A BMAP/SM/1 Queueing System with Hybrid Operation Mechanism

A single-server queueing system with negative customers is considered in this paper. One positive customer will be removed from the head of the queue if any negative customer is present. The distribution of the interarrival time for the positive customer is assumed to have a rate that tends to a constant as time t tends to infinity. An alternative approach will be proposed to derive a set of equations to find the stationary probabilities. The stationary probabilities will then be used to find the stationary queue length distribution. Numerical examples will be presented and compared to the results found using the analytical method and simulation procedure. The advantage of using the proposed alternative approach will be discussed in this paper.

Many researchers have studied variants of queueing systems with vacations. Most of them have dealt with M/G/1 systems and have explicitly analyzed some of their performance measures, such as queue length, waiting time, and so on. Recently, studies on queueing systems whose arrival processes are not Poissonian have appeared. We consider a single server queueing system with multiple vacations and E-limited service discipline, where messages arrive to the system according to a switched Poisson process. First, we consider the joint probability density functions of the queue length and the elapsed service time or the elapsed vacation time. We derive the equations for these pdf's, which include a finite number of unknown values. Using Rouche's theorem, we determine the values from boundary conditions. Finally, we derive the transform of the stationary queue length distribution explicitly.

In this paper, we consider the discrete-time Geo/G/1 retrial queue with preemptive resume and Bernoulli feedback. This model unifies the FCFS and LCFS preemptive resume disciplines. We analyze the Markov chain underlying the regarded queueing system and present some performance measures of the queueing system in steady-state. Besides, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of preemptive resume and feedback on some crucial performance characteristics of the system.

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

Рассмотрена система с повторными вызовами (RQ-система), на вход которой поступает простейший поток с заданной интенсивностью. Если в момент обращения заявки прибор занят, то происходит вытеснение заявки, стоящей на приборе. Заявка, не успевшая успешно обслужиться, переходит на орбиту, чтобы после случайной экспоненциальной задержки вновь обратиться к прибору для обслуживания. Дообслуживание заявки подразумевает, что в момент обращения с орбиты к прибору заявка встает на ту фазу обслуживания, с которой была прервана. Показано, что асимптотическая характеристическая функция числа заявок на орбите и состояний прибора сходится к трехмерному гауссовскому распределению. Для данного распределения получен вектор средних значений и матрица ковариаций. Найдено стационарное распределение вероятностей состояний прибора.

We consider queueing systems in which the service rate is the controlled variable. The cost depends on the queue length and selected rate. The objective is to choose the service rate dynamically, based on the state of the system so as to minimize the average cost over an infinite horizon. Six classes, either known in the literature or new, are studied in detail: queueing systems with vacations, single-server queueing systems with and without switching costs, and tandem queueing systems with and without service costs. A novel approach is presented here using fuzzy control to solve these problems. Simulation shows that the approach is efficient and promising, especially in cases where analytical solutions do not exist.

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.

A single-server queueing system with stationary Poisson arrival process, finite buffer and impatient customers as a model of TCP Short Transfer is investigated. Service time of a customer by a server has an exponential distribution. If the server is busy and the buffer is full at a customer arrival epoch, the customer may leave the system forever or move to the orbit. Customers staying in the buffer exhibit signs of impatience: they can leave the buffer lying out of the service. Patience time of a customer has an exponential distribution. When this time expires the customer either leaves the system permanently or goes to the orbit of infinite size. Customers staying in the orbit repeat their attempts to get the service later on in a random amount of time. This time is exponentially distributed with the rate depending or independent of the current number of customers in the orbit.Behavior of the system under study is described by two-dimensional asymptotically quasi-Toeplitz Markov chain. Stability conditions and the algorithms for calculating the stationary state distribution of the chain are obtained. Main performance measures of the system are calculated.KeywordsRetrial queueing systemFinite bufferImpatient customersStationary performance measures

We consider a single-server queueing system with a doubly stochastic Poisson arrival flow under heavy traffic conditions. We prove the convergence of the limiting stationary or periodic distribution to the exponential distribution. In a scheme of series, we consider the C-convergence of the waiting time process to a diffusion process with constant coefficients and reflection at the zero boundary. Examples of computation of the diffusion coefficient in terms of characteristics of the arrival flow and service time are given.

