Information theoretic analysis for a general queueing system at equilibrium with application to queues in tandem

Summary In this paper, information theoretic inference methology for system modeling is applied to estimate the probability distribution for the number of customers in a general, single server queueing system with infinite capacity utilized by an infinite customer population. Limited to knowledge of only the mean number of customers and system equilibrium, entropy maximization is used to obtain an approximation for the number of customers in the G¦ G¦1 queue. This maximum entropy approximation is exact for the case of G=M, i.e., the M¦M¦1 queue. Subject to both independent and dependent information, an estimate for the joint customer distribution for queueing systems in tandem is presented. Based on the simulation of two queues in tandem, numerical comparisons of the joint maximum entropy distribution is given. These results serve to establish the validity of the inference technique and as an introduction to information theoretic approximation to queueing networks.This work was supported under a Naval Research Laboratory Fellowship under Grant N00014-83G-0203 and under an ONR Grant N00014-84K-0614 Former address:Westinghouse Defense and Electronics Center, Baltimore, MD, USA     

T-preemptive priority queue and its application to the analysis of an opportunistic spectrum access in cognitive radio networks

We propose a new priority discipline called the T-preemptive priority discipline. Under this discipline, during the service of a customer, at every T time units the server periodically reviews the queue states of each class with different queue-review processing times. If the server finds any customers with higher priorities than the customer being serviced during the queue-review process, then the service of the customer being serviced is preempted and the service for customers with higher priorities is started immediately. We derive the waiting-time distributions of each class in the M/G/1 priority queue with multiple classes of customers under the proposed T-preemptive priority discipline. We also present lower and upper bounds on the offered loads and the mean waiting time of each class, which hold regardless of the arrival processes and service-time distributions of lower-class customers. To demonstrate the utility of the T-preemptive priority queueing model, we take as an example an opportunistic spectrum access in cognitive radio networks, where one primary (licensed) user and multiple (unlicensed) users with distinct priorities can share a communication channel. We analyze the queueing delays of the primary and secondary users in the proposed opportunistic spectrum access model, and present numerical results of the queueing analysis.     

Performance analysis of non-preemptive GE/G/1 priority queueing of LER system with bulk arrivals

M^X/G/1 non-preemptive priority queue with bulk arrivals has already been analysed in many papers. Two problems are known in this area: first, when the number of priorities is greater than 2, the analysis for priority processes is obviously very difficult to handle. Therefore, the earlier work on the subject was restricted mostly to two priority classes; and second, although analytically explicit results are available, they require sophisticated closed-form expressions of the mean queue length. One particular bulk size distribution – the GE distribution – is motivated by an ME (Maximum Entropy) formulation for the behavior of a G/G/1 queue. The choice of a GE distribution is motivated by the fact that measurements of actual inter-arrival traffic or service times may be generally limited and so only few parameters, such as mean and variance, can be computed reliably. Thus this paper we can obtain very simple and analytic closed-form expression for the mean queue length of a GE/G/1 priority queue and a significant increase in performance evaluation has been achieved. We present the four-class priority queues, performance analysis and simulation of the LER (Label Edge Router) system in the ATM-based MPLS (Multiprotocol Label Switching) network. We wish to obtain the boundary conditions of the mean queue lengths and the mean queueing delays for each priority class, since this metric is one of the most important in performance evaluation parameters for improving QoS and system performance of the LER system in ATM-based MPLS network. A significant numerical example is presented and discussed as well. In order to obtain optimizing the performance analysis for EF flow, AF 1 flow, AF 2 flow and BE flow, the optimum ratios of COV (Coefficient of Variation) can be found via many numerical experiments carried out by the authors for queueing network model with HOL (Head of Line) priority rules. However, the ratios of COV value constraints exist. Furthermore, we find that each service class gradually begins to deteriorate when SQVs (Squared Coefficient of Variations), , and traffic intensity is greater than 0.95. We also find the values of maximum allowed burst size for EF flow and AF 1 flow and perform necessary policing actions on EF flow and AF 1 flow at the boundary node of the network. Finally, the four-class GE/G/1 priority queues and performance analysis of the LER system are shown accurate and robust after the comparison between theoretical evaluates and computer simulation results.     

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.     

