Heterogeneous multitrunking queueing systems with thresholds

Queueing systems with multiple servers are commonly used to model telecommunications systems. But, in general, the service rate of each of the servers is not the same. This fact is indeed true in a communication network where one path (server) may be a terrestrial link and the other (server) a satellite link with its inherent propagation delay. In this article we consider a two-server system where arriving customers are first placed in the queue for the faster server until that queue size reaches a certain threshold, whereupon they are diverted to the slower server. Additional arriving customers are assigned to the slower server until the faster server's queue drops to another lower threshold, at which point arrivals are reassigned to the faster server. We develop an exact mathematical model of the steady-state behavior of each queueing system and a simple analytic approximation.     

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.     

Stochastic analysis of a single server retrial queue with general retrial times

Retrial queueing systems are widely used in teletraffic theory and computer and communication networks. Although there has been a rapid growth in the literature on retrial queueing systems, the research on retrial queues with nonexponential retrial times is very limited. This paper is concerned with the analytical treatment of an M/G/1 retrial queue with general retrial times. Our queueing model is different from most single server retrial queueing models in several respectives. First, customers who find the server busy are queued in the orbit in accordance with an FCFS (first‐come‐first‐served) discipline and only the customer at the head of the queue is allowed for access to the server. Besides, a retrial time begins (if applicable) only when the server completes a service rather upon a service attempt failure. We carry out an extensive analysis of the queue, including a necessary and sufficient condition for the system to be stable, the steady state distribution of the server state and the orbit length, the waiting time distribution, the busy period, and other related quantities. Finally, we study the joint distribution of the server state and the orbit length in non‐stationary regime. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 561–581, 1999     

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     

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.     

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     

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.     

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.     

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.     

The threshold policy in the M/G/1 queue with server vacations

This article deals with the M/G/1 queue with server vacations in which the return of the server to service depends on the number of customers present in the system. The main goal is optimization, which is done under the average cost criterion in the multiple- and single-vacation models as well as for the “total cost for one busy cycle” criterion in the multiple-vacation case. Expressions that characterize the optimal number of customers, below which the server should not start a new service period, are exhibited for the various cases. It is found that under the average cost criterion, the expression may be universal in the sense that it may hold for a general class of problems including such that arise in production planning and inventory theory (for the particular cost structure discussed).     

Equilibrium customer strategies and social–profit maximization in the single‐server constant retrial queue

We consider the single‐server constant retrial queue with a Poisson arrival process and exponential service and retrial times. This system has not waiting space, so the customers that find the server busy are forced to abandon the system, but they can leave their contact details. Hence, after a service completion, the server seeks for a customer among those that have unsuccessfully applied for service but left their contact details, at a constant retrial rate. We assume that the arriving customers that find the server busy decide whether to leave their contact details or to balk based on a natural reward‐cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine the customers' behavior, and we identify the Nash equilibrium joining strategies. We also study the corresponding social and profit maximization problems. We consider separately the observable case where the customers get informed about the number of customers waiting for service and the unobservable case where they do not receive this information. Several extensions of the model are also discussed. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Estimation in single server queues

Moment and maximum likelihood estimates (m.l.e.'s) arc investigated for nonparametric and parametric models for a single server queue observed over a random time horizon, namely, up to the nth departure epoch. Also. m.l.e's of the mean interarrival time and mean service time in anM/M/1 queue observed over a fixed time-interval are studied Limit distributions of these estimates are obtained Without imposing steady state assumptions on the queue-size or waiting time processes.     

On the busy period of the M/G/1 retrial queue

The M/G/1 queue with repeated attempts is considered. A customer who finds the server busy, leaves the service area and joins a pool of unsatisfied customers. Each customer in the pool repeats his demand after a random amount of time until he finds the server free. We focus on the busy period L of the M/G/1$ retrial queue. The structure of the busy period and its analysis in terms of Laplace transforms have been discussed by several authors. However, this solution has serious limitations in practice. For instance, we cannot compute the first moments of L by direct differentiation. This paper complements the existing work and provides a direct method of calculation for the second moment of L. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 115–127, 2000     

