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 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.     

Some results on a generalized M/G/1 feedback queue with negative customers

This paper deals with a generalized M/G/1 feedback queue in which customers are either “positive" or “negative". We assume that the service time distribution of a positive customer who initiates a busy period is G _e (x) and all subsequent positive customers in the same busy period have service time drawn independently from the distribution G _b (x). The server is idle until a random number N of positive customers accumulate in the queue. Following the arrival of the N-th positive customer, the server serves exhaustively the positive customers in the queue and then a new idle period commences. This queueing system is a generalization of the conventional N-policy queue with N a constant number. Explicit expressions for the probability generating function and mean of the system size of positive customers are obtained under steady-state condition. Various vacation models are discussed as special cases. The effects of various parameters on the mean system size and the probability that the system is empty are also analysed numerically. AMS Subject Classification: Primary: 60 K 25 · Secondary: 60 K 20, 90 B 22     

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.     

Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks

Two variants of an M/G/1 queue with negative customers lead to the study of a random walkX _n+1=[X _n + _n ]⁺ where the integer-valued _n are not bounded from below or from above, and are distributed differently in the interior of the state-space and on the boundary. Their generating functions are assumed to be rational. We give a simple closed-form formula for , corresponding to a representation of the data which is suitable for the queueing model. Alternative representations and derivations are discussed. With this formula, we calculate the queue length generating function of an M/G/1 queue with negative customers, in which the negative customers can remove ordinary customers only at the end of a service. If the service is exponential, the arbitrarytime queue length distribution is a mixture of two geometrical distributions.Supported by the European grant BRA-QMIPS of CEC DG XIII.     

On the M/G/1 retrial queueing system with linear control policy

Summary This paper is concerned with the study of a newM/G/1 retrial queueing system in which the delays between retrials are exponentially distributed random variables with linear intensityg(n)=α+nμ, when there aren≥1 customers in the retrial group. This new retrial discipline will be calledlinear control policy. We carry out an extensive analysis of the model, including existence of stationary regime, stationary distribution of the embedded Markov chain at epochs of service completions, joint distribution of the orbit size and the server state in steady state and busy period. The results agree with known results for special cases.     

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/M/1 Queue with Impatient Customers of Higher Priority

We introduce a simple approach for the analysis of the M/M/c queues with a single class of customers and constant impatience time by finding simple Markov processes (see (2.1) and (2.15) below), and then by applying this approach we analyze the M/M/1 queues with two classes of customers in which class 1 customers have impatience of constant duration, and class 2 customers have no impatience and lower priority than class 1 customers.     

带有负顾客的N策略工作休假M/M/1排队

考虑带有正、负顾客的N策略工作休假M/M/1排队。负顾客一对一抵消队尾的正顾客(若有),若系统中无正顾客,到达的负顾客自动消失,负顾客不接受服务。在休假期间,服务员并未完全停止工作而是以较低的服务率为顾客服务。用拟生灭过程和矩阵几何解方法,我们给出了稳态队长和稳态等待时间的分布。此外,我们也证明了稳态条件下的队长和等待时间的条件随机分解并得到了附加队长和附加延迟的分布。     

M/G/1/N vacation model with varying E-limited service discipline

We define and analyze anM/G/1/N vacation model that uses a service discipline that we call theE-limited with limit variation discipline. According to this discipline, the server provides service until either the system is emptied (i.e. exhausted) or a randomly chosen limit ofl customers has been served. The server then goes on a vacation before returning to service the queue again. The queue length distribution and the Laplace-Stieltjes transforms of the waiting time, busy period and cycle time distributions are found. Further, an expression for the mean waiting time is developed. Several previously analyzed service disciplines, including Bernoulli scheduling, nonexhaustive service and limited service, are special cases of the general varying limit discipline that is analyzed in this paper.     

The M/G/1 retrial queue with bernoulli schedule

We consider anM/G/1 retrial queue with infinite waiting space in which arriving customers who find the server busy join either (a) the retrial group with probabilityp in order to seek service again after a random amount of time, or (b) the infinite waiting space with probabilityq(=1–p) where they wait to be served. The joint generating function of the numbers of customers in the two groups is derived by using the supplementary variable method. It is shown that our results are consistent with known results whenp=0 orp=1.     

Optimal control of a removable and non-reliable server in an M/M/1 queueing system with exponential startup time

We study a single removable and non-reliable server in the N policy M/M/1 queueing system. The server begins service only when the number of customers in the system reaches N (N1). After each idle period, the startup times of the server follow the negative exponential distribution. While the server is working, it is subject to breakdowns according to a Poisson process. When the server breaks down, it requires repair at a repair facility, where the repair times follow the negative exponential distribution. The steady-state results are derived and it is shown that the probability that the server is busy is equal to the traffic intensity. Cost model is developed to determine the optimal operating N policy at minimum cost.     

Bayesian prediction inM/M/1 queues

Simple queues with Poisson input and exponential service times are considered to illustrate how well-suited Bayesian methods are used to handle the common inferential aims that appear when dealing with queue problems. The emphasis will mainly be placed on prediction; in particular, we study the predictive distribution of usual measures of effectiveness in anM/M/1 queue system, such as the number of customers in the queue and in the system, the waiting time in the queue and in the system, the length of an idle period and the length of a busy period.     

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。     

Continuity theorems for the M/M/1/n queueing system

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution. Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.     

A NEW APPROACH FOR THE STUDY OF THE M X /G/1 QUEUE USING RENEWAL ARGUMENTS

The M ^X /G/1 queueing system as well as several of its variants have long ago been studied by considering the embedded discrete-time Markov chain at service completion epochs. Alternatively other approaches have been proposed such as the theory of regenerative processes, the supplementary variable method, properties of the busy period, etc. In this note we study the M ^X /G/1 queue via a simple new method that uses renewal arguments. This approach seems quite powerful and may become fruitful in the investigation of other queueing systems as well.     

AnM/M/c queue with impatient customers

This paper considers theM/M/c queue in which a customer leaves when its service has not begun within a fixed interval after its arrival. The loss probability can be expressed in a simple formula involving the waiting time probabilities in the standardM/M/c queue. The purpose of this paper is to give a probabilistic derivation of this formula and to outline a possible use of this general formula in theM/M/c retrial queue with impatient customers. This research was supported by the INTAS 96-0828 research project and was presented at the First International Workshop on Retrial Queues, Universidad Complutense de Madrid, Madrid, September 22–24, 1998.     

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．     

Consecutive customer losses in oscillating GIX/M//n systems with state dependent services rates

We address the probability that k or more Consecutive Customer Losses take place during a busy period of a queue, the so-called k-CCL probability, for oscillating GI ^X /M//n systems with state dependent services rates, also denoted as GI ^X /M(m)−M(m)//n systems, in which the service rates oscillate between two forms according to the evolution of the number of customers in the system. We derive an efficient algorithm to compute k-CCL probabilities in these systems starting with an arbitrary number of customers in the system that involves solving a linear system of equations. The results derived are illustrated for specific sets of parameters.