Transient error approximation in a Lévy queue
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Abstract
Motivated by a capacity allocation problem within a finite planning period, we conduct a transient analysis of a singleserver queue with Lévy input. From a cost minimization perspective, we investigate the error induced by using stationary congestion measures as opposed to timedependent measures. Invoking recent results from fluctuation theory of Lévy processes, we derive a refined cost function, that accounts for transient effects. This leads to a corrected capacity allocation rule for the transient singleserver queue. Extensive numerical experiments indicate that the cost reductions achieved by this correction can be significant.
Keywords
Singleserver queue Transient analysis Lévy processes Capacity allocationMathematics Subject Classification
60K25 60G511 Introduction
The issue of matching a service system’s capacity to stochastic demand induced by its clients arises in many practical settings. Typically, the resources available to satisfy demand are scarce and hence expensive. This forces the manager to consider a tradeoff between the system efficiency and the quality of service perceived by its clients. In this paper, we focus on this tradeoff in the context of the M / G / 1 queue, in which the variable amenable for optimization is the server speed \(\mu \).
The timedependent behavior of the singleserver queue received much attention in queueing theory. First efforts to analyze the timedependent properties of the M / G / 1 queue date back to the 1950s and 1960s; for example, [7, 10, 17, 28, 29]. The analyses in these papers mostly yield implicit expressions for performance characteristics through Laplace transforms, integrodifferential equations and infinite convolutions. More specifically, there is vast literature on the transient analysis of the M / M / 1 queue, with the goal to derive explicit expressions for queue length characteristics; see, for example, [3, 9, 23, 24]. These works provide a variety of explicit expressions for the transient dynamics, although the complexity of the resulting expressions, typically involving Bessel functions, exposes the intricate intractability of the matter. Consequently, approximation methods for insightful quantification of the dynamics based on numerical [20] or asymptotic methods have become prevalent in more recent literature. The asymptotic methods either exploit knowledge of the evolution of the queueing process as time t grows large [3, 21, 22], or as the arrival rate \(\lambda \) is increased to infinity [1, 2, 11]. It is noteworthy that a substantial contribution to the transient literature is made by Abate and Whitt [1, 2, 3, 4], who exploit the existence of a decomposition of the mean transient queue length and obtain expressions for the moments of the queue length and virtual waiting through probabilistic arguments in several queueing models. More recently, asymptotic methods have been used to justify the application of stationary performance measures in Markovian environments or to refine them; see, for example, [12, 30]. Other approximative methods known as uniform acceleration expansions [19] have been developed to reveal the asymptotic behavior of the singleserver queue as a function of t, which are moreover able to capture timevarying arrival rates.
The majority of the works mentioned above do reflect on the error imposed by usage of steadystate performance metrics instead of the correct timedependent counterpart. However, no light has been shed on the accumulation of this error over a finite period of time. To the best of our knowledge, the only work that addresses this issue is the paper by Steckley and Henderson [27], who compute an approximation for the error accumulated between the steadystate and transient delay probability. Our analysis on the other hand is centered around the mean workload, which requires a different approach. In addition, the focus in [27] is on performance measures only, while the main goal of our paper is to investigate the quality of staffing rules.
Although the M / G / 1 queue serves as the leading example in our analysis, we choose to use a more general framework for the arrival process of the queue. Namely, we let the server face a Lévy process. This gives the advantage that once we have obtained the results, we can apply them to broader queue input classes, such as Brownian motion and the Gamma process.
To shed light on the influence of the transience of the queueing process on traditional staffing questions, we will study the capacity allocation problem in the context of cost minimization in which the objective function is \(\varPi _T(\mu )\), i.e., a function of both \(\mu \) and T. We investigate how the invalidity of the stationary assumption is echoed through the operational cost accounting for congestionrelated penalties.
