The interdeparture time distribution for each class in the ∑Mi/Gi/1 queue with set-up times and repeated server vacations

This paper studies the interdeparture time distribution of one class of customers who arrive at a single server queue where customers of several classes are served and where the server takes a vacation whenever the system becomes empty or is empty when the server returns from a vacation. Furthermore, the first customer in the busy period is allowed to have an exceptional service time (set-up time), depending on the class to which this customer belongs. Batches of customers of each class arrive according to independent Poisson processes and compete with each other on a FIFO basis. All customers who belong to the same class are served according to a common generally distributed service time. Service times, batch sizes and the arrival process are all assumed to be mutually independent. Successive vacation times of the server form independent and identically distributed sequences with a general distribution.For this queueing model we obtain the Laplace transform of the interdeparture time distribution for each class of customers whose batch size is geometrically distributed. No explicit assumptions of the batch size distributions of the other classes of customers are necessary to obtain the results.The paper ends by showing how the mathematical results can be used to evaluate a protocol that controls access to a shared medium of an ATM passive optical network. The numerical results presented in the last section of this paper show that the bundle spacing principle that is used by the permit distribution algorithm of this protocol introduces high delays and in many cases also more variable interdeparture times for the ATM cells of individual connections. An alternative algorithm is proposed that does not cope with these performance short comings and at the same time conserves the good properties of the protocol.     

The departure process of an N/G/1 queue

The departure process of an N/G/1 queue is investigated. The arrival process called an N process is a versatile point process and includes, for example, a Markov-modulated Poisson process, which is comprised of models of packetized voice and video traffic arrival processes. The first passage analysis yields LSTs of distributions of the interdeparture times. Emphasis is on the interdeparture times of an N/D/1 queue. Numerical examples show that correlation of interarrival times is likely to be preserved in interdeparture times, and that the departure of a voice packet multiplexer can be expected to be smoothed for a normal load. The result in this paper enables evaluation of the smoothing effect of burst traffic through nodes in Asynchronous Transfer Mode networks.     

Interdeparture times from a queuing system with preemptive resume priority

We derive the Laplace-Stieltjes transform of the limiting interdeparture times distribution for each class of customers of a queuing system with preemptive resume priority, Poisson inputs and general service times. Numerical results and an application to queuing networks are also given.     

The <Emphasis Type="Italic">MAP/M/N</Emphasis> retrial queueing system with time-phased batch arrivals

We consider a retrial queueing system with batch arrival of customers. Unlike standard batch arrival, where a whole batch enters the system simultaneously, we assume that customers of a batch (session) arrive one by one in exponentially distributed time intervals. Service time is exponentially distributed. The batch arrival flow is MAP. The number of customers in a session is geometrically distributed. The number of sessions that can enter the system simultaneously is a control parameter. We analyze the joint probability distribution of the number of sessions and customers in the system using the techniques of multidimensional asymptotically quasi-Toeplitz Markov chains.     

A note on the departure process in an (S− 1, S) inventory system

Steady-state distribution functions are derived for interdeparture time (of filled requests) in an (S?1,S) inventory system in which (demand) arrival process is Poisson, replenishment times are independently and identically distributed negative exponential variates, and service is on a first-come-first-served basis. The departure process proves to be identical to the arrival process. Some implications of this property in the study of queueing networks with (S ? 1,S) inventory stations are discussed.     

Predicting departure times in multi-stage queueing systems

We develop an approximation model for the state-dependent sojourn time distribution of customers or orders in a multi-stage, multi-server queueing system, when interarrival and service times can take on general distributions. The model can be used to make probabilistic statements about the departure time of a customer or order, given the number and location of customers currently in process or waiting, and these probabilities can be recomputed while waiting at any point during the sojourn time. The model uses phase-type distributions and a new method to estimate the remaining processing times of customers in service when the sojourn time distribution is computed.     

The <Emphasis Type="Italic">MAP</Emphasis> + <Emphasis Type="Italic">MAP/PH</Emphasis>/1/<Emphasis Type="Italic">N</Emphasis> queuing system with single and batch arrivals of customers

The single-server queuing system with finite buffer was considered. The customers may arrive one-by-one or in batches. Arrivals of single customers and their batches obey the Markov input processes. The customers from a batch taken for servicing come one at a time at the exponentially distributed time intervals. The numbers of customers in batches are distributed geometrically. The time of customer servicing has a phase-type distribution. The numbers of batches and single customers that may be simultaneously accepted by the system are controllable parameters. The joint distribution of the number of batches and the number of customers in system, loss probabilities, distribution of the time of batch sojourn, and problems of optimization were analyzed.     

