Transient behaviour of a double-channel Markovian queue with limited waiting space

For a double channel Markovian queue with finite waiting space the difference equations satisfied by the Laplace transforms of the state probabilities at finite time are solved and the state probabilities are obtained in a simple closed form which can be easily used to find the important parameters of the system.     

A simple method for computing state probabilities of theM/G/1 andGI/M/1 finite waiting space queues

This paper presents a simple method for computing steady state probabilities at arbitrary and departure epochs of theM/G/1/K queue. The method is recursive and works efficiently for all service time distributions. The only input required for exact evaluation of state probabilities is the Laplace transform of the probability density function of service time. Results for theGI/M/1/K –1 queue have also been obtained from those ofM/G/1/K queue.     

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.     

Individual blocking probabilities in the loss system G1 + ... + GN/M/1/0

Explicit formulas for the individual call loss probabilities are derived which arise when a finite collection of independent general stationary traffic streams with exponentially distributed service times are offered simultaneously to a single server. The formulas show a modified insensitivity property of the given model.     

On M/M/1 queues with a smart machine

This paper discusses a class of M/M/1 queueing models in which the service time of a customer depends on the number of customers served in the current busy period. It is particularly suited for applications in which the server has kind of learning ability and warms up gradually. We present a simple and computationally tractable scheme which recursively determines the stationary probabilities of the queue length. Other performance measures such as the Laplace transform of the busy period are also obtained. For the firstN exceptional services model which can be considered as a special case of our model, we derive a closed-formula for the generating function of the stationary queue length distribution. Numerical examples are also provided.     

Analysis of the finite source MAP/PH/N retrial G-queue operating in a random environment

Finite source retrial G-queues are good mathematical models of communication systems and networks, so their investigation is important for theory and applications. In this paper, we analyze the MAP/PH/N retrial queue with finite number of sources and MAP arrivals of negative customers operating in a finite state Markovian random environment. The arrival of a negative customer with equal probability goes to any busy server to remove the customer being in service. The multi-dimensional Markov chain describing the behavior of the system is investigated. The algorithms for calculating the stationary state probabilities are elaborated. Main performance measures are obtained. Illustrative numerical examples are presented.     

MAP 1, MAP2 /M/c retrial queue with guard channels and its application to cellular networks

In this paper, we investigate the impact of retrial phenomenon on loss probabilities and compare loss probabilities of several channel allocation schemes giving higher priority to hand-off calls in the cellular mobile wireless network. In general, two channel allocation schemes giving higher priority to hand-off calls are known; one is the scheme with the guard channels for hand-off calls and the other is the scheme with the priority queue for hand-off calls. For mathematical unified model for both schemes, we consider theMAP ₁,MAP ₂ /M/c/b, ∞ retrial queue with infinite retrial group, geometric loss, guard channels and finite priority queue for hand-off class. We approximate the joint distribution of two queue lengths by Neuts' method and also obtain waiting time distribution for hand-off calls. From these results, we obtain the loss probabilities, the mean waiting time and the mean queue lengths. We give numerical examples to show the impact of the repeated attempt and to compare loss probabilities of channel allocation schemes.     

On Random Sampling Without Replacement from a Finite Population

We consider the three progressively more general sampling schemes without replacement from a finite population: simple random sampling without replacement, Midzuno sampling and successive sampling. We (i) obtain a lower bound on the expected sample coverage of a successive sample, (ii) show that the vector of first order inclusion probabilities divided by the sample size is majorized by the vector of selection probabilities of a successive sample, and (iii) partially order the vectors of first order inclusion probabilities for the three sampling schemes by majorization. We also show that the probability of an ordered successive sample enjoys the arrangement increasing property and for sample size two the expected sample coverage of a successive sample is Schur convex in its selection probabilities. We also study the spacings of a simple random sample from a linearly ordered finite population and characterize in several ways a simple random sample.     

两个修理工的M/M/2可修排队系统

该文研究两个修理工的M/M/2可修排队系统, 系统有两个相同的服务台, 服务台忙时与闲时故障率不同. 文中给出系统的稳态状态概率, 系统的稳态可用度及系统的稳态平均队长, 并给出系统稳态概率存在的条件.     

Numerical analysis for bulk-arrival queueing systems: Root-finding and steady-state probabilities inGI r /M/1 queues

Queueing theorists have presented, as solutions to many queueing models, probability generating functions in which state probabilities are expressed as functions of the roots of characteristic equations, evaluation of the roots in particular cases being left to the reader. Many users have complained that such solutions are inadequate. Some queueing theorists, in particular Neuts [6], rather than use Rouché's theorem to count roots and an equation-solver to find them, have developed new algorithms to solve queueing problems numerically, without explicit calculation of roots. Powell [7] has shown that in many bulk service queues arising in transportation models, characteristic equations can be solved and state probabilities can be found without serious difficulty, even when the number of roots to be found is large. We have slightly modified Powell's method, and have extended his work to cover a number of bulk-service queues discussed by Chaudhry et al. [1] and a number of bulk-arrival queues discussed in the present paper.     

