System delay versus system content for discrete-time queueing systems subject to server interruptions

This paper concerns discrete-time queueing systems operating under a first-come-first-served queueing discipline, with deterministic service times of one slot and subject to independent server interruptions. For such systems, we derive a relationship between the probability generating functions of the system content during an arbitrary slot and of the system delay of an arbitrary customer. This relationship is valid regardless of the nature of the arrival process. From this relationship we derive a relationship between the first- and second-order moments of the distributions involved. It is shown that the relationship also applies to subsystems of the queueing system being discussed, and to the waiting time and queue content of a multi-server queueing system with geometric service times and uninterrupted servers.     

The discrete-time preemptive repeat identical priority queue

Priority queueing systems come natural when customers with diversified delay requirements have to wait to get service. The customers that cannot tolerate but small delays get service priority over customers which are less delay-sensitive. In this contribution, we analyze a discrete-time two-class preemptive repeat identical priority queue with infinite buffer space and generally distributed service times. Newly arriving high-priority customers interrupt the on-going service of a low-priority customer. After all high-priority customers have left the system, the interrupted service of the low-priority customer has to be repeated completely. By means of a probability generating functions approach, we analyze the system content and the delay of both types of customers. Performance measures (such as means and variances) are calculated and the impact of the priority scheduling is discussed by means of some numerical examples.     

考虑等待提示策略的呼叫中心流体近似方法

向顾客公布需等待的排队时间用以缓解系统拥挤是目前呼叫中心运营管理的重要手段之一。为了有效刻画等待提示策略下顾客行为变化对呼叫系统性能的影响,采用流体近似方法建立了呼叫排队系统模型。首先,通过排队分析构造等待提示影响下排队行为框架,包含带有心理行为变化特征的多种行为要素概率函数;其次,利用流体方法构建了考虑顾客重拨影响的连续排队模型,并求解了稳态条件下的系统性能指标;最后,通过与仿真模型的对比,验证了该流体近似方法的有效性与精确性。研究结果对于带有等待时间提示的呼叫中心运营具有重要指导作用。     

Delay and partial system contents for a discrete-time G-D-c queue

We consider a discrete-time multiserver queueing system with infinite buffer size, constant service times of multiple slots and a first-come-first-served queueing discipline. A relationship between the probability distributions of the partial system contents and the packet delay is established. The relationship is general in the sense that it doesn’t require knowledge of the exact nature of the arrival process. By means of the relationship, results for the distribution of the partial system contents for a wide variety of discrete-time queueing models can be transformed into corresponding results for the delay distribution. As a result, a separate full analysis of the packet delay becomes unnecessary.      

Delay analysis of a two-class batch-service queue with class-dependent variable server capacity

In this paper, we analyse the delay of a random customer in a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common single-server first-come-first-served queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the length of a sequence of same-class customers. This type of batch server can be found in telecommunications systems and production environments. We first determine the steady state partial probability generating function of the queue occupancy at customer arrival epochs. Using a spectral decomposition technique, we obtain the steady state probability generating function of the delay of a random customer. We also show that the distribution of the delay of a random customer corresponds to a phase-type distribution. Finally, some numerical examples are given that provide further insight in the impact of asymmetry and variance in the arrival process on the number of customers in the system and the delay of a random customer.     

Delay analysis of a discrete-time multiclass slot-bound priority system

This paper introduces a new priority mechanism in discrete-time queueing systems that compromises between first-come-first-served (FCFS) and head-of-line priority. In this scheduling discipline—which we dubbed slot-bound priority—customers of different priority classes entering the system during the same time-slot are served in order of their respective priority class. Customers entering during different slots are served on a FCFS basis. In this paper we study the delay in an N-class discrete-time queueing system under slot-bound priority. General independent arrivals and class-specific general service time distributions are assumed. Expressions for the probability generating function of the delay of a random type-j customer are derived, from which the respective moments are easily obtained. The tail behaviour of these distributions is analyzed as well, and some numerical examples show the effect slot-bound priority can have on the performance measures.     

Performance analysis of discrete-time queue GI/G/1 with negative arrivals

In this paper, we consider a discrete-time GI/G/1 queueing model with negative arrivals. By deriving the probability generating function of actual service time of ordinary customers, we reduced the analysis to an equivalent discrete-time GI/G/1 queueing model without negative arrival, and obtained the probability generating function of buffer contents and random customer delay.     

