Perfect sampling of Jackson Queueing Networks

We consider open Jackson networks with losses with mixed finite and infinite queues and analyze the efficiency of sampling from their exact stationary distribution. We show that perfect sampling is possible, although the underlying Markov chain may have an infinite state space. The main idea is to use a Jackson network with infinite buffers (that has a product form stationary distribution) to bound the number of initial conditions to be considered in the coupling from the past scheme. We also provide bounds on the sampling time of this new perfect sampling algorithm for acyclic or hyperstable networks. These bounds show that the new algorithm is considerably more efficient than existing perfect samplers even in the case where all queues are finite. We illustrate this efficiency through numerical experiments. We also extend our approach to non-monotone networks such as queueing networks with negative customers.On considère les réseaux de Jackson avec perte comportant des files finies et infinies, et l'on s'intéresse à l'efficacité des techniques d'échantillonnage de leur distribution stationnaire exacte. Nous démontrons que la simulation parfaite est possible même si la chaîne de Markov sous-jacente a un espace d'états potentiellement infini. L'idée principale est d'utiliser un réseau de Jackson aux files infinies (qui admet une distribution de forme-produit) pour borner les conditions initiales à considérer dans l'algorithme de simulation parfaite. Nous donnons également des bornes sur le temps d'échantillonnage de ce nouvel algorithme dans le cas des réseaux acycliques, ainsi que pour des réseaux hyperstables. Ces bornes prouvent que le nouvel algorithme est considérablement plus efficace que les échantillonneurs parfaits acuels, même dans le cas où toutes les files sont finies. Nous illustrons cette efficacité par des expériences numériques. Enfin, nous généralisons notre approche au cas des réseaux non-monotones comme les réseaux aux clients négatifs

Perfect sampling of Jackson Queueing Networks

We consider open Jackson networks with losses with mixed finite and infinite queues and analyze the efficiency of sampling from their exact stationary distribution. We show that perfect sampling is possible, although the underlying Markov chain may have an infinite state space. The main idea is to use a Jackson network with infinite buffers (that has a product form stationary distribution) to bound the number of initial conditions to be considered in the coupling from the past scheme. We also provide bounds on the sampling time of this new perfect sampling algorithm for acyclic or hyperstable networks. These bounds show that the new algorithm is considerably more efficient than existing perfect samplers even in the case where all queues are finite. We illustrate this efficiency through numerical experiments. We also extend our approach to non-monotone networks such as queueing networks with negative customers.On considère les réseaux de Jackson avec perte comportant des files finies et infinies, et l'on s'intéresse à l'efficacité des techniques d'échantillonnage de leur distribution stationnaire exacte. Nous démontrons que la simulation parfaite est possible même si la chaîne de Markov sous-jacente a un espace d'états potentiellement infini. L'idée principale est d'utiliser un réseau de Jackson aux files infinies (qui admet une distribution de forme-produit) pour borner les conditions initiales à considérer dans l'algorithme de simulation parfaite. Nous donnons également des bornes sur le temps d'échantillonnage de ce nouvel algorithme dans le cas des réseaux acycliques, ainsi que pour des réseaux hyperstables. Ces bornes prouvent que le nouvel algorithme est considérablement plus efficace que les échantillonneurs parfaits acuels, même dans le cas où toutes les files sont finies. Nous illustrons cette efficacité par des expériences numériques. Enfin, nous généralisons notre approche au cas des réseaux non-monotones comme les réseaux aux clients négatifs

Steady-state simulation of reflected Brownian motion and related stochastic networks

This paper develops the first class of algorithms that enable unbiased estimation of steady-state expectations for multidimensional reflected Brownian motion. In order to explain our ideas, we first consider the case of compound Poisson (possibly Markov modulated) input. In this case, we analyze the complexity of our procedure as the dimension of the network increases and show that, under certain assumptions, the algorithm has polynomial-expected termination time. Our methodology includes procedures that are of interest beyond steady-state simulation and reflected processes. For instance, we use wavelets to construct a piecewise linear function that can be guaranteed to be within $\varepsilon$ distance (deterministic) in the uniform norm to Brownian motion in any compact time interval.Comment: Published at http://dx.doi.org/10.1214/14-AAP1072 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cooperative dynamics of loyal customers in queueing networks

We consider queueing networks (QN's) with feedback loops roamed by "intelligent” agents, able to select their routing on the basis of their measured waiting times at the QN nodes. This is an idealized model to discuss the dynamics of customers who stay loyal to a service supplier, provided their service time remains below a critical threshold. For these QN's, we show that the traffic flows may exhibit collective patterns typically encountered in multi-agent systems. In simple network topologies, the emergent cooperative behaviors manifest themselves via stable macroscopic temporal oscillations, synchronization of the queue contents and stabilization by noise phenomena. For a wide range of control parameters, the underlying presence of the law of large numbers enables us to use deterministic evolution laws to analytically characterize the cooperative evolution of our multi-agent systems. In particular, we study the case where the servers are sporadically subject to failures altering their ordinary behavio