A tandem fluid network with Lévy input in heavy traffic
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Abstract
In this paper we study the stationary workload distribution of a fluid tandem queue in heavy traffic. We consider different types of Lévy input, covering compound Poisson, \(\alpha \)stable Lévy motion (with \(1<\alpha <2\)), and Brownian motion. In our analysis, we separately deal with Lévy input processes with increments that have finite and infinite variance. A distinguishing feature of this paper is that we do not only consider the usual heavy traffic regime, in which the load at one of the nodes goes to unity, but also a regime in which we simultaneously let the load of both servers tend to one, which, as it turns out, leads to entirely different heavy traffic asymptotics. Numerical experiments indicate that under specific conditions the resulting simultaneous heavy traffic approximation significantly outperforms the usual heavy traffic approximation.
Keywords
Queueing theory Tandem queue Lévy processes Fluid queue Heavy traffic Steadystate distribution Heavy tailsMathematics Subject Classification
60K251 Introduction
In this paper we study a fluid tandem queue that consists of two servers in series. A spectrally positive Lévy process serves as the input process of the first queue (also: upstream queue). The first server empties the upstream queue at a deterministic rate \(r_1\), immediately feeding the second (also: downstream) queue. The downstream server leaks at some deterministic rate \(r_2\); to make the system nontrivial we throughout assume \(r_2<r_1\). After the fluid has passed the second server, it leaves the system. We are interested in the stationary workloads in both queues in heavy traffic regimes that we specify below.
The heavy traffic regime was first considered in [1]: one lets the load of the system tend to one, while simultaneously scaling the workload in such a way that a nondegenerate limiting distribution is obtained. Kingman’s approach was mainly based on manipulating Laplace–Stieltjes transforms; this approach we also follow in our paper. Another approach relies on the functional central limit theorem in combination with the continuous mapping theorem; see, for example, [2]. In [3], both approaches are compared, and the traditional heavy traffic results, which assume the increments of the input process have a finite variance, are generalized to the infinite variance case. For excellent surveys, we refer to [4] and the book [5]. Tandem queueing systems in which both queues are experiencing heavy traffic conditions have been studied before. Harrison [6] has focused on the classical setting of a GI/G/1type tandem in which discrete entities (‘customers’) receive service in each server and move to the next queue (or leave the system) only after its full service has been completed. In such queueing systems, the correlation between both queues is typically negative, as the first queue being relatively large could be a consequence of long service times in that queue, which in turn result in long interarrival times in the second queue, and hence a relative small number of customers in the second queue. Harrison manages to quantify the resulting (negative) covariance between the populations in both queues in heavy traffic. Importantly, in the fluid setting considered in our work, this reasoning does not hold. More specifically, for the types of models we study, the correlation between both workloads is positive: large workloads in the upstream queue likely correspond to large workloads in the downstream queue.
Fluid tandem queues with spectrally positive Lévy input were initially scrutinized in a series of papers starting with [7] and the followup paper [8]. The results concerning the joint distribution of the steady state of the workloads were studied in a more general network setting in, for example, [9]. These results play an important role for our analysis and are therefore summarized in Sect. 2.2. An extensive account of Lévydriven networks can be found in Chaps. 12 and 13 of [10].

Regime I, If only the downstream server has a load that tends to unity (whereas the first queue does not operate under heavy traffic);

Regime II, If the up and downstream server have loads that simultaneously tend to unity.

For Regime I, we find that the steadystate distribution of the workload in the second queue is similar to the one of the first queue. Moreover, the up and downstream workloads are asymptotically independent in the heavy traffic limit.

