Static Priority Policies Research Articles

Scheduling policies to minimize abandonment costs in infomercial call centers

Infomercial call centers answer calls in an order that reduces the number of abandoned calls to minimize lost revenue. We study a multi-class queuing system with a single server and customer abandonment, motivated by the case of infomercial call centers. We fully characterize the structure of the server’s optimal scheduling policy that minimizes the long-run average customer abandonment cost. We show that the optimal service policy is a static priority policy. We derive sufficient conditions under which the so-called b μ –rule is optimal. Under the b μ –rule, it is optimal to give priority to the customer type that has a higher service rate ( μ ) and higher abandonment cost (b), i.e., higher index b μ . We numerically test the robustness of our proposed policy in more general settings, including multi-server infomercial call centers.

On the Optimality of Greedy Policies in Dynamic Matching

Hindsight Optimality in Two-Way Matching Networks In “On the Optimality of Greedy Policies in Dynamic Matching”, Kerimov, Ashlagi, and Gurvich study centralized dynamic matching markets with finitely many agent types and heterogeneous match values. A matching policy is hindsight optimal if the policy can (nearly) maximize the total value simultaneously at all times. The article establishes that suitably designed greedy policies are hindsight optimal in two-way matching networks. This implies that there is essentially no positive externality from having agents waiting to form future matches. Proposed policies include the greedy longest-queue policy, with a minor variation, as well as a greedy static priority policy. The matching networks considered in this work satisfy a general position condition. General position is a weak (but necessary) condition that holds when the static-planning problem (a linear program that optimizes the first-order matching rates) has a unique and nondegenerate optimal solution.

Fluid Limits for Multiclass Many-Server Queues with General Reneging Distributions and Head-of-the-Line Scheduling

We describe a fluid model with time-varying input that approximates a multiclass many-server queue with general reneging distribution and multiple customer classes (specifically, the multiclass G/GI/N+GI queue). The system dynamics depend on the policy, which is a rule for determining when to serve a given customer class. The class of admissible control policies are those that are head-of-the-line (HL) and nonanticipating. For a sequence of many-server queues operating under admissible HL control policies and satisfying some mild asymptotic conditions, we establish a tightness result for the sequence of fluid scaled queue state descriptors and associated processes and show that limit points of such sequences are fluid model solutions almost surely. The tightness result together with the characterization of distributional limit points as fluid model solutions almost surely provides a foundation for the analysis of particular HL control policies of interest. We leverage these results to analyze a set of admissible HL control policies that we introduce, called weighted random buffer selection (WRBS), and an associated WRBS fluid model that allows multiple classes to be partially served in the fluid limit (which is in contrast to previously analyzed static priority policies).

On the Optimality of Greedy Policies in Dynamic Matching

We study centralized dynamic matching markets with finitely many agent types and heterogeneous match values. A network topology describes the pairs of agent types that can form a match and the value generated from each match. A matching policy is hindsight optimal if the policy can (nearly) maximize the total value simultaneously at all times. We find that suitably designed greedy policies are hindsight optimal in two-way matching networks. This implies that there is essentially no positive externality from having agents waiting to form future matches. We first show that the greedy longest-queue policy with a minor variation is hindsight optimal. Importantly, the policy is greedy relative to a residual network, which includes only non-redundant matches with respect to the static optimal matching rates. Moreover, when the residual network is acyclic (e.g., as in two-sided networks), we prescribe a greedy static priority policy that is also hindsight optimal. The priority order of this policy is robust to arrival rate perturbations that do not alter the residual network. Hindsight optimality is closely related to the lengths of type-specific queues. Queue-lengths cannot be smaller (in expectation) than Ο(1/e), where e is the general position gap that quantifies the stability in the network. The greedy longest-queue policy achieves this lower bound.

Optimal Control of Markov-Modulated Multiclass Many-Server Queues

We study multiclass many-server queues for which the arrival, service, and abandonment rates are all modulated by a common finite-state Markov process. We assume that the system operates in the “averaged” Halfin–Whitt regime, which means that it is critically loaded in the average sense although not necessarily in each state of the Markov process. We show that, under any static priority policy, the Markov-modulated diffusion-scaled queueing process is exponentially ergodic. This is accomplished by employing a solution to an associated Poisson equation to construct a suitable Lyapunov function. We establish a functional central limit theorem for the diffusion-scaled queueing process and show that the limiting process is a controlled diffusion with piecewise linear drift and constant covariance matrix. We address the infinite-horizon discounted and long-run average (ergodic) optimal control problems and establish asymptotic optimality.

Needy boarding patients in emergency departments: An exploratory case study using discrete-event simulation

Boarding patients and the extra workload they introduce are a major concern in emergency departments. Currently, research on boarding has focused on the beds that these patients occupy while they wait for admission in the inpatient ward. Little attention has been given to the check-ups these patients will inevitably need while waiting. These check-ups also confront the physicians with a challenging task: prioritizing between boarding patients and patients currently under treatment in the emergency department. This article firstly explores the limited previous research on boarding patients. Secondly, this paper quantitatively demonstrates that needy boarding patients can significantly impact system performance and hence should be accounted for in the analysis and planning of an emergency department through discrete-event simulation. Next, three static priority policies (first-come, first-served and always prioritizing either boarding patients or the other patients) and one dynamic priority policy (using accumulating priorities) are evaluated on various performance measures. We find, for our case study, that system performance is optimized by applying a priority policy that favours patients currently under treatment in the emergency department.

