Numerical method for the analysis of queuing models with priority jumps1

An algorithmic approach to study queuing models with common finite and infinite buffer and jump priorities is developed. It is assumed that upon arrival of a low-priority call, one call of such kind can be transferred with some probability to the end of the queue of high-priority calls. The transition probability depends on the state of the queue of heterogeneous calls. The algorithms are proposed to calculate the quality of service metrics of such queuing models.     

Information theoretic approximations for M/G/1 and G/G/1 queuing systems

Summary This paper presents new results concerning the use of information theoretic inference techniques in system modeling and concerning the widespread applicability of certain simple queuing theory formulas. For the case when an M/G/1 queue provides a reasonable system model but when information about the service time probability density is limited to knowledge of a few moments, entropy maximization and cross-entropy minimization are used to derive information theoretic approximations for various performance distributions such as queue length, waiting time, residence time, busy period, etc. Some of these approximations are shown to reduce to exact M/M/1 results when G = M. For the case when a G/G/1 queue provides a reasonable system model, but when information about the arrival and service distributions is limited to the average arrival and service rates, it is shown that various well known M/M/1 formulas are information theoretic approximations. These results not only provide a new method for approximating the performance distributions, but they help to explain the widespread applicability of the M/M/1 formulas.     

Performance analysis of a non-preemptive priority queuing system subjected to a correlated Markovian interruption process

In this paper, we consider a discrete-time queuing system with head-of-line non-preemptive priority scheduling and a single server subjected to server interruptions. We model the server interruptions by a correlated Markovian on/off process with geometrically distributed on and off periods. Two classes of traffic are considered, namely high-priority and low-priority traffic. In the first part of the paper, we derive an expression for the functional equation describing the transient evolution of this priority queuing system. This functional equation is then manipulated and transformed into a mathematical tractable form. This allows us to derive the joint probability generating function (pgf) of the system contents. From this pgf, closed-form expressions for various performance measures, such as mean and variance of system contents and customer delay can be derived. Finally, we illustrate our solution technique with some numerical examples, whereby we demonstrate the negative effect of correlation in the interruption process on the performance of both classes. Some numerical results illustrating the impact of second-order characteristics of the arrival process on mean delays are also presented. The proposed approach which is purely based on pgfs is entirely analytical and enables the derivation of not only steady-state but transient performance measures, as well. The paper presents new insights into the performance analysis of discrete-time queues with service interruption and it also covers some previously published results as a special case.

Scope and purpose

In this contribution, we consider a practical queuing model, with HOL priority scheduling, two classes of traffic, and a server which is subjected to a correlated Markovian interruption process. We first derive a non-linear functional equation relating the joint pgf of the system state vector between two consecutive slots. Then we outline a solution technique to solve for this functional equation. This allows us to derive the joint pgf of the system contents of both classes, from which various performance measures related to mean system contents and customer delays are derived. We also demonstrate how the proposed approach allows for derivation of transient performance measures, as well. It should be noted that detailed coverage of the transient analysis of the system is beyond the scope of this paper.To our best knowledge, this is the first initiative that aims to explore the performance of queuing systems with priority scheduling when the shared server is subjected to service interruption. The paper also generalizes the results of Walraevens et al. (Analysis of a single-server ATM queue with priority scheduling, Computers & Operations Research 2003;30(12):1807–30) by incorporating service interruption into their original queuing model. By means of numerical results, the paper also demonstrates the effect of correlation in the service interruption process on the performance of both classes of customers. The impact of second-order characteristics of the arrival process on mean delays is also investigated.     

Analysis of the busy period in threshold control system

Consideration was given to the controllable Markov queuing system with nonhomogeneous servers and threshold policy of server activation depending on the queue length. By the busy period is meant the time interval between the instant of customer arrival to the empty system and the instant of service completion when the system again becomes free. The system busy period and the number of serviced customers were analyzed. Recurrent relations for the distribution density in terms of the Laplace transform and generating function, as well as formulas for calculation of the corresponding arbitrary-order moments, were obtained.     

Finite capacity priority queues with potential health applications

The queuing processes of interest in this paper are that of waiting lines with two priorities and multiple service channels. The arrival process is assumed Poison and the service time distribution is negative exponential. Arriving units enter service if there is at least one idle channel, otherwise they join a finite queue and are served according to a non-preemptive priority discipline. If a low priority arriving unit finds the queue full, it is not allowed to enter the system and is considered “blocked” or lost. In the first model a high priority arrival may displace a low priority unit from the full queue and may be “blocked” if the queue consists of high priority units only. In the second model the high priority unit may still displace a low priority unit from the full queue but it will never be “blocked” and may wait “outside” the system if the system is full. Thus far there has been no discussion of such models in queuing theory literature. In this paper analytical expressions for average waiting times have been obtained for the two models. Two potential applications of the models are described and the usefulness of the models is illustrated by numerical examples.     

