Mean Buffer Contents in Discrete-Time Single-Server Queues with Heterogeneous Sources

We consider discrete-time single-server queues fed by independent, heterogeneous sources with geometrically distributed idle periods. While being active, each source generates some cells depending on the state of the underlying Markov chain. We first derive a general and explicit formula for the mean buffer contents in steady state when the underlying Markov chain of each source has finite states. Next we show the applicability of the general formula to queues fed by independent sources with infinite-state underlying Markov chains and discrete phase-type active periods. We then provide explicit formulas for the mean buffer contents in queues with Markovian autoregressive sources and greedy sources. Further we study two limiting cases in general settings, one is that the lengths of active periods of each source are governed by an infinite-state absorbing Markov chain, and the other is the model obtained by the limit such that the number of sources goes to infinity under an appropriate normalizing condition. As you will see, the latter limit leads to a queue with (generalized) M/G/∞ input sources. We provide sufficient conditions under which the general formula is applicable to these limiting cases.AMS subject classification: 60K25, 60K37, 60J10This revised version was published online in June 2005 with corrected coverdate     

Queue Length Distribution in a FIFO Single-Server Queue with Multiple Arrival Streams Having Different Service Time Distributions

This paper considers the queue length distribution in a class of FIFO single-server queues with (possibly correlated) multiple arrival streams, where the service time distribution of customers may be different for different streams. It is widely recognized that the queue length distribution in a FIFO queue with multiple non-Poissonian arrival streams having different service time distributions is very hard to analyze, since we have to keep track of the complete order of customers in the queue to describe the queue length dynamics. In this paper, we provide an alternative way to solve the problem for a class of such queues, where arrival streams are governed by a finite-state Markov chain. We characterize the joint probability generating function of the stationary queue length distribution, by considering the joint distribution of the number of customers arriving from each stream during the stationary attained waiting time. Further we provide recursion formulas to compute the stationary joint queue length distribution and the stationary distribution representing from which stream each customer in the queue arrived.     

Joint Distributions for Interacting Fluid Queues

Motivated by recent traffic control models in ATM systems, we analyse three closely related systems of fluid queues, each consisting of two consecutive reservoirs, in which the first reservoir is fed by a two-state (on and off) Markov source. The first system is an ordinary two-node fluid tandem queue. Hence the output of the first reservoir forms the input to the second one. The second system is dual to the first one, in the sense that the second reservoir accumulates fluid when the first reservoir is empty, and releases fluid otherwise. In these models both reservoirs have infinite capacities. The third model is similar to the second one, however the second reservoir is now finite. Furthermore, a feedback mechanism is active, such that the rates at which the first reservoir fills or depletes depend on the state (empty or nonempty) of the second reservoir.The models are analysed by means of Markov processes and regenerative processes in combination with truncation, level crossing and other techniques. The extensive calculations were facilitated by the use of computer algebra. This approach leads to closed-form solutions to the steady-state joint distribution of the content of the two reservoirs in each of the models.     

An Arrival Time Approach to M/G/1-type Queues with Generalized Vacations

We propose a simple way, called the arrival time approach, of finding the queue length distributions for M/G/1-type queues with generalized server vacations. The proposed approach serves as a useful alternative to understanding complicated queueing processes such as priority queues with server vacations and MAP/G/1 queues with server vacations.     

Analysis of the asymmetric shortest queue problem

In this paper we study a system consisting of two parallel servers withdifferent service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. On arrival a job joins the shortest queue and in case both queues have equal lengths, he joins the first queue with probabilityq and the second one with probability 1 −q, whereq is an arbitrary number between 0 and 1. In a previous paper we showed for the symmetric problem, that is for equal service rates andq = 1/2, that the equilibrium distribution of the lengths of the two queues can be exactly represented by an infinite sum of product form solutions by using an elementary compensation procedure. The main purpose of the present paper is to prove a similar product form result for the asymmetric problem by using a generalization of the compensation procedure. Furthermore, it is shown that the product form representation leads to a numerically efficient algorithm. Essentially, the method exploits the convergence properties of the series of product forms. Because of the fast convergence an efficient method is obtained with upper and lower bounds for the exact solution. For states further away from the origin the convergence is faster. This aspect is also exploited in the paper.     

