Queueing models for police-fire merger analysis

Cities with under 100,000 in population expend a significant portion of their budgets on emergency services. One option that a number of these cities have considered for improving service and cutting costs is training personnel to handle both police and fire roles. In this paper we describe a hierarchy of models that we have used to assess the performance viability of a merger as well as to design specific deployment plans. The modeling environment is more complex than a traditional police or fire system. We need to model the response pattern of four or more patrol units along with the simultaneous dispatch of fire equipment from one or more fire stations. The major contribution of the paper is the manner in which a series of models is linked together to forecast a wide range of performance measures under differing dispatch assumptions. We use a queueing model of police patrol to calculate steady state probabilities and expected delays without preemption. We then model two types of preemptive dispatch strategies utilized in responding initially to a major fire by superimposing a binomial distribution on the basic queueing model. There is also a travel time simulation model to calculate conditional expected response time statistics. The queueing models and the travel time simulation are then combined to estimate unconditional expected values. Lastly, we describe a simulation model used to address transient performance issues that are of concern during a major fire.     

Isogeometric Reissner-Mindlin shell analysis - employing different control meshes for displacements and rotations

The main purpose of this paper is the development and implementation of a method for the reduction of the so-called locking effect in the isogeometric Reissner-Mindlin shell formulation. In [1] an isogeometric Reissner-Mindlin shell formulation with an exact interpolation of the director vector based on continuum mechanics was introduced. The numerical examples showed that the accuracy and efficiency increased. However, there are only few effective concepts for the prevention of locking effects for low polynomial degrees. In the work of Beirão da Veiga [2], shear locking is prevented for a Reissner-Mindlin plate formulation by using suitable solution spaces. Here, the method is extended to the Reissner-Mindlin shell formulation. Different control meshes are used for displacements and rotations. Furthermore, the basis functions in the direction of the relevant rotation are one degree less than the ones which are chosen for the displacements. That leads to control meshes with different number and location of the control points. The aim is to avoid shear locking due to the coupling of shear strains and curvature since the compatibility requirement for pure bending is then fulfilled. The accuracy and efficiency of this method are investigated for different examples. In addition, the results are compared to the analytical solutions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fluid analysis of an input control problem

A two-station network with controllable inputs and sequencing control, proposed by Wein (Oper. Res. 38:1065–1078, 1990), is analyzed. A control is sought to minimize holding cost subject to a throughput constraint. In a Lagrangian formulation, input vanishes in the fluid limit. Several alternative fluid models, including workload formulations, are analyzed to develop a heuristic policy for the stochastic network. Both the fluid heuristic and Wein’s diffusion solution are compared with the optimal policy by solving the dynamic program. Examples with up to six customer classes, using Poisson arrival and service processes, are presented. The fluid heuristic does well at sequencing control but the diffusion gives additional, and better, information on input control. The fluid analysis, in particular whether the fluid priorities are greedy, aids in determining whether the fluid heuristic contains useful information.     

Computational method for optimal control of switched systems with input and state constraints

In this paper, we consider an optimal control problem of switched systems with input and state constraints. Since the complexity of such constraint and switching laws, it is difficult to solve the problem using standard optimization techniques. In addition, although conjugate gradient algorithms are very useful for solving nonlinear optimization problem, in practical implementations, the existing Wolfe condition may never be satisfied due to the existence of numerical errors. And the mode insertion technique only leads to suboptimal solutions, due to only certain mode insertions being considered. Thus, based on an improved conjugate gradient algorithm and a discrete filled function method, an improved bi-level algorithm is proposed to solve this optimization problem. Convergence results indicate that the proposed algorithm is globally convergent. Three numerical examples are solved to illustrate the proposed algorithm converges faster and yields a better cost function value than existing bi-level algorithms.     

Queueing models for the analysis of communication systems

Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks; for instance, to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discrete-time models come natural. We start this paper with a review of suitable discrete-time queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter, etc.). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival process as well as non-FCFS scheduling are taken into account. Focus is on delay performance measures, such as the mean delay experienced by both types of packets and probability tails of these delays.     