MRE hierarchical decomposition of general queueing network models

Summary The principle of Minimum Relative Entropy (MRE), given fully decomposable subset and aggregate mean queue length, utilisation and flow-balance constraints, is used in conjunction with asymptotic connections to infinite capacity queues, to derive new analytic approximations for the conditional and marginal state probabilities of single class general closed queueing network models (QNMs) in the context of a multilevel variable aggregation scheme. The concept of subparallelism is applied to preserve the flow conservation and a universal MRE hierarchical decomposition algorithm is proposed for the approximate analysis of arbitrary closed queueing networks with single server queues and general service-times. Heuristic criteria towards an optimal coupling of network's units at each level of aggregation are suggested. As an illustration, the MRE algorithm is implemented iteratively by using the Generalised Exponential (GE) distributional model to approximate the service and asymptotic flow processes in the network. This algorithm captures the exact solution of separable queueing networks, while for general queueing networks it compares favourably against exact solutions and known approximations.This work is sponsored by the Science and Engineering Research Council (SERC), UK, under grant GR/F29271     

Approximating queue lengths inM(t)/G/1 queue using the maximum entropy principle

Using the discrete time approach a model is developed for obtaining the expected queue length of theM(t)/G/1 queue. This type of queue occurs in different forms in transportation and traffic systems and in communications and manufacturing systems. In order to cut down the very high computational efforts required to evaluate the performance measures in such queues by exact methods, the Maximum Entropy Principle is used to approximate the expected queue length which is one of the most commonly used performance measures. A procedure is then developed for reducing the error encountered when this approximation is adopted. The results from this paper will encourage the practitioners to use the appropriate time-varying queueing models when the need arises instead of resorting to very poor approximations.     

Entropy maximization and NT vacation M/G/1 model with a startup and unreliable server: comparative analysis on the first two moments of system size

We consider the control policy of an M/G/1 queueing system with a startup and unreliable server, in which the length of the vacation period is controlled either by the number of arrivals during the idle period, or by a timer. After all the customers have been served in the queue, the server immediately takes a vacation and operates an NT vacation policy: the server reactivates as soon as the number of arrivals in the queue reaches a predetermined threshold N or when the waiting time of the leading customer reaches T units. In such a variant vacation system, the steady-state probabilities cannot be obtained explicitly. Thus, the maximum entropy principle is used to derive the approximate formulas for the steady-state probability distributions of the queue length. A comparitive analysis of two approximation approaches, using the first and the second moments of system size, is studied. Both solutions are compared with the exact results under several service time distributions with specific parameter values. Our numerical investigations demonstrate that the use of the second moment of system size for the available information is, in general, sufficient to obtain more accurate estimations than that of the first moment.     

Entropy maximisation and open queueing networks with priorities and blocking

A review is carried out on the characterisation and algorithmic implementation of an extended product-form approximation, based on the principle of maximum entropy (ME), for a wide class of arbitrary finite capacity open queueing network models (QNMs) with service and space priorities. A single server finite capacity GE/GE/1/N queue with R (R>1) distinct priority classes, compound Poisson arrival processes (CPPs) with geometrically distributed batches and generalised exponential (GE) service times is analysed via entropy maximisation, subject to suitable GE-type queueing theoretic constraints, under preemptive resume (PR) and head-of-line (HOL) scheduling rules combined with complete buffer sharing (CBS) and partial buffer sharing (PBS) management schemes stipulating a sequence of buffer thresholds {N=(N₁,…,N_R),0<N_i≤N_i−1,i=2,…,R}. The GE/GE/1/N queue is utilised, in conjunction with GE-type first two moment flow approximation formulae, as a cost-effective building block towards the establishment of a generic ME queue-by-queue decomposition algorithm for arbitrary open QNMs with space and service priorities under repetitive service blocking with random destination (RS-RD). Typical numerical results are included to illustrate the credibility of the ME algorithm against simulation for various network topologies and define experimentally pessimistic GE-type performance bounds. Remarks on the extensions of the ME algorithm to other types of blocking mechanisms, such as repetitive service blocking with fixed destination (RS-FD) and blocking-after-service (BAS), are included.     

