Transient solutions for multi-server queues with finite buffers

Transient solutions for M/M/c queues are important for staffing call centers, police stations, hospitals and similar institutions. In this paper we show how to find transient solutions for M/M/c queues with finite buffers by using eigenvalues and eigenvectors. To find the eigenvalues, we create a system of difference equations where the coefficients depend on a parameter x. These difference equations allow us to search for all eigenvalues by changing x. To facilitate the search, we use Sturm sequences for locating the eigenvalues. We also show that the resulting method is numerically stable.     

Tandem queues with subexponential service times and finite buffers

We focus on tandem queues with subexponential service time distributions. We assume that number of customers in front of the first station is infinite and there is infinite room for finished customers after the last station but the size of the buffer between two consecutive stations is finite. Using (max, +) linear recursions, we investigate the tail asymptotics of transient response times and waiting times under both communication blocking and manufacturing blocking schemes. We also discuss under which conditions these results can be generalized to the tail asymptotics of stationary response times and waiting times.     

Waiting and sojourn times in a multi-server queue with mixed priorities

We consider a multi-server queue with K priority classes. In this system, customers of the P highest priorities (P<K) can preempt customers with lower priorities, ejecting them from service and sending them back into the queue. Service times are assumed exponential with the same mean for all classes. The Laplace–Stieltjes transforms of waiting times are calculated explicitly and the Laplace–Stieltjes transforms of sojourn times are provided in an implicit form via a system of functional equations. In both cases, moments of any order can be easily calculated. Specifically, we provide formulae for the steady state means and the second moments of waiting times for all priority classes. We also study some approximations of sojourn-time distributions via their moments. In a practical part of our paper, we discuss the use of mixed priorities for different types of Service Level Agreements, including an example based on a real scheduling problem of IT support teams.      

Admission control for a multi-server queue with abandonment

In a M/M/N+M queue, when there are many customers waiting, it may be preferable to reject a new arrival rather than risk that arrival later abandoning without receiving service. On the other hand, rejecting new arrivals increases the percentage of time servers are idle, which also may not be desirable. We address these trade-offs by considering an admission control problem for a M/M/N+M queue when there are costs associated with customer abandonment, server idleness, and turning away customers. First, we formulate the relevant Markov decision process (MDP), show that the optimal policy is of threshold form, and provide a simple and efficient iterative algorithm that does not presuppose a bounded state space to compute the minimum infinite horizon expected average cost and associated threshold level. Under certain conditions we can guarantee that the algorithm provides an exact optimal solution when it stops; otherwise, the algorithm stops when a provided bound on the optimality gap is reached. Next, we solve the approximating diffusion control problem (DCP) that arises in the Halfin–Whitt many-server limit regime. This allows us to establish that the parameter space has a sharp division. Specifically, there is an optimal solution with a finite threshold level when the cost of an abandonment exceeds the cost of rejecting a customer; otherwise, there is an optimal solution that exercises no control. This analysis also yields a convenient analytic expression for the infinite horizon expected average cost as a function of the threshold level. Finally, we propose a policy for the original system that is based on the DCP solution, and show that this policy is asymptotically optimal. Our extensive numerical study shows that the control that arises from solving the DCP achieves a very similar cost to the control that arises from solving the MDP, even when the number of servers is small.     

Heavy-traffic limits for many-server queues with service interruptions

We establish many-server heavy-traffic limits for G/M/n+M queueing models, allowing customer abandonment (the +M), subject to exogenous regenerative service interruptions. With unscaled service interruption times, we obtain a FWLLN for the queue-length process, where the limit is an ordinary differential equation in a two-state random environment. With asymptotically negligible service interruptions, we obtain a FCLT for the queue-length process, where the limit is characterized as the pathwise unique solution to a stochastic integral equation with jumps. When the arrivals are renewal and the interruption cycle time is exponential, the limit is a Markov process, being a jump-diffusion process in the QED regime and an O–U process driven by a Levy process in the ED regime (and for infinite-server queues). A stochastic-decomposition property of the steady-state distribution of the limit process in the ED regime (and for infinite-server queues) is obtained.     

A Poisson–Charlier approximation for nonstationary queues

In this paper, we develop a new approximation for nonstationary multiserver queues with abandonment. Our method uses the Poisson–Charlier polynomials, which are a discrete orthogonal polynomial sequence that is orthogonal with respect to the Poisson distribution. We show that by appealing to the Poisson–Charlier polynomials that we can estimate the mean, variance, and probability of delay of our nonstationary queueing system with good accuracy. Lastly, we provide a numerical example that illustrates that our approximations are effective.     

