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Register a new userAttractiveness of Brownian queues in tandem
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Leandro Pinto Rodrigues Pimentel
Publication date
2018
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English
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Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering in the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between the Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. The result is also interpreted in the related context of Brownian particles acting under one sided reflection.
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Let $X$ be the constrained random walk on ${mathbb Z}_+^2$ taking the steps $(1,0)$, $(-1,1)$ and $(0,-1)$ with probabilities $lambda < (mu_1 eq mu_2)$; in particular, $X$ is assumed stable. Let $tau_n$ be the first time $X$ hits $partial A_n = {x:x(1)+x(2) = n }$ For $x in {mathbb Z}_+^2, x(1) + x(2) < n$, the probability $p_n(x)= P_x( tau_n < tau_0)$ is a key performance measure for the queueing system represented by $X$. Let $Y$ be the constrained random walk on ${mathbb Z} times {mathbb Z}_+$ with increments $(-1,0)$, $(1,1)$ and $(0,-1)$. Let $tau$ be the first time that the components of $Y$ equal each other. We derive the following explicit formula for $P_y(tau < infty)$: [ P_y(tau < infty) = W(y)= rho_2^{y(1)-y(2)} + frac{mu_2 - lambda}{mu_2 - mu_1} rho_1^{ y(1)-y(2)} rho_1^{y(2)} + frac{mu_2-lambda}{mu_1 -mu_2} rho_2^{y(1)-y(2)} rho_1^{y(2)}, ] where, $rho_i = lambda/mu_i$, $i=1,2$, $y in {mathbb Z}times{ mathbb Z}_+$, $y(1) > y(2)$, and show that $W(n-x_n(1),x_n(2))$ approximates $p_n(x_n)$ with relative error {em exponentially decaying} in $n$ for $x_n = lfloor nx rfloor$, $x in {mathbb R}_+^2$, $0 < x(1) + x(2) < 1$. The steps of our analysis: 1) with an affine transformation, move the origin $(0,0)$ to $(n,0)$ on $partial A_n$; let $n earrow infty$ to remove the constraint on the $x(2)$ axis; this step gives the limit {em unstable} /{em transient} constrained random walk $Y$ and reduces $P_{x}(tau_n < tau_0)$ to $P_y(tau < infty)$; 2) construct a basis of harmonic functions of $Y$ and use it to apply the superposition principle to compute $P_y(tau < infty).$ The construction involves the use of conjugate points on a characteristic surface associated with the walk $X$. The proof that the relative error decays exponentially uses a sequence of subsolutions of a related HJB equation on a manifold.
Exponential single server queues with state dependent arrival and service rates are considered which evolve under influences of external environments. The transitions of the queues are influenced by the environments state and the movements of the environment depend on the status of the queues (bi-directional interaction). The structure of the environment is constructed in a way to encompass various models from the recent Operation Research literature, where a queue is coupled e.g. with an inventory or with reliability issues. With a Markovian joint queueing-environment process we prove separability for a large class of such interactive systems, i.e. the steady state distribution is of product form and explicitly given: The queue and the environment processes decouple asymptotically and in steady state. For non-separable systems we develop ergodicity criteria via Lyapunov functions. By examples we show principles for bounding throughputs of non-separable systems by throughputs of two separable systems as upper and lower bound.
A many-server queueing system is considered in which customers with independent and identically distributed service times enter service in the order of arrival. The state of the system is represented by a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The limit process describing the total number in system is shown to be an Ito diffusion with a constant diffusion coefficient that is insensitive to the service distribution. The limit of the sequence of (centered and scaled) age processes is shown to be a Hilbert space valued diffusion that can also be characterized as the unique solution of a stochastic partial differential equation that is coupled with the Ito diffusion. Furthermore, the limit processes are shown to be semimartingales and to possess a strong Markov property.
We consider a processor sharing queue where the number of jobs served at any time is limited to $K$, with the excess jobs waiting in a buffer. We use random counting measures on the positive axis to model this system. The limit of this measure-valued process is obtained under diffusion scaling and heavy traffic conditions. As a consequence, the limit of the system size process is proved to be a piece-wise reflected Brownian motion.
Let $X$ be the constrained random walk on ${mathbb Z}_+^2$ with increments $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$; $X$ represents, at arrivals and service completions, the lengths of two queues working in parallel whose service and interarrival times are exponentially distributed with arrival rates $lambda_i$ and service rates $mu_i$, $i=1,2$; we assume $lambda_i < mu_i$, $i=1,2$, i.e., $X$ is assumed stable. Without loss of generality we assume $rho_1 =lambda_1/mu_1 ge rho_2 = lambda_2/mu_2$. Let $tau_n$ be the first time $X$ hits the line $partial A_n = {x in {mathbb Z}^2:x(1)+x(2) = n }$. Let $Y$ be the same random walk as $X$ but only constrained on ${y in {mathbb Z}^2: y(2)=0}$ and its jump probabilities for the first component reversed. Let $partial B ={y in {mathbb Z}^2: y(1) = y(2) }$ and let $tau$ be the first time $Y$ hits $partial B$. The probability $p_n = P_x(tau_n < tau_0)$ is a key performance measure of the queueing system represented by $X$ (probability of overflow of a shared buffer during systems first busy cycle). Stability of $X$ implies $p_n$ decays exponentially in $n$ when the process starts off $partial A_n.$ We show that, for $x_n= lfloor nx rfloor$, $x in {mathbb R}_+^2$, $x(1)+x(2) le 1$, $x(1) > 0$, $P_{(n-x_n(1),x_n(2))}( tau < infty)$ approximates $P_{x_n}(tau_n < tau_0)$ with exponentially vanishing relative error. Let $r = (lambda_1 + lambda_2)/(mu_1 + mu_2)$; for $r^2 < rho_2$ and $rho_1 eq rho_2$, we construct a class of harmonic functions from single and conjugate points on a characteristic surface of $Y$ with which $P_y(tau < infty)$ can be approximated with bounded relative error. For $r^2 = rho_1 rho_2$, we obtain $P_y(tau < infty) = r^{y(1)-y(2)} +frac{r(1-r)}{r-rho_2}left( rho_1^{y(1)} - r^{y(1)-y(2)} rho_1^{y(2)}right).$
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