This paper considers retrial queueing (RQ) systems with Poisson (elementary) andMarkovmodulated Poisson processes. The study is performed by the method of asymptotic cumulants using the theory of vector characteristic functions and the matrix form of equations, which makes it possible to obtain asymptotic results for the whole class of models. The analysis of a single-line RQ system performed revealed the range of applicability of asymptotic results for prelimit situations.

We consider a single-line queueing system (QS) with Poisson input flow of varying intensity, which depends on the number of demands in the system. The job size (length) distribution for a demand depends on the number of demands in the system at the arrival moment. The service rate also depends on the number of calls in the QS. If the job size for a new arrival is larger than the remaining job size for the currently processed demand, then the arrival is put at the beginning of the queue with a certain probability, which depends on the total number of demands in the system. Otherwise, it occupies the server and displaces the currently processed demand, which is put at the beginning of the queue. The probability distribution of stationary states of the QS is found and necessary and sufficient conditions for this distribution to be invariant with respect to the job size distribution with a fixed mean are obtained.

This paper presents a unifying algorithmic analysis for a general class of single server queueing systems with a state dependent Markovian input process and a phase-type service time distribution including single server queues with random and quasirandom input. Using regenerative analysis we develop numerically stable and efficient recursion schemes to compute the state probabilities. The computation of the waiting times is based on the state probabilities.

Sleep modes are important for user equipment to reduce power consumption or to reduce battery reloading. To keep the user equipment connected, it has to be activated for uplink or downlink data transfer which causes overhead and additional delay. Wireless or mobile equipment has to be monitored repeatedly for location updates. These classes of problems can be modeled by queuing systems with modified busy and idle periods. In this paper, we consider a generalized infinite-buffer, single-server queuing systems with activation overhead and under various types of sleep modes during inactive periods. For the special assumption of Markovian arrival processes of data units and generally distributed activation times and vacation (sleep) periods, the models can be analyzed exactly by a mean value analysis method which is based on regenerative stochastic processes, renewal theory and classical queuing theorems. The method allows also for dynamic sleep periods which are dependent on each other as in the case of the standard IEEE 802.16m for wireless equipment. Explicit results are derived for average values of data unit delays, resource utilization and power- saving margins dependent on given parameters for activation times and wakeup signaling frequencies. The models include the special cases of M/G/1 queuing systems with a modified service time of the first arrival initiating a busy period and M/G/1 queuing systems with vacation periods, respectively, which are well known from literature providing general solutions for the state and delay distributions. Finally, all models are extended to the exact analysis of Batch Poisson arrival processes.

Consider a single-server queueing system with a Poisson arrival process at rate λ and positive service requirements independently distributed with common distribution functionB(z) and finite expectationwhere βλ < 1, i.e. anM/G/1 system. When the queue discipline is first come first served, or last come first served without pre-emption, the stationary departure process is Poisson if and only ifG=M(i.e.B(z) = 1 − exp (−z/β)); see (8), (4) and (2). In this paper it is shown that when the queue discipline is last come first served with pre-emption the stationary departure process is Poisson whatever the form ofB(z). The method used is adapted from the approach of Takács (10) and Shanbhag and Tambouratzis (9).

We find an equilibrium in a single-server queueing system with retrials and strategic timing of the customers. We consider a set of customers, each of which must decide when to arrive to a queueing system during a fixed period of time. In this system, after completion of service, the server seeks a customer blocked in a virtual orbit (orbital customer) to be served next, unless a new customer captures the server. We develop, in detail, a setting with two and three customers in the set, and formulate and discuss the problem for the general case with an arbitrary number of customers. The numerical examples for the system with two and three customers included as well.