A fluid queue driven by a Markovian queue

We consider an infinite buffer fluid queue receiving its input from the output of a Markovian queue with finite or infinite waiting room. The input is characterized by a Markov modulated rate process. We derive a new approach for the computation of the stationary buffer content. This approach leads to a numerically stable algorithm for which the precision of the result can be given in advance. This revised version was published online in June 2006 with corrections to the Cover Date.     

对一类等待空间有限的抢占优先权排队的分析

讨论M/M/1抢占优先权排队模型, 且假设低优先权顾客的等待空间有限. 该模型可以用有限位相拟生灭过程来描述. 由矩阵解析方法, 对该拟生灭过程进行了分析, 并得到排队模型平稳队长的计算公式, 最后还用数值 结果说明了方法的有效性.     

The GI/M/1 queue with exponential vacations

In this paper, we give a detailed analysis of the GI/M/1 queue with exhaustive service and multiple exponential vacation. We express the transition matrix of the imbedded Markov chain as a block-Jacobi form and give a matrix-geometric solution. The probability distribution of the queue length at arrival epochs is derived and is shown to decompose into the distribution of the sum of two independent random variables. In addition, we discuss the limiting behavior of the continuous time queue length processes and obtain the probability distributions for the waiting time and the busy period.     

On multiserver feedback retrial queue with finite buffer

This paper deals with a multiserver feedback retrial queueing system with finite waiting position and constant retrial rate. This system is analyzed as a quasi-birth-and-death (QBD) process and the necessary and sufficient condition for stability of the system is investigated. Some important system performance measures are obtained using matrix geometric method. The effect of various parameters on the system performance measures are illustrated numerically. Finally, the algorithmic development of the full busy period for the model under consideration is discussed.     

On the loss process in a batch arrival queue

In this paper, we investigate the loss process in a finite-buffer queue with batch arrivals and total rejection discipline. In such a model, if the buffer has insufficient capacity to accept all the customers included in an arriving batch, the whole batch is blocked and lost. This scheme is especially useful in performance evaluation of buffering processes in IP (internet protocol) networks. The main result of this paper is a closed-form formula for the joint distribution of the length of the first lost series of batches and the time of the first loss. Moreover, the limiting distribution (as the buffer size grows to infinity) is shown.     

A queue with instantaneous tri-route decision process

This paper focuses on the study of several random processes associated with M/G1 queue with instantaneous tri-route decision process. The stationary distribution of the output process is derived. Some particular queues with feedback and without feedback are also analysed. Some operating characteristics are studied for this queue. Optimum service rate is obtained. A numerical study is carried out to test the feasibility of the queueing model.     

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.     

Occupation times for the finite buffer fluid queue with phase-type ON-times

In this short communication we study a fluid queue with a finite buffer. The performance measure we are interested in is the occupation time over a finite time period, i.e., the fraction of time the workload process is below some fixed target level. Using an alternating renewal sequence, we determine the double transform of the occupation time; the occupation time for the finite buffer M/G/1 queue with phase-type jumps follows as a limiting case.     

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.     

Kitting process in a stochastic assembly system

In small-lot, multi-product, multi-level assembly systems, kitting (or accumulating) components required for assembly plays a crucial role in determining system performance, especially when the system operates in a stochastic environment. This paper analyzes the kitting process of a stochastic assembly system, treating it as an assembly-like queue. If components arrive according to Poisson processes, we show that the output stream departing the kitting operation is a Markov renewal process. The distribution of time between kit completions is also derived. Under the special condition of identical component arrival streams having the same Poisson parameter, we show that the output stream of kits approximates a Poisson process with parameter equal to that of the input stream. This approximately decouples assembly from kitting, allowing the assembly operation to be analyzed separately.     

Decay rate for a PH/M/2 queue with shortest queue discipline

In this paper, we consider a PH/M/2 queue in which each server has its own queue and arriving customers join the shortest queue. For this model, it has been conjectured that the decay rate of the tail probabilities for the shortest queue length in the steady state is equal to the square of the decay rate for the queue length in the corresponding PH/M/2 model with a single queue. We prove this fact in the sense that the tail probabilities are asymptotically geometric when the difference of the queue sizes and the arrival phase are fixed. Our proof is based on the matrix analytic approach pioneered by Neuts and recent results on the decay rates. AMS subject classifications: 60K25 · 60K20 · 60F10 · 90B22     

On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds

We develop tight bounds and a fast parallel algorithm to compute the Markov renewal kernel. Knowledge of the kernel allows us to solve Markov renewal equations numerically to study non-steady state behavior in a finite state Markov renewal process. Computational error and numerical stability for computing the bounds in parallel are discussed using well-known results from numerical analysis. We use our algorithm and computed bounds to study the expected number of departures as a function of time for a two node overflow queueing network.     

