Perturbation analysis of a multiclass queueing system with admission control

Product form was first introduced by Jackson [Operations Research, 1957; Management Science, 1963] who considered only negative exponential service distributions with queue length dependent service rates. Posner and Bernholtz [Operations Research, 1968] showed that one has product form in the case where there are many classes of customers, service times are exponential with class queue length dependent service rates, and lag times (the time for a customer to travel between two servers is a random variable with continuously differentiable distribution function). Baskett, Chandy, Muntz and Palacios [JACM, 1975] extended these results to networks with multiple classes of customers and service time distributions which have rational Laplace transforms vanishing at ∞. Chandy, Howard and Towsley [JACM, 1977] extended this further to the case where the service time distribution functions are continuously differentiable. They also introduced the concept of station balance and established some important relationships between station balance, local balance and product form. In particular, for the closed queueing network they showed that if each queue in the network satisfies local balance in isolation with Poisson arrivals then one has product formsolution for the network. This paper extends previous results to arbitrary service with finite mean (the nondifferentiable case included). This is done by introducing supplementary variables for the remaining service requirements at the various queues in the system. By using generalized function (in the sense of Schwartz) a new technique is presented for handling queueing networks in which service distributions may be general.

The authors consider an M/G/1 queue with a time-varying arrival rate. The objective is to obtain infinitesimal perturbation analysis (IPA) gradient estimates for various performance measures of interest with respect to certain system parameters. In particular, the authors consider the mean system time over n arrivals and an arrival rate alternating between two values. By choosing a convenient sample path representation of this system, they derive an unbiased IPA gradient estimator which, however, is not consistent, and investigate the nature of this problem. >

We consider a queueing system with C customer classes under a nonpreemptive service discipline. The goal is to find gradient estimators for the stationary average sojourn time per customer of each class under admission control. Due to the service discipline, IPA (infinitesimal perturbation analysis) estimators are not applicable. We present the idea of harmonic gradient (HG) estimation, based on the Fourier decomposition of periodic functions. The canonical estimators can be used to obtain consistent estimators for all the control variables in a single run. However, the large number of values for each parameter required in the estimation can greatly affect the performance. We then describe the implementations of the phantom RPA (rare perturbation analysis) method. This method requires evaluating, in parallel, the dynamics of as many phantom systems as customers in each busy period. Since this number is random, the implementation of the method can be rather complex. We use the Fourier decomposition ideas to construct a hybrid estimator that we call the phantom HG method. We then give simulation results to compare the performance of the estimators and their complexity.

We propose a gradient updating procedure for using both present and past data to improve the convergence properties of a stochastic approximation algorithm. This procedure utilizes second derivatives estimated by perturbation analysis techniques. Experimental evidence piovided by simulation runs appear to confirm the improvement in convergence rate gained by this modified algorithm.

On the efficient generation of discrete event sample paths under different system parameter values

We developed an admission control scheme for CDMA cellular system that supports multiple classes of traffic with different QoS requirements and incorporates adaptive power control and antenna array multiuser receiver. A relation of the cell capacity and other-cell interference level is first established via simulation. For multi-class traffic cellular system a notion of bandwidth utility is used to characterize cell capacity. The CAC scheme is then implemented distributively in each cell based on estimation of other-cell interference. The admission control decision is made by comparing capacity bandwidth utility and the sum of current bandwidth utility and new call bandwidth utility. In the simulation it is demonstrated that the CAC scheme can adapt to various traffic distribution. Capacity determined by the CAC algorithm agrees well with that given by simulation. Therefore it can achieve high bandwidth utilization while guarantee QoS requirements of all the users.

We consider a model of a make-to-stock manufacturing system. External demand is met from the finished goods inventory; unsatisfied demand is backlogged. We adopt a base-stock production policy which produces if inventory falls below a certain threshold and idles otherwise. We set this threshold to guarantee stockout or delay probabilities to stay below given constants (service level constraints). These can be set by solving a utility maximization problem which trades-off quality of service with expected inventory costs. We combine analytical (large deviations) and simulation-based (perturbation analysis) techniques. We demonstrate that there is a natural synergy between these two approaches.

This talk considers loss-related performance measures in high-speed networks from the standpoint of gradient estimation by the Infinitesimal Perturbation Analysis (IPA) technique. The underlying model consists of an interconnected system of continuous flow models (CFMs) processing data in a fluid form, and the performance measures of interest are related to the average loss volume caused by buffer overflow over a period of time. The main results presented are: (i) an easily implementable, distributed algorithm for computing the IPA derivative, and (ii) analysis of the algorithm that proves the unbiasedness of the IPA estimator. 'Me distributed algorithm can naturally be implemented in real-time at a network's nodes, and hence potentially can be used for on-line control. The analysis highlights some of the special structure of CFMs, and suggests a general scope of 1PRs unbiasedness, far beyond the specific performance measures discussed. The talk will conclude with remarks about the potential applicability of IPA as a design and analysis tool for networks of CFMs.

