Demetrios Fakinos

A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes

Abstract We consider a continuous-time Markov chain { N (t), t⩾0} with state space S. A point process {X(t), t⩾0} , using a function A:[0,∞)→S in the following way, influences the evolution of { N (t)} : whenever an event of {X(t)} occurs at time t, the Markov chain of interest goes immediately at state A(t) and evolves according to the dynamics of { N (t)} until the next event of {X(t)} and so on. We study the transient and the limiting distribution of the resulting process and prove some stochastic comparison facts. Several examples that demonstrate the applicability of the model are also included.

Product form stationary distributions for queueing networks with blocking and rerouting

In this paper we study Markovian queueing networks in which the service and the routing characteristics have a particular form which leads to a product form stationary distribution for the number of customers in the various queues of the network. We show that if certain transitions are prohibited due to blocking conditions, then the form of the stationary distribution is preserved under a certain rerouting protocol. Several examples are presented which illustrate the wide applicability of the model.

On the Stationary Distribution of the GI X /M Y /1 Queueing System

For the GI X /M/1 queue, it has been recently proved that there exist geometric distributions that are stochastic lower and upper bounds for the stationary distribution of the embedded Markov chain at arrival epochs. In this note we observe that this is also true for the GI X /M Y /1 queue. Moreover, we prove that the stationary distribution of its embedded Markov chain is asymptotically geometric. It is noteworthy that the asymptotic geometric parameter is the same as the geometric parameter of the upper bound. This fact justifies previous numerical findings about the quality of the bounds.

Product form stationary distributions for the MGk group-arrival group-departure loss system under a general acceptance policy

In this paper we investigate under what circumstances the stationary distribution of the numbers of groups of various sizes in the M|G|k group-arrival group-departure loss system under a quite general acceptance policy, can be obtained in a closed product form. Also an algorithm is provided for computing the stationary distribution of the number of customers in the system.

A new approach for the study of the MX/G/1 queue using renewal arguments

The M X /G/1 queueing system as well as several of its variants have long ago been studied by considering the embedded discrete-time Markov chain at service completion epochs. Alternatively other approaches have been proposed such as the theory of regenerative processes, the supplementary variable method, properties of the busy period, etc. In this note we study the M X /G/1 queue via a simple new method that uses renewal arguments. This approach seems quite powerful and may become fruitful in the investigation of other queueing systems as well.

The G/G/1 (LCFS/P) queue with service depending on queue size

Abstract This paper studies a general single server queue, assuming that customers have service times depending on the queue size and also that they are served in accordance with a last-come-first-served queue discipline with preemption and arbitrary restarting policy. Expressions are given for the queue size limiting distribution when the system is considered at arrival (or departure) epochs and in continuous time, by using very simple arguments.

Duality relations for certain single server queues

In this paper, relations are given between the joint distribution of several variables in a GI/G/1 queue and the joint distribution of variables associated with the busy cycle in the dual queue, that is in the queue which results from the original when the interarrival times and the service times are interchanged. It is assumed that the primal queue has the preemptive-resume last-come-first-served queue discipline while the dual queue may have any queue discipline which is work conserving. These relations generalize a result given recently for M/G/1 and GI/M/1 queues.

A generalization of symmetric queues

Well known insensitivity results for symmetric queues are extended to the case where the relevant parameters are functions of the numbers of customers of various types present in the queue.