随机环境中的M/MY/1排队系统

文献［４］讨论了随机环境中的Ｍ／Ｍ／１排队模型，本文提出和讨论随机环境中的Ｍ／Ｍｙ／１排队模型，在统计平衡条件下给出了队长和等待队长的平稳分布以及平均队长和平均等待队长，得到了等待时间和逗留时间分布以及平均等待时间和平均逗留时间。     

Customer delays in M/M/c repair systems with spares

A service center to which customers bring failed items for repair is considered. The items are exchangeable in the sense that a customer is ready to take in return for the failed item he brought to the center any good item of the same kind. This exchangeability feature makes it possible for the service center to possess spares. The focus of the article is on customer delay in the system—the time that elapses since the arrival of a customer with a failed item and his departure with a good one—when repaired items are given to waiting customers on a FIFO basis. An algorithm is developed for the computation of the delay distribution when the item repair system operates as an M/M/c queue.     

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].     

The random order service G/M/m queue

The waiting time in the random order service G/M/m queue is studied. For the Laplace transform we obtain a simpler representation than previously available. For the moments, an explicit recursive algorithm is given and carried out numrically for some cases. This gives rise to the conjecture that the waiting-time distributio can be approximated by the one for M/M/m after a suitable change of scale.     

M/G/1(RVT,P(j))排队系统的另一分析方法及其应用

M/G/1(RVT,P(j))表示服务员具有随机长度休息时间(RVT)的、且一休息时间结束时有 j 个顾客等待的概率为 P(j)的、修正的 M/G/1 排队系统。我们用嵌入 Markov 链的技术已详细地分析过这一排队系统,这里提供另一分析方法。最后,应用这个排队系统的分析结果,对时隙 ALOHA 卫星公用信道的分组碰撞概率计算公式作了推导。     

The queue G/M/m/N: Busy period and cycle,waiting time under reverse and random order service

The busy period, busy cycle, and the numbers of customers served and lost therein, of the G/M/m queue with balking is studied via the embedded Markov chain approach. It is shown that the expectations of the two discrete variables give the loss probability. For the special case G/M/1/N a closed expression in terms of contour integrals is obtained for the Laplace transform of these four variables. This yields as a byproduct the LIFO waiting time distribution for the G/M/m/N queue. The waiting time under random order service for this queue is also studied.     

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     

The effects of information on a queue with balking and phase‐type service times

This article generalizes the models in Guo and Zipkin, who focus on exponential service times, to systems with phase‐type service times. Each arriving customer decides whether to stay or balk based on his expected waiting cost, conditional on the information provided. We show how to compute the throughput and customers' average utility in each case. We then obtain some analytical and numerical results to assess the effect of more or less information. We also show that service‐time variability degrades the system's performance. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

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     

Matrix-geometric solution of discrete time MAP/PH/1 priority queue

We use the matrix-geometric method to study the discrete time MAP/PH/1 priority queue with two types of jobs. Both preemptive and non-preemptive cases are considered. We show that the structure of the R matrix obtained by Miller for the Birth-Death system can be extended to our Quasi-Birth-Death case. For both preemptive and non-preemptive cases the distributions of the number of jobs of each type in the system are obtained and their waiting times are obtained for the non-preemptive. For the preemptive case we obtain the waiting time distribution for the high priority job and the distribution of the lower priority job's wait before it becomes the leading job of its priority class. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 23–50, 1998     

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.     

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.     

Optimal policies for M/M/m queue with two different kinds of (N,T)‐policies

In this paper, two different kinds of (N, T)‐policies for an M/M/m queueing system are studied. The system operates only intermittently and is shut down when no customers are present any more. A fixed setup cost of K > 0 is incurred each time the system is reopened. Also, a holding cost of h > 0 per unit time is incurred for each customer present. The two (N, T)‐policies studied for this queueing system with cost structures are as follows: (1) The system is reactivated as soon as N customers are present or the waiting time of the leading customer reaches a predefined time T, and (2) the system is reactivated as soon as N customers are present or the time units after the end of the last busy period reaches a predefined time T. The equations satisfied by the optimal policy (N*, T*) for minimizing the long‐run average cost per unit time in both cases are obtained. Particularly, we obtain the explicit optimal joint policy (N*, T*) and optimal objective value for the case of a single server, the explicit optimal policy N* and optimal objective value for the case of multiple servers when only predefined customers number N is measured, and the explicit optimal policy T* and optimal objective value for the case of multiple servers when only predefined time units T is measured, respectively. These results partly extend (1) the classic N or T policy to a more practical (N, T)‐policy and (2) the conclusions obtained for single server system to a system consisting of m (m ≥ 1) servers. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 240–258, 2000     