Queueing analysis and optimal control of and systems

We first consider a finite-buffer single server queue where arrivals occur according to batch Markovian arrival process (BMAP). The server serves customers in batches of maximum size ‘b’ with a minimum threshold size ‘a’. The service time of each batch follows general distribution independent of each other as well as the arrival process. We obtain queue length distributions at various epochs such as, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue length, mean waiting time, probability of blocking, etc. have been obtained. Total expected cost function per unit time is also derived to determine the optimal value N^* of N at a minimum cost for given values of a and b. Secondly, we consider a finite-buffer single server queue where arrivals occur according to BMAP and service process in this case follows a non-renewal one, namely, Markovian service process (MSP). Server serves customers according to general bulk service rule as described above. We derive queue length distributions and important performance measures as above. Such queueing systems find applications in the performance analysis of communication, manufacturing and transportation systems.     

Queue size distribution in a discrete-time D-BMAP/G/1 retrial queue

We consider a discrete-time batch Markovian arrival process (D-BMAP)/G/1 retrial queue. We find the light-tailed asymptotics for the stationary distributions of the number of customers at embedded epochs and at arbitrary time. Using these tail asymptotics we propose a method for calculating the stationary distributions of the number of customers at embedded epochs and at arbitrary time. Numerical examples are presented to illustrate our results.     

Triggered Concurrent Batch Arrivals and Batch Departures in Queueing Networks

We consider a queueing network with concurrent batch arrivals and batch services. In addition to the batch movement assumptions made in the regular queueing literature, in this paper we assume that the batch movement can also be triggered by arrivals and/or departures. Specifically, an arriving or departing batch may induce another event to occur before they are routed. This triggered event may either be the addition of a batch of customers to the network, or the removal of a batch of customers from the network. This special feature makes it useful in the study of discrete event dynamic systems with concurrent or simultaneous sequential activities. By assuming that there is an additional arrival process to the network, we show that its stationary distribution has a product form, which depends on a set of non-standard nonlinear traffic equations. Furthermore, this simple product form solution is a stochastic upper bound for the network without the additionally introduced arrivals. Besides, we show that the network satisfies a class of non-standard local balance equations.     

Analysis of a finite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service

We consider a finite capacity single server queue in which the customers arrive according to a Markovian arrival process. The customers are served in batches following a ‘general bulk service rule’. The service times, which depend on the size of the batch, are generally distributed. We obtain, in steady-state, the joint distribution of the random variables of interest at various epochs. Efficient computational procedures in the case of phase type services are presented. An illustrative numerical example to bring out the qualitative nature of the model is presented.     

Reducing congestion in bulk-service finite-buffer queueing system using batch-size-dependent service

In the design and analysis of any queueing system, one of the main objectives is to reduce congestion which can be achieved by controlling either arrival-rates or service-rates. This paper adopts the latter approach and analyzes a single-server finite-buffer queue where customers arrive according to the Poisson process and are served in batches of minimum size a with a maximum threshold limit b. The service times of the batches are arbitrarily distributed and depends on the size of the batches undergoing service. We obtain the joint distribution of the number of customers in the queue and the number with the server, and distributions of the number of customers in the queue, in the system, and the number with the server. Various performance measures such as the average number of customers in the queue (system) and with the server etc. are obtained. Several numerical results are presented in the form of tables and graphs and it is observed that batch-size-dependent service rule is more effective in reducing the congestion as compared to the one when service rates of the batches remain same irrespective of the size of the batch. This model has potential application in manufacturing, computer-communication network, telecommunication systems and group testing.     

A finite capacity BMAP K /G K /1 queue with the generalized foreground-background processor-sharing discipline

A queueing system with a batch Markov arrival process, several types of customers, generalized foreground-background processor-sharing discipline with minimal served length, and separate finite buffers for customers of different types or a common finite buffer for customers of all types is studied. Mathematical relations for computing the stationary joint distributions of the number of customers of all types in the system are derived.     

First and Second Derivative Estimators for Closed Jackson-Like Queueing Networks Using Perturbation Analysis Techniques

We consider a closed Jackson—like queueing network with arbitrary service time distributions and derive an unbiased second derivative estimator of the throughput over N customers served at some node with respect to a parameter of the service distribution at that node. Our approach is based on observing a single sample path of this system, and evaluating all second-order effects on interdeparture times as a result of the parameter perturbation. We then define an estimator as a conditional expectation over appropriate observable quantities, as in Smoothed Perturbation Analysis (SPA). This process recovers the first derivative estimator along the way (which can also be derived using other techniques), and gives new insights into event order change phenomena which are of higher order, and on the type of sample path information we need to condition on for higher-order derivative estimation. Despite the complexity of the analysis, the final algorithm we obtain is relatively simple. Our estimators can be used in conjunction with other techniques to obtain rational approximations of the entire throughput response surface as a function of system parameters.     