Furthermore, we establish a result on the strict convexity of the function \(\varPi _T(\mu )\), for almost all values of T (with a few minor exceptions for certain deterministic initial states), which is an essential property for convergence of both cost function and corresponding minimizer to their stationary counterparts.
Building upon the insights gained through the analysis of this optimality gap, we reflect on the parameter settings of the underlying queueing process in which our refined capacity sizing rule yields significant improvement and in which cases it has little effect. Special emphasis is put on the relationship between the accuracy of the standard procedure and the length of the planning period.
The remainder of the paper is structured as follows. Section 2 is devoted to the model description and presents some preliminary results. The main result will be given in Sect. 3, and results regarding the optimization problem will be discussed in Sect. 4, followed by the validation of our novel techniques through numerical experiments in Sect. 5. We will give some concluding remarks and topics for further research in Sect. 6. We have deferred all proofs to the appendices.
2 Model description
2.1 A queueing model with Lévy input
Lemma 1
The proof of Lemma 1 follows directly by differentiation of the Laplace transform of \(Q_\mu (\infty )\) and is therefore straightforward.
2.2 Finite horizon
Since the time horizon of our analysis is limited to \(t\le T\), the process may not approach the steadystate distribution sufficiently close to appropriately use its steadystate properties for capacity allocation. To overcome this disparity, we propose a way to include the influence of this transient phase in the capacity allocation problem.
2.3 Cost structure
Proposition 1
3 Analysis of the objective function
From (2.7) it is evident that, for finding an explicit characterization of \(\varPi _{T}(\mu )\), it suffices to study the term \(\varOmega _T(\mu )\) in more detail. We start by stating the main result of this section, which describes the leading order behavior of \(\varOmega _T(\mu )\) as T increases.
Theorem 1
Note that this expression provides an approximation of the actual cost function \(\varPi _T(\mu )\). We elaborate on the implications of this additional information on the optimization problem in Sect. 4.
In the remainder of this section, we provide a detailed description of the steps taken to obtain this outcome. We assume a fixed service rate \(\mu \) throughout the analysis in this section and therefore omit the subscript \(\mu \). Proofs of the intermediate results can be found in Appendix 2.
3.1 Constructing a coupling
Before starting our analysis of the correction term \(\varOmega _{T}(\mu )\), we introduce some auxiliary notation. By \(Q^A(t)\) we denote the workload process as described in Sect. 2.1 with initial state A and \(\mathbb {E}_A\) the expectation with respect to any nonnegative random variable A, which is independent of the netinput process X. To be able to compare \(\mathbb {E}[Q(t)]\) and \(\mathbb {E}[Q(\infty )]\) as in \(\varOmega _T(\mu )\), we will use a coupling technique. Observe that by the definition of the stationary distribution \(Q(\infty ) \,{\buildrel d \over =}\,Q^{Q(\infty )}(t)\) for all \(t \ge 0\) and therefore \(\mathbb {E}[Q(\infty )] = \mathbb {E}_{Q(\infty )}[Q^{Q(\infty )}(t)]\). Furthermore, \(\mathbb {E}[Q(t)] = \mathbb {E}_{Q(0)}[Q^{Q(0)}(t)]\). Hence, quantifying the difference between the transient and stationary mean is equivalent to comparing the workload processes of two queues starting in two different (random) states at \(t=0\).
We starting our analysis for two queues starting in two deterministic states \(x,y\ge 0\), respectively. At the end of our analysis, we will obtain the original form by replacing x with Q(0) and y with \(Q(\infty )\).
3.2 Difference process and leading order behavior of the correction term
Lemma 2
This leaves us with two unknowns: \(\mathbb {E}[\tau ^y(0)]\) and \(\varPsi _T^{xy,0}\). The next lemma gives an equivalent form for the latter.
Lemma 3
Since the term \(\mathbb {E}[\tau ^z(0)]\), for several values of z, appears in many of the preliminary results, we devote our attention to this in the next subsection.