Stochastic Flow Models (SFMs) form a class of hybrid systems used as abstractions of complex Discrete Event Systems (DES) for the purpose of deriving performance sensitivity estimates through Infinitesimal Perturbation Analysis (IPA) techniques when these cannot be applied to the original DES. In this paper, we establish explicit connections between gradient estimators obtained through a SFM and those obtained in the underlying DES, thus providing analytical evidence for the effectiveness of these estimators which has so far been limited to empirical observations. We consider DES for which analytical expressions of IPA (or finite difference) estimators are available, specifically G/G/1 and G/G/1/K queueing systems. In the case of the G/G/1 system, we show that, when evaluated on the same sample path of the underlying DES, the IPA gradient estimators of states, event times, and various performance metrics derived through SFMs are, under certain conditions, the same as those of the associated DES or their expected values are asymptotically the same under large traffic rates. For G/G/1/K systems without and with feedback, we show that SFM-based derivative estimates capture basic properties of finite difference estimates evaluated on a sample path of the underlying DES.

Using Infinitesimal Perturbation Analysis of Stochastic Flow Models to Recover Performance Sensitivity Estimates of Discrete Event Systems

We propose an admission and routing control policy for a network of service facilities in a stochastic setting in order to maximize a long run average reward. Queueing and reneging before entering the network is allowed; we introduce orbiting as an approximation to the queueing. Once a customer has entered the network, it incurs no more waiting. Our control policy is easy to implement and we prove that it performs well in steady state as long as the capacity request sizes are relatively small compared to the capacity of the service facilities. The policy is a target tracking policy: a linear program provides a target operating point and an exponential penalty function is used to translate the optimal deterministic point into a feasible admission and routing policy. This translation essentially transforms the admission and routing control problem into a problem of load balancing via the construction of fictitious systems. Simulation studies are included to illustrate that our policy also performs well when request sizes are moderate or large with respect to the capacity.

A manufacturing system with two tandem machines producing one part type is considered in this work. The machines are unreliable, each having two states, up and down. Both surplus controls and Kanban systems are considered. Algorithms for approximating the optimal threshold values are developed. First, perturbation analysis techniques are employed to obtain consistent gradient estimates based on a single simulation run. Then, iterative algorithms of the stochastic optimization type are constructed. It is shown that the algorithms converge to the optimal threshold values in an appropriate sense. Numerical examples are provided to demonstrate the performance of the algorithms.

The conceptual design and prototypical development of a performance management procedure for computer communication networks are presented. The procedure aims to improve the network performance by online adjustment of protocol parameters. The techniques of perturbation analysis in discrete-event dynamical systems, stochastic approximation, and learning automata are used in formulating the performance management procedure. The prototype procedure is evaluated on a test apparatus which consists of a number of host computers interconnected by an IEEE 802.4-based MAP network, and test results are presented and discussed. The conceptual design offers a step forward to bridging the gap between management standards and users' demands for efficient network operations, since most standards, such as ISO and IEEE standards, address only the architecture and interfaces for network management. The proposed concept of performance management can be used as an offline tool for selecting default parameter settings of a network. >

We present a two-phase parallel computing method for obtaining an implementable receding horizon control solution for constrained nonlinear systems. The phase 1 method solves a feasibility problem to find an approximate open-loop admissible control and horizon pair. The phase 2 method successively improves the admissible control to obtain an implementable open-loop receding horizon control solution. We briefly sketch an approach to realizing this two-phase method using VLSI array processors. Solution times for simulation examples estimated on the basis of current VLSI technology confirm that our controller is well suited to the stabilization of real-time processing, fast, constrained nonlinear systems. >

We study geolocation based on biased range estimates. Positive bias arises using time delay ranging methods in a multipath fading environment, when the line of sight direct path is severely attenuated. We model the range measurement as contaminated with Gaussian noise and an additive nonnegative bias term, and consider deterministic and random bias cases. We develop weighted least squares (WLS) and maximum likelihood (ML) geolocation estimators, and show that in general they yield biased geolocation estimates. A perturbation analysis technique is applied to find bias and mean square error (MSE) expressions for the WLS and MLE algorithms. MLE generally outperforms WLS, because MLE exploits knowledge of the range measurement bias distribution. The location error expressions are functions of the measurement bias and variance, as well as the network geometry. These results are useful to study achievable geolocation performance, and are applied to optimize sensor placement for improving the overall geolocation accuracy. We also develop the Cramér-Rao bound (CRB) on geolocation for our model. The CRB is a bound on unbiased estimation, whereas the geolocation algorithms may be biased, and we show how the estimators approach the CRB in certain cases. Numerical examples are presented to verify the analysis and study some cases of interest.