A Finite Buffer Fluid Queue Driven by a Markovian Queue

We consider a finite buffer fluid queue receiving its input from the output of a Markovian queue with finite or infinite waiting room. The input flow into the fluid queue is thus characterized by a Markov modulated input rate process and we derive, for a wide class of such input processes, a procedure for the computation of the stationary buffer content of the fluid queue and the stationary overflow probability. This approach leads to a numerically stable algorithm for which the precision of the result can be specified in advance.     

Fluid queue driven by anM/M/1 queue

A fluid queue receiving its input from the output of a precedingM/M/1 queue is considered. The input can be characterized as a Markov modulated rate process and the well known spectral decomposition technique can be applied. The novel features in this system relate to the nature of the spectrum, which is shown to be composed of a continuous part and one or two discrete points depending on whether the load of the fluid queue is less or greater than the output to input rate ratio. Explicit expressions of the generalized eigenvectors are given in terms of Chebyshev polynomials of the second kind, and the resolution of unity is determined. The solution for the buffer content distribution is obtained as a simple integral expression. Numerical examples are given.     

Simple analysis of a fluid queue driven by an M/M/1 queue

In this note we consider the fluid queue driven by anM/M/1 queue as analysed by Virtamo and Norros [Queueing Systems 16 (1994) 373–386]. We show that the stationary buffer content in this model can be easily analysed by looking at embedded time points. This approach gives the stationary buffer content distribution in terms of the modified Bessel function of the first kind of order one. By using a suitable integral representation for this Bessel function we show that our results coincide with the ones of Virtamo and Norros.     

A Tandem Queue with Blocking and Markovian Arrival Process

Queueing networks with blocking have proved useful in modelling of data communications and production lines. We study such a network consisting of a sequence of two service stations with an infinite queue allowed before the first station and no intermediate queue allowed between them. This restriction results in the blocking of the first station whenever a unit having completed its service in that station cannot enter into the second one due to the presence of another unit there. The input of units to the network is the MAP (Markovian Arrival Process). At the first station, service requirements are of phase type whereas service times at the second station are arbitrarily distributed. The focus is on the embedded process at departures. The essential tool in our analysis is the general theory on Markov renewal processes of M/G/1-type.     

A second-order fluid queue with subordinator input and Markov-modulated linear output

We consider an infinite capacity second-order fluid queue with subordinator input and Markovmodulated linear release rate. The fluid queue level is described by a generalized Langevin stochastic differential equation （SDE）. Applying infinitesimal generator, we obtain the stationary distribution that satisfies an integro-differential equation. We derive the solution of the SDE and study the transient level＇s convergence in distribution. When the coefficients of the SDE are constants, we deduce the system transient property.     

Threshold control by a single-server retrial queue with batch arrivals and group services

A controlled single-server retrial queueing system is investigated. Customers arrive according to batch Markovian arrival process. The system has several operation modes which are controlled by means of a threshold strategy. The stationary distribution is calculated. Optimization problem is considered and a numerical example is presented.     

Analysis of a single-server queue interacting with a fluid reservoir

We consider a single-server queueing system with Poisson arrivals in which the speed of the server depends on whether an associated fluid reservoir is empty or not. Conversely, the rate of change of the content of the reservoir is determined by the state of the queueing system, since the reservoir fills during idle periods and depletes during busy periods of the server. Our interest focuses on the stationary joint distribution of the number of customers in the system and the content of the fluid reservoir, from which various performance measures such as the steady-state sojourn time distribution of a customer may be obtained. We study two variants of the system. For the first, in which the fluid reservoir is infinitely large, we present an exact analysis. The variant in which the fluid reservoir is finite is analysed approximatively through a discretization technique. The system may serve as a mathematical model for a traffic regulation mechanism - a two-level traffic shaper - at the edge of an ATM network, regulating a very bursty source. We present some numerical results showing the effect of the mechanism. This revised version was published online in June 2006 with corrections to the Cover Date.     

The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process

We consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process (BMAP). The server serves until system emptied and after that server takes a vacation. The server will take a maximum number H of vacations until either he finds at least one customer in the queue or the server has exhaustively taken all the vacations. We obtain queue length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue lengths and mean waiting times, etc. have been obtained. Several other vacation queueing models like, single and multiple vacation model, queues with exceptional first vacation time, etc. can be considered as special cases of our model.     