In Regime II, we establish the interesting feature that the workloads do not decouple in heavy traffic, i.e. some dependence between the up and downstream workloads remains. Moreover, the marginal steadystate distribution of the downstream queue is crucially different from the one obtained in Regime I. This has practical implications: as verified through a set of experiments, Regime II approximations tend to outperform those based on Regime I, particularly when the load of both servers is large.
In Regime I, we prove that the stationary workload of the downstream queue has an exponential distribution (for the case of finite variance) or MittagLeffler distribution (for infinite variance). Remarkably, the same distributions (up to some factor) were found for single fluid queues; apparently, the fact that there is an additional fluid server that modifies the process hardly affects the limiting distribution. In addition, similar results were also found for waiting times in nonfluid single GI/G/1 queues; see [11] for the case of infinite variance.
The paper is organized as follows. In Sect. 2 we introduce our framework of queueing models with Lévy input; we subsequently explain the fluid Lévy tandem queueing model that we consider and recall results that play a key role throughout the paper. As mentioned above, there is a dichotomy between the case of finite (Sect. 3) and infinite variance (Sect. 4). In Sect. 3 we first consider Brownian input, for which all computations can be done explicitly, and then turn to general spectrally positive Lévy input. This section also includes numerical experiments that indicate that the Regime II approximation typically outperforms the Regime I approximation. Section 4, which focuses on infinite variance input, covers results for compound Poisson input and \(\alpha \)stable input. Finally, in the Appendix, we state Tauberian theorems that are used in Sect. 4.
2 Lévy driven queues
In this section we briefly introduce the fluid tandem queueing model, and we state some results that are important for the remainder of the paper.
2.1 A fluid tandem queueing model
In Regime I, the upstream server will have a fixed load of \(\rho _1 = {{{\mathrm{{\mathbb {E}}}}}J_1}/{r_1}<1\) as \(\epsilon \downarrow 0\), whereas the load of the downstream server will tend to one: \(\rho _2={{{\mathrm{{\mathbb {E}}}}}J_1}/{r_2}\uparrow 1\) as \(\epsilon \downarrow 0\). In Regime II, on the contrary, both the up and downstream server will have loads that tend to one: \(\rho _1,\rho _2\uparrow 1\) as \(\epsilon \downarrow 0\). To avoid the workload from increasing indefinitely, we scale the workloads so as to obtain a nondegenerate limit. Only the queues for which the load is increasing, an appropriate scaling is required. The specific way in which the workloads should be scaled depends on the type of input (more specifically, it matters whether the increments have finite variance or not); this will be pointed out in detail later in the paper. In addition, note that \(r_i\) contains a term \({{\mathrm{{\mathbb {E}}}}}J_i\), which negates the drift of the input process. Therefore, the drift of the input process is not important and can be assumed to be zero in the remainder of the paper.
2.2 Useful results on transforms
To ensure stability, it is required that the average input rate is less than the speed of the slowest server, i.e. \({{\mathrm{{\mathbb {E}}}}}J_1 < r_2 \). Therefore, it is possible to define a random variable \((Q_0^{(1)},Q_0^{(2)})\), so that the resulting bivariate process \(\{(Q_t^{(1)},Q_t^{(2)}),t\ge 0\}\) is stationary. We write \(Q^{(i)}\) for a random variable with distribution equal to \(Q^{(i)}_t\), for a fixed t, when the process is initiated as mentioned above.
The theorems stated below, which uniquely characterize the distributions of the \(Q^{(i)}\), play a crucial role throughout the paper. The following assertions are Theorems 3.2, 12.11, and 12.3, respectively, copied from the book [10] (mostly using their notation). Closely related results were originally developed in [7], cf. Eq. (4.12) in their paper. Theorem 2.1 gives the Laplace–Stieltjes tranform (LST) for the stationary workload if there is only one server and can be considered to be a generalization of the wellknown Pollaczek–Khinchine formula. The LST for the joint stationary workload in the fluid tandem system is presented in Theorem 2.2, which also provides us with the LST for the downstream queue only (Corollary 2.3).
Theorem 2.1
Theorem 2.2
Corollary 2.3
Remark 2.4
Throughout the remainder of the paper, we assume \(J\in \mathcal {S}^+\) and \({{\mathrm{{\mathbb {E}}}}}J_1<\infty \). It is straightforward to extend our results to spectrally negative input processes \(J\in \mathcal {S}^\), by making use of Laplace–Stieltjes transforms for \(\mathcal {S}^\)processes, which can be found in, for example, Theorem 12.12 of [10].
3 Input processes with finite variance
In this section we consider the fluid tandem queue for various types of input processes that have increments with finite variance. Since, for Brownian input, an explicit analysis can be performed, we consider this case first (Sect. 3.1). Using appropriate expansions, we show in Sect. 3.2 how these results extend to spectrally positive Lévy processes. In both cases, we establish Regime I and Regime II results. Finally, in Sect. 3.3, we provide a numerical comparison between the Regime I and Regime II approximations.
3.1 Brownian input
3.1.1 Regime I
Proposition 3.1
In particular, this implies that the distribution of \(\epsilon Q^{(2)}\) converges to an exponential distribution with rate \(2/\sigma ^2\), which is equal to the distribution of the total workload. Moreover, it turns out that \(Q^{(1)}\) and \(\epsilon Q^{(2)}\) are asymptotically independent in the limit \(\epsilon \downarrow 0\). Asymptotic independence should not be very surprising. In a prelimit setting, there is a positive correlation between both buffer contents (cf. [7], Corollary 4.2). It follows from, for example, Eq. (4.11) in the same paper that the correlation tends to zero as the load in (only) the second node increases to one. Furthermore, one should realize that \(Q^{(2)}\) is scaled by a factor \(\epsilon \), whereas \(Q^{(1)}\) is not. It turns out that, due to the asymmetry in the spatial scaling, we obtain asymptotic independence.
Although this asymptotic independence is an interesting finding from a theoretical point of view, it has the intrinsic drawback that the original dependency structure is lost. Another drawback of this approximation is that it leads to significant errors if \(\rho _1\) is large as well, as will be illustrated in Sect. 3.3. This prompts us to consider Regime II.
3.1.2 Regime II
3.2 General input
We now extend the results for the Brownian case in the previous section to spectrally positive Lévy input. Again we consider both regimes, starting with Regime I.
3.3 Regime I
In this section we prove the following main result.
Proposition 3.2
Let the input process \(J\in \mathcal {S}^+\) be such that \({{\mathrm{Var}}}J_1 = \sigma ^2 <\infty \). Then, in Regime I, the stationary workloads of the up and downstream queue are asymptotically independent, with \(Q^{(1)}\) given by Theorem 2.1, and \(Q^{(2)} \mathop {=}\limits ^\mathrm{d} \mathrm{Exp}(\frac{2}{\sigma ^2})\).
To prove this proposition, we require the following lemma.
Lemma 3.3
Proof of Lemma 3.3
At first glance, it may be unclear why \(\psi \) in Lemma 3.3 has this specific form. However, in case of, for example, compound Poisson input, this shape arises naturally, as is demonstrated in Example 3.4 below. We first prove the main result.
Proof of Proposition 3.2
Example 3.4

