Nonpreemptive Multi-Server Priority Queues

Hokstad recently published an approximate method for calculating the behaviour of an M/G/m queue. This note applies his results to the nonpreemptive priority situation with two priority classes having the same service-time distribution. Laplace transforms and the first two moments of the waiting-time distributions are given.     

带有阈值转换和启动时间的优先权排队

在诸如ＩＳＤＮ的通信网络中,多种信息共用一条线路,为了满足不同类型信息的服务质量要求,带有阈值转换的优先权排队系统应是一种合适的模型。本文研究单服务员、两类顾客的带有阈值转换和启动时间的优先权排队系统,首先,分别就抢占和非抢占情形讨论了具有泊松到达、服务时间和启动时间均有指数贩系统,然后就非抢占情况进上步考虑了服务时间和启动时间有一般分布的系统,求出了系统中两类顾客队长的稳态联合概率母函数,藉助这     

Many-Sources Delay Asymptotics with Applications to Priority Queues

In this paper, we study discrete-time priority queueing systems fed by a large number of arrival streams. We first provide bounds on the actual delay asymptote in terms of the virtual delay asymptote. Then, under suitable assumptions on the arrival process to the queue, we show that these asymptotes are the same. As an application of this result, we then consider a priority queueing system with two queues. Using the earlier result, we derive an upper bound on the tail probability of the delay. Under certain assumptions on the rate function of the arrival process, we show that the upper bound is tight. We then consider a system with Markovian arrivals and numerically evaluate the delay tail probability and validate these results with simulations.     

Tail Asymptotics for HOL Priority Queues Handling a Large Number of Independent Stationary Sources

In this paper we study the asymptotics of the tail of the buffer occupancy distribution in buffers accessed by a large number of stationary independent sources and which are served according to a strict HOL priority rule. As in the case of single buffers, the results are valid for a very general class of sources which include long-range dependent sources with bounded instantaneous rates. We first consider the case of two buffers with one of them having strict priority over the other and we obtain asymptotic upper bound for the buffer tail probability for lower priority buffer. We discuss the conditions to have asymptotic equivalents. The asymptotics are studied in terms of a scaling parameter which reflects the server speed, buffer level and the number of sources in such a way that the ratios remain constant. The results are then generalized to the case of M buffers which leads to the source pooling idea. We conclude with numerical validation of our formulae against simulations which show that the asymptotic bounds are tight. We also show that the commonly suggested reduced service rate approximation can give extremely low estimates.     

Robust Optimal Service Analysis of Single-Server Re-Entrant Queues

We generalize the analysis of J.A. Ball, M.V. Day, and P. Kachroo (Mathematics of Control, Signals, and Systems, vol. 12, pp. 307–345, 1999) to a fluid model of a single server re-entrant queue. The approach is to solve the Hamilton-Jacobi-Isaacs equation associated with optimal robust control of the system. The method of staged characteristics is generalized from Ball et al. (1999) to construct the solution explicitly. Formulas are developed allowing explicit calculations for the Skorokhod problem involved in the system equations. Such formulas are particularly important for numerical verification of conditions on the boundary of the nonnegative orthant. The optimal control (server) strategy is shown to be of linear-index type. Dai-type stability properties are discussed. A modification of the model in which new customers are allowed only at a specified entry queue is considered in 2 dimensions. The same optimal strategy is found in that case as well.     

Predicting Time-Dependent Distributions of Queues and Delays for Road Traffic at Roundabouts and Priority Junctions

Vehicle queues and delays at busy road junctions have to be treated time-dependently when the traffic demand and the available capacity are approximately equal. Existing methods allow the queue length at a given time to be directly estimated as an average over all possible evolutions of the queueing system consistent with the given initial conditions and the time-dependent arrival and service rates. The paper describes the development of methods to predict the underlying distributions. Estimates of the variance and the overall frequency distribution for queue length and delay are obtained by simulating an M/M/1 queueing model with parameters varying with time. Predictive models are developed to represent the simulation results. They require as input values of parameters describing the duration of the peak and the time-average traffic intensities and capacities.     