Coupling method for asymptotic analysis of queues with regenerative input and unreliable server

First we define a regenerative flow and describe its properties. Then a single-server queueing system with regenerative input flow and an unreliable server are considered. By applying coupling we establish the ergodicity condition and prove the limit theorem in the heavy traffic situation (traffic coefficient $\rho <1, \rho \uparrow 1$ ). The asymptotic analysis of the super-heavy traffic situation ( $\rho \ge 1$ ) is also realized.     

Stability analysis for networked control systems with sampling,transmission protocols and input delays

In this paper, the stability problem is investigated for networked control systems. Input delays and multiple communication imperfections containing time-varying transmission intervals and transmission protocols are considered. A unified framework based on the hybrid systems with memory is proposed to model the whole networked control system. Hybrid systems with memory are used to model hybrid systems affected by delays and permit multiple jumps at a jumping instant. The stability analysis depends on the Lyapunov–Krasovskii functional approaches for hybrid systems with memory and the proposed stability theorem does not need strict decrease of the Lyapunov–Krasovskii functional during jumps. Based on the developed stability theorems, stability conditions for networked control systems are established. An explicit formula is given to compute the maximal allowable transmission interval. In the special case that the networked control system contains linear dynamics, an explicit Lyapunov functional is constructed and stability conditions in terms of linear matrix inequalities (LMI) are proposed. Finally, an example of a chemical batch reactor is given to illustrate the effectiveness of the proposed results.     

Quadratic control for linear periodic systems

We propose a new quadratic control problem for linear periodic systems which can be finite or infinite dimensional. We consider both deterministic and stochastic cases. It is a generalization of average cost criterion, which is usually considered for time-invariant systems. We give sufficient conditions for the existence of periodic solutions.Under stabilizability and detectability conditions we show that the optimal control is given by a periodic feedback which involves the periodic solution of a Riccati equation. The optimal closed-loop system has a unique periodic solution which is globally exponentially asymptotically stable. In the stochastic case we also consider the quadratic problem under partial observation. Under two sets of stabilizability and detectability conditions we obtain the separation principle. The filter equation is not periodic, but we show that it can be effectively replaced by a periodic filter. The theory is illustrated by simple examples.This work was done while this author was a visiting professor at the Scuola Normale Superiore, Pisa.     

Asymptotic analysis of Neumann periodic optimal boundary control problem

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.     

Hamilton-Jacobi theory for periodic control problems

The general problem of periodic optimization is considered in this paper. The contribution consists in stating sufficient conditions for the optimality of a control satisfying the periodicity constraint. The result has been achieved by means of the classical Hamilton-Jacobi equation, suitably modified in order to consider the peculiar constraint of the problem. Finally, an interesting application of the theory to the case of linear, time-invariant systems is given.This work was supported by CNR (Consiglio Nazionale delle Ricerche), Rome, Italy.     

Lyapunov method for the stability analysis of uncertain fractional-order systems under input saturation

The robust stability and stabilization of a class of fractional order systems under input saturation is studied using the Lyapunov method. In the problem formulation the Lipchitz and bounded conditions are adopted. Moreover, sufficient conditions, in the form of a Linear Matrix Inequality, for stabilizing the system via a state feedback control, are derived. Two numerical examples show the effectiveness of the proposed technique.     

Queueing systems with leadtime constraints: A fluid-model approach for admission and sequencing control

We study how multi-product queueing systems should be controlled so that sojourn times (or end-to-end delays) do not exceed specified leadtimes. The network dynamically decides when to admit new arrivals and how to sequence the jobs in the system. To analyze this difficult problem, we propose an approach based on fluid-model analysis that translates the leadtime specifications into deterministic constraints on the queue length vector. The main benefit of this approach is that it is possible (and relatively easy) to construct scheduling and multi-product admission policies for leadtime control. Additional results are: (a) While this approach is simpler than a heavy-traffic approach, the admission policies that emerge from it are also more specific than, but consistent with, those from heavy-traffic analysis. (b) A simulation study gives a first indication that the policies also perform well in stochastic systems. (c) Our approach specifies a “tailored” admission region for any given sequencing policy. Such joint admission and sequencing control is “robust” in the following sense: system performance is relatively insensitive to the particular choice of sequencing rule when used in conjunction with tailored admission control. As an example, we discuss the tailored admission regions for two well-known sequencing policies: Generalized Processor Sharing and Generalized Longest Queue. (d) While we first focus on the multi-product single server system, we do extend to networks and identify some subtleties.     