The BMAP/PH/N retrial queue with Markovian flow of breakdowns

We consider a multi-server retrial queue with the Batch Markovian Arrival Process (BMAP). The servers are identical and independent of each other. The service time distribution of a customer by a server is of the phase (PH) type. If a group of primary calls meets idle servers the primary calls occupy the corresponding number of servers. If the number of idle servers is insufficient the rest of calls go to the orbit of unlimited size and repeat their attempts to get service after exponential amount of time independently of each other. Busy servers are subject to breakdowns and repairs. The common flow of breakdowns is the MAP. An event of this flow causes a failure of any busy server with equal probability. When a server fails the repair period starts immediately. This period has PH type distribution and does not depend on the repair time of other broken-down servers and the service time of customers occupying the working servers. A customer whose service was interrupted goes to the orbit with some probability and leaves the system with the supplementary probability. We derive the ergodicity condition and calculate the stationary distribution and the main performance characteristics of the system. Illustrative numerical examples are presented.     

Performance measures of a multi-layer Markovian fluid model

Our goal is to model the behaviour of the fluid in a buffer with threshold controls with a wide range of behaviour possible at the boundaries. To model this, we consider a class of Markovian fluid flow models with several layers, each with their own parameters, separated by boundaries. The behaviour of the fluid at each boundary is modelled by parameters unique to that boundary. We derive the Laplace-Stieltjes transforms of time-related performance measures of this model. This is illustrated with numerical examples. All results are obtained via techniques within the fluid flow environment, and useful physical interpretations are presented.     

A disaster queue with Markovian arrivals and impatient customers

We consider a single server queueing system in which arrivals occur according to a Markovian arrival process. The system is subject to disastrous failures at which times all customers in the system are lost. Arrivals occurring during the time the system undergoes repair are stored in a buffer of finite capacity. These customers can become impatient after waiting a random amount of time and leave the system. However, these customers do not become impatient once the system becomes operable. When the system is operable, there is no limit on the number of customers who can be admitted. The structure of this queueing model is of GI/M/1-type that has been extensively studied by Neuts and others. The model is analyzed in steady state by exploiting the special nature of this type queueing model. A number of useful performance measures along with some illustrative examples are reported.     

Response time in a tandem queue with blocking, Markovian arrivals and phase-type services

A novel approach for obtaining the response time in a discrete-time tandem-queue with blocking is presented. The approach constructs a Markov chain based on the age of the leading customer in the first queue. We also provide a stability condition and carry out several numerical examples.     

马氏环境中的马氏链与马氏双链

本文研究了马氏环境中的马氏链,利用马氏双链的性质,得到了马氏环境中的马氏链回返于小柱集上的概率的若干估计式.     

The erlangization method for Markovian fluid flows

For applications of stochastic fluid models, such as those related to wildfire spread and containment, one wants a fast method to compute time dependent probabilities. Erlangization is an approximation method that replaces various distributions at a time t by the corresponding ones at a random time with Erlang distribution having mean t. Here, we develop an efficient version of that algorithm for various first passage time distributions of a fluid flow, exploiting recent results on fluid flows, probabilistic underpinnings, and some special structures. Some connections with a familiar Laplace transform inversion algorithm due to Jagerman are also noted up front.     

On a synchronization queue with two finite buffers

In this paper, we consider a synchronization queue (or synchronization node) consisting of two buffers with finite capacities. One stream of tokens arriving at the system forms a Poisson process and the other forms a PH-renewal process. The tokens are held in the buffers until one is available from each flow, and then a group-token is instantaneously released as a synchronized departure. We show that the output stream of a synchronization queue is a Markov renewal process, and that the time between consecutive departures has a phase type distribution. Thus, we obtain the throughput of this synchronization queue and the loss probabilities of each type of tokens. Moreover, we consider an extended synchronization model with two Poisson streams where a departing group-token consists of several tokens in each buffer. This revised version was published online in June 2006 with corrections to the Cover Date.     

The MAP/PH/N retrial queue in a random environment

We consider the MAP/PH/N retrial queue with a finite number of sources operating in a finite state Markovian random environment. Two different types of multi-dimensional Markov chains are investigated describing the behavior of the system based on state space arrangements. The special features of the two formulations are discussed. The algorithms for calculating the stationary state probabilities are elaborated, based on which the main performance measures are obtained, and numerical examples are presented as well.     

Optimal multithreshold control for a BMAP/G/1 queue with N service modes

This paper deals with the problem of the optimal service rate control in the system with BMAP (Batch Markovian Arrival Process) arrival stream. An algorithm for the computation of the embedded stationary queue length distribution is developed. The procedure for the cost criteria calculation is elaborated for any fixed parameters of the multithreshold control policy. This revised version was published online in June 2006 with corrections to the Cover Date.     

Age-based Markovian approximation of the G/M/1 queue

We extend the approach of Koole et al. (2012) [15] and Legros et al. (2018) [20] for the G/M/1 queue. The idea is to provide a Markovian approximation where a state represents the oldest customer's wait. This modeling is made possible by creating states with negative wait, representing an estimate of the time at which a new customer would arrive when the system is empty. We apply this method for performance evaluation and routing optimization. Finally, we further extend the model to the G/M/1+G queue.