A difference equation approach for analysing a batch service queue with the batch renewal arrival process

We consider an infinite buffer single server queue wherein customers arrive according to the batch renewal arrival process and are served in batches following the random serving capacity rule. The service-batch times follow exponential distribution. This model has been studied in the past using the embedded Markov chain technique and probability generating function. In this paper we provide an alternative yet simple methodology to carry out the whole analysis which is based on the supplementary variable technique and the theory of difference equations. The procedure used here is simple in the sense that it does not require the complicated task of constructing the transition probability matrix. We obtain explicit expressions of the steady-state system-content distribution at pre-arrival and arbitrary epochs in terms of roots of the associated characteristic equation. We also present few numerical results in order to illustrate the computational procedure.     

On performance comparison of MR/GI/1 queues

In this paper we are interested in the effect that dependencies in the arrival process to a queue have on queueing properties such as mean queue length and mean waiting time. We start with a review of the well known relations used to compare random variables and random vectors, e.g., stochastic orderings, stochastic increasing convexity, and strong stochastic increasing concavity. These relations and others are used to compare interarrival times in Markov renewal processes first in the case where the interarrival time distributions depend only on the current state in the underlying Markov chain and then in the general case where these interarrivai times depend on both the current state and the next state in that chain. These results are used to study a problem previously considered by Patuwo et al. [14].Then, in order to keep the marginal distributions of the interarrivai times constant, we build a particular transition matrix for the underlying Markov chain depending on a single parameter,p. This Markov renewal process is used in the Patuwo et al. [14] problem so as to investigate the behavior of the mean queue length and mean waiting time on a correlation measure depending only onp. As constructed, the interarrival time distributions do not depend onp so that the effects we find depend only on correlation in the arrival process.As a result of this latter construction, we find that the mean queue length is always larger in the case where correlations are non-zero than they are in the more usual case of renewal arrivals (i.e., where the correlations are zero). The implications of our results are clear.     

Departures from queues with changeover times

We consider the M/M ^ij /1 queue as a model of queues with changeover times, i.e., the service is exponential with parameter _ij depending on the previous job type (i) and the current job type (j). It is shown that the departure process is renewal and Poisson iff _ij = (constant). In this case, types of departures are dependent renewal processes. Crosscovariance and crosscorrelations are given.     

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.     

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.     

The discrete-time MAP/PH/1 queue with multiple working vacations

We consider a discrete-time single-server queueing model where arrivals are governed by a discrete Markovian arrival process (DMAP), which captures both burstiness and correlation in the interarrival times, and the service times and the vacation duration times are assumed to have a general phase-type distributions. The vacation policy is that of a working vacation policy where the server serves the customers at a lower rate during the vacation period as compared to the rate during the normal busy period. Various performance measures of this queueing system like the stationary queue length distribution, waiting time distribution and the distribution of regular busy period are derived. Through numerical experiments, certain insights are presented based on a comparison of the considered model with an equivalent model with independent arrivals, and the effect of the parameters on the performance measures of this model are analyzed.     

Performance of the（BMAP1, BMAP2 ）/（PH1, PH2 ）/N Retrial Queueing System with Finite Buffer

This paper consider the （BMAP1, BMAP2）/（PH1, PH2）/N retrial queue with finite-position buffer. The behavior of the system is described in terms of continuous time multi-dimensional Markov chain. Arriving type I calls find all servers busy and join the buffer, if the positions of the buffer are insufficient, they can go to orbit. Arriving type II calls find all servers busy and join the orbit directly. Each server can provide two types heterogeneous services with Phase-type （PH） time distribution to every arriving call （including types I and II calls）, arriving calls have an option to choose either type of services. The model is quite general enough to cover most of the systems in communication networks. We derive the ergodicity condition, the stationary distribution and the main performance characteristics of the system. The effects of various parameters on the system performance measures are illustrated numerically.