Synchronized reneging in queueing systems with vacations

In this paper we present a detailed analysis of queueing models with vacations and impatient customers, where the source of impatience is the absence of the server. Instead of the standard assumption that customers perform independent abandonments, we consider situations where customers abandon the system simultaneously. This is, for example, the case in remote systems where customers may decide to abandon the system, when a transport facility becomes available.     

Markovian bulk-arrival and bulk-service queues with state-dependent control

We study a modified Markovian bulk-arrival and bulk-service queue incorporating state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated and the relationship with our queueing model is examined and exploited. Equilibrium behaviour is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined and the Laplace transform of the queue length distribution is presented. The important questions regarding hitting time and busy period distributions are answered in detail and the Laplace transforms of these distributions are presented. Further properties including expectations of hitting times and busy period are also explored.     

Grassmann algorithms for low rank approximation of matrices with missing values

The problem of approximating a matrix by another matrix of lower rank, when a modest portion of its elements are missing, is considered. The solution is obtained using Newton’s algorithm to find a zero of a vector field on a product manifold. As a preliminary the algorithm is formulated for the well-known case with no missing elements where also a rederivation of the correction equation in a block Jacobi-Davidson method is included. Numerical examples show that the Newton algorithm grows more efficient than an alternating least squares procedure as the amount of missing values increases.     

An approximation for the nonlinear filtering problem,with error bound †

For the standard continuous-time nonlinear filtering problem an approximation approach is derived. The approximate filter is given by the solution to an appropriate discrete-time approximating filtering problem that can be explicitly solved by a finite-dimensional procedure. Furthermore an explicit upper bound for the approximation error is derived. The approximating problem is obtained by first approximating the signal and then using measure transformation to express the original observation process in terms of the approximating signal     

An algorithm for nonlinear optimization problems with binary variables

One of the challenging optimization problems is determining the minimizer of a nonlinear programming problem that has binary variables. A vexing difficulty is the rate the work to solve such problems increases as the number of discrete variables increases. Any such problem with bounded discrete variables, especially binary variables, may be transformed to that of finding a global optimum of a problem in continuous variables. However, the transformed problems usually have astronomically large numbers of local minimizers, making them harder to solve than typical global optimization problems. Despite this apparent disadvantage, we show that the approach is not futile if we use smoothing techniques. The method we advocate first convexifies the problem and then solves a sequence of subproblems, whose solutions form a trajectory that leads to the solution. To illustrate how well the algorithm performs we show the computational results of applying it to problems taken from the literature and new test problems with known optimal solutions.     

Two queues with vastly different arrival rates and processor-sharing factors

We consider a 2-class queueing system, operating under a generalized processor-sharing discipline. The arrival rate to the secondary queue is much smaller than that to the primary queue, while the exponentially distributed service requirements have comparable parameters. The primary queue is assumed to be heavily loaded, so the processor-sharing factor for the secondary queue is assumed to be relatively small. We use singular perturbation analyses in a small parameter measuring the ratio of arrival rates, and the closeness of the system to instability. Two different regimes are analyzed, corresponding to a heavily loaded and a lightly loaded secondary queue, respectively. With suitable scaling of variables, lowest order asymptotic approximations to the joint stationary distribution of the numbers of jobs in the two queues are derived, as well as to the marginal distributions.     

An optimal Gauss–Markov approximation for a process with stochastic drift and applications

We consider a linear stochastic differential equation with stochastic drift. We study the problem of approximating the solution of such equation through an Ornstein–Uhlenbeck type process, by using direct methods of calculus of variations. We show that general power cost functionals satisfy the conditions for existence and uniqueness of the approximation. We provide some examples of general interest and we give bounds on the goodness of the corresponding approximations. Finally, we focus on a model of a neuron embedded in a simple network and we study the approximation of its activity, by exploiting the aforementioned results.     