Optimal Screening Policies for Hepatitis A

This paper presents a simple method of determining optimal screening policies for detecting immunity to Hepatitis A virus in travellers to areas of high endemicity. The method is constrained by the requirement that screening policies be kept simple in order to ensure their wide implementation.     

Optimal Continuous Policies for Repair and Replacement

This paper investigates the problem of finding optimal replacement policies for equipment subject to failures with randomly distributed repair costs, the degree of reliability of the equipment being considered as a state of a Markov process. Algorithms have been devised to find optimal combined policies both for preventive replacement and for replacement in case of failure by using repair-limit strategies.First a simple procedure to obtain an optimal discrete policy is described. Then an algorithm is formulated in order to calculate an optimal continuous policy: it is shown how the optimal repair limit is the solution to an ordinary differential equation, and how the value of the repair limit determines the optimal preventive replacement policy.     

保险公司的最优投资和再保险策略

在保险公司既可以做证券(股票和债券)投资,同时又采取比例再保险策略的情况下,通过对经典的Cramér-Lundberg保险公司盈余过程模型的连续扩散近似,利用动态规划原理分别得出了在破产概率最小和终值期望效用最大两种目标函数下,保险公司的最优投资和最优再保策略的显式解和对应的目标函数值.对两种目标函数下的最优策略做了比较研究.     

Optimal Policies for Investment with Time-Varying Return Distributions

We develop a model in which investors must learn the distribution of asset returns over time. The process of learning is made more difficult by the fact that the distributions are not constant through time. We consider risk-neutral investors who have quadratic utility and are selecting between two risky assets. We determine the time at which it is optimal to update the distribution estimate and hence, alter portfolio weights. Our results deliver an optimal policy for asset allocation, that is, the sequence of time intervals at which it is optimal to switch between assets, based on stochastic optimal control theory. In addition, we determine the time intervals in which asset switching leads to a loss with high probability. We provide estimates of the effectiveness of the optimal policy.     

Optimal Decisions in a Time Priority Queue

We show how the position of a limit order (LO) in the queue influences the decision of whether to cancel the order or let it rest. Using ultra-high-frequency data from the Nasdaq exchange, we perform empirical analysis on various LO book events and propose novel ways for modelling some of these events, including cancellation of LOs in various positions and size of market orders. Based on our empirical findings, we develop a queuing model that captures stylized facts on the data. This model includes a distinct feature which allows for a potentially random effect due to the agent’s impulse control. We apply the queuing model in an algorithmic trading setting by considering an agent maximizing her expected utility through placing and cancelling of LOs. The agent’s optimal strategy is presented after calibrating the model to real data. A simulation study shows that for the same level of standard deviation of terminal wealth, the optimal strategy has a 2.5% higher mean compared to a strategy which ignores the effect of position, or an 8.8% lower standard deviation for the same level of mean. This extra gain stems from posting an LO during adverse conditions and obtaining a good queue position before conditions become favourable.     

Optimal Ordering Policies for Inventory Systems with Emergency Ordering

This paper considers dynamic single- and multi-product inventory problems in which the demands in each period are independent and identically distributed random variables. The problems considered have the following common characteristics. At the beginning of each period two order quantities are determined for each product. A “normal order” quantity with a constant positive lead time of λ _n periods and an “emergency order” quantity with a lead time of λ _e periods, where λ _e = λ _n - 1. The ordering decisions are based on linear procurement costs for both methods of ordering and convex holding and penalty costs. The emergency ordering costs are assumed to be higher than the normal ordering costs. In addition, future costs are discounted.For the single-product problem the optimal ordering policy is shown to be the same for all periods with the exception of the last period in the N-period problem. For the multi-product problem the one- and N-period optimal ordering policy is characterized where it is assumed that there are resource constraints on the total amount that can be ordered or produced in each period.     

