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.      

On extreme values in open queueing networks

This paper presents heavy traffic limit theorems for the extreme virtual waiting time of a customer in an open queueing network. In this paper, functional limit theorems are proved for extreme values of important probability characteristics of the open queueing network investigated as the maximum and minimum of the total virtual waiting time of a customer, and the maximum and minimum of the virtual waiting time of a customer. Also, the paper presents the previous related works for extreme values in queues and the virtual waiting time in heavy traffic.     

About the Sojourn Time Process in Multiphase Queueing Systems

Multiphase queueing systems (MQS) (tandem queues, queues in series) are of special interest both in theory and in practical applications (packet switch structures, cellular mobile networks, message switching systems, retransmission of video images, asembly lines, etc.). In this paper, we deal with approximations of MQS and present a heavy traffic limit theorems for the sojourn time of a customer in MQS. Functional limit theorems are proved for the customer sojourn time – an important probability characteristic of the queueing system under conditions of heavy traffic.      

On the law of the iterated logarithm in open queueing networks

An open queueing network model in heavy traffic is developed. Such models are mathematical models of computer networks in heavy traffic. Laws of the iterated logarithm for the virtual waiting time of the customer in open queueing networks and homogeneous computer networks are proved.     

具有插队和止步行为的M/M/c 排队系统

研究了具有插队和止步行为的M/M/c排队系统. 将到达顾客分为常规顾客和插队顾客, 常规顾客在队尾排队等待服务, 插队顾客总是尽可能的靠近队首插队等待服务. 插队行为由到达顾客的插队概率和队列中等待顾客对插队行为的容忍来描述. 利用负指数分布的性质、Laplace-Stieltjes变换和全概率公式, 给出了处于等待队列位置n的顾客、任意一个常规顾客和任意一个插队顾客的等待时间的表达式. 在此基础上, 讨论了系统相关指标随系统参数的变化情况.     

On the Analysis of the Virtual Waiting Time in Open Queueing Networks

The model of an open queueing network in heavy traffic has been developed. These models are mathematical models of computer networks in heavy traffic. A limit theorem has been presented for the virtual waiting time of a customer in heavy traffic in open queueing networks. Finally, we present an application of the theorem—a reliability model from computer network practice.     

The Versatility of MMAP[K] and the MMAP[K]/G[K]/1 Queue

This paper studies a single server queueing system with multiple types of customers. The first part of the paper discusses some modeling issues associated with the Markov arrival processes with marked arrivals (MMAP[K], where K is an integer representing the number of types of customers). The usefulness of MMAP[K] in modeling point processes is shown by a number of interesting examples. The second part of the paper studies a single server queueing system with an MMAP[K] as its input process. The busy period, virtual waiting time, and actual waiting times are studied. The focus is on the actual waiting times of individual types of customers. Explicit formulas are obtained for the Laplace–Stieltjes transforms of these actual waiting times.     

Laplace Transform and Moments of Waiting Times in Poisson Driven (max,+) Linear Systems

(Max,+) linear systems can be used to represent stochastic Petri nets belonging to the class of event graphs. This class contains various instances of queueing networks like acyclic or cyclic fork-and-join queueing networks, finite or infinite capacity tandem queueing networks with various types of blocking, synchronized queueing networks and so on. It also contains some basic manufacturing models such as kanban networks, assembly systems and so forth.In their 1997 paper, Baccelli, Hasenfuss and Schmidt provide explicit expressions for the expected value of the waiting time of the nth customer in a given subarea of a (max,+) linear system. Using similar analysis, we present explicit expressions for the moments and the Laplace transform of transient waiting times in Poisson driven (max,+) linear systems. Furthermore, starting with these closed form expressions, we also derive explicit expressions for the moments and the Laplace transform of stationary waiting times in a class of (max,+) linear systems with deterministic service times. Examples pertaining to queueing theory are given to illustrate the results.     

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.     

Two-server queue with one server idle below a threshold

A birth-death queueing system with two identical servers, first-come first-served discipline, and Poisson arrivals is considered. Only one of the servers is active when the number of customers in the system does not exceed a prescribed threshold, whereas both are active above the threshold. The problem of determining the equilibrium density of the waiting time is formulated. A generating function is given for the Laplace transform of the density of the waiting time, and it is pointed out that it leads to an explicit expression for this quantity. Explicit expressions are obtained for the first and second moments of the waiting and sojourn times, and they are compared with the corresponding quantities for a single-server system with the same state-dependent mean service rates.     

