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     

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     

On the orbit characteristics of the M/G/ 1 retrial queue

In teletraffic applications of retrial queues only the service zone is observable. Another part of a retrial queue, the orbit, which represents the delay before repeated attempts to get service, cannot be observed. Thus, it is very important to get general results about behavior of the orbit. We investigate two characteristics of the orbit, namely, the orbit busy period and the orbit idle period, which seem to be very useful from this point of view. © 1996 John Wiley & Sons, Inc.     

Computational analysis of MMAP[K]/PH[K]/1 queues with a mixed FCFS and LCFS service discipline

This paper studies a queueing system with a Markov arrival process with marked arrivals and PH‐distribution service times for each type of customer. Customers (regardless of their types) are served on a mixed first‐come‐first‐served (FCFS) and last‐come‐first‐served (LCFS) nonpreemptive basis. That is, when the queue length is N (a positive integer) or less, customers are served on an FCFS basis; otherwise, customers are served on an LCFS basis. The focus is on the stationary distribution of queue strings, busy periods, and waiting times of individual types of customers. A computational approach is developed for computing the stationary distribution of queue strings, the mean of busy period, and the means and variances of waiting times. The relationship between these performance measures and the threshold number N is analyzed in depth numerically. It is found that the variance of the virtual (actual) waiting time of an arbitrary customer can be reduced by increasing N. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 399–421, 2000     

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.     

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     

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].     

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     

Waiting time analysis of the M/G/1 queue with finite retrial group

We consider an M/G/1 retrial queue with finite capacity of the retrial group. First, we obtain equations governing the dynamic of the waiting time. Then, we focus on the numerical inversion of the density function and the computation of moments. These results are used to approximate the waiting time of the M/G/1 queue with infinite retrial group for which direct analysis seems intractable. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

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.     

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.     

Stochastic control of queueing systems

Suppose that the state of a queueing system is described by a Markov process { Y_t, t ≥ 0}, and the profit from operating it up to a time t is given by the function f(Y_t). We operate the system up to a time T, where the random variable T is a stopping time for the process Yt. Optimal stochastic control is achieved by choosing the stopping time T that maximizes Ef(Y_T) over a given class of stopping times. In this paper a theory of stochastic control is developed for a single server queue with Poisson arrivals and general service times.     

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.     

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.     

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.     

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.     

Priority queue with customer upgrades

This article is concerned with a general multi‐class multi‐server priority queueing system with customer priority upgrades. The queueing system has various applications in inventory control, call centers operations, and health care management. Through a novel design of Lyapunov functions, and using matrix‐analytic methods, sufficient conditions for the queueing system to be stable or instable are obtained. Bounds on the queue length process are obtained by a sample path method, with the help of an auxiliary queueing system. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Arrival and departure state distributions in the general bulk‐service queue

In this paper, we give an explicit relation between steady‐state probability distributions of the buffer occupancy at customer entrance and departure epochs, for the classical single‐server system G/G^[N]/1 with batch services and for the finite capacity case. The method relies on level‐crossing arguments. For the particular case of Poisson input, we also express the loss probability in terms of state probabilities at departure epochs, yielding probabilities observed by arriving customers. This work provides the “bulk queue” version of a result established by Burke, who stated the equality between probabilities at arrival and departure epochs for systems with “unit jumps.” © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 107–118, 1999     

On queueing systems with variable service capacities

In some queueing systems the total service capacity utilized at any given time is a variable under the control of a decision maker. Management doctrines are examined which prescribe the actual service capacity as a function of the queue length and the recent history of the system. Steady state probabilities, expected queue lengths and frequencies of change in capacity are evaluated for a wide class of possible control schemes. Optimization procedures are outlined.     

Approximations for heavily loaded G/GI/n + GI queues

Motivated by applications to service systems, we develop simple engineering approximation formulas for the steady‐state performance of heavily loaded G/GI/n+GI multiserver queues, which can have non‐Poisson and nonrenewal arrivals and non‐exponential service‐time and patience‐time distributions. The formulas are based on recently established Gaussian many‐server heavy‐traffic limits in the efficiency‐driven (ED) regime, where the traffic intensity is fixed at ρ > 1, but the approximations also apply to systems in the quality‐and‐ED regime, where ρ > 1 but ρ is close to 1. Good performance across a wide range of parameters is obtained by making heuristic refinements, the main one being truncation of the queue length and waiting time approximations to nonnegative values. Simulation experiments show that the proposed approximations are effective for large‐scale queuing systems for a significant range of the traffic intensity ρ and the abandonment rate θ, roughly for ρ > 1.02 and θ > 2.0. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 187–217, 2016