Note on the article: Maximum entropy analysis of the M/M/1 queueing system with multiple vacations and server breakdowns

Wang et al. [Wang, K. H., Chan, M. C., & Ke, J. C. (2007). Maximum entropy analysis of the M^[x]/M/1 queueing system with multiple vacations and server breakdowns. Computers & Industrial Engineering, 52, 192–202] elaborate on an interesting approach to estimate the equilibrium distribution for the number of customers in the M^[x]/M/1 queueing model with multiple vacations and server breakdowns. Their approach consists of maximizing an entropy function subject to constraints, where the constraints are formed by some known exact results. By a comparison between the exact expression for the expected delay time and an approximate expected delay time based on the maximum entropy estimate, they argue that their maximum entropy estimate is sufficiently accurate for practical purposes. In this note, we show that their maximum entropy estimate is easily rejected by simulation. We propose a minor modification of their maximum entropy method that significantly improves the quality of the estimate.     

Entropy maximised queueing networks with blocking and multiple job classes

The method of entropy maximisation (MEM) is applied in a state space partitioning mode for the approximation of the joint stationary queue length distribution of an M/M/1/N queue with finite capacity, N( > 1), multiple and distinct classes of jobs, R( > 1), under a complete buffer sharing scheme and mixed service disciplines drawn from the first-come-first-served (FCFS), last-come-first-served with (LCFS-PR) or without (LCFS-NPR) preemption and processor sharing (PS) rules. The marginal and aggregate maximum entropy (ME) queue length distributions and the associated blocking probabilities per class are also determined. These ME results in conjunction with the first moments of the effective flows are used, as building blocks, in order to establish a new product-form approximation for arbitrary exponential open queueing networks with multiple classes of jobs under repetitive-service (RS) blocking with random destination (RD). It is verified that the ME approximation reduces to the exact truncated solution of open multi-class reversible queueing networks. Numerical experiments demonstrate a good accuracy level of ME statistics in relation to simulation. Moreover, recent extentions of MEM for arbitrary GE-type queueing networks with RS-RD blocking and multiple classes of jobs are presented.     

A hybrid method for performance analysis of G/G/m queueing networks

Open queueing networks are useful for the performance analysis of numerous real systems. Since exact results exist only for a limited class of networks, decomposition methods have been extensively used for approximate analysis of general networks. This procedure is based on several approximation steps. Successive approximations made in this approach can lead to a considerable error in the output. In particular, there are no general accurate formulas for computing the mean waiting time and the inter-departure variance in general multiple-server queues. This causes the results from decomposition methods when applied to G/G/m queueing networks to be very approximative and to significantly deviate from actual performance values. We suggest substituting some approximate formulae by low-cost simulation estimates in order to obtain more accurate results when benefiting from the speed of an analytical method. Numerical experiments are presented to show that the proposed approach provides improved performance.     

Maximum entropy and the G/G/1/N queue

Summary A new hybrid analytic framework, based on the principle of maximum entropy, is used to derive a closed form expression for the queue length distribution of a G/G/1 finite capacity queue. It is shown that Birth-Death homogeneous recursions for a single resource queue are special cases of maximum entropy one-step transitions which can be applied either in an operational or stochastic context. Furthermore, an equivalence relationship is used to analyse two-stage cyclic queueing networks with general service times, and favourable comparisons are made with global balance and approximate results. Numerical examples provide useful information on how critically system behaviour is affected by the distributional form of interarrival and service patterns. Comments on the implication of the work to the performance analysis and aggregation of computer systems are included.Some of the material included in this paper has been presented to the Performance '86 and ACM Sigmetrics 1986 Joint Conference on Computer Modelling, Measurement and Evaluation, May 28–30, 1986, University of North Carolina, USA     

Maximum entropy analysis of the M[x]/M/1 queueing system with multiple vacations and server breakdowns

We consider a single unreliable sever in an M^[x]/M/1 queueing system with multiple vacations. As soon as the system becomes empty, the server leaves the system for a vacation of exponential length. When he returns from the vacation, if there are customers waiting in the queue, he begins to serve the customers; otherwise, another vacation is taken. Breakdown times and repair times of the server are assumed to obey a negative exponential distribution. Arrival rate varies according to the server’s status: vacation, busy, or breakdown. Using the maximum entropy principle, we develop the approximate formulae for the probability distributions of the number of customers in the system which is used to obtain various system performance measures. We perform a comparative analysis between the exact results and the maximum entropy results. We demonstrate, through the maximum entropy results, that the maximum entropy principle approach is accurate enough for practical purposes.     