Stationary distribution of the closed queuing network with batch transitions of customers

Consideration was given to the closed queuing network with batch jockeying customers. This network extends the Gordon-Newell model to the case of batch customer servicing. The serviced batch discharges a node without changing its size and arrives to another node with a probability depending on its size. Some constraints must be imposed on the servicing process. The form of the stationary distribution was established, and an efficient algorithm to determine it was proposed.     

Stability, queue length, and delay of deterministic and stochasticqueueing networks

We present two types of stability problems: 1) conditions for queueing networks that render bounded queue lengths and bounded delay for customers, and 2) conditions for queueing networks in which the queue length distribution of a queue has an exponential tail with rate &thetas;. To answer these two types of stability problems, we introduce two new notions of traffic characterization: minimum envelope rate (MER) and MER with respect to &thetas;. We also develop a set of rules for network operations such as superposition, input-output relation of a single queue, and routing. Specifically, we show that: 1) the MER of a superposition process is less than or equal to the sum of the MER of each process, 2) a queue is stable in the sense of bounded queue length if the MER of the input traffic is smaller than the capacity, 3) the MER of a departure process from a stable queue is less than or equal to that of the input process, and 4) the MER of a routed process from a departure process is less than or equal to the MER of the departure process multiplied by the MER of the routing process. Similar results hold for MER with respect to &thetas; under a further assumption of independence. For single class networks with nonfeedforward routing, we provide a new method to show that similar stability results hold for such networks under the first come, first served policy. Moreover, when restricting to the family of two-state Markov modulated arrival processes, the notion of MER with respect to &thetas; is shown to be equivalent to the recently developed notion of effective bandwidth in communication networks     

基于排队分析的端到端路径可用带宽的测量

端到端路径可用带宽是衡量网络性能的重要指标.目前,大多数对可用带宽测量的研究都集中在自拥塞方法上,但自拥塞方法会严重影响路径上的原有流量,因此,提出了一种基于模型的方法,将Internet视为一个具有单一服务单元且服务两种流量(探测流量和背景流量)的排队系统,通过分析探测流量离开排队系统的过程,在路径的负载与探测流量离去间隔的变异系数之间建立了对应关系,并在此基础上获得可用带宽值.与自拥塞方法相比,不会造成路径拥塞且具有较好的准确性.     

Computing steady state probabilities in λ(n)/G/1/K queue

In this paper a recursive method is developed to obtain the steady state probability distribution of the number in system at arbitrary and departure time epochs of a single server state-dependent arrival rate queue λ(n)/G/1/K in which the arrival process is Markovian with arrival rates λ(n) which depend on the number of customers n in the system and general service time distribution. It is assumed that there exists an integer K such that λ(n) > 0 for all 0 n < K and λ(n) = 0 for all n K. Numerical results have been presented for many queueing models by suitably defining the function λ(n). These include machine interference model, queues with balking, queues with finite waiting space and machine interference model with finite waiting space. These models have wide application in computer/communication networks.     

Analysis of the resequencing buffer in a homogeneous M/M/2 queue

The resequencing problem is encountered in many practical information systems such as distributed database and communication networks. In these systems customers, such as messages in a computer network, have to be delivered to users in their original order. Therefore, those customers which become out of order due to the randomness of the system are forced to wait in a resequencing buffer so that their delivered order can be guaranteed. The previous work on the resequencing problem mainly concentrated on the delay aspect. From both theoretical and practical viewpoints, however, the queue length characteristics of the resequencing buffer are also significant. We consider the queue length distribution of the resequencing buffer fed by a homogeneous M/M/2 queue. The exact analysis is carried out for the probability mass functions of the queue length in equilibrium and the maximal occupancy which corresponds to the queue length just before the departure instants of customers from the resequencing buffer.     

基于Q学习的供应链分销系统最优订货策略研究

研究由一个制造商和多个分销商组成的分销系统的最优订货策略问题.在外部顾客需求不断变化的情况下,以不断提高分销系统双方合作绩效为目标,基于Q学习算法来确定每个分销商的最优订货批量.实例结果表明,在外部需求不断变化的条件下,该算法能简便地解决供应链企业分销系统合作中的最优订货批量问题.