A sequential scheduling problem with impatient jobs

There are n customers that need to be served. Customer i will only wait in queue for an exponentially distributed time with rate λ_i before departing the system. The service time of customer i has distribution F_i, and on completion of service of customer i a positive reward r_i is earned. There is a single server and the problem is to choose, after each service completion, which currently in queue customer to serve next so as to maximize the expected total return. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 659–663, 2015     

The effect of catastrophes on the strategic customer behavior in queueing systems

We consider the single server Markovian queue subject to Poisson generated catastrophes. Whenever a catastrophe occurs, all customers are forced to abandon the system, the server is rendered inoperative and an exponential repair time is set on. During the repair time new arrivals are allowed to join the system. We assume that the arriving customers decide whether to join the system or balk, based on a natural linear reward‐cost structure with two types of rewards: A (usual) service reward for those customers that receive service and a (compensation) failure reward for those customers that are forced to abandon the system due to a catastrophe. We study the strategic behavior of the customers regarding balking and derive the corresponding (Nash) equilibrium strategies for the observable and unobservable cases. We show that both types of strategic behavior may be optimal: to avoid the crowd or to follow it. The crucial factor that determines the type of customer behavior is the relative value of the service reward to the failure compensation. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Level crossing analysis of priority queues and a conservation identity for vacation models

The two purposes of this article are to illustrate the power and simplicity of level crossing analysis and to present a conservation identity for M/G/1 priority queues with server vacations. To illustrate the use of level crossing analysis we apply it to preemptive (resume) priority M/G/1 queues with single- and multiple-server vacations considered by Kella and Yechiali (1986) and to non-preemptive priority M/M/c queues considered by Kella and Yechiali (1985). The conservation identity presented here states that the ratios of mean waiting times in an M/G/1 queue with and without server vacation policies are independent of the service discipline for first come first served, shortest processing time, shortest processing time within generations and non-preemptive priority service disciplines.     

Scheduling policies for an antiterrorist surveillance system

This article concerns scheduling policies in a surveillance system aimed at detecting a terrorist attack in time. Terrorist suspects arriving at a public area are subject to continuous monitoring, while a surveillance team takes their biometric signatures and compares them with records stored in a terrorist database. Because the surveillance team can screen only one terrorist suspect at a time, the team faces a dynamic scheduling problem among the suspects. We build a model consisting of an M/G/1 queue with two types of customers—red and white—to study this problem. Both types of customers are impatient but the reneging time distributions are different. The server only receives a reward by serving a red customer and can use the time a customer has spent in the queue to deduce its likely type. In a few special cases, a simple service rule—such as first‐come‐first‐serve—is optimal. We explain why the problem is in general difficult and we develop a heuristic policy motivated by the fact that terrorist attacks tend to be rare events. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

An M/M/1 queue with delayed feedback

We present some results for M/M/1 queues with finite capacities with delayed feedback. The delay in the feedback to an M/M/1 queue is modelled as another M-server queue with a finite capacity. The steady state probabilities for the two dimensional Markov process {N(t), M(t)} are solved when N(t) = queue length at server 1 at t and M(t) = queue length at server 2 at t. It is shown that a matrix operation can be performed to obtain the steady state probabilities. The eigenvalues of the operator and its eigenvectors are found. The problem is solved by fitting boundary conditions to the general solution and by normalizing. A sample problem is run to show that the solution methods can be programmed and meaningful results obtained numerically.     

A note on sojourn times in M/G/1 queues with instantaneous,bernoulli feedback

Queueing systems which include the possibility for a customer to return to the same server for additional service are called queueing systems with feedback. Such systems occur in computer networks for example. In these systems a chosen customer will wait in the queue, be serviced and then, with probability p, return to wait again, be serviced again and continue this process until, with probability (1 – p) = q, it departs the system never to return. The time of waiting plus service time, the nth time the customer goes through, we will call his nth sojourn time. The (random) sum of these sojourn times we will call the total sojourn time (abbreviated, sojourn time when there is no confusion which sojourn time we are talking about). In this paper we study the total sojourn time in a queueing system with feedback. We give the details for M/G/1 queues in which the decision to feedback or not is a Bernoulli process. While the details of the computations can be more difficult, the structure of the sojourn time process is unchanged for the M/G/1 queue with a more general decision process as will be shown. We assume the reader is familiar with Disney, McNickle and Simon [1].     

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