Optimal Inventory Policies with Service-Level Constraints

In a practical situation it is often difficult to determine the value of the shortage costs for use in in ventory-control systems. However, in cost-minimization problems including service-level constraints, shortage costs are implicitly prevailing. With the purpose of exploring these relations, a continuous review (Q, r) stock-control system is considered, where the order points and lot sizes are computed simultaneously. Instead of explicitly expressing the shortage cost in the objective function, it is taken into consideration through a service-level constraint. The shadow price of this constraint can in some sense be interpreted as the shortage cost corresponding to the requested service level. By changing the value of the service level, interesting relations between shortage costs and service levels can be viewed for different sets of other inventory parameters. In order to investigate the sensitivity for probabilistic variations in the input data, two different probability distributions are used to describe the lead-time demand.     

Existence of Risk-Sensitive Optimal Stationary Policies for Controlled Markov Processes

In this paper we are concerned with the existence of optimal stationary policies for infinite-horizon risk-sensitive Markov control processes with denumerable state space, unbounded cost function, and long-run average cost. Introducing a discounted cost dynamic game, we prove that its value function satisfies an Isaacs equation, and its relationship with the risk-sensitive control problem is studied. Using the vanishing discount approach, we prove that the risk-sensitive dynamic programming inequality holds, and derive an optimal stationary policy. Accepted 1 October 1997     

两不同部件冷贮备系统的最优备件定购策略分析

本文研究了两不同部件冷贮备系统的最优备件定购策略问题,特别把替失下来的部件所能回收的费用引入模型假设,并且紧急定购备件的交付周期为随机变量,得到了在一定条件限制下的最优通常定购时间,它使得以长期运行单位时间内的期望费用达到最小或者使得系统的稳态可用度达到最大。     

考虑个体保单风险特征的最优奖惩系统

奖惩系统在汽车保险中的应用非常普遍。论文首先介绍和讨论了泊松-伽马假设下的最优奖惩系统及其性质;其次在假设个体保单的索赔频率服从二项分布,而二项分布的一个参数服从贝塔分布的条件下,建立了一种考虑个体保单风险特征信息的最优奖惩系统,其中风险特征信息可以通过广义线性模型的形式引入奖惩系统;然后在假设个体保单的索赔频率服从负二项分布,而负二项分布的一个参数服从贝塔分布的条件下,建立了另一个最优奖惩系统;最后讨论了这两个奖惩系统的性质和应用。     

Martingale Methods for Analysing Single-Server Queues

In this paper we presents a martingale method for analysing queues of M/G/1 type, which have been generalised so that the system passes through a series of phases on which the service behaviour may differ. The analysis uses the process embedded at departures to create a martingale, which makes possible the calculation of the probability generating function of the stationary occupancy distribution. Specific examples are given, for instance, a model of an unreliable queueing system, and an example of a queue-length-threshold overload-control system.     

Joint Distributions for Interacting Fluid Queues

Motivated by recent traffic control models in ATM systems, we analyse three closely related systems of fluid queues, each consisting of two consecutive reservoirs, in which the first reservoir is fed by a two-state (on and off) Markov source. The first system is an ordinary two-node fluid tandem queue. Hence the output of the first reservoir forms the input to the second one. The second system is dual to the first one, in the sense that the second reservoir accumulates fluid when the first reservoir is empty, and releases fluid otherwise. In these models both reservoirs have infinite capacities. The third model is similar to the second one, however the second reservoir is now finite. Furthermore, a feedback mechanism is active, such that the rates at which the first reservoir fills or depletes depend on the state (empty or nonempty) of the second reservoir.The models are analysed by means of Markov processes and regenerative processes in combination with truncation, level crossing and other techniques. The extensive calculations were facilitated by the use of computer algebra. This approach leads to closed-form solutions to the steady-state joint distribution of the content of the two reservoirs in each of the models.     

具有随机风险的公司最优投资策略

本文讨论具有随机风险的公司的最优投资策略问题,公司投资选择是存款、贷款及股票交易、,因市场的不完备性,公司在任一时刻存在概率为正值的破产可能性,本文主要结果是：从贷款利率高于存款利率的实际出发,运用最优随机控制理论,得到使公司生存概率取得最大值的最优投资策略,以及相应的最大生存概率,并并对这些结果给出了严格证明。