Multiclass Queueing Network Research Articles

State space collapse for multi-class queueing networks under SBP service policies

State space collapse for multi-class queueing networks under SBP service policies

Alternative care providers in rheumatoid arthritis patient care: a queueing and simulation analysis

ABSTRACT Patients diagnosed with rheumatoid arthritis require lifelong monitoring by a rheumatologist. Initiation of the disease-modifying anti-rheumatic drug therapy within twelve weeks of the onset of symptoms is crucial to prevent joint damage and functional disability. We examine the impact of the engagement of alternate care providers (ACP) in alleviating delay due to limited rheumatologist capacity. Using queueing theory and discrete-event simulation, we model rheumatologist-only and rheumatologist-with-ACP system configurations as closed, multi-class queueing networks with class switching.Using summary data from an actual rheumatology clinic for illustration, we analyze various parameter conditions to aid clinic managers and policymakers in decisions concerning capacity allocations and feasible patient panel size that impact timeliness of care and resource utilization.Results not only confirm that a substantial increase in RA patient panel size with an ACP involved in the care of follow-up patients but also demonstrates the boundaries for feasible panel sizes and workload allocation

Multiclass Queueing Network Modeling and Traffic Flow Analysis for SDN-Enabled Mobile Core Networks With Network Slicing

The back-haul networks of 5G are formed by heterogeneous links which need to handle massive traffic. The service providers are not able to provide good QoS for their users. The technology like Software Defined Networks(SDN) and Network Slicing helps a little for a service provider to providing QoS for multiple links. The service providers face a challenge in the efficient utilization of resources to fulfill the QoS requirement of users to comply with the growth and thereby increasing the revenue. These problems require an accurate traffic model to determine the steady-state of the system. The proposed model uses an architecture that has the combination of two technologies: SDN and network slicing, which empowers an administrator a flexible, programmable network, and the best management of network resources. Heterogeneous application is well managed by creating multiple logical networks called slicing. The slicing can be modeled using multi-class queuing networks. These technologies encourage service providers to fulfill QoS and revenue growth. To leverage the benefits of these technologies in allocating QoS is to identify the performance of the system, which requires a precise model of traffic to decide the steady-state condition of the framework. In this paper, we focus on SDN and slicing in mobile networks and quantify the performance measure considering an in-band OpenFlow architecture for a single node and homogeneous traffic class, which is further extended to the multi-class heterogeneous class queuing model and analyzed. The results obtained help a service provider to monitor the utilization of resources in every node by every class of core network, which in turn helps to allocate the resources precisely to fulfill QoS requirements

On the modelling and performance measurement of service networks with heterogeneous customers

Service networks are common throughout the modern world, yet understanding how their individual services effect each other and contribute to overall system performance can be difficult. An important metric in these systems is the quality of service. This is an often overlooked measure when modelling and relates to how customers are affected by a service. Presented is a novel perspective for evaluating the performance of multi-class queueing networks through a combination of operational performance and service quality—denoted the “flow of outcomes”. Here, quality is quantified by customers moving between or remaining in classes as a result of receiving service or lacking service. Importantly, each class may have different flow parameters, hence the positive/negative impact of service quality on the system’s operational performance is captured. A fluid–diffusion approximation for networks of stochastic queues is used since it allows for several complex flow dynamics: the sequential use of multiple services; abandonment and possible rejoin; reuse of the same service; multiple customers classes; and, class and time dependent parameters. The scalability of the approach is a significant benefit since, the modelled systems may be relatively large, and the included flow dynamics may render the system analytically intractable or computationally burdensome. Under the right conditions, this method provides a framework for quickly modelling large time-dependent systems. This combination of computational speed and the “flow of outcomes” provides new avenues for the analysis of multi-class service networks where both service quality and operational efficiency interact.

Customer Acquisition and Retention: A Fluid Approach for Staffing

We investigate the trade-off between acquisition and retention efforts when customers are sensitive to the quality of service they receive, i.e., whether they get timely access to a company's resources when requested. We model the problem as a multi-class queueing network with new and returning customers, time-dependent arrivals, and abandonment. We derive its fluid approximation; a system of ordinary linear differential equations with continuous, piecewise smooth, right-hand sides. Based on the fluid model, we propose a novel approach to determine optimal stationary staffing levels for new and returning customer queues in anticipation of future time-varying dynamics. Using system accessibility as a proxy for service quality and staffing levels as a proxy for investment, we demonstrate how to apply our approach to two families of time-varying arrival functions motivated by real-world applications: an advertising campaign and a clinical setting. In a numerical study, we demonstrate that our approach creates staffing policies that maximize throughput while balancing acquisition and retention efforts more effectively (i.e., equitable abandonment from each customer class) than commonly used near-stationary methods such as segmentation-based square-root staffing policies. Our model confirms that acquisition and retention efforts are intimately linked; this has been found in empirical studies but not captured in the operations literature. We suggest that in time-varying environments, focusing on either alone is not sufficient to maintain high levels of throughput and service quality.

