Multiclass Queueing Network Research Articles

Measurement and optimization of robust stability of multiclass queueing networks: Applications in dynamic supply chains

Multiclass queueing networks are an essential tool for modeling and analyzing complex supply chains. Roughly speaking, stability of these networks implies that the total number of customers/jobs in the network remains bounded over time. In this context robustness characterizes the ability of a multiclass queueing network to remain stable, if the expected values of the interarrival and service times distributions are subject to uncertain shifts. A powerful starting point for the stability analysis of multiclass queueing networks is the associated fluid network. Based on the fluid network analysis we present a measure to quantify the robustness, which is indicated by a single number. This number will be called the stability radius. It represents the magnitude of the smallest shift of the expected value of the interarrival and/or service times distributions so that the associated fluid network looses the property of stability. The stability radius is a worst case measure and is a conceptual adaptation from the dynamical systems literature. Moreover, we provide a characterization of the shifts that destabilize the network. Based on these results, we formulate a mathematical program that minimizes the required network capacity, while ensuring a desired level of robustness towards shifts of the expected values of the interarrival times distributions. This approach provides a new view on long-term robust production capacity allocation in supply chains. The capabilities of our method are demonstrated using a real world supply chain.

A simulation model with synchronization manufacturing stations

PurposeThe aim of this paper is to develop an efficient analytical procedure to evaluate performance of the most general pull production systems particularly when multiple‐part‐types are involved. The authors consider a kanban controlled production system that can be modelled as a closed queuing network with different product classes. The production line is decomposed into stages which consist of one or several stations and an output buffer. Each stage is associated with a given number of kanbans. The main idea of this analytical algorithm is to analyze each subnetwork individually using a product form approximation technique. The iterative procedure is used to find the unknown parameters.Design/methodology/approachThe authors design a multiclass queuing network that can be used to represent kanban controlled production systems. To solve this model, three procedures are used: decompose the original network into M subnetworks, convergence of unknown parameters in each subnetwork, and convergence of unknown parameters in the original network. The authors now describe these procedures separately.FindingsThe main contribution of this paper is the formulation of the problem of kanban controlled production systems with several part‐types. The methodology is based on approximate formula with decomposition and is applicable to more general manufacturing environments. The authors' method can be applied to both limited and unlimited demands. The analytical algorithm designed in this work has demonstrated excellent performance in analyzing kanban controlled production systems.Originality/valueThe methodology of this algorithm is based on approximate formula and is applicable to more general manufacturing environments.

Flow-level Convergence and Insensitivity for Multi-class Queueing Networks

We consider a multi-class queueing network as a model of packet transfer in a communication network. We define a second stochastic model as a model of document transfer in a communication network where the documents transferred have a general distribution. We prove the weak convergence of the multi-class queueing process to the document transfer process. Our convergence result allows the comparison of general document size distributions, and consequently, we prove general insensitivity results for the limit queueing process. We discuss how this separation of time-scales method of proving insensitivity may be applied to other insensitive queueing systems.

A cross entropy based multiagent approach for multiclass activity chain modeling and simulation

This paper attempts to model complex destination-chain, departure time and route choices based on activity plan implementation and proposes an arc-based cross entropy method for solving approximately the dynamic user equilibrium in multiagent-based multiclass network context. A multiagent-based dynamic activity chain model is developed, combining travelers’ day-to-day learning process in the presence of both traffic flow and activity supply dynamics. The learning process towards user equilibrium in multiagent systems is based on the framework of Bellman’s principle of optimality, and iteratively solved by the cross entropy method. A numerical example is implemented to illustrate the performance of the proposed method on a multiclass queuing network.

Flow-level convergence and insensitivity for multi-class queueing networks

We consider a multi-class queueing network as a model of packet transfer in a communication network. We define a second stochastic model as a model of document transfer in a communication network where the documents transferred have a general distribution. We prove the weak convergence of the multi-class queueing process to the document transfer process. Our convergence result allows the comparison of general document size distributions, and consequently, we prove general insensitivity results for the limit queueing process. We discuss how this separation of time-scales method of proving insensitivity may be applied to other insensitive queueing systems.

