A functional approximation for the M/G/1/N queue
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Abstract
This paper presents a new approach to the functional approximation of the M/G/1/N built on a Taylor series approach. Specifically, we establish an approximative expression for the remainder term of the Taylor series that can be computed in an efficient manner. As we will illustrate with numerical examples, the resulting Taylor series approximation turns out to be of practical value.
Keywords
Series expansion approach Taylor series M/G/1/N queue Performance measures Deviation matrix1 Introduction
Queueing models are a well-established tool for the analysis of stochastic systems from areas as divers as manufacturing, telecommunication, transport and the service industry. Typically, a queueing model is a simplified representation of the real-world system under consideration. In addition, often there is not sufficient statistical data to determine the service and interarrival time distribution, or, in case the type of distribution is known, there is statical uncertainty on the exact values of the parameters of the distribution. For these reasons, perturbation analysis of queueing systems (PAQS) has been developed. PAQS studies the dependence of the performance of a given queueing system on the underlying distributional assumptions. This paper is concerned with PAQS for the finite capacity single server queue, where we assume that the arrival stream is of Poisson type, an assumption which is often justified in applications, see Chen and Xia (2011).
- 1.
Fast convergence of the series (a Taylor polynomial of small order yields already a satisfying approximation).
- 2.
The ability of computing the remainder term of the Taylor series in an efficient way so that the order of the Taylor polynomial that is sufficient to achieve a desired precision of the approximation can be decided a priori.
The contribution of the paper is as follows. We investigate the use of Taylor polynomials for the numerical evaluation of the M/G/1/N queue. Specifically, our numerical studies show that already a Taylor series of small order yields good approximations (this addresses topic (1) above); and that a simplified and easily computable expression bounding the remainder of the Taylor series can be established (this addresses topic (2) above).
The paper is organized as follows. The embedded Markov chain of the M/G/1/N model is presented in Section 2. Our series expansions approach is detailed in Section 3. Numerical examples are provided in Section 4 for the case of the M/D/1/N queue. A more detailed version of this paper is available as technical report (Abbas et al. 2011), which contains more numerical material on the M/D/1 queue, and, as an additional example, a perturbation analysis of the M/W/1/N queue, with W denoting the Weibull distribution is presented.
We conclude the introduction with a brief discussion of implications of our approach to the numerical approximation of the M/G/1/∞ queue. Numerical approximations for the M/G/1/N queue have a long tradition, and we refer to the excellent survey in Smith (2004). The approach presented in this paper is different from the standard approaches as it is also feasible for traffic loads larger than one. In addition, our approach yields an approximation of a performance functional on an entire interval and allows for an error bound of the approximation that holds uniformly on an interval.
2 The M/G/1/N queue
Consider the M/G/1/N queue, where customers arrive according to a Poisson process with rate λ and demand an independent and identically distributed service time with common distribution function B(t) with mean 1/μ. There can at most be N customers be present at the queue (including the one in service), and customers attempting to enter the queue when there are already N customers present are lost. The service discipline is FCFS.
3 The Taylor series expansion approach
In this section, we present the Taylor series approximation for the M/G/1/N queue. Let B ( ·) have density mapping b ( ·) . Let Θ = ( a , b ) ⊂ ℝ , for 0 < a < b < ∞.
- (A)
For 0 ≤ k ≤ N − 2 it holds that a_{k} is n-times differentiable with respect to θ on Θ.
Example 1
Theorem 1
Proof
The proof for the general case follows by induction with respect to n like in conventional analysis. □
Example 2
The basic idea is that analyticity of π_{θ} implies that of \( \pi_\theta^{(k)} \) for all k and we can again use a Taylor series to approximate \( \pi_{ \theta + x }^{(k+1)} \) in Eq. 9. By doing so we initiate the Taylor series in the tail of original Taylor series, and we expect that the error of this second Taylor approximation step is negligibly small. We explain this approach in the following in more details.
In order to state the precise statement, we introduce the norm \( || x || = \sum_{i=1}^n | x_i| \) on ℝ^{n} .
Theorem 2
Proof
In the numerical examples presented in the following sections, we will show that choosing m = 2 already yields a sufficient precision for approximating the remainder term.