Derive the Takács equation (describing the LST \(\pi \) of the busy period in an M/G/1 queue) with service rate equal to \(r_1\);

Use this Takács equation to express \(\psi \) in terms of \(\pi \);

Expand \(\pi \), which yields an expansion for \(\psi \).
3.4 Regime II
In the following we consider the corresponding Regime II result. It should be noted that the methodology is similar to that for Regime I. However, since the \(\epsilon \) now plays a different role, we cannot use Lemma 3.3, but we develop Lemma 3.7 instead.
Proposition 3.5
Remark 3.6
Note that the result in Proposition 3.5 corresponds to Eq. (5), i.e. the LST we found in case of Brownian input, except now we do take a proper heavy traffic limit, whereas Eq. (5) holds for all \(\epsilon >0\).
Lemma 3.7
Proof of Lemma 3.7
Proof of Proposition 3.5
This result follows from Lemma 3.7 and Theorem 2.2, and taking the limit \(\epsilon \downarrow 0\). The calculations are similar to those in the Brownian case, except there are some additional terms of small order \(\epsilon \) that cancel in the heavy traffic limit. \(\square \)
3.5 Numerical approximations for exponential jobs
Example 3.8
The values in this table correspond to the left and right plot in Figure 3
Figure 3, left plot  Figure 3, right plot  