Information theoretic approximations for theM/G/1 retrial queue

In this paper we present information theoretic approximations for theM/G/1 queue with retrials. Various approximations for this model are obtained according to the available information about the service time probability density and the steady-state distribution of the system state. The results are well-suited for numerical computation.     

A conservation law based approximate analysis for a class of simultaneous resource possession problems

In this paper we analyze a single server queueing model in which there are two types of jobs, one of which must wait in an external queue until a token is available, and only then may join the service queue. The interarrival times and service requirements for both types of jobs are assumed to be independent and exponentially distributed.

We derive the stability condition for such a model where the service queue discipline is either FCFS (First-Come-First-Serve) or PS (Processor-Sharing). We then propose analytic approximations for the mean waiting times for both types of jobs, relying heavily on the M/G/1 conservation law. Numerical results show that our approximations are very accurate (within a few percent of the simulated results) even when the system is heavily loaded. The approximations are also shown to be asymptotically exact as the number of tokens N → ∞.     

Achieving prescribed mean waiting times in a single-server queueing system

A new queueing discipline is proposed, which achieves any prescribed mean waiting time under stationary conditions in the GI|G_n|1 queue. Mean waiting times for customers of each type are obtained for the HM|G_n|1 and GI|HM_n|1 queues. A polynomial-time algorithm is described to determine the parameters of the queueing discipline given the mean waiting times.Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 97–101, November–December, 1991.     

A two-stage approach in solving the state probabilities of the multi-queue M/G/1 model

The M/G/1 model is the fundamental basis of the queueing system in many network systems. Usually, the study of the M/G/1 is limited by the assumption of single queue and infinite capacity. In practice, however, these postulations may not be valid, particularly when dealing with many real-world problems. In this paper, a two-stage state-space approach is devoted to solving the state probabilities for the multi-queue finite-capacity M/G/1 model, i.e. q-M/G/1/K_i with K_i buffers in the ith queue. The state probabilities at departure instants are determined by solving a set of state transition equations. Afterward, an embedded Markov chain analysis is applied to derive the state probabilities with another set of state balance equations at arbitrary time instants. The closed forms of the state probabilities are also presented with theorems for reference. Applications of Little's theorem further present the corresponding results for queue lengths and average waiting times. Simulation experiments have demonstrated the correctness of the proposed approaches.     

Practical Extensions to Cycle Time Approximations for the G/G/m-Queue With Applications

Approximate closed form expressions for the mean cycle time in a G/G/m-queue often serve as practical and intuitive alternatives to more exact but less tractable analyses. However, the G/G/m-queue model may not fully address issues that arise in practical manufacturing systems. Such issues include tools with production parallelism, tools that are idle with work in process, travel to the queue, and the tendency of lots to defect from a failed server and return to the queue even after they have entered production. In this paper, we extend popular approximate mean cycle time formulae to address these practical manufacturing issues. Employing automated data extraction algorithms embedded in software, we test the approximations using parameters gleaned from production tool groups in IBM's 200 mm semiconductor wafer fabricator.     

On approximating the conditional wait process of general multiserver queueing systems

In view of the extremely difficult task of analyzing G/G/K queueing systems, relatively few general results have been established. Most of the literature dealing with the G/G/K system has been concerned with the heavy traffic situation. Of special note are the bounds given by Kingman[7] on the wait process, the weak convergence theorems established by Iglehart and Whitt[4] and Loulou[8], and the many server approximations due to Newell[9].In this paper a presentation will be made of new formulas constructed to approximate the mean and variance of the conditional wait process. Two key concepts play major roles behind the formulas to be presented. They are: (a) the use of a continuous (diffusion-type) process to approximate the steady-state probabilities of the number in the queue, and (b) the substitution of the original process of interdeparture times by a process of independent variables.In order to assess the quality of the suggested approximation for the mean and variance of the conditional wait process, the approximation will be numerically compared with the results obtained from Kingman's[7] and Kendall's[5] exact formula for the special case of the G/M/K system, and will be tested against point estimate simulations.     

Approximations for the waiting time distribution of the M/G/c queue

For the M/G/c queue we present an approximate analysis of the waiting time distribution. The result is given in the form of the defective renewal equation. This integral equation can be numerically solved by a simple recursion procedure. Also, asymptotic results for the waiting times are presented. Numerical results indicate that the approximations are sufficiently accurate for practical purposes.