Stationary Solution of a Fluid Queue Driven by a Queue with Chain Sequence Rates and Controlled Input

A fluid queueing system in which the fluid flow in to the buffer is regulated by the state of the background queueing process is considered. In this model, the arrival and service rates follow chain sequence rates and are controlled by an exponential timer. The buffer content distribution along with averages are found using continued fraction methodology. Numerical results are illustrated to analyze the trend of the average buffer content for the model under consideration. It is interesting to note that the stationary solution of a fluid queue driven by a queue with chain sequence rates does not exist in the absence of exponential timer.

Synchronized Lévy queues

We consider a multivariate Lévy process where the first coordinate is a Lévy process with no negative jumps which is not a subordinator and the others are non-decreasing. We determine the Laplace–Stieltjes transform of the steady-state buffer content vector of an associated system of parallel queues. The special structure of this transform allows us to rewrite it as a product of joint Laplace–Stieltjes transforms. We are thus able to interpret the buffer content vector as a sum of independent random vectors.

The fluid $M/M/1$ catastrophic queue in a random environment

Our main objective in this paper is to investigate the stationary behavior of a fluid disaster queue of $M/M/1$ in a random multi-phase environment. Occasionally, the system experiences a catastrophic failure causing the loss of all current jobs. The system then goes into a process of repair. As soon as the system is repaired, it moves with probability \textit{q${}_{i}$} $\mathrm{\ge}$ 0 to phase \textit{i}. The distribution of the buffer content is determined using the probability generating function. In addition, some numerical results are provided to illustrate the effect of various parameters on the distribution of buffer content.

Stability of the stochastic matching model

We introduce and study a new model that we call the matching model. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be matched. There is a finite set of classes 𝒱 for the items, and the allowed matchings depend on the classes, according to a matching graph on 𝒱. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a matching policy. When the sequence of classes of the arriving items is independent and identically distributed, the sequence of buffer-content is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is nonbipartite.