Dynamic fluid-based scheduling in a multi-class abandonment queue

International audienceWe investigate how to share a common resource among multiple classes of customers in the presence of abandonments. We consider two different models: (1) customers can abandon both while waiting in the queue and while being served, (2) only customers that are in the queue can abandon. Given the complexity of the stochastic optimization problem we propose a fluid model as a deterministic approximation. For the overload case we directly obtain that the c˜µ/θ rule is optimal. For the underload case we use Pontryagin’s Maximum Principle to obtain the optimal solution for two classes of customers; there exists a switching curve that splits the two-dimensional state-space into two regions such that when the number of customers in both classes is sufficiently small the optimal policy follows the c˜µ-rule and when the number of customers is sufficiently large the optimal policy follows the c˜µ/θ-rule. The same structure is observed in the optimal policy of the stochastic model for an arbitrary number of classes. Based on this we develop a heuristic and by numerical experiments we evaluate its performance and compare it to several index policies. We observe that the suboptimality gap of our solution is small

Optimal Control of Parallel Queues for Managing Volunteer Convergence

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/163497/2/poms13224.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/163497/1/poms13224_am.pd

Coupled queues with customer impatience

Motivated by assembly processes, we consider a Markovian queueing system with multiple coupled queues and customer impatience. Coupling means that departures from all constituent queues are synchronised and that service is interrupted whenever any of the queues is empty and only resumes when all queues are non-empty again. Even under Markovian assumptions, the state space grows exponentially with the number of queues involved. To cope with this inherent state space explosion problem, we investigate performance by means of two numerical approximation techniques based on series expansions, as well as by deriving the fluid limit. In addition, we provide closed-form expressions for the first terms in the series expansion of the mean queue content for the symmetric coupled queueing system. By an extensive set of numerical experiments, we show that the approximation methods complement each other, each one being accurate in a particular subset of the parameter space. (C) 2017 Elsevier B.V. All rights reserved

Asymptotically optimal index policies for an abandonment queue with convex holding cost.

International audienceWe investigate a resource allocation problem in a multi-class server with convex holding costs and user impatience under the average cost criterion. In general, the optimal policy has a complex dependency on all the input parameters and state information. Our main contribution is to derive index policies that can serve as heuristics and are shown to give good performance. Our index policy attributes to each class an index, which depends on the number of customers currently present in that class. The index values are obtained by solving a relaxed version of the optimal stochastic control problem and combining results from restless multi-armed bandits and queueing theory. They can be expressed as a function of the steady-state distribution probabilities of a one-dimensional birth-and-death process. For linear holding cost, the index can be calculated in closed-form and turns out to be independent of the arrival rates and the number of customers present. In the case of no abandonments and linear holding cost, our index coincides with the $c\mu$-rule, which is known to be optimal in this simple setting. For general convex holding cost we derive properties of the index value in limiting regimes: we consider the behavior of the index (i) as the number of customers in a class grows large, which allows us to derive the asymptotic structure of the index policies, (ii) as the abandonment rate vanishes, which allows us to retrieve an index policy proposed for the multi-class M/M/1 queue with convex holding cost and no abandonments, and (iii) as the arrival rate goes to either 0 or $\infty$, representing light-traffic and heavy-traffic regimes, respectively. We show that Whittle's index policy is asymptotically optimal in both light-traffic and heavy-traffic regimes. To obtain further insights into the index policy, we consider the fluid version of the relaxed problem and derive a closed-form expression for the fluid index. The latter is shown to coincide with the index values for the stochastic model in asymptotic regimes. For arbitrary convex holding cost the fluid index can be seen as the $Gc\mu/\theta$-rule, that is, including abandonments into the generalized $c\mu$-rule ($Gc\mu$-rule). Numerical experiments for a wide range of parameters have shown that the Whittle index policy and the fluid index policy perform very well for a broad range of parameters

Performance analysis of time-dependent queueing systems: survey and classification

Many queueing systems are subject to time-dependent changes in system parameters, such as the arrival rate or number of servers. Examples include time-dependent call volumes and agents at inbound call centers, time-varying air traffic at airports, time-dependent truck arrival rates at seaports, and cyclic message volumes in computer systems.There are several approaches for the performance analysis of queueing systems with deterministic parameter changes over time. In this survey, we develop a classification scheme that groups these approaches according to their underlying key ideas into (i) numerical and analytical solutions,(ii)approaches based on models with piecewise constant parameters, and (iii) approaches based on mod-ified system characteristics. Additionally, we identify links between the different approaches and provide a survey of applications that are categorized into service, road and air traffic, and IT systems

Dynamic routing policies for multi-skill call centers

We consider the problem of routing calls dynamically in a multiskill call center. Calls from different skill classes are offered to the call center according to a Poisson process. The agents in the center are grouped according to their heterogeneous skill sets that determine the classes of calls they can serve. Each agent group serves calls with independent exponentially distributed service times. We consider two scenarios. The first scenario deals with a call center with no buffers in the system, so that every arriving call either has to be routed immediately or has to be blocked and is lost. The objective in the system is to minimize the average number of blocked calls. The second scenario deals with call centers consisting of only agents that have one skill and fully cross-trained agents, where calls are pooled in common queues. The objective in this system is to minimize the average number of calls in the system. We obtain nearly optimal dynamic routing policies that are scalable with the problem instance and can be computed online. The algorithm is based on one-step policy improvement using the relative value functions of simpler queuing systems. Numerical experiments demonstrate the good performance of the routing policies. Finally, we discuss how the algorithm can be used to handle more general cases with the techniques described in this article. © 2009 Cambridge University Press

The application of Approximate Dynamic Programming techniques

Koole, G.M. [Promotor]Bhulai, S. [Copromotor