Scaling limits for closed product-form queueing networks

We consider a general class of closed product-form queueing networks, consisting of single-server queues and infinite-server queues. Even if a network is of product-form type, performance evaluation tends to be difficult due to the potentially large state space and the dependence between the individual queues. To remedy this, we analyze the model in a Halfin–Whitt inspired scaling regime, where we jointly blow up the traffic loads of all queues and the number of customers in the network. This leads to a closed-form limiting stationary distribution, which provides intuition on the impact of the dependence between the queues on the network's behavior. We assess the practical applicability of our results through a series of numerical experiments, which illustrate the convergence and show how the scaling parameters can be chosen to obtain accurate approximations

Two-choice regulation in heterogeneous closed networks

A heterogeneous closed network with one-server queues with finite capacity and one infinite-server queue is studied. A target application is bike-sharing systems. Heterogeneity is taken into account through clusters whose queues have the same parameters. Incentives to the customer to go to the least loaded one-server queue among two chosen within a cluster are investigated. By mean-field arguments, the limiting queue length stationary distribution as the number of queues gets large is analytically tractable. Moreover, when all customers follow incentives, the probability that a queue is empty or full is approximated. Sizing the system to improve performance is reachable under this policy.Comment: 19 pages, 4 figure

Networks of ⋅/G/∞\cdot/G/\infty Server Queues with Shot-Noise-Driven Arrival Intensities

We study infinite-server queues in which the arrival process is a Cox process (or doubly stochastic Poisson process), of which the arrival rate is given by shot noise. A shot-noise rate emerges as a natural model, if the arrival rate tends to display sudden increases (or: shots) at random epochs, after which the rate is inclined to revert to lower values. Exponential decay of the shot noise is assumed, so that the queueing systems are amenable for analysis. In particular, we perform transient analysis on the number of customers in the queue jointly with the value of the driving shot-noise process. Additionally, we derive heavy-traffic asymptotics for the number of customers in the system by using a linear scaling of the shot intensity. First we focus on a one dimensional setting in which there is a single infinite-server queue, which we then extend to a network setting