A unified algorithm for computing the stationary queue length distributions inM(k)/G/1/N and GI/M(k)/1/N queues

An iterative algorithm is developed for computing numerically the stationary queue length distributions in M/G/1/N queues with arbitrary state-dependent arrivals, or simply M(k)/G/1/N queues. The only input requirement is the Laplace-Stieltjes transform of the service time distribution.In addition, the algorithm can also be used to obtain the stationary queue length distributions in GI/M/1/N queues with state-dependent services, orGI/M(k)/1/N, after establishing a relationship between the stationary queue length distributions inGI/M(k)/1/N and M(k)/G/1/N+1 queues.Finally, we elaborate on some of the well studied special cases, such asM/G/1/N queues,M/G/1/N queues with distinct arrival rates (which includes the machine interference problems), andGI/M/C/N queues. The discussions lead to a simplified algorithm for each of the three cases.     

具有不耐烦顾客和PH分布休假的M/M/1排队系统

研究了具有不耐烦顾客的M/M/1休假排队系统,其中休假时间服从位相分布.当顾客在休假时间到达系统,顾客则会因为等待变得不耐烦.服务员休假结束后立刻开始工作.如果在顾客不耐烦时间段内,系统的休假还没有结束,顾客就会离开系统不再回来.建立的模型为水平相依QBD拟生灭过程,通过利用BrightTaylor算法得到系统的稳态概率解.同时还得到一些重要的性能指标.最后通过数据实例验证了我们的结论.     

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.     

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 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.     

服务台休假的GI/Geom/1离散时间排队

本文研究了具有位相型休假、位相型启动和单重几何休假的离散时间排队,假定 顾客到达间隔服从一般分布,服务时间服从几何分布,运用矩阵解析方法我们得到了这 些排队系统中顾客在到达时刻稳态队长分布及其随机分解.     

Delay moments for FIFO GI/GI/s queues

For stable FIFO GI/GI/s queues, s ≥ 2, we show that finite (k+1)st moment of service time, S, is not in general necessary for finite kth moment of steady-state customer delay, D, thus weakening some classical conditions of Kiefer and Wolfowitz (1956). Further, we demonstrate that the conditions required for E[D ^k]<∞ are closely related to the magnitude of traffic intensity ρ (defined to be the ratio of the expected service time to the expected interarrival time). In particular, if ρ is less than the integer part of s/2, then E[D] < ∞ if E[S^3/2]<∞, and E[D^k]<∞ if E[S^k]<∞, ^k≥ 2. On the other hand, if s-1 < ρ < s, then E[D^k]<∞ if and only if E[S^k+1]<∞, k ≥ 1. Our method of proof involves three key elements: a novel recursion for delay which reduces the problem to that of a reflected random walk with dependent increments, a new theorem for proving the existence of finite moments of the steady-state distribution of reflected random walks with stationary increments, and use of the classic Kiefer and Wolfowitz conditions. This revised version was published online in June 2006 with corrections to the Cover Date.     

GI／G／1排队系统的队长和等待时间的瞬时分布

本是[1，2]的继续,在本中利用马氏骨架过程给出了GI／G／1排队系统的队长的瞬时分布的另一新的计算方法和等待时间的计算方法。     

Two finite-difference methods for solving MAP(t)/PH(t)/1/K queueing models

In this paper two solution methods to the MAP(t)/PH(t)/1/K queueing model are introduced, one based on the Backwards Euler Method and the other on the Uniformization Method. Both methods use finite-differencing with a discretized, adaptive time-mesh to obtain time-dependent values for the entire state probability vector. From this vector, most performance parameters such as expected waiting time and expected number in the system can be computed. Also presented is a technique to compute the entire waiting (sojourn) time distribution as a function of transient time. With these two solution methods one can examine any transient associated with the MAP(t)/PH(t)/1/K model including time-varying arrival and/or service patterns. Four test cases are used to demonstrate the effectiveness of these methods. Results from these cases indicate that both methods provide fast and accurate solutions to a wide range of transient scenarios. This revised version was published online in June 2006 with corrections to the Cover Date.     

