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Register a new userTwo queues with random time-limited polling
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In this paper, we analyse a single server polling model with two queues. Customers arrive at the two queues according to two independent Poisson processes. There is a single server that serves both queues with generally distributed service times. The server spends an exponentially distributed amount of time in each queue. After the completion of this residing time, the server instantaneously switches to the other queue, i.e., there is no switch-over time. For this polling model we derive the steady-state marginal workload distribution, as well as heavy traffic and heavy tail asymptotic results. Furthermore, we also calculate the joint queue length distribution for the special case of exponentially distributed service times using singular perturbation analysis.
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We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, an M/G/$infty$-type queue. A job that is not completed during a visit will have a new service time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, this paper is the first in which an M/G/$infty$-type polling system is analysed. For this polling model, we derive the probability generating function and expected value of the queue lengths, and the Laplace-Stieltjes transform and expected value of the sojourn time of a customer. Moreover, we identify the policy that maximises the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order is emph{independent} of the number of customers present at the various queues at the start of the cycle.
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.
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.
In this paper, we consider a $G_t/G_t/infty$ infinite server queueing model in a random environment. More specifically, the arrival rate in our server is modeled as a highly fluctuating stochastic process, which arguably takes into account some small time scale variations often observed in practice. We show a homogenization property for this system, which yields an approximation by a $M_t/G_t/infty$ queue with modified parameters. Our limiting results include the description of the number of active servers, the total accumulated input and the solution of the storage equation. Hence in the fast oscillatory context under consideration, we show how the queuing system in a random environment can be approximated by a more classical Markovian system.
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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