Optimal admission pricing policies for M/Ek/1 queues

This paper extends the Low-Lippman M/M/1 model to the case of Gamma service times. Specifically, we have a queue in which arrivals are Poisson, service time is Gamma-distributed, and the arrival rate to the system is subject to setting an admission fee p. The arrival rate λ(p) is non-increasing in p. We prove that the optimal admission fee p^* is a non-decreasing function of the customer work load on the server. The proof is for an infinite capacity queue and holds for the infinite horizon continuous time Markov decision process. In the special case of exponential service time, we extend the Low-Lippman model to include a state-dependent service rate and service cost structure (for finite or infinite time horizon and queue capacity). Relatively recent dynamic programming techniques are employed throughout the paper. Due to the large class of functions represented by the Gamma family, the extension is of interest and utility.     

A queuing process with bilevel hysteretic service-rate control

The purpose of this paper is to analyze the effect of a particular control doctrine applied to the service mechanism of a queuing process. A bilevel hysteretic control based on queue length control levels is employed in an M/M/1 queuing system. Expressions are obtained for queue length probabilities, the first two factorial moments of queue length and two figures of merit for describing control performance under the assumption of statistical equilibrium. Computational examples illustrate the effects on queuing processes subject to this type of control. Several cost formulae are considered for comparison of costs when the queue control doctrine is varied. Situations in which hysteretic control is useful are discussed.     

The single server queue in discrete time-numerical analysis III

This paper deals with the stationary analysis of the finite, single server queue in discrete time. The following stntionary distributions and other quantities of practical interest are investigated: (1) the joint density of the queue length and the residual service time, (2) the queue length distribution and its mean, (3) the distribution of the residual service time and its mean, (4) the distribution and the expected value of the number of customers lost per unit of time due to saturation of the waiting capacity, (5) the distribution and the mean of the waiting time, (6) the asymptotic distribution of the queue length following departures The latter distribution is particularly noteworthy, in view of the substantial difference which exists, in general, between the distributions of the queue lengths at arbitrary points of time and those immediately following departures.     

Allocating capacity in parallel queues to improve their resilience to deliberate attack

We develop models that lend insight into how to design systems that enjoy economies of scale in their operating costs, when those systems will subsequently face disruptions from accidents, acts of nature, or an intentional attack from a well‐informed attacker. The systems are modeled as parallel M/M/1 queues, and the key question is how to allocate service capacity among the queues to make the system resilient to worst‐case disruptions. We formulate this problem as a three‐level sequential game of perfect information between a defender and a hypothetical attacker. The optimal allocation of service capacity to queues depends on the type of attack one is facing. We distinguish between deterministic incremental attacks, where some, but not all, of the capacity of each attacked queue is knocked out, and zero‐one random‐outcome (ZORO) attacks, where the outcome is random and either all capacity at an attacked queue is knocked out or none is. There are differences in the way one should design systems in the face of incremental or ZORO attacks. For incremental attacks it is best to concentrate capacity. For ZORO attacks the optimal allocation is more complex, typically, but not always, involving spreading the service capacity out somewhat among the servers. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Workload distribution of discrete‐time parallel queues with two servers

We study discrete‐time, parallel queues with two identical servers. Customers arrive randomly at the system and join the queue with the shortest workload that is defined as the total service time required for the server to complete all the customers in the queue. The arrivals are assumed to follow a geometric distribution and the service times are assumed to have a general distribution. It is a no‐jockeying queue. The two‐dimensional state space is truncated into a banded array. The resulting modified queue is studied using the method of probability generating function (pgf) The workload distribution in steady state is obtained in form of pgf. A special case where the service time is a deterministic constant is further investigated. Numerical examples are illustrated. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 440–454, 2000     

On multipopulation queuing systems with first-come first-served discipline

Steady-state probabilities for a multipopulation single channel multiserver queuing system with first-come first-served queue discipline are established. Special cases lead to simplified formulas which sometimes can be evaluated using existing statistical and/or queuing tables. Optimization aspects are considered involving control of service rates. Several applications are presented.     

A queueing process with varying degree of service

This article describes and presents some results of the analysis of a queueing process where the degree of service given to a customer is subject to control by the management. The control doctrine used is based on the queue length and also on the recent history of the system.     

Markov chain analyses of multiprogrammed computer systems

Most operating systems for large computing facilities involve service disciplines which base, to some extent, the sequencing of object program executions on the amount of running time they require. It is the object of this paper to study mathematical models of such service disciplines applicable to both batch and time-shared processing systems. In particular, Markov queueing models are defined and analyzed for round-robin and foreground-background service disciplines. With the round-robin discipline, the service facility processes each program or job for a maximum of q seconds; if the program's service is completed during this quantum, it leaves the system, otherwise it returns to the end of the waiting line to await another quantum of service. With the foreground-background discipline each new arrival joins the end of the foreground queue and awaits a single quantum of service. If it requires more it is subsequently placed at the end of the background queue which is allocated service only when the foreground queue is empty. The analysis focuses on the efficiency of the above systems by assuming a swap or set-up time (overhead cost) associated with the switching of programs on and off the processor. The analysis leads to generating functions for the equilibrium queue length probabilities, the moments of this latter distribution, and measures of mean waiting times. The paper concludes with a discussion of the results along with several examples.     