Regularly varying tails in a queue with discrete autoregressive arrivals of order p

We consider a discrete time single server queueing system where the service time of a customer is one slot, and the arrival process is governed by a discrete autoregressive process of order p (DAR(p)). For this queueing system, we investigate the tail behavior of the queue size and the waiting time distributions. Specifically, we show that if the stationary distribution of DAR(p) input has a tail of regular variation with index −β−1, then the stationary distributions of the queue size and the waiting time have tails of regular variation with index −β. This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

Transient and stationary waiting times in (max,+)-linear systems with Poisson input

We consider a certain class of vectorial evolution equations, which are linear in the (max,+) semi-field. They can be used to model several Types of discrete event systems, in particular queueing networks where we assume that the arrival process of customers (tokens, jobs, etc.) is Poisson. Under natural Cramér Type conditions on certain variables, we show that the expected waiting time which the nth customer has to spend in a given subarea of such a system can be expanded analytically in an infinite power series with respect to the arrival intensity λ. Furthermore, we state an algorithm for computing all coefficients of this series expansion and derive an explicit finite representation formula for the remainder term. We also give an explicit finite expansion for expected stationary waiting times in (max,+)-linear systems with deterministic queueing services. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on conservation laws for a multiclass service queueing system with setup times

This paper derives a conservation law for mean waiting times in a single-server multi-class service queueing system (M ^X/G/1 type queue) with setup times which may be dependent on multiple customer classes and its arrival batch size by using the work decomposition property in the queueing system with vacations.     

Markovian Controllable Queueing Systems with Hysteretic Policies: Busy Period and Waiting Time Analysis

We study Markovian queueing systems in which the service rate varies whenever the queue length changes. More specifically we consider controllable queues operating under the so-called hysteretic policy which provides a rather versatile class of operating rules for increasing and decreasing service rate at the arrival and service completion times. The objective of this paper is to investigate algorithmically the busy period and the waiting time distributions. Our analysis supplements the classical work of Yadin and Naor (1967) who focused on the steady-state probabilities of the system state. AMS 2000 Subject Classification 60K25, 90B22     

On the Full Idle Time in Multiphase Queueing Systems

The modern queueing theory is a powerful tool for a quantitative and qualitative analysis of communication systems, computer networks, transportation systems, and many other technical systems. The paper is designated to the analysis of queueing systems arising in the network theory and communications theory (such as the so-called multiphase queueing systems, tandem queues, or series of queueing systems). We present heavy traffic limit theorems for the full idle time in multiphase queueing systems. We prove functional limit theorems for values of the full idle time of a queueing system, which is its important probability characteristic. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 367–386, July–September, 2005.     

M/M/1算子的特征值及其应用(英文)

讨论 Ｍ／Ｍ／１算子的谱特征,证明０是 Ｍ／Ｍ／１算子的几何重数为 １的特征值,并且对应的特征向量是正的,作为应用给出了排队论中四个指标：系统中顾客的平均逗留时间,顾客的平均等待时间,顾客总数及等待的顾客总数的计算方法．     

饱和情形开排队网络的强逼近

本文用强逼近理论和斜反射定理，研究了饱和情形下的开排队网络，得到了队长、虚等待时间和顾客在网络中逗留时间的强逼近定理．     

Sojourn and waiting times in a single-server system with state-dependent mean service rate

A birth-death queueing system with asingle server, first-come first-served discipline, Poisson arrivals and state-dependent mean service rate is considered. The problem of determining the equilibrium densities of the sojourn and waiting times is formulated, in general. The particular case in which the mean service rate has one of two values, depending on whether or not the number of customers in the system exceeds a prescribed threshold, is then investigated. A generating function is derived for the Laplace transforms of the densities of the sojourn and waiting times, leading to explicit expressions for these quantities. Explicit expressions for the second moments of the sojourn and waiting times are also obtained.     

Waiting time analysis of a two-stage queueing system with priorities

A two-stage queueing system with two types of customers and non-preemptive priorities is analyzed. There is no waiting space between stages and so the blocking phenomenon is observed. The arrivals follow a Poisson distribution for the high priority customers and a gamma distribution for the low priority customers, while all service times are arbitrarily distributed. We derive expressions for the Laplace transform of the waiting time density of a low priority customer both in the transient and the steady state.     

Optimal server scheduling in nonpreemptive finite-population queueing systems

We consider a finite-population queueing system with heterogeneous classes of customers and a single server. For the case of nonpreemptive service, we fully characterize the structure of the server's optimal service policy that minimizes the total average customer waiting costs. We show that the optimal service policy may never serve some classes of customers. For those classes that are served, we show that the optimal service policy is a simple static priority policy. We also derive sufficient conditions that determine the optimal priority sequence.