Discrete-time bulk-service queue with two heterogeneous servers

This paper analyzes a discrete-time bulk-service queueing system with two heterogeneous servers, i.e., two batch servers working with different service rates. The interarrival times of customers and service times of batches are assumed to be independent and geometrically distributed. Applications can be found in a wide variety of real systems including servers formed with different types of processors as a consequence of system updates, nodes in telecommunications network with links of different capacities, nodes in wireless systems serving different mobile users. We obtain closed-form expressions for the steady-state probabilities at arbitrary epoch with the help of displacement operator method and derive the outside observer’s observation epoch probabilities and waiting time distribution measured in slots. Computational experiences with a variety of numerical results in the form of tables and graphs are discussed. Moreover, some queueing models discussed in the literature are derived as special cases of our model. Finally, it is shown that in the limiting case the results obtained in this paper tend to the continuous-time counterpart.     

A queue with working breakdowns

In this paper, we consider a new class of queueing models with working breakdowns. The system may become defective at any point of time when it is in operation. However, when the system is defective, instead of stopping service completely, the service continues at a slower rate. Using the probability generating function, we give the joint distribution of the server state and the number of customers in the system in steady state. We also derive the necessary and sufficient condition for the existence of the steady state. We study the waiting time distribution of our model. Finally, some performance measures and numerical examples are presented.     

Analysis of a multi-server queue with two priority classes and (M, N)-threshold service schedule I: non-preemptive priority

In this paper, we propose an ( M , N )-threshold non-preemptive priority service schedule for a queueing system consisting of two-parallel queues and a multi-server. The arrival process for each queue is Poisson, and the service times are exponentially distributed with different means. We derive the generating functions of the stationary joint queue-length distribution, and then obtain the mean queue length and the mean waiting time for each queue.     

A discrete-time retrial queue with negative customers and unreliable server

This paper treats a discrete-time single-server retrial queue with geometrical arrivals of both positive and negative customers in which the server is subject to breakdowns and repairs. Positive customers who find sever busy or down are obliged to leave the service area and join the retrial orbit. They request service again after some random time. If the server is found idle or busy, the arrival of a negative customer will break the server down and simultaneously kill the positive customer under service if any. But the negative customer has no effect on the system if the server is down. The failed server is sent to repair immediately and after repair it is assumed as good as new. We analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle, busy or down. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.     

Analysis of a discrete-time queue with general three-state Markovian traffic sources

In this paper, we analyze the performance characteristics of an ATM switch, where cell arrivals are generated according to a three-state Markovian arrival law. This arrival process is an extension for the two-state ON/OFF source model that has already been extensively treated in the literature, and it allows a much more flexible modeling of a wide variety of VBR sources. As a result, for the finite storage capacity case, the system contents distribution, as well as the cell loss ratio are derived. On the other hand, for the infinite storage capacity case, a method for deriving an expression for the probability generating function of the system contents is presented. Furthermore, an approximation (consisting of multiple geometric terms) for the tail distribution of the system contents is given. The accuracy of this approximation as well as the impact of the traffic parameters on the buffer behavior are discussed via some numerical examples.     

Analysis of finite-buffer discrete-time batch-service queue with batch-size-dependent service

Over the last two decades there has been considerable growth in digital communication systems which operate on a slotted system. In several applications, transmission of packets over the network takes place in batches of varying size, and transmission time depends upon the size of the batch. Performance modelling of these systems is usually done using discrete-time queues. In view of this, we consider a single-server queue with finite-buffer in a discrete-time domain where the packets are transmitted in batches (of varying size) according to minimum and maximum threshold limit, usually known as general batch service rule. The transmission time (in number of slots) of these batches depends on the number of packets within the batch under transmission, and is arbitrarily distributed. We obtain, in steady-state, distribution of the number of packets waiting in the queue and in service (those being transmitted in batches). In addition, we also obtain average number of packets waiting in queue, in the system, with the server, rejection probabilities, etc. Finally, computational experiences with a variety of numerical results have been discussed by introducing a cost model which gives optimum value of the lower threshold limit.     