LocTrackJINQS: An Extensible Location-aware Simulation Tool for Multiclass Queueing Networks

This paper presents LocTrackJINQS, a flexible and extensible spatio-temporal simulation tool for systems that involve the flow and processing of customers at multiple service centres. Developed based on the multi-class queueing network simulation package JINQS, LocTrackJINQS retains the abstract model specification power of JINQS while providing additional low-level models of entity movement. Besides traditional performance metrics, LocTrackJINQS produces as output a trace of each entityʼs location in the system over time. It can thus be used to generate synthetic location tracking data for location-based research or applications.

Routing Internet traffic in heterogeneous mesh networks: Analysis and algorithms

In practical wireless mesh networks (WMNs), gateways are subject to hard capacity limits on the aggregate number of flows (in terms of bit rate) that they can support. Thus, if traffic is routed in the mesh network without considering those constraints, as well as the traffic distribution, some gateways or intermediate mesh routers may rapidly get overloaded, and the network resources can be unevenly utilized. To address this problem, in this paper we firstly develop a multi-class queuing network model to analyze feasible throughput allocations, as well as average end-to-end delay, in heterogeneous WMNs. Guided by our analysis, we design a Capacity-Aware Route Selection algorithm ( CARS), which allocates network paths to downstream and upstream Internet flows so as to ensure a more balanced utilization of wireless network resources and gateways’ fixed connections. Through simulations in a number of different network scenarios we show that the CARS scheme significantly outperforms conventional shortest path routing, as well as an alternative routing method that distributes the traffic load on the gateway nodes to minimize its variance.

Supply Network Engineering: An Approach to Robust Capacity Allocation for Stochastic Production Processes

AbstractThe dynamics of supply networks can exhibit considerable complexity. Expected and unexpected events can reduce the production capacity available on average and put the timely satisfaction of customer demand at risk. In this paper we model supply networks as multiclass queueing networks and present an approach to robust capacity allocation with respect to perturbations of the production processes. To this end we consider the fluid model of a multiclass queueing network and use its stability radius to measure the robustness. The stability radius reflects the smallest decrease of production rate that destabilizes the network. Based on results concerning this measure we set up an optimization problem for the capacity allocation.

State space collapse and stability of queueing networks

We study the stability of subcritical multi-class queueing networks with feedback allowed and a work-conserving head-of-the-line service discipline. Assuming that the fluid limit model associated to the queueing network satisfies a state space collapse condition, we show that the queueing network is stable provided that any solution of an associated linear Skorokhod problem is attracted to the origin in finite time. We also give sufficient conditions ensuring this attraction in terms of the reflection matrix of the Skorokhod problem, by using an adequate Lyapunov function. State space collapse establishes that the fluid limit of the queue process can be expressed in terms of the fluid limit of the workload process by means of a lifting matrix.

Strong approximations for a Kumar-Seidman network under a priority service discipline

In this paper, we derive the strong approximations for a four-class two station multi-class queuing network (Kumar-Seidman network) under a priority service discipline. Within a group, jobs are served in the order of arrival, that is, a first-in-first-out disciple, and among groups, jobs are served under a pre-emptiveresume priority disciple. We show that if the input data (i.e., the arrival and service processe) satisfy an approximation (such as the functional law-of-iterated logarithm approximation or the strong approximation), the output data (the departure processes) and the performance measures (such as the queue length, the work load and the sojourn time process) satisfy a similar approximation.