A general approximation for the single product lot sizing model with queueing delays

The objective of this paper is to derive a general approximation for the single product lot sizing model with queueing delays, explicitly including a non-zero setup time. Most research focuses on bulk (batch) arrival and departure processes. In this paper we assume an individual arrival and departure process allowing the modelling of more realistic demand patterns. A general approximation of the expected lead time and the variance of the lead time is derived. The lead time probability distribution is approximated by means of a lognormal distribution. This allows the manufacturer to quote lead times satisfying a specified customer service level as a function of the lot size. The main result is a convex relationship of the expected lead time and the quoted lead time as a function of the lot size. The results are illustrated by means of numerical examples.     

带有止步和中途退出的M/M/R/N同步多重工作休假排队系统

研究了带有止步和中途退出的M/M/R/N同步多重工作休假排队系统,利用马尔可夫过程理论和矩阵解法求出了含有两个逆阵的系统稳态概率的矩阵解,并得到了系统的平均队长、服务员处在工作休假期的概率以及顾客的平均止步率等性能指标.最后通过数值例子分析了系统的参数对平均队长的影响.     

Complete characterisation of the customer delay in a queueing system with batch arrivals and batch service

Whereas the buffer content of batch-service queueing systems has been studied extensively, the customer delay has only occasionally been studied. The few papers concerning the customer delay share the common feature that only the moments are calculated explicitly. In addition, none of these surveys consider models including the combination of batch arrivals and a server operating under the full-batch service policy (the server waits to initiate service until he can serve at full capacity). In this paper, we aim for a complete characterisation—i.e., moments and tail probabilities - of the customer delay in a discrete-time queueing system with batch arrivals and a batch server adopting the full-batch service policy. In addition, we demonstrate that the distribution of the number of customer arrivals in an arbitrary slot has a significant impact on the moments and the tail probabilities of the customer delay.     

Queueing systems with different types of server interruptions

We consider a queueing system with disruptive and non-disruptive server interruptions. Both disruptive and non-disruptive interruptions may start when there is a customer in service. The customer repeats its service after a disruptive interruption, and continues its service after a non-disruptive interruption. Using a transform approach, we obtain various performance measures such as the moments of the queue content and waiting times. We illustrate our approach by means of some numerical examples.     

A modified HOL priority scheduling discipline: Performance analysis

In this paper, we introduce and analyze a modified HOL (head-of-the-line) priority scheduling discipline. The modification is incorporated to cope with the so-called starvation problem of regular HOL priority queues. We consider a discrete-time single-server queueing system with two priority queues of infinite capacity and with the introduced priority scheme. We show that the use of probability generating functions is suitable for analyzing the system contents and the packet delay. Some performance measures (such as means and variances) of these stochastic quantities will be derived. Furthermore, approximate expressions of the tail probabilities are obtained from the probability generating functions, by means of the dominant-singularity method. These expressions, together with their characteristics, constitute one of the main contributions of this paper. Finally, the impact and significance of the m-HOL (modified HOL) priority scheduling on these performance measures is illustrated by some numerical examples.     

Analysis of a Discrete-Time Queueing System with a Single Server and Heterogeneous Markovian Arrivals

We consider a discrete-time queueing system with a single deterministic server, heterogeneous Markovian arrivals and finite capacity. Most existing techniques model the queueing system using a direct bivariate Markov chain which requires a state space that grows rapidly as the number of customer types increases. In this paper, we define renewal cycles in terms of the input process and model the system occupancy level on each renewal cycle using a one-dimensional Markov chain. We derive the exact joint steady-state probability distribution of both states of input and system occupancy with a considerably reduced state space, which leads to the efficient calculation of overall/individual performance measures such as loss probability and average delay.     