Transient analysis of multi-server queues with Markov-modulated Poisson arrivals and overload control

This paper studies the transient behavior of a Markov-molulated Poisson arrival queue under overload control. The queue has finite or infinite buffer capacity with multiple exponential servers. A Markov-modulated Poisson process is used to represent an aggregated voice or video packet arrival process in integrated service networks. By overload control, we mean to properly adapt the arrival process once the buffer contents exceed a designated level. The probability distribution of queue length as a function of time is obtained. The temporal effect of the overload control is measured in two forms. While in overload, we measure the amount of time for the queue to fall into underload. While in underload, we measure the amount of time for the queue to rise to overload. A proper design of the control will not only reduce the fall time but also increase the rise time. We also explore the transient queueing behavior as affected by time stochastic properties of the underlying Markov chain for the arrival process.     

Performance analysis of a discrete-time queuing system with a correlated train arrival process

In this paper, we present an exact transient and steady-state discrete-time queuing analysis of a statistical multiplexer with a finite number of input links and whose arrival process is correlated and consists of a train of a fixed number of fixed-length packets. The functional equation describing this queuing model is manipulated and transformed into a mathematical tractable form. This allows us to derive the transient probability generating function (pgf) of the buffer occupancy. From this transient pgf, time-dependent performance measures such as transient probability of empty buffer, transient mean of buffer occupancy and instantaneous packet overflow probabilities are derived. By applying the final-value theorem, the corresponding exact expressions for the steady-state pgf of the queue length and packet arrivals are derived. We also show how the transient analysis provides insights into the derivation of the system's busy period distribution. Closed-form expressions for the mean packet and message delays are also provided. The paper presents significant results on the transient and steady-state analysis of statistical multiplexers with N input links and correlated train arrivals.     

Approximate method for analysis of queuing models with jump priorities

An approximate method for analysis of the queuing models with jump priorities was developed. It is assumed that upon arrival of a low-priority call one call of such kind might be immediately transferred to the end of the queue of the high-priority calls. The transfer probabilities depend on the state of the queue of heterogeneous calls. Consideration was given both to the models with separate and common queues. Algorithms to calculate the quality of service metrics of such queuing models were described, and the results of the numerical experiments were presented.     

具有两类请求的云计算中心服务器数量的优化

为提高云计算中心的服务质量,节约系统成本,针对具有两类用户请求的云计算中心,提出云计算中心的服务器数量的优化方案。首先,建立了具有两类用户请求的排队模型,分析系统的稳态概率分布、平均队长等性能指标;然后,建立了云计算中心的能耗模型;最后,联合系统的等待成本和能耗成本,构建系统的成本函数,对系统的服务器数量进行优化,从而使系统的成本最小。数值分析结果表明最优服务器数量是用户请求到达率的非减函数,为了使系统成本最小,云计算中心需要动态调整服务器的数量。     

A multi-class queuing network analysis methodology for improving hospital emergency department performance

This paper derives an open queuing network model of an emergency department (ED) design intended to increase the capacity of an ED to treat patients. The methodology captures hospital-specific differences in patient acuity mix, arrival patterns and volumes, and efficiencies of processes in a single common computational model. A spreadsheet implementation of the resulting queuing equations is used by managers, in real time, to size ED areas using waiting time and overflow probability as quality of service targets. Non-homogeneous arrival patterns, non-exponential service time distributions, and multiple patient types are all incorporated. The methodology has been applied to a fleet of hospitals for validation. Results from one of them are used to demonstrate the methodology.     

Access control to two multiserver loss queues in series

We consider admission policies to two multiserver loss queues in series with two types of traffic. Both are generated according to independent Poisson processes with constant arrival rates. The first type requires service at the first queue and with a positive probability enters the second queue; the second type requires service at only the second queue. The service time distribution is exponential at either station. We show that under appropriate conditions the optimal admission policy that maximizes the expected total discounted reward over an infinite horizon is given by a switching curve. We characterize the form and shape of this curve and its variation with system parameters     

An Efficient Computational Method for Solving Nonlinear Matrix Equation and Its Application in Queuing Analysis

The matrix analytic analysis of queues with complex arrival,vacation and service characteristics requires the solution of nonlinear matrix equation.The complexity and large dimensionality of the model require an effcient and smart algorithm for the solution.In this paper,we propose and efficient Adaptive Newton-Kantorovich(ANK) method for speeding up the algorithm solving the nonlinear matrix equation which is an inevitable step in the analysis of the queue with embedded Markov chain such as BMAP/SMSP/1/∞ queue or its discrete version.BMAP/SMSP/1/∞ is a queuing model with a Semi Markov Service time Process (SMSP) and a Batch Markovian Arfival Process(BMAP).The numerical result is presented for the discrete case of N-MMBP/D/1 queue which arises in analyzing traffic aspect of computer communication network,where MMBP is Markov Modulated Bermoulli Process.The comparisons of Adaptive Newton-Kantorovich(ANK)with Modified Newton-Kantorovich(MNK) show that ANK saves 30% of CPU tim when the number of user N is 50.     

