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

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

Stationary distributions of the multi-type ASEP

We give a recursive construction of the stationary distribution of multi-type asymmetric simple exclusion processes on a finite ring or on the infinite line $Z$. The construction can be interpreted in terms of "multi-line diagrams" or systems of queues in tandem. Let $q$ be the asymmetry parameter of the system. The queueing construction generalises the one previously known for the totally asymmetric ($q=0$) case, by introducing queues in which each potential service is unused with probability $q^k$ when the queue-length is $k$. The analysis is based on the matrix product representation of Prolhac, Evans and Mallick. Consequences of the construction include: a simple method for sampling exactly from the stationary distribution for the system on a ring; results on common denominators of the stationary probabilities, expressed as rational functions of $q$ with non-negative integer coefficients; and probabilistic descriptions of "convoy formation" phenomena in large systems.Comment: 54 pages, 4 figure

Alternative proof and interpretations for a recent state-dependent importance sampling scheme

Recently, a state-dependent change of measure for simulating overflows in the two-node tandem queue was proposed by Dupuis et al. (Ann. Appl. Probab. 17(4):1306–1346, 2007), together with a proof of its asymptotic optimality. In the present paper, we present an alternative, shorter and simpler proof. As a side result, we obtain interpretations for several of the quantities involved in the change of measure in terms of likelihood ratios

Local stability in a transient Markov chain

We prove two propositions with conditions that a system, which is described by a transient Markov chain, will display local stability. Examples of such systems include partly overloaded Jackson networks, partly overloaded polling systems, and overloaded multi-server queues with skill based service, under first come first served policy.Comment: 6 page

Stationary distributions of multi-type totally asymmetric exclusion processes

We consider totally asymmetric simple exclusion processes with n types of particle and holes ($n$-TASEPs) on $\mathbb {Z}$ and on the cycle $\mathbb {Z}_N$. Angel recently gave an elegant construction of the stationary measures for the 2-TASEP, based on a pair of independent product measures. We show that Angel's construction can be interpreted in terms of the operation of a discrete-time $M/M/1$ queueing server; the two product measures correspond to the arrival and service processes of the queue. We extend this construction to represent the stationary measures of an n-TASEP in terms of a system of queues in tandem. The proof of stationarity involves a system of n 1-TASEPs, whose evolutions are coupled but whose distributions at any fixed time are independent. Using the queueing representation, we give quantitative results for stationary probabilities of states of the n-TASEP on $\mathbb {Z}_N$, and simple proofs of various independence and regeneration properties for systems on $\mathbb {Z}$.Comment: Published at http://dx.doi.org/10.1214/009117906000000944 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org