Busy period analysis of the state dependent M/M/1/K queue

In this paper, we study the transient behavior of a state dependent M/M/1/K queue during the busy period. We derive in closed-form the joint transform of the length of the busy period, the number of customers served during the busy period, and the number of losses during the busy period. For two special cases called the threshold policy and the static policy we determine simple expressions for their joint transform.     

Busy period analysis of the level dependent PH/PH/1/K queue

In this paper, we study the transient behavior of a level dependent single server queuing system with a waiting room of finite size during the busy period. The focus is on the level dependent PH/PH/1/K queue. We derive in closed form the joint transform of the length of the busy period, the number of customers served during the busy period, and the number of losses during the busy period. We differentiate between two types of losses: the overflow losses that are due to a full queue and the losses due to an admission controller. For the M/PH/1/K, M/PH/1/K under a threshold policy, and PH/M/1/K queues, we determine simple expressions for their joint transforms.     

Services within a Busy Period of an M/M/1 Queue and Dyck Paths

We analyze the service times of customers in a stable M/M/1 queue in equilibrium depending on their position in a busy period. We give the law of the service of a customer at the beginning, at the end, or in the middle of the busy period. It enables as a by-product to prove that the process of instants of beginning of services is not Poisson. We then proceed to a more precise analysis. We consider a family of polynomial generating series associated with Dyck paths of length 2n and we show that they provide the correlation function of the successive services in a busy period with n+1 customers.     

On the M/G/1 queue with D-policy

This paper deals with the M/G/1 queue with D-policy, i.e., the server is turned off at the end of a busy period and turned on when the cumulative amount of work firstly exceeds some fixed value D. We first concentrate on the computation of the steady-state probabilities. The first moments and relationships among the busy period, the number of customers served and other performance measures are investigated. Some variants of the main model and the special case of the M/M/1 are also studied.     

带启动时间的多级适应性休假的M/G/1排队

本研究带启动时间的多级适应性休假的M/G/1间排队。给出稳态队长分布和母函数、等待时间分布和其LST及其随机分解结果，推导出忙期、假期和启动期的母函数。带有启动时间的单重休假和多重休假是本中模型的两个极端情况。     

A direct approach to sojourn times in a busy period of an M/M/1 queue

In this paper we present a direct approach to obtaining joint distributions of various quantities of interest in a busy period in an M/M/1 queue. These quantities are: the sojourn times and waiting times of all the customers in the busy period, the busy period length and the number of customers served in a busy period. Since the evolution of the total workload process between two successive customer arrivals is deterministic, this work gives statistic of the complete evolution of the workload process within a busy period. This work was done when the author was post doctoral fellow with the MAESTRO group at INRIA, Sophia Antipolis, France, and was supported by project no. 2900-IT-1 from the Centre Franco-Indien pour la Promotion de la Recherche Avancee (CEFIPRA).     

A Markov-modulated M/G/1 queue II: Busy period and time for buffer overflow

We consider an M/G/1 queue where the arrival and service processes are modulated by a two state Markov chain. We assume that the arrival rate, service time density and the rates at which the Markov chain switches its state, are functions of the total unfinished work (buffer content) in the queue. We compute asymptotic approximations to performance measures such as the mean residual busy period, mean length of a busy period, and the mean time to reach capacity.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

Rejection rules in theM/G/1 queue

We consider aM/G/1 queue modified such that an arriving customer may be totally or partially rejected depending on a r.v. (the barricade) describing his impatience and on the state of the system. Three main variants of this scheme are studied. The steady-state distribution is expressed in terms of Volterra equations and the relation to storage processes, dams and queues with state-dependent Poisson arrival rate is discussed. For exponential service times, we further find the busy period Laplace transform in the case of a deterministic barricade, whereas for exponential barricade it is shown by a coupling argument that the busy period can be identified with a first passage time in an associated birth-death process.     

