Stein’s method for diffusive limits of queueing processes

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.

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

$$\begin{aligned} {\mathbf {E}}\left[ \max _{i=1,\ldots ,n}X_i\right] \le c\, {\log n \over \log \log n}{.} \end{aligned}$$

(47)

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

$$\begin{aligned} \exp \left( u{\mathbf {E}}\left[ \max _{i=1,\ldots ,n}Z_i\right] \right)\le & {} {\mathbf {E}}\left[ \max _{i=1,\ldots ,n}\exp (uZ_i)\right] \le \sum _{i=1}^n{\mathbf {E}}\left[ \exp (uZ_1)\right] \\\le & {} n\exp \left( \varPsi _{Z_i}(u)\right) . \end{aligned}$$

After some quick algebra, this readily implies that

$$\begin{aligned} {\mathbf {E}}\left[ \max _{i=1,\ldots ,n}Z_i\right]&\le \inf _{u\in {\mathbb {R}}}\left( \frac{\log n+\nu \left( e^u-u-1\right) }{u}\right) \\&= \frac{\log n+\nu \left( e \frac{a}{W(a)}-1-W(a)-1\right) }{1+W(a)}, \end{aligned}$$

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

$$\begin{aligned} {\mathbf {E}}\left[ \max _{i=1,\ldots ,n}X_i\right] \le \nu e \frac{a}{W(a)}-\nu +\nu =\frac{\log {(n/e^{\nu })}}{W(\log (n/e^{\nu })/e^{\nu })}{.} \end{aligned}$$

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 $n\ge \exp \left( e^{\nu +1}+\nu \right)$,

$$\begin{aligned} {\mathbf {E}}\left[ \max _{i=1,\ldots ,n}X_i\right] \le \frac{\log {(n/e^{\nu })}}{\log (\log (n/e^{\nu })/e^{\nu }) -\log \log (\log (n/e^{\nu })/e^{\nu })} \le c\, {\log n \over \log \log n}, \end{aligned}$$

which completes the proof. $\square$

B Proof of Proposition 2

Fix n throughout this section and denote, for all $i=0,...,n-1$ and $(x,z) \in {\mathbb {R}}^2$,

$$\begin{aligned} \alpha _{i}(x,z)={\mathbf {1}}_{C_{{t^n_i}}}(x,z),\quad \quad \beta _i(x,z)=\int _{t^n_i}^{t^n_{i+1}}{\mathbf {1}}_{C_{u}}(x,z)\text { d}u. \end{aligned}$$

Proof of (i)

Recall (42), and fix two indexes $0\le i <j \le n-1$. We have that

$$\begin{aligned}&\int \int u^{\sharp }_iu^{\sharp }_j\text { d}\nu ^\sharp _n= \int \int \left( \alpha _{i+1}- \alpha _i\right) \left( \alpha _{j+1}-\alpha _j \right) \text { d}\nu ^\sharp _n\nonumber \\&\qquad +\mu \int \int \beta _i\left( \alpha _{j+1}-\alpha _j\right) \text { d}\nu ^\sharp _n\nonumber \\&\qquad +\mu \int \int \beta _j\left( \alpha _{i+1}-\alpha _i\right) \text { d}\nu ^\sharp _n{+\mu ^2 \int \int \beta _j\beta _j\text { d}\nu ^\sharp _n}\nonumber \\&\quad =: I_1 +I_2 +I_3+I_4, \end{aligned}$$

(48)

where straightforward computations show that

$$\begin{aligned} I_1&=\lambda n\left( 2 e^{-\mu (t^n_j-t^n_i)} - e^{-\mu (t^n_j-t^n_{i+1})} - e^{-\mu (t^n_j-t^n_{i-1})}\right) ;\\ I_2&={\lambda n\over \mu }\left( 2 e^{-\mu (t^n_j-t^n_i)} - e^{-\mu (t^n_j-t^n_{i+1})} - e^{-\mu (t^n_j-t^n_{i-1})}\right) - \lambda \left( e^{-\mu t_{j+1}^n} - e^{-\mu t^n_j}\right) ;\\ I_3&={\lambda n\over \mu }\left( -2 e^{-\mu (t^n_j-t^n_i)} + e^{-\mu (t^n_j-t^n_{i+1})} + e^{-\mu (t^n_j-t^n_{i-1})}\right) ;\\ I_4&={\lambda n\over \mu }\left( -2 e^{-\mu (t^n_j-t^n_i)} + e^{-\mu (t^n_j-t^n_{i+1})} + e^{-\mu (t^n_j-t^n_{i-1})}\right) + \lambda \left( e^{-\mu t_{j+1}^n} - e^{-\mu t^n_j}\right) . \end{aligned}$$

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

$$\begin{aligned} I_{i,j,k}:= & {} \int _{{\mathbb {R}}^2} |u^{\sharp }_iu^{\sharp }_{j}u^{\sharp }_{k}|\text { d}\nu ^\sharp _n\le \int \left| (\alpha _{i+1} - \alpha _{i})(\alpha _{j+1} - \alpha _{j})(\alpha _{k+1}-\alpha _{k})\right| \text { d}\nu ^\sharp _n\nonumber \\&+ \int \left| (\alpha _{i+1} - \alpha _i)(\alpha _{j+1} - \alpha _j)\mu \beta _k\right| \text { d}\nu ^\sharp _n\nonumber \\&+ \int \left| (\alpha _{j+1} - \alpha _j)(\alpha _{k+1} - \alpha _k)\mu \beta _i\right| \text { d}\nu ^\sharp _n\nonumber \\&+ \int \left| (\alpha _{i+1} - \alpha _i)(\alpha _{k+1} - \alpha _k)\mu \beta _j\right| \text { d}\nu ^\sharp _n+\int \left| \left( \alpha _{i+1} - \alpha _i\right) \mu ^2\beta _j\beta _k\right| \text { d}\nu ^\sharp _n\nonumber \\&+\int \left| \left( \alpha _{j+1} - \alpha _j\right) \mu ^2\beta _i\beta _k\right| \text { d}\nu ^\sharp _n+\int \left| \left( \alpha _{k+1} - \alpha _k\right) \mu ^2\beta _i\beta _j\right| \text { d}\nu ^\sharp _n\nonumber \\&+\int \left| \mu ^3\beta _i\beta _j\beta _k\right| \text { d}\nu ^\sharp _n=: \sum _{l=1}^8 I_{i,j,k}^l. \end{aligned}$$