带比例关系的实带状矩阵特征值反问题

研究了通过矩阵A的顺序主子矩阵A_((k))=(aij)_(i,j=1)^(n-k+1)的特征值{λ_i(n-k+1)的特征值{λ_i^((k)))}_(i=1)((k)))}_(i=1)^(n-k+1)k=1,2,…,r+1来构造一个带比例关系的实带状矩阵的特征值反问题.对当特征值{λ_i(n-k+1)k=1,2,…,r+1来构造一个带比例关系的实带状矩阵的特征值反问题.对当特征值{λ_i^((k))}_(i=1)((k))}_(i=1)^(n-k+1)中有多重特征值出现时,应当如何来构造这类矩阵进行了讨论,并给出了问题的具体算法及数值例子.     

Factorization and Stochastic Decomposition Properties in Bulk Queues with Generalized Vacations

This paper considers a class of stationary batch-arrival, bulk-service queues with generalized vacations. The system consists of a single server and a waiting room of infinite capacity. Arrivals of customers follow a batch Markovian arrival process. The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in groups of fixed size B. For this class of queues, we show that the vector probability generating function of the stationary queue length distribution is factored into two terms, one of which is the vector probability generating function of the conditional queue length distribution given that the server is on vacation. The special case of batch Poisson arrivals is carefully examined, and a new stochastic decomposition formula is derived for the stationary queue length distribution.AMS subject classification: 60K25, 90B22, 60K37     

A Communication Multiplexer Problem: Two Alternating Queues with Dependent Randomly-Timed Gated Regime

Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling instants and at an arbitrary moment, mean busy period duration and mean cycle time length, expected number of messages transmitted during a busy period and mean waiting times. Finally, we present graphs of numerical results comparing the mean waiting times in the two queues as functions of the relative loads, showing the effect of the RTG regime.     

Analytic Computation Schemes for the Discrete-Time Bulk Service Queue

In commonly used root-finding approaches for the discrete-time bulk service queue, the stationary queue length distribution follows from the roots inside or outside the unit circle of a characteristic equation. We present analytic representations of these roots in the form of sample values of periodic functions with analytically given Fourier series coefficients, making these approaches more transparent and explicit. The resulting computational scheme is easy to implement and numerically stable. We also discuss a method to determine the roots by applying successive substitutions to a fixed point equation. We outline under which conditions this method works, and compare these conditions with those needed for the Fourier series representation. Finally, we present a solution for the stationary queue length distribution that does not depend on roots. This solution is explicit and well-suited for determining tail probabilities up to a high accuracy, as demonstrated by some numerical examples.AMS subject classification: 42B05, 60K25, 68M20     

同输入M/M/∞排队群的联合队长过程

本提出一个偏微分方程方法，用这一方法研究同输入M/M／∞排队群中的联合队长分布。在任意初始条件下，给出了瞬时联合队长分布的多元母函数，也讨论了稳态队长的联合分布及各排队系统之间的相关性。     

Numerical Computation and Tail Asymptotic for Stationary Indices of a Discrete-Time Vacation Queue

In this paper we study a Geo/Geo/1 queue with T-IPH vacations, where T-IPH denotes the discrete-time phase type distribution defined on a birth and death process with countably many states. Both the multiple and single vacation strategies are considered. For each case, based on the system of stationary equations and using complex analysis method, we firstly give the probability generating functions (PGFs) of stationary distributions for queue length and sojourn time. Moreover, by analysis the PGFs, recursive and asymptotic formulas for additional queue length and additional delay are also given. Finally, we further give some numerical examples to show the effectiveness of the method.     

Palm calculus for a process with a stationary random measure and its applications to fluid queues

We consider a process associated with a stationary random measure, which may have infinitely many jumps in a finite interval. Such a process is a generalization of a process with a stationary embedded point process, and is applicable to fluid queues. Here, fluid queue means that customers are modeled as a continuous flow. Such models naturally arise in the study of high speed digital communication networks. We first derive the rate conservation law (RCL) for them, and then introduce a process indexed by the level of the accumulated input. This indexed process can be viewed as a continuous version of a customer characteristic of an ordinary queue, e.g., of the sojourn time. It is shown that the indexed process is stationary under a certain kind of Palm probability measure, called detailed Palm. By using this result, we consider the sojourn time processes in fluid queues. We derive the continuous version of Little's formula in our framework. We give a distributional relationship between the buffer content and the sojourn time in a fluid queue with a constant release rate.     