x  Simul  R1  R2  x  Simul  R1  R2 
1  0.975  0.990  0.984  1  0.884  0.990  0.888 
20  0.800  0.817  0.810  5  0.750  0.951  0.753 
40  0.653  0.668  0.662  10  0.654  0.904  0.656 
80  0.436  0.446  0.442  15  0.584  0.859  0.585 
100  0.356  0.364  0.362  20  0.527  0.817  0.528 
150  0.215  0.220  0.219  25  0.480  0.777  0.480 
200  0.129  0.133  0.132  30  0.439  0.739  0.439 
250  0.077  0.080  0.080  35  0.403  0.702  0.403 
300  0.047  0.048  0.049  40  0.372  0.668  0.372 
400  0.017  0.018  0.019  45  0.344  0.635  0.343 
500  0.006  0.006  0.008  50  0.318  0.603  0.318 
The values in this table correspond to the left and right plot in Figure 4
Figure 4, left plot  Figure 4, right plot  

x  Simul  R1  R2  x  Simul  R1  R2 
1  0.498  0.777  0.551  0.5  0.719  0.945  0.761 
2  0.362  0.605  0.387  1.0  0.630  0.894  0.665 
3  0.273  0.471  0.283  1.5  0.565  0.846  0.594 
4  0.210  0.367  0.210  2.0  0.512  0.800  0.537 
5  0.163  0.286  0.157  2.5  0.468  0.757  0.489 
6  0.128  0.223  0.119  3.0  0.429  0.716  0.447 
7  0.101  0.173  0.090  3.5  0.395  0.677  0.410 
8  0.080  0.135  0.069  4.0  0.365  0.640  0.378 
9  0.064  0.105  0.053  4.5  0.338  0.606  0.349 
10  0.051  0.082  0.040  5.0  0.313  0.573  0.323 
In addition, we estimated the probabilities by simulation. The results are gathered in Tables 1 and 2, and are plotted in Figs. 3 and 4. Observe from Fig. 3 that the Regime II approximation is substantially more accurate than the Regime I approximation when \(\rho _2\) is high (in this case \(\rho _2=0.99\)). By comparing the two plots in Fig. 3, we see that increasing \(\rho _1\) negatively affects the performance of the Regime I approximation. Figure 4 shows that the Regime II approximation works remarkably well even when relatively low loads are imposed on both servers. Our experiments reveal that it is only reasonable to use Regime I approximations in a tandem queue when the load of the first server \(\rho _1\) is low; in all other cases, it is outperformed by the Regime II approximation. If \(\rho _1\) is high, then there is a stronger dependence between the up and downstream workloads (cf. Eq. (9), noting that \(\rho _1\) increases as \(\gamma \) decreases). Apparently, the dependence between both workloads, which is ignored in Regime I, has a crucial impact.
4 Heavytailed input
In this section we consider spectrally positive Lévy input processes with increments that have infinite variance. Unlike in the finite variance case, the precise form of the heavy traffic limit depends on the specific features of the Lévy input process. In Sect. 4.1, we consider compound Poisson input with heavytailed jumps, and in Sect. 4.2, we consider \(\alpha \)stable Lévy input (where \(1<\alpha <2\)). Note that \(\alpha \)stable Lévy motion can be regarded as a generalization of Brownian motion. Indeed, for \(\alpha =2\), an \(\alpha \)stable Lévy motion reduces to a Brownian motion.
Remark 4.1
We only consider Regime I results, because we have not managed to compute Regime II results here. In the finite variance case we relied on the existence of the inverse function of \(\phi \) in the Brownian case to construct \(\psi (s\epsilon (r_1r_2))\) as in Lemma 3.7. However, for heavytailed input, there is in general no inverse function of \(\phi \) available, except for some special cases, such as \(\frac{3}{2}\)stable Lévy motion.
4.1 Compound Poisson
In this section we consider spectrally positive compound Poisson input processes with heavytailed jumps.
Remark 4.2
In [11] a heavy traffic problem for heavytailed input was studied in a GI/G/1 setting. In their paper the correct scaling function \(\Delta (\epsilon )\) was also found by letting it be the zero of an appropriate equation. We follow a similar approach.
Proposition 4.3
Proof
Example 4.4
4.2 \(\alpha \)Stable Lévy motion
In this subsection we prove the following result. It entails that the workloads are asymptotically independent in the heavy traffic limit and that the marginals correspond to scaled MittagLeffler distributed random variables.
Proposition 4.5
Proof of Proposition 4.5
In the case \(\alpha =\frac{3}{2}\), \(\psi \) can be calculated explicitly and the result can be obtained without the use of Tauberian theorems. We include this in the paper, as the calculations potentially contain clues as to how Regime II results can be eventually obtained.
Example 4.6
4.3 Numerical heavy traffic approximations
This table corresponds to the left and right plot in Fig. 5
\(\rho _2=0.95\)  \(\rho _2=0.99\)  