离散系统最速控制综合函数

针对自抗扰控制技术中的fhan()函数,指出其只是离散系统最速控制综合函数的一种简化.详细推导了离散系统真正的最速控制综合函数——fsun(),完善了自抗扰控制技术的理论框架.理论分析和数值仿真表明,采用fsun()函数后的二阶离散系统,状态变量达到稳态时位置和速度曲线均不会出现超调,且达到稳态时所用的步数少于函数fhan().     

ℓ1 robust performance of discrete-time systems with structured uncertainty

This paper addresses the robust performance problem when a linear time-invariant system is subjected to both norm bounded time-varying uncertainty and worst-case external bounded input. The tightest upper bounds for steady-state robust performance measures over classes of finite memory and fading memory perturbations are computed in the regulation and tracking problems. Since the classes of finite memory and fading memory perturbations are inappropriate to the purposes of model validation and adaptive control, approximating subclasses of bounded memory perturbations and perturbations with exponentially decreasing impulse responses are considered. It is shown that the robust performance bounds obtained are nonconservative in the case of bounded memory perturbations. The worst-case steady-state robust performance measures for SISO system under coprime factor perturbations are computed for illustration.     

Null controllability with vanishing energy for discrete-time systems

In this paper null controllability with vanishing energy for discrete-time systems is considered. As in the case of continuous time systems necessary and sufficient conditions in terms of an algebraic Riccati equation are given. Then necessary and sufficient conditions involving the eigenvalues of the state matrix are given. Reachability and controllability with vanishing energy are also considered and necessary and sufficient conditions for them are given. Finally applications to sampled-data systems, systems with impulse control and periodic systems are discussed.     

Analysis of discrete-time batch service renewal input queue with multiple working vacations

This paper investigates a discrete-time single server batch service queue with multiple working vacations wherein arrivals occur according to a discrete-time renewal process. The server works with a different service rate rather than completely stopping during the vacation period. The service is performed in batches and the server takes a vacation when the system does not have any waiting customers at a service completion epoch or a vacation completion epoch. We present a recursive method, using the supplementary variable technique to obtain the steady-state queue-length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. The displacement operator method is used to solve simultaneous non-homogeneous difference equations. Some performance measures and waiting-time distribution in the system have also been discussed. Finally, numerical results showing the effect of model parameters on key performance measures are presented.     

Analysis of the discrete-time Geo/G/1 working vacation queue and its application to network scheduling

In this paper we present an exact steady-state analysis of a discrete-time Geo/G/1 queueing system with working vacations, where the server can keep on working, but at a slower speed during the vacation period. The transition probability matrix describing this queuing model can be seen as an M/G/1-type matrix form. This allows us to derive the probability generating function (PGF) of the stationary queue length at the departure epochs by the M/G/1-type matrix analytic approach. To understand the stationary queue length better, by applying the stochastic decomposition theory of the standard M/G/1 queue with general vacations, another equivalent expression for the PGF is derived. We also show the different cases of the customer waiting to obtain the PGF of the waiting time, and the normal busy period and busy cycle analysis is provided. Finally, we discuss various performance measures and numerical results, and an application to network scheduling in the wavelength division-multiplexed (WDM) system illustrates the benefit of this model in real problems.     

Optimal control of discrete-time bilinear systems with applications to switched linear stochastic systems

This paper aims at characterizing the most destabilizing switching law for discrete-time switched systems governed by a set of bounded linear operators. The switched system is embedded in a special class of discrete-time bilinear control systems. This allows us to apply the variational approach to the bilinear control system associated with a Mayer-type optimal control problem, and a second-order necessary optimality condition is derived. Optimal equivalence between the bilinear system and the switched system is analyzed, which shows that any optimal control law can be equivalently expressed as a switching law. This specific switching law is most unstable for the switched system, and thus can be used to determine stability under arbitrary switching. Based on the second-order moment of the state, the proposed approach is applied to analyze uniform mean-square stability of discrete-time switched linear stochastic systems. Numerical simulations are presented to verify the usefulness of the theoretic results.     

A discrete-time on–off source queueing system with negative customers

This paper studies a discrete-time single-server infinite-capacity queueing system with correlated arrivals, geometrically distributed service times and negative customers. Positive customers are generated by a Bernoulli bursty source, with geometrically distributed lengths of the on-periods and off-periods. Negative customers arrive to the system according to a geometrical arrival process which is independent of the positive arrival process. A negative customer removes a positive customer in service if any, but has no effect on the system if it finds the system empty. We analyze the Markov chain underlying the queueing system and evaluate the performance of the system based on generating functions technique. Closed-form expressions of some performance measures of the system are obtained, such as stationary probability generating functions of queue length, unfinished work, sojourn time distribution and so on. Finally, the effect of several parameters on the system is shown numerically.     

