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).     

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.     

Busy period analysis for the GI/M/1 queue with exponential vacations

We consider the busy period in the GI/M/1 queue with multiple exponential vacations. We derive the transform of the joint distribution of the length of a busy period, the number of customers served during the busy period, and the residual interarrival time at the instant the busy period ends.     

Tail asymptotics of the queue size distribution in the M/M/m retrial queue

We consider an M/M/m retrial queue and investigate the tail asymptotics for the joint distribution of the queue size and the number of busy servers in the steady state. The stationary queue size distribution with the number of busy servers being fixed is asymptotically given by a geometric function multiplied by a power function. The decay rate of the geometric function is the offered load and independent of the number of busy servers, whereas the exponent of the power function depends on the number of busy servers. Numerical examples are presented to illustrate the result.     

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.     

具有可利用服务员的M/G/1排队模型

本文考虑了具有可利用服务员的M／G／1有有限容量的排队模型．当工作量超过k(k是常数或者随机变量),可利用服务员参与工作,一直到工作量少于或等于k．可利用服务员的速率依赖于目前工作量．应用Level-crossing方法,获得了工作量的平稳分布．应用Kolmogorov向后微分方程方法,构造更新方程以获得忙期的Laplace变换．     

The busy period of an M/M/1 queue with balking and reneging

It has been shown by (R.O. Al-Seedy, A.A. El-Sherbiny, S.A. El-Shehawy, S.I. Ammar, Transient solution of the M/M/c queue with balking and reneging, Comput. Math. Appl. 57 (2009) 1280–1285) that a generating function technique can be successfully applied to derive the transient solution for an M/M/c queueing system. In this paper, we further illustrate how this technique can be used to obtain the busy period density function of an M/M/1 queue with balking and reneging. Finally, numerical calculations are presented.     

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.     

同步N—策略多重休假M／M／c排队

研究了具有同步N-策略多重休假的M／M／c排队系统．在休假时间服从相型（PH）分布的假设下，给出了系统的稳态指标．证明在已知服务台全忙并且系统中顾客数大于或等于N的条件下，条件随机变量可分解成独立随机变量之和，其中一个是无休假经典M／M／c系统中的对应条件变量，另一个是休假引起的附加随机变量     

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.     

每个忙期中第一个顾客被拒绝服务的M／M／1排队模型的另一个特征值

研究每个忙期中第一个顾客被拒绝服务的M／M／1排队模型主算子在左半复平面中的特征值,证明2√λμ-λ-μ是该主算子的几何重数为1的特征值。     

A new approach to the busy period of the M/M/1 queue

In this paper, we provide a new approach to the computation of the Laplace transform of the length of the busy period of the M/M/1 queue with constrained workload (finite dam), without the use of complex analysis. This revised version was published online in June 2006 with corrections to the Cover Date.     

每个忙期中第一个顾客被拒绝服务的M／M/1排队模型的其他特征值

研究每个忙期中第一个顾客被拒绝服务的M／M／1排队模型的主算子在左半复平面中的特征值,证明对一切θ∈（0,1）,（2√λμ-λ—μ）θ是该主算子的几何重数为1的特征值．     

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.     

休假时间服从T-SPH分布的M/M/1多重休假排队

本文研究休假时间服从T-SPH分布的M/M/1多重休假排队,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,并得到了平稳队长和平稳等待时间的随机分解结果以及附加队长和附加延迟的母函数和LST的具体形式.     

N策略工作休假M/M/1排队

考虑策略工作休假M/M/1排队,简记为M/M/1(N-WV)。在休假期间,服务员并未完全停止工作而是以较低的速率为顾客服务。用拟生灭过程和矩阵几何解方法,我们给出了有直观概率意义的稳态队长和稳态条件等待时间的分布。此外,我们也得到了队长和等待时间的条件随机分解结构及附加队长和附加延迟的分布。     

Heavy traffic approximations for busy period in an M/G/∞ queue

A necessary and sufficient condition is given for busy period in an M/G/∞ queue to be asymptotically exponential with mean equal to expected busy period, as traffic intensity increases. The case where this condition fails is also investigated, and the results applied.     

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.     

带有Bernoulli控制策略的M/M/1多重休假排队模型

研究具有Bernoulli控制策略的M/M/1多重休假排队模型: 当系统为空时, 服务台依一定的概率或进入闲期, 或进入普通休假状态, 或进入工作休假状态. 对该模型, 应用拟生灭(QBD)过程和矩阵几何解的方法, 得到了过程平稳队长的具体形式, 在此基础上, 还得到了平稳队长和平稳逗留时间的随机分解结果以及附加队长分布和附加延迟的LST的具体形式. 结果表明, 经典的M/M/1排队, M/M/1多重休假排队, M/M/1多重工作休假排队都是该模型的特殊情形.     

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.