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.     

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.     

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.     

Workload and busy period for M/GI/1 with a general impatience mechanism

The paper deals with the workload and busy period for the $M/GI/1$ M / G I / 1 system with impatience under FCFS discipline. The customers may become impatient during their waiting for service with generally distributed maximal waiting times and also during their service with generally distributed maximal service times depending on the time waited for service. This general impatience mechanism was originally introduced by Kovalenko (1961) and considered by Daley (1965), too. It covers the special cases of impatience on waiting times as well as impatience on sojourn times, for which Boxma et al. (2010, 2011) gave new results and outlined special cases recently. Our unified approach bases on the vector process of workload and busy time. Explicit representations for the LSTs of workload and busy period are given in case of phase-type distributed impatience.     

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     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

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.     

Workload and Waiting Time Analyses of MAP/G/1 Queue under D-policy

We study the workload (unfinished work) and the waiting time of the queueing system with MAP arrivals under D-policy. The D-policy stipulates that the idle server begin to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We first set up the system equations for workload and obtain the steady-state distributions of workloads at an arbitrary idle and busy points of time. We then proceed to obtain the waiting time distribution of an arbitrary customer based on the workload results. The M/G/1/D-policy queue will be investigated as a special case.     

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．     

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.     

Continuous time control of the arrival process in an m/g/1 queue

The problem of continuously controlling the arrival process in an M/G/1 queue is studied. The control is exercised by keeping the facility open or closed for potential arrivals, and is based on the residual workload process. The reward structure includes a reward rate R when the server is busy, and a holding cost rate cx when the residual workload is x. The economic criterion used is long run average return. A control limit policy is shown to be optimal. An iterative method for calculating this control limit policy is suggested.     

A controlled M / G / 1 workload process with an application to perishable inventory systems

We derive the stationary distribution of the regenerative process W(t), t ≥ 0, whose cycles behave like an M / G / 1 workload process terminating at the end of its first busy period or when it reaches or exceeds level 1, and restarting with some fixed workload . The result is used to obtain the overflow distribution of this controlled workload process; we derive and , where T is the duration of the first cycle. W(t) can be linked to a certain perishable inventory model, and we use our results to determine the distribution of the duration of an empty period.D. Perry was supported by a Mercator Fellowship of the Deutsche Forschungsgemeinschaft.     

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.     

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.     

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。     

The M/M/c with critical jobs

We consider theM/M/c queue, where customers transfer to a critical state when their queueing (sojourn) time exceeds a random time. Lower and upper bounds for the distribution of the number of critical jobs are derived from two modifications of the original system. The two modified systems can be efficiently solved. Numerical calculations indicate the power of the approach.     

Optimal control of the N policy M/G/1 queueing system with server breakdowns and general startup times

This paper deals with an N policy M/G/1 queueing system with a single removable and unreliable server whose arrivals form a Poisson process. Service times, repair times, and startup times are assumed to be generally distributed. When the queue length reaches N(N ? 1), the server is immediately turned on but is temporarily unavailable to serve the waiting customers. The server needs a startup time before providing service until there are no customers in the system. We analyze various system performance measures and investigate some designated known expected cost function per unit time to determine the optimal threshold N at a minimum cost. Sensitivity analysis is also studied.     

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.     

The stationary workload of the <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 queue with impatient customers

We consider a G/M/1 queue in which the patience time of the customers is constant. The stationary distribution of the workload of the server, or the virtual waiting time, is derived by the level crossing argument. To this end, we obtain the expected downcrossings of a level in the workload process during a busy cycle and then the expected length of a busy cycle. For both the expectations, we use the dual property between the M/G/1 and G/M/1 queue.