First passage time
Theorem 2
This immediately induces an expression for \(\mathbb {E}[\tau ^w(0)]\) and henceforth \(\varPsi ^{z,0}\).
Corollary 1
Randomization
3.3 Truncation error
Proposition 2
Remark
In the case where the netinput process X is lighttailed, that is, there exists \(u>0\) such that \(\mathbb {E}[\mathrm{e}^{u X(1)}] < \infty \), it can be shown that the truncation error is of order \(\mathrm{e}^{\beta T}/T\) for some \(\beta >0\). See Appendix 2 for details.
4 Optimization
Throughout this section, we assume that \(u_2, u_3 <\infty \) and \(\mathbb {E}[Q(0)^2] <\infty \).
Lemma 4

convex in \(\mu \), if \(Q(0)\equiv x\), \(T<x/\mu \) and \(\sigma =0\),

strictly convex in \(\mu \), otherwise.
Building upon strict convexity of both \(\varPi _T(\mu )\) and \(\varPi _\infty (\mu )\) for \(\mu >\lambda \), we derive the following convergence result.
Proposition 3
The next result describes a refinement of \(\mu _T^\star \) in terms of \(\mu _\infty ^\star \).
Proposition 4
Proposition 5
5 Numerical experiments
5.1 Influence of \(\varOmega _{T}(\mu )\)
 1.M / M / 1 queue: U(t) is a unit rate compound Poisson process with exponentially distributed increments. We have \(u_2 = 2\), \(u_3=3\), so that$$\begin{aligned} \hat{C}_{T}(\mu ) = \frac{\lambda }{\mu \lambda } + \frac{1}{T(\mu \lambda )} \left( \frac{x^2}{2}  \frac{\lambda ^2}{(\mu \lambda )^2}  \frac{\lambda }{\mu \lambda } \right) . \end{aligned}$$(5.1)
 2.\(M/\mathrm{Pareto}/1\) queue: U(t) is a unit rate compound Poisson process with Pareto increments. The Pareto distribution deserves special attention due to its heavy tailed nature, having tail probability \(\bar{F}(x) = (x/k)^{\gamma }\), if \(x\ge k\), and 1 otherwise. It is wellknown that heavytailed service times lead to long relaxation time. For our purposes, we fix shape parameter \(\gamma = 16/5\) and scale parameter \(k=11/16\), so that \(\beta = 1\), \(u_2 = 121/96\), \(u_3 = 1331/256\) and \(u_k=\infty \) for all \(k>3\). Hence,$$\begin{aligned} \hat{C}_{T}(\mu ) = \frac{121\lambda }{192(\mu \lambda )} + \frac{1}{2T(\mu \lambda )} \left( x^2  \frac{(121\lambda /96)^2}{2(\mu \lambda )^2}  \frac{ 1331\lambda /256 }{2(\mu \lambda )}\right) . \end{aligned}$$(5.2)
 3.Reflected Brownian motion: U(t) is Brownian motion with drift 1 and infinitesimal variance \(\sigma ^2\). We have \(u_2 = \sigma ^2\), \(u_3=0\), so that$$\begin{aligned} \hat{C}_{T}(\mu ) = \frac{\lambda \sigma ^2}{2(\mu \lambda )} + \frac{1}{2T(\mu \lambda )} \left( x^2  \frac{\lambda ^2\sigma ^4}{2(\mu \lambda )^2}\right) . \end{aligned}$$(5.3)
For the M / M / 1 and \(M/\mathrm{Pareto}/1\) queues, we obtained the function \(C_{T}(\mu )\) through simulation and the results are accurate up until a 95% confidence interval of width \(10^{3}\). For reflected Brownian motion, we used the explicit distribution function given in [13] for double numerical integration. The results for several values of T and two different starting states are depicted in Figs. 4, 5 and 6. These plots also include the approximated functions \(\hat{C}_{T}(\mu )\). We make a few observations based on these figures.