具有不同到达率的带有启动时间及不耐烦策略的多级适应性休假M/G/1排队

研究了具有不同到达率的带有启动时间及不耐烦策略的多级适应性休假M/G/1排队模型,给出了稳态队长的母函数,等待时间的LST及其随机分解结果,并推导出忙期、全忙期及在线期均值.     

Second-Order Tail Behavior of the Busy Period Distribution of Certain GI/G/1 Queues

We consider the stable GI/G/1 queue in which the service time distribution has a dominated-varying tail. Under simple assumptions, we obtain the first- and second-order tail behavior of the busy period distribution in this queue.     

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.     

Markov-modulatedPH/G/1 queueing systems

We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated E_K/G/1 queueing system as a special case.     

Tail asymptotics of the waiting time and the busy period for the {{\varvec{M/G/1/K}}} queues with subexponential service times

We study the asymptotic behavior of the tail probabilities of the waiting time and the busy period for the $M/G/1/K$ queues with subexponential service times under three different service disciplines: FCFS, LCFS, and ROS. Under the FCFS discipline, the result on the waiting time is proved for the more general $GI/G/1/K$ queue with subexponential service times and lighter interarrival times. Using the well-known Laplace–Stieltjes transform (LST) expressions for the probability distribution of the busy period of the $M/G/1/K$ queue, we decompose the busy period into a sum of a random number of independent random variables. The result is used to obtain the tail asymptotics for the waiting time distributions under the LCFS and ROS disciplines.     

The M/G/1 Queue with Finite Workload Capacity

We consider the M/G/1 queueing system in which customers whose admission to the system would increase the workload beyond a prespecified finite capacity limit are not accepted. Various results on the distribution of the workload are derived; in particular, we give explicit formulas for its stationary distribution for M/M/1 and in the general case, under the preemptive LIFO discipline, for the joint stationary distribution of the number of customers in the system and their residual service times. Furthermore, the Laplace transform of the length of a busy period is determined. Finally, for M/D/1 the busy period distribution is derived in closed form.     

M/G/1 queue with single working vacation

In this paper, an M/G/1 queue with single working vacation is analyzed. Using the method of supplementary variable and the matrix-analytic method, we obtain the queue length distribution and service status at the arbitrary epoch under steady state conditions. Further, we derive expected busy period and expected busy cycle. Finally, server special cases are presented.     

Large deviations of the waiting time in the GI/G/1 queue with random order service

We consider the GI/G/1 queue where customers are served in random order and the service time distribution has a finite exponential moment. We derive the large deviations result for the waiting time distribution by showing that the asymptotic decay rate of the waiting time distribution is the same as that of the busy period distribution.     

G/M/1 queues with delayed vacations

We considerG/M/1 queues with multiple vacation discipline, where at the end of every busy period the server stays idle in the system for a period of time called changeover time and then follows a vacation if there is no arrival during the changeover time. The vacation time has a hyperexponential distribution. By using the methods of the shift operator and supplementary variable, we explicitly obtain the queue length probabilities at arrival time points and arbitrary time points simultaneously.     

带启动时间的多级适应性休假的Geom/G/1排队

本文研究带启动时间的多级适应性Geom／G／1离散时间排队．给出稳态队长和等待时间分布的母函数及其随机分解结果，推导出忙期和全假期的母函数和均值。     

M/M/m/m防空系统射击效能的排队概率特性

研究了具有消失制的M/M/m/m防空系统的射击效能,利用排队论及随机运筹学的有关知识,在模型的条件与假设下给出了其平稳状态的队长的分布律πk,平均工作的防空武器数E,敌机的突防概率πm,忙期长度等指标.     

Global and local asymptotics for the busy period of an M/G/1 queue

We consider an M/G/1 queue with subexponential service times. We give a simple derivation of the global and local asymptotics for the busy period. Our analysis relies on the explicit formula for the joint distribution for the number of customers and the length of the busy period of an M/G/1 queue.