to each queue. The difficult super- by numerical transform inversion, given the Laplace position operation (for component streams assumed to transform of the service-time distribution, as discussed be mutually independent) is already covered by Sec- in section 13 of Abate and Whitt (1992). tion 4.3.3 here and has shown to be effective for approx- To show that this approach for approximating the imating the mean workload in Section 5.1. IDC Id t has promise, we apply it to the series queue For splitting, as in QNA we assume independent example( ) in Section 5.2. Recall that the arrival process splitting, with each customer routed in a given direc- is an H2 renewal process, while the service distribution tion according to independent Bernoulli random vari- at the first eight nodes is Erlang E2 with traffic intensity ables. For independent splitting, we can express the 0.6 and the ninth node has a traffic intensity of 0.95. The split counting process B t given the original counting IDCs for H2 and E2 are given in examples 3.1 and 3.2 of process A t by the random( ) sum ( ) Fendick and Whitt (1989). From (42), we iteratively obtain the approximation A t X( ) for the IDC of the departures from the ninth queue B t Xi , (40) ( ) i1 I t w8 t w t I t + 1 w8 t w t I t . (45) 9, d, ρ( ) ≈ ρ1 ( ) ρ9 ( ) a( ) ( − ρ1 ( ) ρ9 ( )) s ( ) where Xi is i.i.d. and independent of A t with { } ( ) P Xi 1 p 1 P Xi 0 . Under those regularity con- This framework decomposes the IDC of the departure ditions,( ) we can apply− ( the) conditional variance formula from the ninth queue into combinations of the IDC Whitt and You: Dependence in Single-Server Queues 198 Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS

Figure 6. (Color online) Left: Contrasting the IDW of Departure Process from the Ninth Queue from Simulation and the IDW Approximation Obtained from the Candidate RQNA Framework for the Example in Section 5.2; Right: Simulation Estimation of the Steady-State Mean Workload, the RQ Approximation in Section 5.2, and the RQNA Approximation

6 6 Simulation RQ 5 RQNA approximation 5 RQNA approximation Simulation

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Sriram K, Whitt W (1986) Characterizing superposition arrival pro- Whitt W (2002) Stochastic-Process Limits (Springer, New York). cesses in packet multiplexers for voice and data. IEEE J. Selected Whitt W, You W (2016) Time-varying robust queueing. Working Areas Comm. 4(6):833–846. paper, Columbia University, New York. Suresh S, Whitt W (1990) The heavy-traffic bottleneck phenomenon in open queueing networks. Oper. Res. Lett. 9(6):355–362. Whitt W (1982) Approximating a point process by a renewal process: Ward Whitt is a professor in the Industrial Engineering Two basic methods. Oper. Res. 30(1):125–147. and Operations Research Department at Columbia Univer- Whitt W (1983) The queueing network analyzer. Bell Laboratories sity. A major focus of his early work was the Queueing Net- Tech. J. 62(9):2779–2815. work Analyzer performance analysis software tool, which is Whitt W (1984a) On approximations for queues, I. AT&T Bell Labora- described in a 1983 paper in the Bell Labs Technical Journal. tories Tech. J. 63(1):115–137. His new research explores ways to develop more effective Whitt W (1984b) On approximations for queues, III: Mixtures of exponential distributions. AT&T Bell Laboratories Tech. J. 63(1): approximations, drawing on new robust optimization meth- 163–175. ods as well as previous heavy-traffic limits and indices of Whitt W (1985) Queues with superposition arrival processes in heavy dispersion. traffic. Stochastic Processes Their Appl. 21(1):81–91. Wei You is a doctoral student in the Industrial Engineer- Whitt W (1989) Planning queueing simulations. Management Sci. ing and Operations Research Department at Columbia Uni- 35(11):1341–1366. Whitt W (1995) Variability functions for parametric-decomposition versity. His primary research focus is on queueing theory, approximations of queueing networks. Management Sci. 41(10): applied probability, and their applications to service systems 1704–1715. using stochastic modeling, optimization, and simulation.