State-space collapse in stationarity and its application to a multiclass single-server queue in heavy traffic

Recently Gamarnik and Zeevi (Ann. Appl. Probab. 16:56---90, 2006) and Budhiraja and Lee (Math. Oper. Res. 34:45---56, 2009) established that, under suitable conditions, a sequence of the stationary scaled queue lengths in a generalized Jackson queueing network converges to the stationary distribution of multidimensional reflected Brownian motion in the heavy-traffic regime. In this work we study the corresponding problem in multiclass queueing networks (MQNs). In the first part of this work we consider the MQNs for which the fluid stability is valid, state-space collapse is exhibited under suitable initial conditions and a heavy traffic limit theorem holds. For such MQNs we establish that, under the assumption of the tightness of a sequence of stationary scaled workloads, the sequence converges to the stationary distribution of semimartingale reflecting Brownian motion in the heavy-traffic regime. The key to the proof is to show that state-space collapse occurs in the heavy-traffic regime in stationarity under the assumption of tightness. In the second part, using the result obtained, it is shown that such a convergence of stationary workload holds for a multiclass single-server queue with feedback routing, where the tightness is proved by the Lyapunov function method developed in (Gamarnik and Zeevi in Ann. Appl. Probab. 16:56---90, 2006).

Proportional fairness and its relationship with multi-class queueing networks

We consider multi-class single-server queueing networks that have a product form stationary distribution. A new limit result proves a sequence of such networks converges weakly to a stochastic flow level model. The stochastic flow level model found is insensitive. A large deviation principle for the stationary distribution of these multi-class queueing networks is also found. Its rate function has a dual form that coincides with proportional fairness. We then give the first rigorous proof that the stationary throughput of a multi-class single-server queueing network converges to a proportionally fair allocation. This work combines classical queueing networks with more recent work on stochastic flow level models and proportional fairness. One could view these seemingly different models as the same system described at different levels of granularity: a microscopic, queueing level description; a macroscopic, flow level description and a teleological, optimization description.

Positive Harris recurrence and diffusion scale analysis of a push pull queueing network

We consider a push pull queueing network with two servers and two types of job which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull network was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the Kumar–Seidman Rybko–Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform fluid and diffusion scale analysis of this network under such policies, to show fluid stability, positive Harris recurrence, and to obtain a diffusion limit for the network. On the diffusion scale the network is empty, and the departures of the two types of job are highly negatively correlated Brownian motions. Using similar methods we also derive a diffusion limit of a re-entrant line with an infinite supply of work.

Incentives for Quality Through Endogenous Routing

We study how rework routing together with wage and piece-rate compensation can strengthen incentives for quality. Traditionally, rework is assigned back to the agent who generates the defect (in a self-routing scheme) or to another agent dedicated to rework (in a dedicated routing scheme). In contrast, a novel cross-routing scheme allocates rework to a parallel agent performing both new jobs and rework. The agent who passes quality inspection or completes rework receives the piece rate paid per job. We compare the incentives of these rework-allocation schemes in a principal-agent model with embedded quality control and routing in a multiclass queueing network. We show that conventional self-routing of rework cannot induce first-best effort. Dedicated routing and cross-routing, however, strengthen incentives for quality by imposing an implicit punishment for quality failure. In addition, cross-routing leads to workload-allocation externalities and a prisoner's dilemma, thereby creating the greatest incentives for quality. Firm profitability depends on demand levels, revenues, and quality costs. When the number of agents increases, the incentive effect of cross-routing reduces monotonically and approaches that of dedicated routing.

Networks of queues with multiple customer types: application in emergency departments

Emergency Departments (EDs) have become a useful way to access hospitals and they are the subject of study of many researchers. The research developed in this paper aims to improve the performance of the ED of Habib Bourguiba Sfax Hospital through an analytical method. A network of queues with multiple customer types is proposed. Different indicators of performance are used. We illustrate here multiclass networks of queues. The ED of Habib Bourguiba Sfax Hospital presents two types of patients: urgent cases and non-urgent cases. The discipline of the department is First-Come, First-Served (FCFS). Following William's (1991) procedure, we obtain results based on the diagnosis of the specialist (first step/evaluation) and then diagnosis of the general physician (second step/additional evaluation.) The two evaluations are commonly the procedures that patients have to endure and more often than not this process makes them wait a long time for care service; this finding confirms the results given by the Witness simulation model.