Analysis of the M/G/1 queue with discriminatory random order service policy

We consider an M/G/1 queue with different classes of customers and discriminatory random order service (DROS) discipline. The DROS discipline generalizes the random order service (ROS) discipline: when the server selects a customer to serve, all customers waiting in the system have the same selection probability under ROS discipline, whereas customers belonging to different classes may have different selection probabilities under DROS discipline. For the M/G/1 queue with DROS discipline, we derive equations for the joint queue length distributions and for the waiting time distributions of each class. We also obtain the moments of the queue lengths and the waiting time of each class. Numerical results are given to illustrate our results.     

H-infinity measurement-feedback control for discrete-time systems with a single measurement delay

H-infinity control problem for linear discrete-time systems with instantaneous and delayed measurements is studied. A necessary and sufficient condition for the existence of the H-infinity controller is derived by applying reorganized innovation analysis approach in Krein space. The measurement-feedback controller is designed by performing two Riccati equations. The presented approach does not require the state augmentation.     

A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems

A powerful approach for analyzing the stability of continuous-time switched systems is based on using optimal control theory to characterize the “most unstable” switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific “most unstable” switching law. For discrete-time switched systems, the variational approach received considerably less attention. This approach is based on using a first-order necessary optimality condition in the form of a maximum principle (MP), and typically this is not enough to completely characterize the “most unstable” switching law. In this paper, we provide a simple and self-contained derivation of a second-order necessary optimality condition for discrete-time bilinear control systems. This provides new information that cannot be derived using the first-order MP. We demonstrate several applications of this second-order MP to the stability analysis of discrete-time linear switched systems.     

A repetitive controller for discrete-time passive systems

This work proposes and studies a new repetitive controller for discrete-time systems which are required to track or to attenuate periodic signals. The main characteristic of the proposed controller is its passivity. This fact implies closed-loop stable behavior when it is used with discrete-time passive plants. The work also discusses the energetic structure, the frequency response and the time response of the proposed controller structure. A numerical example is included to illustrate its practical use.     

A globally stable discrete-time controller for current-fed induction motors

In this short paper we present a discrete-time field oriented controller (FOC) for current-fed induction motors which insures global asymptotic speed regulation as well as rotor flux norm tracking. To the best of our knowledge, this is the first time such a result is rigorously established for a controller implemented in discrete-time. To insure global stability a condition on the reference for the rotor flux norm, which is time-varying, is imposed. This condition disappears as the sampling period goes to zero, hence allowing for independent speed regulation and rotor flux norm tracking. One important feature of our scheme is that, compared with the first-difference approximation of the classical indirect FOC, the additional computational burden is negligible. It is also shown that the result can easily be extended to the case of tracking time-varying references in speed or position.     

A formula for the optimal cost in the general discrete-time LEQG problem

This paper is aimed at deriving an explicit formula for the optimal cost for discrete-time linear exponential-of-quadratic Gaussian (LEQG) control problems. We make direct calculations for the general case with cross terms in the cost and noise covariance matrices using an information-state approach.     

Analysis of a tandem queue with state-dependent general blocking: A GSMP perspective

In this paper, we study a tandem queue where there is a finite number of buffer positions at each stage. The blocking scheme is general in the sense that it can model a number of classical blocking schemes, including communication, manufacturing and kanban blockings as special cases. The system considered here differs from the conventional system in two aspects: (1) departure of jobs from the system is determined by an external arrival process of another queue lying parallel to the tandem queue; (2) the control parameters of the blocking scheme are state-dependent in that they may change values depending on the state of the system.

We model this system as a generalized semi-Markov process (GSMP), we study the structural properties of its scheme, and establish monotonicity and convexity properties of event times with respect to both the clock times and the integer blocking parameters. In particular, we demonstrate that the state-dependent scheme is “better”, in certain sense, than the corresponding static scheme. Our results also recover the structural properties previously established for the classical blockings.