Control of systems with flexible multi-server pools: a shadow routing approach

A general model with multiple input flows (classes) and several flexible multi-server pools is considered. We propose a robust, generic scheme for routing new arrivals, which optimally balances server pools’ loads, without the knowledge of the flow input rates and without solving any optimization problem. The scheme is based on Shadow routing in a virtual queueing system. We study the behavior of our scheme in the Halfin–Whitt (or, QED) asymptotic regime, when server pool sizes and the input rates are scaled up simultaneously by a factor r growing to infinity, while keeping the system load within \(O(\sqrt{r}\,)\) of its capacity.The main results are as follows. (i) We show that, in general, a system in a stationary regime has at least \(O(\sqrt{r}\,)\) average queue lengths, even if the so called null-controllability (Atar et al., Ann. Appl. Probab. 16, 1764–1804, 2006) on a finite time interval is possible; strategies achieving this \(O(\sqrt{r}\,)\) growth rate we call order-optimal. (ii) We show that some natural algorithms, such as MaxWeight, that guarantee stability, are not order-optimal. (iii) Under the complete resource pooling condition, we prove the diffusion limit of the arrival processes into server pools, under the Shadow routing. (We conjecture that result (iii) leads to order-optimality of the Shadow routing algorithm; a formal proof of this fact is an important subject of future work.) Simulation results demonstrate good performance and robustness of our scheme.     

Metaheuristics for the multi-objective FJSP with sequence-dependent set-up times,auxiliary resources and machine down time

This paper investigates the application of particle swarm optimization (PSO) to the multi-objective flexible job shop scheduling problem with sequence-dependent set-up times, auxiliary resources and machine down time. To achieve this goal, alternative particle representations and problem mapping mechanisms were implemented within the PSO paradigm. This resulted in the development of four PSO-based heuristics. Benchmarking on real customer data indicated that using the priority-based representation resulted in a significant performance improvement over the existing rule-based algorithms commonly used to solve this problem. Additional investigation into algorithm scalability led to the development of a priority-based differential evolution algorithm. Apart from the academic significance of the paper, the benefit of an improved production schedule can be generalized to include cost reduction, customer satisfaction, improved profitability, and overall competitive advantage.     

On the optimality of threshold control in queues with model uncertainty

We consider a single-stage queuing system where arrivals and departures are modeled by point processes with stochastic intensities. An arrival incurs a cost, while a departure earns a revenue. The objective is to maximize the profit by controlling the intensities subject to capacity limits and holding costs. When the stochastic model for arrival and departure processes are completely known, then a threshold policy is known to be optimal. Many times arrival and departure processes can not be accurately modeled and controlled due to lack of sufficient calibration data or inaccurate assumptions. We prove that a threshold policy is optimal under a max–min robust model when the uncertainty in the processes is characterized by relative entropy. Our model generalizes the standard notion of relative entropy to account for different levels of model uncertainty in arrival and departure processes. We also study the impact of uncertainty levels on the optimal threshold control.     

A smoothing sample average approximation method for stochastic optimization problems with CVaR risk measure

This paper is concerned with solving single CVaR and mixed CVaR minimization problems. A CHKS-type smoothing sample average approximation (SAA) method is proposed for solving these two problems, which retains the convexity and smoothness of the original problem and is easy to implement. For any fixed smoothing constant ε, this method produces a sequence whose cluster points are weak stationary points of the CVaR optimization problems with probability one. This framework of combining smoothing technique and SAA scheme can be extended to other smoothing functions as well. Practical numerical examples arising from logistics management are presented to show the usefulness of this method.     

An optimal replenishment policy for an EOQ model with partial backlogging

We study an inventory system where demand on the stockout period is partially backlogged. The backlogged demand ratio is a mixture of two exponential functions. The shortage cost has two significant costs: the unit backorder cost (which includes a fixed cost and a cost proportional to the length of time for which the backorder exists) and the cost of lost sales. A general procedure to determine the optimal policy and the minimum inventory cost for all the parameter values is developed. This model generalizes several inventory systems analyzed by different authors. Numerical examples are used to illustrate the theoretical results.     

An aggregate deformation homotopy method for min-max-min problems with max-min constraints

In this paper, the constrained min-max-min problem, which is an essentially nonsmooth and nonconvex problem, is considered. Based on a twice aggregate function with a modification, an aggregate deformation homotopy method is established. Under some suitable assumptions, a smooth path from a randomly given point to a solution of the generalized KKT system is proven to exist. By numerically tracing the smooth path, a globally convergent algorithm for some solution of the problem is given. Some numerical results are given to show the feasibility of the method.     

High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.