Abstract
Donsker’s theorem is perhaps the most famous invariance principle result for Markov processes. It states that, when properly normalized, a random walk behaves asymptotically like a Brownian motion. This approach can be extended to general Markov processes whose driving parameters are taken to a limit, which can lead to insightful results in contexts like large distributed systems or queueing networks. The purpose of this paper is to assess the rate of convergence in these so-called diffusion approximations, in a queueing context. To this end, we extend the functional Stein method, introduced for the Brownian approximation of Poisson processes, to two simple examples: the single-server queue and the infinite-server queue. By doing so, we complete the recent applications of Stein’s method to queueing systems, with results concerning the whole trajectory of the considered process, rather than its stationary distribution.
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Appendices
A Moment bound for Poisson variables
By following closely Chapter 2 in [3], we show hereafter a moment bound for the maximum of n Poisson variables. (Notice that, contrary to Exercise 2.18 in [3], we do not assume here that the Poisson variables are independent.)
Proposition 4
Let \(n\in {{\mathbb {N}}}\) and let \(X_i,\, i=1,\ldots ,n\), be Poisson random variables of parameter \(\nu \). Then, for some c depending only on \(\nu \), we have that
Proof
Denote, for all i, \(Z_i=X_i-\nu \), and by \(\varPsi _{Z_i}\) the moment generating function of \(Z_i\). By Jensen’s inequality and the monotonicity of \(\exp (.)\) we get that
After some quick algebra, this readily implies that
where W is the so-called Lambert function, solving the equation \(W(x)e^{W(x)}=x\) over \([-1/e,\infty ]\), and \(a=\frac{\log (n/e^{\nu })}{e^\nu }\). This entails in turn that
We conclude by observing that \(W(z)\ge \log (z) - \log \log (z)\) for all \(z >e\). Therefore, there exists \(c>0\) such that, for \(n\ge \exp \left( e^{\nu +1}+\nu \right) \),
which completes the proof. \(\square \)
B Proof of Proposition 2
Fix n throughout this section and denote, for all \(i=0,...,n-1\) and \((x,z) \in {\mathbb {R}}^2\),
Proof of (i)
Recall (42), and fix two indexes \(0\le i <j \le n-1\). We have that
where straightforward computations show that
Adding up the above in (48) yields the result. \(\square \)
Proof of (ii)
For all \(0 \le i,j,k \le n-1\), we write
It can be easily retrieved that
and the other cases can be treated similarly. Also, simple computations show that if \(i<j\)
whereas if \(i=j\), the above integral is upper bounded by
It readily follows that in all cases, \(I_{i,j,k}^2,I_{i,j,k}^3\) and \(I_{i,j,k}^4\) are less than \({c\, n^{-1}}\) for some constant c. Reasoning similarly, we also obtain that, for all i, j, k,
so that in all cases \(I_{i,j,k}^5,I_{i,j,k}^6\) and \(I_{i,j,k}^7\) are less than \({c \, n^{-2}}\) for some c. Finally, observing that, for all u, v, w,
we can similarly bound \(I_{i,j,k}^8\) by \(c\,{n^{-2}}\) for all i, j, k. To summarize, all the \(I_{i,j,k}\) are less than \(c\,{n^{-2}}\) for some c, except for the \(I^1_{i,i,i}\), \(i=1,...,n\), which are bounded by a constant but are only n in number, and all terms where one index appears twice, which are less than \(c\,n^{-1}\) for some c, but are only \(n^2\) in number. Hence, (ii) follows. \(\square \)
Proof of (iii)
We have, for all \(0\le i \le n-1\),
where straightforward calculations show that
Recalling (37), adding up the \(J_j\), \(j=1,...,6\), concludes the proof. \(\square \)
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Besançon, E., Decreusefond, L. & Moyal, P. Stein’s method for diffusive limits of queueing processes. Queueing Syst 95, 173–201 (2020). https://doi.org/10.1007/s11134-020-09658-8
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DOI: https://doi.org/10.1007/s11134-020-09658-8