First, we indeed note the pointwise convergence of \(\hat{C}_{T}(\mu )\) to \(\hat{C}_{\infty }(\mu )\) as T grows, for all \(\mu \) in all three cases. However, the difference between the stationary costs and those for small values of T can be significant. This is most clear in the plots with \(x=2.5\) and when \(\mu \) is close to \(\lambda \), i.e., it is in heavy traffic. In these scenarios, it is evident that refinements to the stationary cost function are needed. \(\hat{C}_{T}(\mu )\) does a fairly good job at providing such correction, especially for moderate values of \(\mu \).
Finally, observe that \(\hat{C}_{T}(\mu )\rightarrow \infty \) as \(\mu \) approaches \(\lambda \) from above. This divergence is clear from the expressions in (5.1)(5.3). Our correction term relies on the premise that under the coupling scheme, the sample paths of the two queues starting from different states have hit with high probability. This is equivalent to stating that the ‘largest’ of the two queues is emptied at least once before time T. However, as \(\mu \) approaches \(\lambda \), the system enters heavy traffic, and hence the hitting time of the zero barrier is set to run off to infinity. Consequently, this causes our approximation to be inaccurate for small values of \(\mu \).
5.2 Validation of corrected staffing rule
In this section we examine whether the corrected staffing rule \(\tilde{\mu }_T^\star \) as in (4.5) indeed yields a significant cost reduction over the choice of \(\mu _\infty ^\star \) by comparing their true costs \(\varPi _{T}(\tilde{\mu }_T^\star )\) and \(\varPi _{T}(\mu _\infty ^\star )\). We conduct this comparison for different values of the parameters, \(\alpha \), T and starting state x through numerical experiments. The three models on which we do our calculations are the M / M / 1 queue, the M / Pareto / 1 queue and the reflected Brownian motion, as introduced in the previous subsection. We again focus on \(\lambda =1\) only.
M / M / 1 queue
Comparison of costs for the M / M / 1 queue for steadystate and corrected staffing rules and relative cost improvement (r.c.i.)
\(x = 0\)  \(x = 2\sqrt{\alpha }\)  

\(\alpha \)  T  \(\mu _\infty ^\star \)  \(\varPi _T(\mu _\infty ^\star )\)  \(\tilde{\mu }_T^\star \)  \(\varPi _T(\tilde{\mu }_T^\star )\)  r.c.i.  \(\mu _\infty ^\star \)  \(\varPi _T(\mu _\infty ^\star )\)  \(\tilde{\mu }_T^\star \)  \(\varPi _T(\tilde{\mu }_T^\star )\)  r.c.i. 
0.1  1  4.162  0.620  2.688  0.536  0.136  4.162  0.682  2.688  0.536  0.214 
2  4.162  0.669  3.425  0.641  0.041  4.162  0.700  3.425  0.641  0.085  
5  4.162  0.706  3.867  0.703  0.005  4.162  0.719  3.867  0.703  0.022  
10  4.162  0.719  4.015  0.719  0.001  4.162  0.726  4.015  0.719  0.010  
1  1  2.000  2.309  0.000  0.500  0.783  2.000  3.500  0.500  2.750  0.214 
2  2.000  2.461  0.750  1.480  0.398  2.000  3.218  1.250  3.125  0.029  
5  2.000  2.675  1.500  2.400  0.103  2.000  3.043  1.700  2.968  0.025  
10  2.000  2.810  1.750  2.726  0.030  2.000  3.007  1.850  2.980  0.009  
2  1  1.707  3.744  0.000  0.500  0.866  1.707  5.889  0.000  3.328  0.435 
2  1.707  3.924  0.146  1.232  0.686  1.707  5.547  0.854  4.682  0.156  
5  1.707  4.209  1.083  3.343  0.206  1.707  5.114  1.366  4.910  0.040  
10  1.707  4.424  1.395  4.108  0.071  1.707  4.945  1.536  4.868  0.016 
M/Pareto/1 queue
Comparison of costs for the \(M/\mathrm{Pareto}/1\) queue for steadystate and corrected staffing rules and relative cost improvement (r.c.i.)