Optimal Policy in Single-Server Multi-Class Queuing Systems with Abandonment

We study a multi-class queuing system with a single server and customer abandonment. We fully characterize the structure of the server's optimal scheduling policy that minimizes the long-run average customer abandonment cost. We show that the optimal service policy is a static priority policy. We derive sufficient conditions under which the so-called b.mu-rule is optimal. Under the b.mu-rule, it is optimal to give priority to the customer type that has higher service rate (mu) and higher abandonment cost (b), i.e., higher index b.mu.

Asymptotically optimal control of N-systems with $$H_2^*$$ H 2 ∗ service times under many-server heavy traffic

We address a control problem for a queueing system, known as the "N-system," under the Halfin---Whitt heavy-traffic regime. It has two customer classes and two server pools: servers from one pool can serve both customer classes, while servers from the other pool can only serve one class. The service time is assumed to follow a special case of hyper-exponential distributions, called the $$H^*_2$$H2ź distribution, and customers are impatient. We consider an expected infinite-horizon discounted cost function, with linear holding and abandonment cost components. A static priority policy is proposed and is shown to be asymptotically optimal in a certain parameter regime, using weak convergence techniques.

Determining the conditions for reverse triage in emergency medical services using queuing theory

In emergency health care, there are situations where the less wounded are treated in preference to the more severely wounded, so called reverse triage. This may arise in situations such as war where soldiers are required to return to the battlefield as quickly as possible, or disaster situations where medical resources are limited in order to conserve resources for those likely to survive without requiring advanced medical care. This article is to study the reverse triage in time-critical systems, where conditions of patients may deteriorate while waiting for treatment. A queueing model with transfers and abandonments is developed to study the health care system. Using smoothed rate truncation method, sufficient conditions of reverse triage are derived, under which non-critical patients are treated with priority. For other conditions, the optimal policy is very complicated. We suggest a dynamic control policy according to the system state rather than any static priority policies.

Stability analysis of N-model systems under a static priority rule

We consider the stability of N-model systems that consist of two customer classes and two server pools. Servers in one of the pools can serve both classes, but those in the other pool can serve only one of the classes. The standard fluid models in general are not sufficient to establish the stability region of these systems under static priority policies. Therefore, we use a novel and a general approach to augment the fluid model equations based on induced Markov chains. Using this new approach, we establish the stability region of these systems under a static priority rule with thresholds when the service and interarrival times have phase-type distributions. We show that, in certain cases, the stability region depends on the distributions of the service and interarrival times (beyond their mean), on the number of servers in the system, and on the threshold value. We also show that it is possible to expand the stability region in these systems by increasing the variability of the service times (without changing their mean) while keeping the other parameters fixed. The extension of our results to parallel server systems and general service time distributions is also discussed.

Asymptotically optimal control of many-server heterogeneous service systems with $H_{2}^{*}$ service times

Optimal control of many-server heterogenous service systems with service times that have a special hyper-exponential distribution, denoted by $H_{2}^{*}$ , which is a mixture of an exponential distribution and a unit point mass at 0, is considered. A static priority policy that assigns priorities to server pools based on their service time distributions is proposed. This policy is shown to be asymptotically optimal in the many-server heavy-traffic regime in minimizing the total number of customers.

Applications of machine learning approach on multi-queue message scheduling

Due to limited resource contentions and deadline constraints, messages on the controller area network (CAN) are competing for service from the common resources. This problem can be resolved by assigning priorities to different message classes to satisfy time-critical applications. Actually, because of the fluctuation of network traffic or an inefficient use of resources, these static or dynamic priority policies may not guarantee flexibility for different kinds of messages in real-time scheduling. Consequently, the message transmission which cannot comply with the timing requirements or deadlines may deteriorate system performance significantly. In this paper, we have proposed a controller-plant model, where the plant is analogous to a message queue pool (MQP) and the message scheduling controller (MSC) is responsible to dispatch resources for queued messages according to the feedback information from the MQP. The message scheduling controller, which is realized by the radial basis function (RBF) network, is designed with machine learning algorithm to compensate the variations in plant dynamics. The MSC with the novel hybrid learning schemes can ensure a low and stable message waiting time variance (or a uniform distribution of waiting time) and lower transmission failures. A significant emphasis of the MSC is the variable structure of the RBF model to accommodate to complex scheduling situations. Simulation experiments have shown that several variants of the MSC significantly improve overall system performance over the static scheduling strategies and the dynamic earliest-deadline first (EDF) algorithms under a wide range of workload characteristics and execution environments.

