GI/M/1 Queue with Server Vacations

A queueing model with server vacations is studied in which it is assumed that the interarrival time has a general distribution, the service-time distribution is exponential and vacations are independently and indentically distributed with a general distribution. Using the embedded Markov chain technique, the equilibrium probability distributions of system size have been obtained at pre-arrival and at random epochs separately. Finally, the distribution of waiting time of a customer in the queue (excluding service) has been derived.     

具有延误休假时间的排队系统队长分布

考虑具延误休假时间的N-策略M／G／1／∞排队系统,研究队长的瞬态和稳态性质．通过引进“服务员忙期”和使用全概率分解技术,导出在任意时刻t瞬态队长分布的L变换的递推表达式和稳态队长分布的递推表达式,以及平稳队长的随机分解．     

A Numerical Method for Evaluating Delay to a Customer in a Time-Inhomogeneous,Single Server Queue with Batch Arrivals

Using the method of discrete time analysis, a numerical method is developed for evaluating the distribution of the delay encountered by a customer in a time-inhomogeneous, single server queue with batch arrivals.     

Equilibrium Joining Strategies of Positive Customers in a Markovian Queue with Negative Arrivals and Working Vacations

Methodology and Computing in Applied Probability - We study the economic analysis of a single server Markovian queueing system with positive and negative customers and multiple working vacations....     

The Finite Capacity GI/M/1 Queue with Server Vacations

We consider the GI/M/1/K queue where the server takes exponentially distributed vacations when there are no customers left to serve in the queue. We obtain the queue length distribution at arrival epochs and random epochs for the multiple vacation case. We present heuristic algorithms to compute the blocking probability for this system. Several numerical examples are presented to analyze the behaviour of the blocking probability and to test the performance of the heuristics.     

A Queue with Semi-Markovian Batch Plus Poisson Arrivals with Application to the MPEG Frame Sequence

We consider a queueing system with a single server having a mixture of a semi-Markov process (SMP) and a Poisson process as the arrival process, where each SMP arrival contains a batch of customers. The service times are exponentially distributed. We derive the distributions of the queue length of both SMP and Poisson customers when the sojourn time distributions of the SMP have rational Laplace–Stieltjes transforms. We prove that the number of unknown constants contained in the generating function for the queue length distribution equals the number of zeros of the denominator of this generating function in the case where the sojourn times of the SMP follow exponential distributions. The linear independence of the equations generated by those zeros is discussed for the same case with additional assumption. The necessary and sufficient condition for the stability of the system is also analyzed. The distributions of the waiting times of both SMP and Poisson customers are derived. The results are applied to the case in which the SMP arrivals correspond to the exact sequence of Motion Picture Experts Group (MPEG) frames. Poisson arrivals are regarded as interfering traffic. In the numerical examples, the mean and variance of the waiting time of the ATM cells generated from the MPEG frames of real video data are evaluated.     

The Infinite Server Queue with Arrivals Generated by a Non-Homogeneous Compound Poisson Process

This paper considers the infinite server queue with arrivals generated by a non-homogeneous compound Poisson process. In such a system, customers arrive in groups of variable size, the arrival epochs of groups being points of a non-homogeneous Poisson process, and they are served without delay. The service times of customers who belong to the same group need not be independent nor identically distributed. Assuming that the system is initially empty, the transient distribution of the queue size and of the counting departure process is obtained. Also the limiting queue size distribution (when it exists) is determined and it is found to be insensitive to the form the service time distribution functions.     

推广的多级适应性休假Mx/G/1排队系统

用LS变换和L变换研究了推广的多级适应性休假Mx/G/1排队系统中队长的瞬态分布,并且进一步用LS变换的终值定理和洛比达法则得到了队长平稳分布的概率母函数,这个结果可以应用到很多模型.     

带启动时间的同步多重休假的GI/M/c排队

本文研究带启动时间的同步多重休假的GI/M/c排队,通过矩阵几何解方法,给出了稳态队长,等待时间的分布函数及其条件随机分解结果。     

服务员假期中以概率p进入的M/G/1排队系统的随机分解

该文研究M/G/1多重休假排队系统，其中在服务员休假中到达顾客以概率p(0≤p≤1)进入。通过引进“服务员忙期”和使用拉普拉斯变换或拉普拉斯— —司梯阶变换，我们获得队长瞬态分布的拉普拉斯变换和稳态分布的递推表达式，进一步得到稳态队长分布的随机分解和在特殊情况下相应的一些结果。     

Waiting-Time Distributions in Polling Systems with Simultaneous Batch Arrivals

We study the delay in asymmetric cyclic polling models with general mixtures of gated and exhaustive service, with generally distributed service times and switch-over times, and in which batches of customers may arrive simultaneously at the different queues. We show that (1–)X _i converges to a gamma distribution with known parameters as the offered load tends to unity, where X _i is the steady-state length of queue i at an arbitrary polling instant at that queue. The result is shown to lead to closed-form expressions for the Laplace–Stieltjes transform (LST) of the waiting-time distributions at each of the queues (under proper scalings), in a general parameter setting. The results show explicitly how the distribution of the delay depends on the system parameters, and in particular, on the simultaneity of the arrivals. The results also suggest simple and fast approximations for the tail probabilities and the moments of the delay in stable polling systems, explicitly capturing the impact of the correlation structure in the arrival processes. Numerical experiments indicate that the approximations are accurate for medium and heavily loaded systems.     

启动延迟N-策略批到达多重休假排队

目前N-策略批到达排队系统稳态队长分布难以给出解析解.提出一种新的递归算法研究顾客批到达,服务台延迟启动且多重休假的N-策略休假排队系统稳态队长分布.首先采用条件随机分解的方法得到稳态队长分布的概率母函数;然后采用递归算法推导附加队长分布的解析表达式;最后推导出稳态队长分布的递推关系式.     

An Alternative Model for Queueing Systems with Single Arrivals,Batch Services and Customer Coalescence

In this paper we consider a queueing system with single arrivals, batch services and customer coalescence and we use it as a building block for constructing queueing networks that incorporate such characteristics. Chao et al. (1996) considered a similar model and they proved that it possesses a geometric product form stationary distribution, under the assumption that if the number of units present at a service completion epoch is less than the required number of units, then all the units coalesce into an incomplete (defective) batch which leaves the system. We drop this assumption and we study a model without incomplete batches. We prove that the stationary distribution of such a queue has a nearly geometric form. Using quasi-reversibility arguments we construct a network model with such queues which provides relevant bounds and approximations for the behaviour of assembly processes. Several issues about the validity of these bounds and approximations are also discussed.     

部分服务台休假的M/M/c排队的等待时间

我们证明了Erlang分布的若干有趣性质，使用这些性质，给出部分服务台休假的排队系统中等待时间分布的一个简洁而直观的表达式．     

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。     

An Explicit Solution for the Value Function of a Priority Queue

We consider a multiclass preemptive-resume priority queue with Poisson arrivals and general service times. We derive explicit expressions for the discounted expected and long-run average weighted queue lengths and switching costs, the latter one only in the case of exponential service times. We illustrate our results with numerical calculations.     

Iterative Methods for Queuing Systems with Batch Arrivals and Negative Customers

In this paper, we are interested in solving the stationary probability distributions of Markovian queuing systems having batch arrivals and negative customers by using the Preconditioned Conjugate Gradient Squared (PCGS) method. The preconditioner is constructed by exploiting the near-Toeplitz structure of the generator matrix of the system. We proved that under some mild conditions the preconditioned linear systems have singular values clustered around one when the size of the queue tends to infinity. Numerical results indicated that the convergence rate of the proposed method is very fast.     

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