\(x = 0\)  \(x = 11/4\cdot \sqrt{\alpha /3}\)  

\(\alpha \)  T  \(\mu _\infty ^\star \)  \(\varPi _T(\mu _\infty ^\star )\)  \(\tilde{\mu }_T^\star \)  \(\varPi _T(\tilde{\mu }_T^\star )\)  r.c.i.  \(\mu _\infty ^\star \)  \(\varPi _T(\mu _\infty ^\star )\)  \(\tilde{\mu }_T^\star \)  \(\varPi _T(\tilde{\mu }_T^\star )\)  r.c.i. 
0.1  1  3.510  0.524  1.759  0.461  0.120  3.510  0.573  2.010  0.562  0.019 
2  3.510  0.555  2.635  0.539  0.029  3.510  0.580  2.760  0.574  0.010  
5  3.510  0.580  3.160  0.578  0.003  3.510  0.591  3.210  0.589  0.002  
10  3.510  0.590  3.335  0.590  0.000  3.510  0.596  3.360  0.595  0.001  
1  1  1.794  2.076  0.000  0.500  0.759  1.794  2.989  0.000  2.088  0.302 
2  1.794  2.190  0.511  1.291  0.411  1.794  2.790  0.610  2.588  0.072  
5  1.794  2.345  1.281  2.108  0.101  1.794  2.638  1.320  2.607  0.012  
10  1.794  2.441  1.537  2.371  0.029  1.794  2.597  1.557  2.585  0.005  
2  1  1.561  3.427  0.000  0.500  0.854  1.561  5.087  0.000  2.745  0.460 
2  1.561  3.567  0.032  1.050  0.706  1.561  4.832  0.172  3.417  0.293  
5  1.561  3.779  0.950  3.012  0.203  1.561  4.499  1.006  4.313  0.041  
10  1.561  3.935  1.255  3.356  0.147  1.561  4.351  1.284  4.304  0.011 
Just as in the results for the M / M / 1 queue, we observe a higher reduction for larger value of \(\alpha \) and T. Also, again \(\tilde{\mu }_T < \mu _\infty ^\star \). Hence, the conclusions for the \(M/\mathrm{Pareto}/1\) queue are similar to those of the M / M / 1 queue.
Reflected Brownian motion
Comparison of costs for RBM with \(\sigma = 1\) for steadystate and corrected staffing rules and relative cost improvement (r.c.i.)
\(x = 0\)  \(x = \sqrt{2\alpha } \)  

\(\alpha \)  T  \(\mu _\infty ^\star \)  \(\varPi _T(\mu _\infty ^\star )\)  \(\tilde{\mu }_T^\star \)  \(\varPi _T(\tilde{\mu }_T^\star )\)  r.c.i.  \(\mu _\infty ^\star \)  \(\varPi _T(\mu _\infty ^\star )\)  \(\tilde{\mu }_T^\star \)  \(\varPi _T(\tilde{\mu }_T^\star )\)  r.c.i. 
0.1  1  3.236  0.525  2.901  0.518  0.013  3.236  0.565  3.124  0.564  0.001 
2  3.236  0.536  3.068  0.534  0.003  3.236  0.556  3.180  0.556  0.000  
5  3.236  0.543  3.169  0.542  0.000  3.236  0.551  3.214  0.551  0.000  
10  3.236  0.545  3.203  0.545  0.000  3.236  0.549  3.225  0.549  0.000  
1  1  1.500  3.420  0.000  0.833  0.756  1.500  4.741  1.000  3.984  0.160 
2  1.500  3.539  0.750  2.386  0.326  1.500  4.579  1.250  4.293  0.063  
5  1.500  3.707  1.200  3.363  0.093  1.500  4.335  1.400  4.274  0.014  
10  1.500  3.820  1.350  3.705  0.030  1.500  4.190  1.450  4.175  0.004  
2  1  1.500  3.420  0.000  0.833  0.756  1.500  4.741  1.000  3.984  0.160 
2  1.500  3.539  0.750  2.386  0.326  1.500  4.579  1.250  4.293  0.063  
5  1.500  3.707  1.200  3.363  0.093  1.500  4.335  1.400  4.274  0.014  
10  1.500  3.820  1.350  3.705  0.030  1.500  4.190  1.450  4.175  0.004 
Comparison of costs for RBM with \(\sigma = 2\) for steadystate and corrected staffing rules and relative cost improvement (r.c.i.)