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     

AnM/M/c queue with impatient customers

This paper considers theM/M/c queue in which a customer leaves when its service has not begun within a fixed interval after its arrival. The loss probability can be expressed in a simple formula involving the waiting time probabilities in the standardM/M/c queue. The purpose of this paper is to give a probabilistic derivation of this formula and to outline a possible use of this general formula in theM/M/c retrial queue with impatient customers. This research was supported by the INTAS 96-0828 research project and was presented at the First International Workshop on Retrial Queues, Universidad Complutense de Madrid, Madrid, September 22–24, 1998.     

M/C2/c/K queuing model and optimization for a geriatric care center

Queuing systems with finite buffers are reasonable models for many manufacturing, telecommunication, and healthcare systems. Although some approximations exist, the exact analysis of multi‐server and finite‐buffer queues with general service time distribution is unknown. However, the phase‐type assumption for service time is a frequently used approach. Because the Cox distribution, a kind of phase‐type distribution, provides a good representation of data with great variability, it has a vast area of application in modeling service times. The research focus is twofold. First, a theoretical structure of a multi‐server and finite‐buffer queuing system in which the service time is modeled by the two‐phase Cox distribution is studied. It is focused on finding an efficient solution to the stationary probabilities using the matrix‐geometric method. It is shown that the stationary probability vector can be obtained with the matrix‐geometric method by using level‐dependent rate matrices, and the mean queue length is computed. Second, an empirical analysis of the model is presented. The proposed methodology is applied in a case study concerning the geriatric patients. Some numerical calculations and optimizations are performed by using geriatric data. Copyright © 2015 John Wiley & Sons, Ltd.     

M/M/∞ queues in semi-Markovian random environment

In this paper we investigate an M/M/∞ queue whose parameters depend on an external random environment that we assume to be a semi-Markovian process with finite state space. For this model we show a recursive formula that allows to compute all the factorial moments for the number of customers in the system in steady state. The used technique is based on the calculation of the raw moments of the measure of a bidimensional random set. Finally the case when the random environment has only two states is deeper analyzed. We obtain an explicit formula to compute the above mentioned factorial moments when at least one of the two states has sojourn time exponentially distributed. Part of this research took place while the author was still post-doc at EURANDOM, Eindhoven, The Netherlands. The work was supported by the Spanish Ministry of Education and Science by the Grant MTM2007-63140.     

A finite model of mobility

The Markov chains with stationary transition probabilities have not proved satisfactory as a model of human mobility. A modification of this simple model is the ‘duration specific’ chain incorporating the axiom of cumulative inertia: the longer a person has been in a state the less likely he is to leave it. Such a process is a Markov chain with a denumerably infinite number of states, specifying both location and duration of time in the location. Here we suggest that a finite upper bound be placed on duration, thus making the process into a finite state Markov chain. Analytic representations of the equilibrium distribution of the process are obtained under two conditions: (a) the maximum duration is an absorbing state, for all locations; and (b) the maximum duration is non‐absorbing. In the former case the chain is absorbing, in the latter it is regular.     

Bayesian estimation of finite time ruin probabilities

In this paper, we consider Bayesian inference and estimation of finite time ruin probabilities for the Sparre Andersen risk model. The dense family of Coxian distributions is considered for the approximation of both the inter‐claim time and claim size distributions. We illustrate that the Coxian model can be well fitted to real, long‐tailed claims data and that this compares well with the generalized Pareto model. The main advantage of using the Coxian model for inter‐claim times and claim sizes is that it is possible to compute finite time ruin probabilities making use of recent results from queueing theory. In practice, finite time ruin probabilities are much more useful than infinite time ruin probabilities as insurance companies are usually interested in predictions for short periods of future time and not just in the limit. We show how to obtain predictive distributions of these finite time ruin probabilities, which are more informative than simple point estimations and take account of model and parameter uncertainty. We illustrate the procedure with simulated data and the well‐known Danish fire loss data set. Copyright © 2009 John Wiley & Sons, Ltd.     

Transition probabilities for Markov chains having fuzzy states

We consider a Markov Chain in which the states are fuzzy subsets defined on some finite state space. Building on the relationship between set-valued Markov chains to the Dempster-Shafer combination rule, we construct a procedure for finding transition probabilities from one fuzzy state to another. This construction involves Dempster-Shafer type mass functions having fuzzy focal elements. It also involves a measure of the degree to which two fuzzy sets are equal. We also show how to find approximate transition probabilities from a fuzzy state to a crisp state in the original state space     

Analysis of the GIX/M/c model

In this paper, the GI^X/M/c queueing model is analyzed. An explicit expression of the generating function of equilibrium probabilities of customer numbers in the system for the model is derived. Based on the generating function, it is proved that the equilibrium probabilities are given by a linear combination of some geometric terms. Due to this result, other interesting measures are also considered without difficulty. Examples and numerical results are given.     

Steady-state analysis of M/M/c/c-type retrial queueing systems with constant retrial rate

The paper deals with a research of bivariate Markov process \(\{X(t), t\ge 0\}\) whose state space is a lattice semistrip \(S(X)=\{0,1,{\ldots },c\} \times Z_{+}\). The process \(\{X(t), t\ge 0\}\) describes the service policy of a multi-server retrial queue in which the rate of repeated flow does not depend on the number of sources of retrial calls. In this class of queues, a vector–matrix representation of steady-state distribution was obtained. This representation allows to write down the stationary probabilities through the model parameters in closed form and to propose the closed formulas of its main performance measures. The investigative techniques use an approximation of the initial model by means of the truncated one and the direct passage to the limit.