Remark 1
Taylor series approaches for performance approximation have been studied in the literature before, see, e.g. Girish and Hu (1996, 1997) and Gong and Hu (1992). However, no a priori knowledge on the quality of the approximation of these approach could be established.
The Taylor series approximation developed above applies to differentiable Markov kernels. This extends the case of linear θ dependence that has been studied in the literature so far; see, for example, Cao (1998), Heidergott et al. (2010), Kirkland et al. (1998), Leder et al. (2010) and Schweitzer (1968). An interesting property of the linear-dependence case is that the remainder term can be bounded in an efficient way, see Heidergott et al. (2007).
4 Applications to the M/D/1/N queue
Consider the M/D/1/5 queue with arrival rate λ and deterministic service time c = θ. The elements of P are provided in Example 1.
Lemma 1
The transition probability matrix P of the embedded chain of the M/D/1/N queue is infinitely often differentiable with respect to c.
Proof
By Eq. 6 differentiability properties of P can be deduced from that of the α_{j} entries. By Example 1, all higher-order derivatives exist for a_{j}, which proves the claim. □
The relative absolute error in predicting the loss probability for various traffic rates
Δ | ρ = 0.5 (θ = 2) | ρ = 1 (θ = 1) | ρ = 1.2 (θ = 0.833) | |||
---|---|---|---|---|---|---|
k = 2 | k = 3 | k = 2 | k = 3 | k = 2 | k = 3 | |
10^{ − 3} × | 10^{ − 4} × | |||||
0.01 | 0.000155 | 0.000002 | 0.000001 | 0.000054 | 0.000001 | 0.000024 |
0.02 | 0.001114 | 0.000028 | 0.000014 | 0.000823 | 0.000011 | 0.000403 |
0.03 | 0.003385 | 0.000130 | 0.000046 | 0.003922 | 0.000037 | 0.002087 |
0.04 | 0.007237 | 0.000356 | 0.000107 | 0.011670 | 0.000085 | 0.006719 |
0.05 | 0.012774 | 0.000796 | 0.000204 | 0.026837 | 0.000161 | 0.016646 |
0.06 | 0.019988 | 0.001475 | 0.000346 | 0.052442 | 0.000268 | 0.034916 |
0.07 | 0.028795 | 0.002446 | 0.000536 | 0.091600 | 0.000410 | 0.065248 |
0.09 | 0.039068 | 0.003742 | 0.000782 | 0.147396 | 0.000590 | 0.112003 |
0.09 | 0.050650 | 0.005381 | 0.001089 | 0.222798 | 0.000810 | 0.180128 |
0.1 | 0.063373 | 0.007376 | 0.001456 | 0.320589 | 0.001073 | 0.275112 |
We conclude the discussion of the M/D/1/N queue by providing a bound on the error of the Taylor series approximation for \( \pi_{\theta + \Delta }^\ast ( N) \).
Lemma 2
Proof
The remainder term vs. the bound for the remainder at ρ = 1 for k = 2 and m = 2
Δ | Remainder | |
---|---|---|
Bound | True | |
10^{ − 3} × | 10^{ − 3} × | |
0.01 | 0.000257 | 0.000242 |
0.02 | 0.002241 | 0.001995 |
0.03 | 0.008188 | 0.006927 |
0.04 | 0.020899 | 0.016862 |
0.05 | 0.043757 | 0.033776 |
0.06 | 0.080739 | 0.059782 |
0.07 | 0.136433 | 0.097121 |
0.08 | 0.216045 | 0.148148 |
0.09 | 0.325420 | 0.215323 |
0.1 | 0.471053 | 0.301201 |
5 Conclusion
We have presented a new approach to the functional approximation of finite queues. As illustrated by the numerical examples for the M/D/1/N queue, the convergence rate of the Taylor series is such that already a Taylor polynomial of degree 2 or 3 yields good numerical results. We established an approximation for the remainder term of the Taylor series that provides an efficient way of computing (approximately) the remainder term and thereby provides an algorithmic way of deciding which order of the Taylor polynomial is sufficient to achieve a desired precision of the approximation. This implies that the proposed Taylor series approximation can be of practical value. Future research will be on investigating the behavior of the series expansion for multi-server queues.