\(x = 0\)  \(x = 2\sqrt{2\alpha } \)  

\(\alpha \)  T  \(\mu _\infty ^\star \)  \(\varPi _T(\mu _\infty ^\star )\)  \(\tilde{\mu }_T^\star \)  \(\varPi _T(\tilde{\mu }_T^\star )\)  r.c.i.  \(\mu _\infty ^\star \)  \(\varPi _T(\mu _\infty ^\star )\)  \(\tilde{\mu }_T^\star \)  \(\varPi _T(\tilde{\mu }_T^\star )\)  r.c.i. 
0.1  1  5.472  0.950  4.801  0.936  0.015  5.472  1.030  5.249  1.029  0.001 
2  5.472  0.972  5.137  0.968  0.003  5.472  1.012  5.360  1.012  0.000  
5  5.472  0.985  5.338  0.985  0.000  5.472  1.002  5.427  1.002  0.000  
10  5.472  0.990  5.405  0.990  0.000  5.472  0.998  5.450  0.998  0.000  
1  1  2.414  3.176  0.293  1.546  0.513  2.414  4.633  1.707  4.228  0.087 
2  2.414  3.356  1.354  2.690  0.199  2.414  4.375  2.061  4.247  0.029  
5  2.414  3.573  1.990  3.411  0.045  2.414  4.094  2.273  4.073  0.005  
10  2.414  3.689  2.202  3.646  0.012  2.414  3.966  2.344  3.962  0.001  
2  1  2.000  4.839  0.000  1.339  0.723  2.000  7.481  1.000  5.967  0.202 
2  2.000  5.078  0.500  2.773  0.454  2.000  7.158  1.500  6.585  0.080  
5  2.000  5.414  1.400  4.726  0.127  2.000  6.670  1.800  6.549  0.018  
10  2.000  5.639  1.700  5.409  0.041  2.000  6.380  1.900  6.349  0.005 
5.3 Discussion
Based upon these numerical results in Tables 1, 2, 3 and 4, we make a few remarks. The three models roughly exhibit similar behavior as T, x and \(\alpha \) are varied.
Nonsurprisingly, we note that \(\tilde{\mu }_T\) approaches \(\mu _\infty ^\star \) with increasing T, which also implies that the cost reduction achieved by the corrected staffing rule vanishes as \(T\rightarrow \infty \). Also, we observe that in all scenarios examined, the cost reduction increases with \(\alpha \). This can be explained through investigation of the objective function \(\varPi _T\) as a function of \(\mu \). Namely, for \(\alpha \) small, the curve is relatively flat around the true optimum \(\mu _T^\star \). Hence, in this case a moderate deviation from \(\mu _T^\star \) will likely not lead to a significant cost increase. However, as \(\alpha \) becomes larger, i.e., server efficiency is valued more than minimization of congestion, the curve becomes more sharp around \(\mu _T^\star \), and hence more accurate approximations of \(\mu _T^\star \) are required to achieve an acceptable cost level. Hence, the corrected staffing rule (4.5) proves particularly useful in these cases.