(49)

It can be easily retrieved that

$$\begin{aligned} I_{i,i,i}^1&=n\left( \frac{\lambda }{n}-\frac{\lambda }{\mu }\left( 1-e^{-\frac{\mu T}{n}}\right) \left( 1-e^{\mu T{i+1 \over n}}\right) \right) \le {\lambda \over \mu };\\ I_{i,j,k}^1&=0,\quad 1\le i<j<k \le n;\\ I_{i,i,k}^1&={\lambda n\over \mu }\left( e^{\mu t^n_{i+1}} - e^{\mu t^n_{i}}\right) \left( e^{-\mu t^n_{k}} - e^{-\mu t^n_{k+1}} \right) \le {\lambda T^2\over \mu n},\quad i=j<k, \end{aligned}$$

and the other cases can be treated similarly. Also, simple computations show that if $i<j$

$$\begin{aligned} \mu \int \left| (\alpha _{i+1} - \alpha _i)(\alpha _{j+1} - \alpha _j)\beta _k\right| \text { d}\nu ^\sharp _n\le \lambda \left( e^{\mu t^n_{i+1}} - e^{\mu t^n_i}\right) \left( e^{-\mu t^n_j} - e^{-\mu t^n_{j+1}} \right) \le {\lambda T^2\over n^2}, \end{aligned}$$

whereas if $i=j$, the above integral is upper bounded by

$$\begin{aligned} 2\lambda T\left( 2+e^{-\mu t^n_{i+1}}-e^{-\mu t^n_i}-2e^{-\frac{\mu T}{n}}\right) \le \frac{2\lambda \mu T^2}{n}T^2. \end{aligned}$$

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,

$$\begin{aligned} \mu ^2\int \left| (\alpha _{i+1} - \alpha _i)\mu ^2\beta _j\beta _k\right| \text { d}\nu ^\sharp _n&\le \frac{\mu ^2}{n}\left( \frac{\lambda }{n}-\frac{\lambda }{\mu }\left( 1-e^{-\frac{\mu T}{n}}\right) \left( 1-e^{\mu T{i+1 \over n}}\right) \right) \\&\le {\lambda \over \mu n^2}T, \end{aligned}$$

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,

$$\begin{aligned} \int \int {\mathbf {1}}_{C_u}{\mathbf {1}}_{C_v}{\mathbf {1}}_{C_w}\lambda \mu e^{-\mu y}\text { d}x\text { d}y=\frac{\lambda }{\mu }(e^{-\mu (\max (u,v,w)-\min (u,v,w))}-e^{-\mu \max (u,v,w)}), \end{aligned}$$

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 $0\le i \le n-1$,

$$\begin{aligned}&\int \int u^{\sharp }_iu^{\sharp }_i\text { d}\nu ^\sharp _n=\int \int \alpha _{i+1}\text { d}\nu ^\sharp _n+\int \int \alpha _{i}\text { d}\nu ^\sharp _n-2 \int \int \alpha _{i+1}\alpha _{i}\text { d}\nu ^\sharp _n\nonumber \\&\qquad +2\mu \int \int \beta _i\alpha _{i+1}\text { d}\nu ^\sharp _n-2\mu \int \int \beta _i\alpha _{i}\text { d}\nu ^\sharp _n+\mu ^2\int \int \beta _i\beta _i\text { d}\nu ^\sharp _n\nonumber \\&\quad =J_1+J_2+J_3+J_4+J_5+J_6, \end{aligned}$$

(50)

where straightforward calculations show that

$$\begin{aligned} J_1&=\frac{\lambda n }{\mu }\left( 1-e^{-\mu t^n_{i+1}}\right) ;\quad J_2=\frac{\lambda n}{\mu }\left( 1-e^{-\mu t^n_i}\right) ;\\ J_3&=-2\frac{\lambda n}{\mu }\left( e^{-\frac{\mu T}{n}}-e^{-\mu t^n_{i+1}}\right) ;\quad J_4 =2\frac{\lambda n}{\mu }(1-e^{-\frac{\mu T}{n}})-2\lambda e^{-\mu t^n_{i+1}};\\ J_5&=-2\frac{\lambda n}{\mu }(1-e^{-\frac{\mu T}{n}})-2\frac{\lambda n}{\mu }(e^{-\mu t^n_{i+1}}-e^{-\mu t^n_i});\\ J_6&=\lambda \left( 2+2e^{-\mu t^{n}_{i+1}}+\frac{2n}{\mu }(e^{-\mu t_{i+1}^{n}}-e^{-\mu t^n_i}+e^{\frac{-\mu T}{n}}-1)\right) . \end{aligned}$$

Recalling (37), adding up the $J_j$, $j=1,...,6$, concludes the proof. $\square$