Rejoinder on: Queueing models for the analysis of communication systems

In this rejoinder, we respond to the comments and questions of three discussants of our paper on queueing models for the analysis of communication systems. Our responses are structured around two main topics: discrete-time modeling and further extensions of the presented queueing analysis.     

Comments on: Queueing models for the analysis of communication systems

The paper under discussion is a well-written exposition on the performance modeling of communication systems by discrete-time queueing systems, and their analysis. It basically consists of two parts: a review of the literature, focusing on the modelling of information streams and on scheduling disciplines (Sects. 2, 3), and a demonstration of some key methods for the analysis of discrete-time queueing systems, focusing on a particular two-class discrete-time queue with correlated arrivals and two priority classes (Sects. 4–6). In Sect. 1 of the present note, we make some introductory comments. In Sect. 2, realizing that the literature review in Bruneel et al. (TOP, 2014) is authoritative and extensive, we focus on a few adjacent topics which fall outside the scope of Bruneel et al. (TOP, 2014) but which in our view may also be of some interest. Finally, in Sect. 3, we discuss the analysis in Sects. 4–6 of Bruneel et al. (TOP, 2014).     

Sweep method in analysis optimal control for rendez-vous problems

This paper deals with determining the sufficient conditions of minimum for the class of problems in which the necessary conditions of optimum are satisfied in the strengthened form Legendre-Clebsch. To this paper, we shall use the sweep method which analysis the conditions of existence of the conjugated points on the optimal trajectory. The study we have done evaluates the command variation on the neighboring optimal trajectory. The sufficient conditions of minimum are obtained by imposing the positivity of the second variation. The results that this method offers are applied to the problem o the orbital rendez-vous for the linear case of the equations of movement.     

Limits of Bernstein-Bezier curves for periodic control nets

Ifn given control pointsb ₀,...b _n?1∈? ^d are repeated periodically byb _1+kn =b _j for allk∈?, the uniform limit of the Bernstein-Bezier polynomial curves of degreer with control pointsb ₀,…b _r forr»∞ is a Poisson curve (after a suitable reparametrization). This fact reveals some interesting self-similar structures in case of regularn-gons in the plane.     

Queueing models,block hessenberg matrices and the method of neuts

We introduce a numerical method to compute the stationary probability vector of queueing models whose infinitesimal generator is of block Hessenberg form. It is shown that the stationary probability vector is equal to the first column of the inverse of the coefficient matrix. Furthermore, it is shown that the first column of the inverse of an upper (or lower) Hessenberg matrix may be obtained in a relatively small number of operations. Together, these results allow us to define a powerful algorithm for solving certain queueing models. The efficiency of this algorithm is discussed and a comparison with the method of Neuts is undertaken. A relationship with the method of Gaussian elimination is established and used to develop some stability results.This work was supported in part by NSF Grant MCS-83-00438.     

Optimal regulation policies for a multipurpose reservoir with seasonal input and return function

Optimal regulation policies for a multipurpose reservoir are characterized analytically for the case where the input and return functions are subject to seasonal variations and the return function is piecewise linear, concave, and separable in reservoir volume and discharge. Efficient upper and lower bounds on optimal policies are derived for two important special cases. Necessary and sufficient conditions are obtained for a finite-time horizon optimal policy to be also optimal for the infinite-time horizon.This research was supported in part by the Office of Water Resources Research, Grant No. A-043-CAL, through the Water Resources Center, University of California.