Approximation for the departure process of a queue in a network

A simple renewal process is identified to approximate the complex departure process of a queue often found in queueing network models. The arrival process to the queue is the superposition or merging of several independent component-renewal processes that are approximations of departure processes from other queues and external arrival processes; there is a single server with exponential service times, and the waiting space is infinite. The departure process of this queue is of interest because it is the arrival process to other queues in the network. The approximation proposed is a hybrid; the mean and variance of the approximating departure intervals is a weighted average of those determined by basic methods in Whitt [41] with the weighting function empirically determined using simulation. Tandem queueing systems with superposition arrival processes and exponential service times are used to evaluate the approximation. The departure process of the first queue in the tandem is approximated by a renewal process, the tandem system is replaced by two independent queues, and the second queue is solved analytically. When compared to simulation estimates, the average absolute error in hybrid approximations of the expected number in the second queue is 6%, a significant improvement over 22–41% in the basic methods.     

无标度网络上的队列资源重分配策略

在实际的通信系统中,由于物理条件的限制,每个节点都具有有限长度的队列。研究了在有限队列资源的条件下,队列资源对无标度数据流动力学的影响。提出了一种队列资源重分配策略,策略中节点的队列长度与节点的介数成正比。仿真结果表明,在无标度网络中使用最短路径路由的条件下,所提出的重分配策略可以有效地改进网络的容量。同时对有限队列资源条件下的网络容量进行了理论分析。     

State-dependent gap acceptance

This paper considers a traffic problem in which vehicles queue up according to a Poisson process on a single-lane entrance ramp prior to merging into a major stream of vehicular traffic. In order to then prevent the ramp queue from becoming too large, a model is proposed which considers a lowering of the critical gap as the ramp queue size increases. With the critical gap assumed to be a nonincreasing function of the number of vehicles on the ramp at instances that correspond to departure times of lead vehicles from the ramp queue, the resultant model is an M/G/1 queue with state-dependent service times. Some general results are obtained for this model and a specific case discussed in moderate detail.     

On a single-server queue with state-dependent service

This paper discusses a class of queueing models in which the service time of a customer al a single server facility is dependent on the queue size at the onset of its service. The Laplace transform for the wait in queue distribution is derived and the utilization of the server is given when the arrival is a homogeneous Poisson process.     

一个具有相互独立、不同分布服务时间序列的排队模型

讨论的排队模型 ,放宽了GI/G/1系统中“服务时间独立同分布”的限制 ,只要求各服务时间相互独立 ,因而较GI/G/1排队模型能更合理地拟合实际问题 .在此较宽的条件下 ,利用补充变量的方法 ,求得了该排队系统队长的瞬时分布     

Waiting time analysis for a queueing system with time‐limited service and exponential timer

We consider a single‐queue with exhaustive or gated time‐limited services and server vacations, in which the length of each service period at the queue is controlled by a timer, i.e., the server serves customers until the timer expires or the queue becomes empty, whichever occurs first, and then takes vacations. The customer whose service is interrupted due to the timer expiration may be attended according to nonpreemptive or preemptive service disciplines. For the M/G/1 exhaustive/gated time‐limited service queueing system with an exponential timer and four typical preemptive/nonpreemptive service disciplines, we derive the Laplace—Stieltjes transforms and the moment formulas for waiting times and sojourn times through a unified approach, and provide some new results for these time‐limited service disciplines. © John Wiley & Sons, Inc. Naval Research Logistics 48: 638–651, 2001.     

On queues with dependent interarrival and service times

AnM/G/1 queueing system is studied in which the service time required by a customer is dependent on the interarrival time between his arrival and that of his predecessor Assuming the two variables are “associated,” we prove that the expected delay in this system is less than or equal to than of a conventional M/G/1 queue This conclusion has been verified via simulation by Mitchell and Paulson [9] for a special class of dependent M/M/1 queue. Their model is a special case of the one we consider here. We also study another modified GI/G/1 queue. where the arrival process and/or the service process are individually “associated”.     

Machine maintenance with workload considerations

Machine maintenance is modeled in the setting of a single‐server queue. Machine deterioration corresponds to slower service rates and failure. This leads to higher congestion and an increase in customer holding costs. The decision‐maker decides when to perform maintenance, which may be done pre‐emptively; before catastrophic failures. Similar to classic maintenance control models, the information available to the decision‐maker includes the state of the server. Unlike classic models, the information also includes the number of customers in queue. Considered are both a repair model and a replacement model. In the repair model, with random replacement times, fixed costs are assumed to be constant in the server state. In the replacement model, both constant and variable fixed costs are considered. It is shown in general that the optimal maintenance policies have switching curve structure that is monotone in the server state. However, the switching curve policies for the repair model are not always monotone in the number of customers in the queue. Numerical examples and two heuristics are also presented. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A queue with waiting time dependent service times

Sufficient conditions are developed for the ergodicity of a single server, first-come-first-serve queue with waiting time dependent service times.     

On the discrete-time queue length distribution under markov-dependent phases

The technique of probability generating functions has been applied to solve the steady state behavior of a discrete-time, single-channel, queueing problem wherein the arrivals to the queue at consecutive time-marks are statistically independent, but the service is accomplished in phases which are Markov-dependent. Special cases of importance have been discussed. In the end, mean number of phases, its special cases, the mean queue lengths, and the variances have been ascertained.     

Optimal control for entry of many classes of customers to an M/M/1 queue

A method is developed for determining the optimal policy for entry of customers from many independent classes of Poisson arrivals to a first-come, first-serve (for customers admitted to the queue) single-server queue with exponential service times. The solution technique utilizes a semi-Markov formulation or the decision problem.     

A note on a computational model for a data/voice communication queueing system

Certain types of communication nodes can be viewed as multichannel queueing systems with two types of arrival streams. Data arrivals are characterized by high arrival and service rates and have the ability to queue if all service channels are busy. Voice arrivals have small arrival and service rates and do not have the ability to wait when the channels are full. Computational procedures are presented for obtaining the invariant probabilities associated with the queueing model.