Dynamic Control of N-Systems with Many Servers: Asymptotic Optimality of a Static Priority Policy in Heavy Traffic

We consider a class of parallel server systems that are known as N-systems. In an N-system, there are two customer classes that are catered by servers in two pools. Servers in one of the pools are cross-trained and can serve customers from both classes, whereas all of the servers in the other pool can serve only one of the customer classes. A customer reneges from his queue if his waiting time in the queue exceeds his patience. Our objective is to minimize the total cost that includes a linear holding cost and a reneging cost. We prove that, when the service speed is pool dependent, but not class dependent, a cμ-type greedy policy is asymptotically optimal in many-server heavy traffic.

Design and control of agile automated CONWIP production lines

AbstractIn this article, we study the design and control of manufacturing cells with a mix of manual and automated equipment, operating under a CONWIP pull protocol, and staffed by a single agile (cross‐trained) worker. For a three‐station line with one automated station, we fully characterize the structure of the optimal control policy for the worker and show that it is a static priority policy. Using analytical models and extensive simulation experiments, we also evaluate the effectiveness of practical heuristic control policies and provide managerial insights on automation configuration design of the line. This characterization of the worker control policy enables us to develop managerial insights into the design issues of how best to locate and concentrate automation in the line. Finally, we show that, in addition to ease of control and greater design flexibility, the CONWIP protocol also offers higher efficiency and robustness than does the push protocol. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2009

Optimal server scheduling in nonpreemptive finite-population queueing systems

We consider a finite-population queueing system with heterogeneous classes of customers and a single server. For the case of nonpreemptive service, we fully characterize the structure of the server's optimal service policy that minimizes the total average customer waiting costs. We show that the optimal service policy may never serve some classes of customers. For those classes that are served, we show that the optimal service policy is a simple static priority policy. We also derive sufficient conditions that determine the optimal priority sequence.

Delay performance of some scheduling strategies in an input queuing ATM switch with multiclass bursty traffic

Considers an N/spl times/N nonblocking, space division, input queuing ATM cell switch, and a class of Markovian models for cell arrivals on each of its inputs. The traffic at each input comprises geometrically distributed bursts of cells, each burst destined for a particular output. The inputs differ in the burstiness of the offered traffic, with burstiness being characterized in terms of the average burst length. We analyze burst delays where some inputs receive traffic with low burstiness and others receive traffic with higher burstiness. Three policies for head-of-the-line contention resolution are studied: two static priority policies [shorter-expected-burst-length-first (SEBF), longer-expected-burst-length-first (LEBF)] and random selection (RS). Direct queuing analysis is used to obtain approximations for asymptotic high and low priority mean burst delays with the priority policies. Simulation is used for obtaining mean burst delays for finite N and for the random selection policy. As the traffic burstiness increases, the asymptotic analysis can serve as a good approximation only for large switch sizes. Qualitative performance comparisons based on the asymptotic analysis are, however, found to continue to hold for finite switch sizes. It is found that the SEBF policy yields the best delay performance over a wide range of loads, while RS lies in between. SEBF drastically reduces the delay of the less bursty traffic while only slightly increasing the delay of the more bursty traffic. LEBF causes severe degradation in the delay of less bursty traffic, while only marginally improving the delays of the more bursty traffic. RS can be an adequate compromise if there is no prior knowledge of input traffic burstiness.

Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Closed Network

We consider a multiclass closed queueing network with two single-server stations. Each class requires service at a particular station, and customers change class after service according to specified probabilities. There is a general service time distribution for each class. The problem is to schedule the two servers to maximize the long-run average throughput of the network. By assuming a large customer population and nearly balanced loading of the two stations, the scheduling problem can be approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved exactly and the solution is interpreted in terms of the queueing network to obtain a scheduling rule. (We conjecture, quite naturally, that the resulting scheduling rule is asymptotically optimal under heavy traffic conditions, but no attempt is made to prove that.) The scheduling rule is a static priority policy that computes an index for each class and awards higher priority at station 1 (respectively, station 2) to classes with the smaller (respectively, larger) values of this index. An analytical comparison of this rule to any other static policy is also obtained. An example is given that illustrates the procedure and demonstrates its effectiveness.

Necessary optimality conditions for priority policies in stochastic weighted flowtime scheduling problems

Conditions implied by the optimality of static priority policies in a special class of stochastic scheduling problems will be derived.

Necessary optimality conditions for priority policies in stochastic weighted flowtime scheduling problems

Conditions implied by the optimality of static priority policies in a special class of stochastic scheduling problems will be derived.

On the optimality of static priority policies in stochastic scheduling on parallel machines

n jobs are to be preemptively scheduled for processing on n machines. The machines may have differing speeds and the jobs have processing requirements which are distributed as independent exponential random variables with different means. Holding cost g(U) is incurred per unit time that the set of uncompleted jobs is U and it is desired to minimize the total expected holding cost which is incurred until all jobs are complete. We show that if g satisfies certain simple conditions then the optimal policy is one which takes the jobs in the order 1, 2, ···, n and assigns each uncompleted job in turn to the fastest available machine. In the special case in which the objective is to minimize the expected weighted flowtime, where there is a holding cost of wi while job i is incomplete, the sufficient condition is simply w1 ≧ … ≧ wn and λ1 w1 ≧ … ≧ λn wn.