Fuzzy-Based Dynamic Distributed Queue Scheduling for Packet Switched Networks

Addressing the problem of queue scheduling for the packet-switched system is a vital aspect of congestion control. In this paper, the fuzzy logic based decision method is adopted for queue scheduling in order to enforce some level of control for traffic of different quality of service requirements using predetermined values. The fuzzy scheduler proposed in this paper takes into account the dynamic nature of the Internet traffic with respect to its time-varying packet arrival process that affects the network states and performance. Three queues are defined, viz low, medium and high priority queues. The choice of prioritizing packets influences how queues are served. The fuzzy scheduler not only utilizes queue priority in the queue scheduling scheme, but also considers packet drop susceptibility and queue limit. Through simulation it is shown that the fuzzy scheduler is more appropriate for the dynamic nature of Internet traffic in a packet-switched system as compared with some existing queue scheduling methods. Results show that the scheduling strategy of the proposed fuzzy scheduler reduces packet drop, provides good link utilization and minimizes queue delay as compared with the priority queuing (PQ), first-in-first-out (FIFO), and weighted fair queuing (WFQ).     

An application of modern control theory to a time-dependent queuing system for optimal operation

‘ Bang-bang ’ optimal closed-loop service policy for a time dependent M/M/l queuing system is derived using optimal control theory. The policy is based on probabilistic (and not stochastic) behaviour of the queue. Computational results are obtained for an illustrative example with non-bang-bang service policy using the conjugate gradient algorithm with bounded control variables. It is interesting to note that the optimal service policy is not sensitive to the arrival rate but to the mean service cost of a customer.     

QoS enhancements and performance analysis for delay sensitive applications

This paper presents a comprehensive system modeling and analysis approach for both predicting queuing delay and controlling average queuing delay of a single buffer to a required value in a multiple traffic source network environment. This approach could effectively enhance the QoS performance of delay sensitive applications. A discrete-time analytical model that approximates the multi-source arrival process with a binomial distribution has been developed to analyze the relationship between the queuing threshold and average queuing delay. A control strategy with dynamic queue thresholds based on the analytical result is then used to control the average queuing delay to a required value within the buffer. Packet dropping is treated as implicit congestion feedback to the arrival process for rate adjustment. The feasibility of the system has been validated by comparing theoretical analysis with a diverse set of simulation results. Following from the simulation results, a set of statistical analyses has been performed to evaluate the efficiency and accuracy of the proposed scheme. In addition, a user-friendly graphical user interface has been developed to allow user-configuration of the simulation process and display simulation results.     

On the Invariance of Stationary State Probabilities of a Non-Product-Form Single-Line Queueing System

We consider a single-line queueing system (QS) with Poisson input flow of varying intensity, which depends on the number of demands in the system. The job size (length) distribution for a demand depends on the number of demands in the system at the arrival moment. The service rate also depends on the number of calls in the QS. If the job size for a new arrival is larger than the remaining job size for the currently processed demand, then the arrival is put at the beginning of the queue with a certain probability, which depends on the total number of demands in the system. Otherwise, it occupies the server and displaces the currently processed demand, which is put at the beginning of the queue. The probability distribution of stationary states of the QS is found and necessary and sufficient conditions for this distribution to be invariant with respect to the job size distribution with a fixed mean are obtained.     

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.     

基于排队论的呼叫中心ACD技术研究

本文首先阐述了ACD的排队理论;其次对先到先服务、优先级服务的排队策略,基于负载均衡、座席技能级别、客户信息和经验的路由算法进行了详细地分析;最后提出可根据呼叫到达时间、主叫号码、DNIS、用户可接受的等待时间、客户等级多项参数进行线性加权确定优先级的排队算法策略,根据系统规模、服务效率、客户信息等来综合地确定路由分配方法,真正实现合理的排队和智能的路由分配。     

Performance analysis of parallel processing systems

A bulk arrival M/sup x//M/c queuing system is used to model a centralized parallel processing system with job splitting. In such a system, jobs wait in a central queue, which is accessible by all the processors, and are split into independent tasks that can be executed on separate processors. The job response-time consists of three components: queuing delay, service time, and synchronization delay. An expression for the mean job response-time is obtained for this centralized parallel-processing system. Centralized and distributed parallel-processing systems (with and without job-splitting) are considered and their performances compared. Furthermore, the effects of parallelism and overheads due to job-splitting are investigated.<>     

On the analysis of exponential queuing systems with randomly changing arrival rates: Stability conditions and finite buffer scheme with a resume level

This paper considers a single server exponential queue with random fluctuations in the intensity of the arrival process. The motivation being the modelling of random changes in traffic patterns. This random intensity model does not obey the independence assumption made in queuing theory. Necessary and sufficient conditions for the stability or ergodicity of the queuing process are obtained via analytic techniques using Jury's stability criteria, often used to study discrete time control systems. The effect of such fluctuations is then studied for a finite resume level queue which is often used in flow control. Exact performance measures are computed and are compared with existing results.