x  Simul  ML  diff (%)  Simul  ML  diff (%) 
10  0.744  0.775  4.2  0.943  0.949  0.64 
20  0.676  0.705  4.3  0.924  0.929  0.54 
40  0.597  0.622  4.2  0.900  0.903  0.33 
60  0.546  0.569  4.2  0.879  0.883  0.46 
80  0.508  0.530  4.3  0.863  0.867  0.46 
100  0.478  0.499  4.4  0.850  0.853  0.35 
150  0.424  0.443  4.5  0.823  0.824  0.12 
200  0.387  0.404  4.4  0.801  0.802  0.12 
300  0.335  0.351  4.8  0.766  0.766  0.00 
400  0.300  0.315  5.0  0.739  0.737  −0.27 
500  0.274  0.288  5.1  0.716  0.714  −0.28 
5 Discussion and concluding remarks
In this paper we considered two types of heavy traffic regimes for a twonode fluid tandem queue with spectrally positive Lévy input. In Regime I, only the second server experiences heavy traffic. In this case, the load of the first server has no influence on the steadystate distribution of the workload in the second server. In Regime II, where both servers experience heavy traffic, the dependence structure between both workloads is preserved. In the case where the increments of the Lévy input process have finite variance, we have obtained Regime I and II results, whereas for the infinite variance case we established Regime I results.
The numerical experiments led to the interesting insight that (for finite variance input processes) the Regime II approximation performs typically better than the Regime I approximation, particularly when the load of the first server is high as well. This leads us to wonder if results of this kind carry over to a more general setting.
An open problem concerns Regime II results in the case where the increments of the input process have infinite variance. It is not clear how such results can be established. In the finite variance case we could define an inverse Laplace exponent that was in line with the exact inverse for Brownian motion. However, in the case of heavytailed input, for example for \(\alpha \)stable Lévy motion, there is no explicit inverse Laplace exponent for all \(1<\alpha <2\), and hence a fundamentally different approach needs to be developed.
Another direction for further research concerns stochasticprocess limits. In the singlenode case there is convergence to reflected Brownian motion (in the finite variance case) and to a reflected stable process (in the infinite variance case), and the question is whether we can establish the counterpart of such results for the downstream node in a tandem system, or even for the joint distribution of both workloads.
Notes
Acknowledgments
The research for this paper is partly funded by the NWO Gravitation Project NETWORKS, Grant Number 024.002.003. The research of Onno Boxma was also partly funded by the Belgian Government, via the IAP Bestcom Project.
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