6 Conclusion and further research
Motivated by the timevarying nature of queues in practical applications, we studied the impact that the transient phase has on traditional capacity allocation questions. By defining a cost minimization problem in which the objective function contains a correction accounting for the transient period, we identified the leading and secondorder behavior of the cost function as a function of the interval length T. As a byproduct, this result yields an approximation for the actual cost function, which is a refinement to its stationary counterpart. Our numerical experiments in Sect. 5.1 demonstrate the improved accuracy achieved by this approximation in a number of settings. By perturbation analysis of the optimization problem, this furthermore gives rise to a correction to the steadystate optimal capacity allocation of order 1 / T. The necessity of the refined capacity allocation level is substantiated by the numerics in Sect. 5.2, which show the cost reduction that can be achieved in a number of settings, compared to settings in which stationary metrics are used. Particularly for small values of T and large values of \(\alpha \), this reduction is significant. Additionally, these results also indicate that it is relatively safe to use the stationary cost when T is moderate, or \(\alpha \) is small. The latter reflects the scenario in which QoS is much more valued than service efficiency. This observation links to the flat nature of the cost function around its optimal value for \(\alpha \) small, a statement on the optimality gap that we formally proved in Proposition 4.
Besides the validation of our theoretical results of Sects. 3 and 4, the numerical results also reveal some phenomena that require more investigation.
As noted, our corrected capacity allocation level \(\tilde{\mu }_T^\star \) is in most cases studied less than the steadystate optimal value \(\mu _{\infty }^\star \). This implies that congestion levels tend to be higher under our staffing scheme then under stationary staffing. A possible explanation for this may be the fact that the planning period under consideration is finite. Clearly, in the setting we analyzed, anything that happens after time T is neglected. Therefore, it might be beneficial from the cost perspective to end the period with a higher expected congestion level, as it does not need to be canceled out in the future. Related to this observation, it would be interesting to look at the setting in which staffing decisions need to be made in consecutive periods of equal length, in which the arrival rate changes at the start of each period. This case requires careful consideration of the correlation among the staffing decisions within the separate periods.
Another question that arises concerns the translation of our (qualitative) findings to more general queues, in particular the M / G / s queue. Whereas in our analysis the central decision variable is the server speed \(\mu \), the variable of interest in multiserver queues is typically the number of servers. It may well be that similar explicit corrections to staffing levels can be deduced to account for transience. Since our analysis heavily relies on the comparability of the sample paths of two singleserver queues, which is due to the equal negative drift for the two processes, another approach must be taken to tackle this extension.
The analysis and findings for the singleserver queue with Lévy input presented in this paper may serve as a stepping stone for investigation of these more elaborate problems.
References
 1.Abate, J., Whitt, W.: Transient behavior of regulated Brownian motion, I: starting at the origin. Adv. Appl. Probab. 19(3), 560–598 (1987)CrossRefGoogle Scholar
 2.Abate, J., Whitt, W.: Transient behavior of regulated Brownian motion, II: nonzero initial conditions. Adv. Appl. Probab. 19(3), 599–631 (1987)CrossRefGoogle Scholar
 3.Abate, J., Whitt, W.: Transient behavior of the M/M/1 queue: starting at the origin. Queueing Syst. Theory Appl. 2(1), 41–65 (1987)CrossRefGoogle Scholar