A Law of the Iterated Logarithm for Extreme Queue Length in Multiphase Queues

The object of this research in queueing theory is the Law of the Iterated Logarithm (LIL) under the conditions of heavy traffic in Multiphase Queueing Systems (MQS). In this paper, the LIL is proved for extreme values of important probabilistic characteristics of the MQS investigated as well as maxima and minima of the summary queue length of customers and maxima and minima of the queue length of customers. Also, the paper presents a survey on the works for extreme values in queues and the queues in heavy traffic.      

对M/T-SPH/1排队平稳队长的分析

研究了M/T-SPH/l排队模型,利用拟生灭过程和算子几何解的方法给出了平稳队长分布的概率母函数.在此基础上,指出该分布不是一个离散PH分布,但在一定条件下却是一个几何尾部分布.     

A New Unified Approach for the Simulation of a Wide Class of Directional Distributions

The need for effective simulation methods for directional distributions has grown as they have become components in more sophisticated statistical models. A new acceptance–rejection method is proposed and investigated for the Bingham distribution on the sphere using the angular central Gaussian distribution as an envelope. It is shown that the proposed method has high efficiency and is also straightforward to use. Next, the simulation method is extended to the Fisher and Fisher–Bingham distributions on spheres and related manifolds. Together, these results provide a widely applicable and efficient methodology to simulate many of the standard models in directional data analysis. An R package simdd, available in the online supplementary material, implements these simulation methods.     

A Discrete-Time Geo/G/1 retrial queue with the server subject to starting failures

This paper studies a discrete-time Geo/G/1 retrial queue where the server is subject to starting failures. We analyse the Markov chain underlying the regarded queueing system and present some performance measures of the system in steady-state. Then, we give two stochastic decomposition laws and find a measure of the proximity between the system size distributions of our model and the corresponding model without retrials. We also develop a procedure for calculating the distributions of the orbit and system size as well as the marginal distributions of the orbit size when the server is idle, busy or down. Besides, we prove that the M/G/1 retrial queue with starting failures can be approximated by its discrete-time counterpart. Finally, some numerical examples show the influence of the parameters on several performance characteristics. This work is supported by the DGINV through the project BFM2002-02189.     

A two-queue,one-server model with priority for the longer queue

The queueing model studied consists of one server and two queues. Each queue has its own Poisson arrival stream and service time distribution. After a service completion, the server proceeds with a customer from the longer queue, if the queues are unequal; if the queues are equal, the server chooses with some probability a customer from one of the queues. The model is of practical interest in performance analysis, but also of theoretical interest because the functional equation to be solved has not yet been studied in the queueing literature. A basic analysis of this functional equation is presented. Some numerical results are given to assess the influence of the present service discipline. Some new properties of L.S. transforms of service time distributions are discussed in the appendix.Dr. T. Katayama has formulated the present problem and brought it to the author's attention during his visit in October/November 1984 to the NTT-Electr. Comm. Lab.'s Musashino, Tokyo 180.     

A Unified Approach to Combinatorial Formulas for Schubert Polynomials

Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams for permutations known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are certain operations on rc-graphs, which correspond to the coplactic operations on tableaux, and thus define a crystal graph structure on rc-graphs; a new definition of these operations is presented. One application of these operations is a straightforward, purely combinatorial proof of a recent formula (due to Buch, Kresch, Tamvakis, and Yong), which expresses Schubert polynomials in terms of products of Schur polynomials. In spite of the fact that it refers to many objects and results related to them, the paper is mostly self-contained.     

A single server model for packetwise transmission of messages

We study a discrete-time single-server queue where batches of messages arrive. Each message consists of a geometrically distributed number of packets which do not arrive at the same instant and which require a time unit as service time. We consider the cases of constant spacing and geometrically distributed (random) spacing between consecutive packets of a message. For the probability generating function of the stationary distribution of the embedded Markov chain we derive in both cases a functional equation which involves a boundary function. The stationary mean number of packets in the system can be computed via this boundary function without solving the functional equation. In case of constant (random) spacing the boundary function can be determined by solving a finite-dimensional (an infinite-dimensional) system of linear equations numerically. For Poisson- and Bernoulli-distributed arrivals of messages numerical results are presented. Further, limiting results are derived.     

利用关系矩阵求传递闭包的一种方法

介绍了一种利用关系矩阵求有限集合上二元关系的传递闭包的方法 ,该方法简便、实用 .还可用此方法计算有向图的可达性矩阵 .