具有第二次可选服务的带反馈的N-策略Mx/G/1(E,MV)排队系统分析

研究N策略下的批量到达的具有第二次可选择服务且两次服务均可反馈的多重休假排队系统。建立了休假、反馈、可选服务多类型相结合的排队模型。本文采用补充变量法，首先建立了系统稳态下的状态转移方程，通过求解得到了稳态下系统队长的概率母函数，进而计算出稳态下系统的平均队长。对稳态队长进行分析之后，我们又给出了稳态队长的随机分解定理，其中给出了附加队长的明确概率解释。     

On queueing with customer impatience until the beginning of service

We study queueing systems where customers have strict deadlines until the beginning of their service. An analytic method is given for the analysis of a class of such queues, namely, models. These are queues with state-dependent Poisson arrival process, exponential service times, multiple servers, FCFS service discipline, and general customer impatience. The state of the system is viewed to be the number of customers in the system. The principal measure of performance is the probability measure induced by the offered waiting time. Other measures of interest are the probability of missing deadline and the probability of blocking. Closed-form solutions are derived for the steady-state probabilities of the state process and some important modeling variables and parameters. The efficacy of our method is illustrated through a numerical example. This revised version was published online in June 2006 with corrections to the Cover Date.     

Implementation of Markovian Queueing Network Model with Multiple Closed Chains

Mathematical strategy portrays the performance evaluation of computer and communication system and it deals with the stochastic properties of the multiclass Markovian queueing system with class-dependent and server-dependent service times. An algorithm is designed where the job transitions are characterized by more than one closed Markov chain. Generating functions are implemented to derive closed form of solutions and product form solution with the parameters such as stability, normalizations constant and marginal distributions. For such a system with N servers and L chains, the solutions are considerably more complicated than those for the systems with one sub-chain only. In Multi-class queueing network, a job moves from a queue to another queue with some probability after getting a service. A multiple class of customer could be open or closed where each class has its own set of queueing parameters. These parameters are obtained by analyzing each station in isolation under the assumption that the arrival process of each class is a state-dependent Markovian process along with different service time distributions. An algorithmic approach is implemented from the generating function representation for the general class of Networks. Based on the algorithmic approach it is proved that how open and closed sub-chain interact with each other in such system. Specifically, computation techniques are provided for the calculation of the Markovian model for multiple chains and it is shown that these algorithms converge exponentially fast.     

Many-Sources Delay Asymptotics with Applications to Priority Queues

In this paper, we study discrete-time priority queueing systems fed by a large number of arrival streams. We first provide bounds on the actual delay asymptote in terms of the virtual delay asymptote. Then, under suitable assumptions on the arrival process to the queue, we show that these asymptotes are the same. As an application of this result, we then consider a priority queueing system with two queues. Using the earlier result, we derive an upper bound on the tail probability of the delay. Under certain assumptions on the rate function of the arrival process, we show that the upper bound is tight. We then consider a system with Markovian arrivals and numerically evaluate the delay tail probability and validate these results with simulations.     

具有批到达的滞后排队系统分析

本文考虑一个基本模型的M^[x]/G^[r]/1/∞的排队系统，引入了(r,N）－策略（也称为滞后系统），首先分析了嵌入马氏链{Qn,n∈N}的平稳分布，得到其概率母函数Ｐ(z)，再利用半再生过程理论得到原系统的稳态队长分布的概率母函数π（ｚ）。     

Analysis of the GIX/M/c model

In this paper, the GI^X/M/c queueing model is analyzed. An explicit expression of the generating function of equilibrium probabilities of customer numbers in the system for the model is derived. Based on the generating function, it is proved that the equilibrium probabilities are given by a linear combination of some geometric terms. Due to this result, other interesting measures are also considered without difficulty. Examples and numerical results are given.     

Analyzing the discrete-timeG (G)/Geo/1 queue using complex contour integration

In this paper, a discrete-time single-server queueing system with an infinite waiting room, referred to as theG ^(G)/Geo/1 model, i.e., a system with general interarrival-time distribution, general arrival bulk-size distribution and geometrical service times, is studied. A method of analysis based on integration along contours in the complex plane is presented. Using this technique, analytical expressions are obtained for the probability generating functions of the system contents at various observation epochs and of the delay and waiting time of an arbitrary customer, assuming a first-come-first-served queueing discipline, under the single restriction that the probability generating function for the interarrival-time distribution be rational. Furthermore, treating several special cases we rediscover a number of well-known results, such as Hunter's result for theG/Geo/1 model. Finally, as an illustration of the generality of the analysis, it is applied to the derivation of the waiting time and the delay of the more generalG ^(G)/G/1 model and the system contents of a multi-server buffer-system with independent arrivals and random output interruptions.Both authors wish to thank the Belgian National Fund for Scientific Research (NFWO) for support of this work.