 4.Abate, J., Whitt, W.: Transient behavior of the M/G/1 workload process. Oper. Res. 42(4), 750–764 (1994)CrossRefGoogle Scholar
 5.Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)Google Scholar
 6.Awad, H.P., Glynn, P.W.: On the theoretical comparison of lowbias steadystate estimators. ACM Trans. Model. Comput. Simul. 17(1), 4 (2007)Google Scholar
 7.Benes, V.E.: On queues with Poisson arrivals. Ann. Math. Stat. 28(3), 670–677 (1957)CrossRefGoogle Scholar
 8.Bertoin, J.: Lévy processes. Cambridge University Press, Cambridge (1996)Google Scholar
 9.Cohen, J.W.: The Single Server Queue. NorthHolland Pub. Co., Amsterdam (1969)Google Scholar
 10.Gaver, D.P.: Imbedded Markov chain analysis of a waitingline process in continuous time. Ann. Math. Stat. 30(3), 698–720 (1959)CrossRefGoogle Scholar
 11.Gaver, D.P.: Diffusion approximations and models for certain congestion problems. J. Appl. Probab. 5(3), 607–623 (1968)CrossRefGoogle Scholar
 12.Green, L.V., Kolesar, P.: The pointwise stationary approximation for queues with nonstationary arrivals. Manag. Sci. 37(1), 84–97 (1991)CrossRefGoogle Scholar
 13.Harrison, J.M.: Brownian Motion and Stochastic Flow Systems. Wiley, New York (1985)Google Scholar
 14.Janssen, A.J.E.M., van Leeuwaarden, J.S.H., Mathijsen, B.W.J.: Novel heavytraffic regimes for largescale service systems. SIAM J. Appl. Math. 75(2), 787–812 (2015)CrossRefGoogle Scholar
 15.Janssen, A.J.E.M., van Leeuwaarden, J.S.H., Zwart, A.P.: Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula. Adv. Appl. Prob. 40(1), 122–143 (2008)CrossRefGoogle Scholar
 16.Janssen, A.J.E.M., van Leeuwaarden, J.S.H., Zwart, A.P.: Refining squareroot safety staffing by expanding Erlang C. Oper. Res. 59(6), 1512–1522 (2011)CrossRefGoogle Scholar
 17.Kendall, D.G.: Some problems in the theory of queues. J. R. Stat. Soc. 113(2), 151–185 (1951)Google Scholar
 18.Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006)Google Scholar
 19.Massey, W.A., Whitt, W.: Uniform acceleration expansions for Markov chains with timevarying rates. Ann. Appl. Prob. 8(4), 1130–1155 (1998)Google Scholar
 20.Neuts, M.F.: The single server queue with Poisson input and semiMarkov service times. J. Appl. Probab. 3(1), 202–230 (1966)CrossRefGoogle Scholar
 21.Newell, G.F.: Applications of Queueing Theory. Chapman and Hall, London (1982)CrossRefGoogle Scholar
 22.Odoni, A.R., Roth, E.: An empirical investigation of the transient behavior of stationary queueing systems. Oper. Res. Int. J. 31(3), 432–455 (1983)Google Scholar
 23.Pegden, C.D., Rosenshine, M.: Some new results for the M/M/1 queue. Manag. Sci. 28(7), 821–828 (1982)CrossRefGoogle Scholar
 24.Prabhu, N.U.: Timedependent results in storage theory. J. Appl. Probab. 1(1), 1–46 (1964)CrossRefGoogle Scholar
 25.Randhawa, R.S.: Optimality gap of symptotically derived prescriptions in queueing systems: \(o(1)\)optimality. Queueing Syst. Theory Appl. 83(1), 131–155 (2016)Google Scholar
 26.Sato, K.I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)Google Scholar
 27.Steckley, S.G., Henderson, S.G.: The error in steadystate approximations for the timedependent waiting time distribution. Stoch. Models 23(2), 307–332 (2007)CrossRefGoogle Scholar
 28.Takaćs, L.: Investigation of waiting time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hung. 6(1), 101–129 (1955)CrossRefGoogle Scholar
 29.Takaćs, L.: The time dependence of a singleserver queue with Poisson input and general service times. Ann. Math. Stat. 33(4), 1340–1348 (1962)CrossRefGoogle Scholar
 30.Whitt, W.: The pointwise stationary approximation is apsymptotically correct as the rates increase. Management Science, (1991)Google Scholar
 31.Zhang, B., van Leeuwaarden, J.S.H., Zwart, A.P.: Staffing call centers with impatient customers: refinements to manyserver asymptotics. Oper. Res. 60(2), 461–474 (2012)CrossRefGoogle Scholar
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