Optimal control of an M/G/1 queue with impatient priority customers

An optimal operating policy is characterized for the infinite‐horizon average‐cost case of a single server queueing control problem. The server may be turned on at arrival epochs or off at departure epochs. Two classes of customers, each of them arriving according to an independent Poisson processes, are considered. An arriving 1‐customer enters the system if the server is turned on upon his arrival, or if the server is on and idle. In the former case, the 1‐customer is selected for service ahead of those customers waiting in the system; otherwise he leaves the system immediately. 2‐Customers remain in the system until they complete their service requirements. Under a linear cost structure, this paper shows that a stationary optimal policy exists such that either (1) leaves the server on at all times, or (2) turns the server off when the system is empty. In the latter case, we show that the stationary optimal policy is a threshold strategy, this feature being commonplace in most of priority queueing systems and inventory models. However, the optimal policy in our model is determined by two thresholds instead of one. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 201–209, 2001     

A componentwise index of service measurement in multi‐component systems

In this paper we present a componentwise delay measure for estimating and improving the expected delays experienced by customers in a multi‐component inventory/assembly system. We show that this measure is easily computed. Further, in an environment where the performance of each of the item delays could be improved with investment, we present a solution that aims to minimize this measure and, in effect, minimizes the average waiting time experienced by customers. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 50: 2003     

Discrete‐time analysis of MAP/PH/1 multiclass general preemptive priority queue

We use the matrix‐geometric method to study the MAP/PH/1 general preemptive priority queue with a multiple class of jobs. A procedure for obtaining the block matrices representing the transition matrix P is presented. We show that the special upper triangular structure of the matrix R obtained by Miller [Computation of steady‐state probabilities for M/M/1 priority queues, Oper Res 29(5) (1981), 945–958] can be extended to an upper triangular block structure. Moreover, the subblock matrices of matrix R also have such a structure. With this special structure, we develop a procedure to compute the matrix R. After obtaining the stationary distribution of the system, we study two primary performance indices, namely, the distributions of the number of jobs of each type in the system and their waiting times. Although most of our analysis is carried out for the case of K = 3, the developed approach is general enough to study the other cases (K ≥ 4). © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 662–682, 2003.     

Probabilistic priorities in the M/G/1 queue

A simple method is presented for deriving the mean and variance of the queueing time distribution in an M/G/1 queue when the priorities assigned to customers have an assignment probability distribution. Several examples illustrate the results. The mean and variance of the queueing time distribution for the longest service time discipline are derived, and its disadvantages are discussed.     

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”.     

Bilateral phase-type distributions

In this article we define a class of distributions called bilateral phase type (BPH), and study its closure and computational properties. The class of BPH distributions is closed under convolution, negative convolution, and mixtures. The one-sided version of BPH, called generalized phase type (GPH), is also defined. The class of GPH distributions is strictly larger than the class of phase-type distributions introduced by Neuts, and is closed under convolution, negative convolution with nonnegativity condition, mixtures, and formation of coherent systems. We give computational schemes to compute the resulting distributions from the above operations and extend them to analyze queueing processes. In particular, we present efficient algorithms to compute the steady-state and transient waiting times in GPH/GPH/1 queues and a simple algorithm to compute the steady-state waiting time in M/GPH/1 queues.     

有限服务能力的损失制排队系统M/M/1分析

提出了随机服务系统的服务能力问题,给出有限服务能力的损失制M／M／1模型的描述和其稳态解,并求出了评价系统运行的几个主要数量指标。