First-passage-time and busy-period distributions of discrete-time Markovian queues:Geom(n)/Geom(n)/1/N

This paper gives simple explicit solutions of various first-passage-time distributions for a general class of discrete-time queueing models under arbitrary initial conditions, state-dependent transition probabilities and the finite waiting room. Explicit closed form expressions are obtained in terms of roots. These expressions are then used to get numerical as well as graphical results. Explicit closed-form expressions are also deduced for the continuous-time models including the busy-period distributions. The analysis is then extended to cover the case of two absorbing states.     

推广的M~x/G(M/G)/1(M/G)可修排队系统(I)── 一些排队指标

考虑M     

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.     

Computing steady-state queueing-time distributions of single-server queues:GI X /M/1

This paper presents a computationally efficient method to find the steady-state distributions of actual queueing times of the first customer, as well as of a randomly selected customer, of an arrival group for the queueing systemGI ^X /M/1, and hence the queueing-time distribution of a customer for the systemGI/E _X /1. The distribution of virtual queueing time is also obtained. Approximate analysis based on one or more roots is also discussed. Though the exact detailed as well as approximate computations for a variety of interarrival-time distributions such as generalized Erlang, mixed generalized Erlang, hyperexponential, generalized hyperexponential, and deterministic have been carried out, only representative results in the form of tables have been appended. The results obtained should prove useful to queueing theorists, practitioners, and others.     

Explicit solutions for the steady state distributions in M/PH/1 queues with workload dependent balking

We consider an M/PH/1 queue with balking based on the workload. An arriving customer joins the queue and stays until served only if the system workload is below a fixed level at the time of arrival. The steady state workload distribution in such a system satisfies an integral equation. We derive a differential equation for Phase type service time distribution and we solve it explicitly, with Erlang, Hyper-exponential and Exponential distributions as special cases. We illustrate the results with numerical examples.     

服务中断的带优先反馈排队系统弱极限定理

利用了概率论中的弱极限定理,研究了服务中断的带优先反馈排队系统弱极限定理,得到了队长,虚等待时间及逗留时间的弱极限定理.     

A sample path analysis of M/GI/1 queues with workload restrictions

A simple random time change is used to analyze M/GI/1 queues with workload restrictions. The types of restrictions considered include workload bounds and rejection of jobs whose waiting times exceed a (possibly random) threshold. Load dependent service rates and vacations are also allowed and in each case the steady state distribution of the workload process for the system with workload restrictions is obtained in terms of that of the corresponding M/ GI/1 queue without restrictions. The novel sample path arguments used simplify and generalize previous results.     

同步休假GI／M／c排队的稳态理论

本文研究同步多重休假的GI／M／c排队系统,休假时间服从指数分布,使用发展了矩阵几何解决方法,给出了系统的平衡条件、稳态队长及等等时间分布。证明了队长和等等时间的条件随机分解定理,并讨论了由休假引起的附加队长和附加延迟的位相（PH）结构。     

Bounds for the quasi-stationary distribution of some specialized Markov chains

The quasi-stationary distributions of Markov chains have been investigated by many papers and are known to have considerable practical importance in, e.g., biological, chemical and applied probability models. However, computation of the quasi-stationary distributions is often nontrivial, which has limited its use in practice despite the usefulness of its own, except for some simple cases. This paper develops some bounds, which are relatively easy to calculate, for the quasi-stationary distribution of some specialized Markov chains.     

Transient Behaviour of an M/M/1/N queue

For a simple 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 have been obtained. The method economizes in algebra and the simple closed form of the state probabilities is used to obtain important parameters.     

Asymptotic Results and a Markovian Approximation for the M(n)/M(n)/s+GI System

In this paper for the M(n)/M(n)/s+GI system, i.e. for a s-server queueing system where the calls in the queue may leave the system due to impatience, we present new asymptotic results for the intensities of calls leaving the system due to impatience and a Markovian system approximation where these results are applied. Furthermore, we present a new proof for the formulae of the conditional density of the virtual waiting time distributions, recently given by Movaghar for the less general M(n)/M/s+GI system. Also we obtain new explicit expressions for refined virtual waiting time characteristics as a byproduct.