Near optimal control of queueing networks over a finite time horizon

We propose a method for the control of multi-class queueing networks over a finite time horizon. We approximate the multi-class queueing network by a fluid network and formulate a fluid optimization problem which we solve as a separated continuous linear program. The optimal fluid solution partitions the time horizon to intervals in which constant fluid flow rates are maintained. We then use a policy by which the queueing network tracks the fluid solution. To that end we model the deviations between the queuing and the fluid network in each of the intervals by a multi-class queueing network with some infinite virtual queues. We then keep these deviations stable by an adaptation of a maximum pressure policy. We show that this method is asymptotically optimal when the number of items that is processed and the processing speed increase. We illustrate these results through a simple example of a three stage re-entrant line.

A switching rule for plastics identification in electronics recycling

The looming obsolescence of 3 billion consumer electronics by 2010 calls for effective recycling of 10 billion pounds of high-value engineering thermoplastics. Because general shredding operations generate mixed plastics of low value, a switching rule is needed rapidly to identify plastics for sortation by colour and type to increase plastics-to-plastics recycling value. The current paper develops a heuristic rule to determine the thresholds for a switching rule that reduces set-up time while managing queue space. Our recycling system is modelled as a multiclass queuing network in a simulation model. Results indicate that laser identification probe utilization, setups, and queue space are sensitive to the switching rule. Our work contributes important insights into the relationship between traffic intensity, setup time, and queue capacity, and generalizes to multiclass queuing systems in other applications.

A practical approach for performance-driven UML modelling of handheld devices – A case study

In this article, we present a performance engineering enhanced modelling methodology for designing embedded devices and describe the experiences we have gained in applying this methodology during the design of a DVB-H enabled handheld device. The methodology uses UML 2.0 to model the system following a strict separation of architectural and behavioural aspects of the system. For this purpose we employ the new composite structure diagram and show its advantages over already established approaches. This methodology specially aims on an easy application by non performance experts. From the model, a multiclass queueing network is generated for the analysis of the system performance. The configuration of hardware resources and resource demands is done using the standard SPT Profile which is extended where necessary. This makes queueing theory accessible to system designers even if they are not familiar with the underlying mathematics. In this way the acceptance of developers to use performance engineering in their daily work is increased. Special attention has been put on an easy evaluation of design alternatives. We describe our implementation and its seamless integration into a UML 2.0 CASE tool.

A multi-class queuing network analysis methodology for improving hospital emergency department performance

This paper derives an open queuing network model of an emergency department (ED) design intended to increase the capacity of an ED to treat patients. The methodology captures hospital-specific differences in patient acuity mix, arrival patterns and volumes, and efficiencies of processes in a single common computational model. A spreadsheet implementation of the resulting queuing equations is used by managers, in real time, to size ED areas using waiting time and overflow probability as quality of service targets. Non-homogeneous arrival patterns, non-exponential service time distributions, and multiple patient types are all incorporated. The methodology has been applied to a fleet of hospitals for validation. Results from one of them are used to demonstrate the methodology. Scope and purpose Population growth, closure of emergency departments (EDs) nearby, or seasonal peak variation can cause ED patient wait times to increase dramatically. Prolonged waits induce patients to leave the ED before receiving treatment, creating a public health problem. The purpose of this paper is to introduce a new paradigm of ED care that reduces “walk-aways”, and increases ED access, through an operational research method that customizes to any hospital through the use of hospital-specific data elements.

A Paradox for Admission Control of Multiclass Queueing Network with Differentiated Service

In this paper we present counter-intuitive examples for the multiclass queueing network, where each station may serve more than one job class with differentiated service priority and each job may require service sequentially by more than one service station. In our examples, the network performance is improved even when more jobs are admitted for service.