G-Networks with Adders

Queueing networks are used to model the performance of the Internet, of manufacturing and job-shop systems, supply chains, and other networked systems in transportation or emergency management. Composed of service stations where customers receive service, and then move to another service station till they leave the network, queueing networks are based on probabilistic assumptions concerning service times and customer movement that represent the variability of system workloads. Subject to restrictive assumptions regarding external arrivals, Markovian movement of customers, and service time distributions, such networks can be solved efficiently with “product form solutions” that reduce the need for software simulators requiring lengthy computations. G-networks generalise these models to include the effect of “signals” that re-route customer traffic, or negative customers that reject service requests, and also have a convenient product form solution. This paper extends G-networks by including a new type of signal, that we call an “Adder”, which probabilistically changes the queue length at the service center that it visits, acting as a load regulator. We show that this generalisation of G-networks has a product form solution

Arrival first queueing networks with applications in kanban production systems

In this paper we introduce a new class of queueing networks called {\it arrival first networks}. We characterise its transition rates and derive the relationship between arrival rules, linear partial balance equations, and product form stationary distributions. This model is motivated by production systems operating under a kanban protocol. In contrast with the conventional {\em departure first networks}, where a transition is initiated by service completion of items at the originating nodes that are subsequently routed to the destination nodes (push system), in an arrival first network a transition is initiated by the destination nodes of the items and subsequently those items are processed at and removed from the originating nodes (pull system). These are similar to the push and pull systems in manufacturing systems

Traffic signal control using queueing theory

Traffic signal control has drawn considerable attention in the literatures thanks to its ability to improve the mobility of urban networks. Queueing models are capable of capturing performance or effectiveness of a queueing system. In this report, SOCPs (second order cone program) are proposed based on different queueing models as pre-timed signal control techniques to minimize total travel delay. Stochastic programs are developed in order to handle the uncertainties in the arrival rates. In addition, the superiority of the proposed model over Webster’s model has been validated in a microscopic traffic simulation software named CORSIM.Statistic

Zero-automatic queues and product form

We introduce and study a new model: 0-automatic queues. Roughly, 0-automatic queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The salient result is that all stable 0-automatic queues have a product form stationary distribution and a Poisson output process. When considering the two simplest and extremal cases of 0-automatic queues, we recover the simple M/M/1 queue, and Gelenbe's G-queue with positive and negative customers

Reversibility in Queueing Models

In stochastic models for queues and their networks, random events evolve in time. A process for their backward evolution is referred to as a time reversed process. It is often greatly helpful to view a stochastic model from two different time directions. In particular, if some property is unchanged under time reversal, we may better understand that property. A concept of reversibility is invented for this invariance. Local balance for a stationary Markov chain has been used for a weaker version of the reversibility. However, it is still too strong for queueing applications. We are concerned with a continuous time Markov chain, but dose not assume it has the stationary distribution. We define reversibility in structure as an invariant property of a family of the set of models under certain operation. The member of this set is a pair of transition rate function and its supporting measure, and each set represents dynamics of queueing systems such as arrivals and departures. We use a permutation {\Gamma} of the family menmbers, that is, the sets themselves, to describe the change of the dynamics under time reversal. This reversibility is is called {\Gamma}-reversibility in structure. To apply these definitions, we introduce new classes of models, called reacting systems and self-reacting systems. Using those definitions and models, we give a unified view for queues and their networks which have reversibility in structure, and show how their stationary distributions can be obtained. They include symmetric service, batch movements and state dependent routing.Comment: Submitted for publicatio