Physician Staffing in Emergency Rooms (ERs): Opening the Black-box of ER Care via a Multi-Class Multi-Stage Network

Problem Definition and Relevance: Characterized by time-varying arrivals, multi-stage service, and multi-class patient population, emergency rooms (ERs) are complex healthcare delivery systems, where optimizing the staffing levels of physicians is a challenge. In collaboration with Mayo Clinic, we study a staffing and an associated routing problem for ER physicians and propose a new staffing rule to meet tail probability of delay (TPoD) type service targets. Methodology: We capture the time-varying patient flow in the ER with a multi-class multi-stage queuing network, describing the ER care as sequences of treatment queues, where physicians serve, and groups of diagnostic medical processes. Treatment stations in ER are busy-server queues, where waiting before service is common, but experience negligible abandonments. Motivated by these queues, we propose a new staffing algorithm, translating the offered load into staffing decisions for efficiency-driven queues with TPoD targets and perfectly patient customers. Results: We analytically show the asymptotic effectiveness of our staffing rule on stabilizing TPoD for M/M/s queues, operating in efficiency-driven mode, and numerically demonstrate its robustness and optimality in various time-varying ER settings via realistic and data-driven simulation experiments. We further show that as the service complexity of an ER increases, hybrid routing rules, using pre-determined (static) priorities and (dynamic) current system state jointly, become necessary to meet TPoD targets. Managerial Implications: Instead of treating the entire patient length of stay as a black-box service model, our multi-stage network model provides more managerial control over the internal components of ER service and is easily implementable in practice.

Stochastic Monotonicity of Markovian Multiclass Queueing Networks

Multiclass queueing networks (McQNs) extend the classical concept of the Jackson network by allowing jobs of different classes to visit the same server. Although such a generalization seems rather natural, from a structural perspective, there is a significant gap between the two concepts. Nice analytical features of Jackson networks, such as stability conditions, product–form equilibrium distributions, and stochastic monotonicity, do not immediately carry over to the multiclass framework. The aim of this paper is to shed some light on this structural gap, focusing on monotonicity properties. To this end, we introduce and study a class of Markov processes, which we call Q-processes, modeling the time evolution of the network configuration of any open, work-conservative McQN having exponential service times and Poisson input. We define a new monotonicity notion tailored for this class of processes. Our main result is that we show monotonicity for a large class of McQN models, covering virtually all instances of practical interest. This leads to interesting properties that are commonly encountered for “traditional” queueing processes, such as (i) monotonicity with respect to external arrival rates and (ii) star-convexity of the stability region (with respect to the external arrival rates); such properties are well known for Jackson networks but had not been established at this level of generality. This research was partly motivated by the recent development of a simulation-based method that allows one to numerically determine the stability region of a McQN parameterized in terms of the arrival-rates vector.

Justifying diffusion approximations for multiclass queueing networks under a moment condition

Multiclass queueing networks (MQN) are, in general, difficult objects to study analytically. The diffusion approximation refers to using the stationary distribution of the diffusion limit as an approximation of the diffusion-scaled process (say, the workload) in the original MQN. To validate such an approximation amounts to justifying the interchange of two limits, $t\to\infty$ and $k\to\infty$, with $t$ being the time index and $k$, the scaling parameter. Here, we show this interchange of limits is justified under a $p^{*}$th moment condition on the primitive data, the interarrival and service times; and we provide an explicit characterization of the required order ($p^{*}$), which depends naturally on the desired order of moment of the workload process.

Performance Analysis of Network Coding with IEEE 802.11 DCF in Multi-Hop Wireless Networks

Network coding is an effective idea to boost the capacity of wireless networks, and a variety of studies have explored its advantages in different scenarios. However, there is not much analytical study on throughput and end-to-end delay of network coding in multi-hop wireless networks considering the specifications of IEEE 802.11 Distributed Coordination Function. In this paper, we utilize queuing theory to propose an analytical framework for bidirectional unicast flows in multi-hop wireless mesh networks. We study the throughput and end-to-end delay of inter-flow network coding under the IEEE 802.11 standard with CSMA/CA random access and exponential back-off time considering clock freezing and virtual carrier sensing, and formulate several parameters such as the probability of successful transmission in terms of bit error rate and collision probability, waiting time of packets at nodes, and retransmission mechanism. Our model uses a multi-class queuing network with stable queues, where coded packets have a non-preemptive higher priority over native packets, and forwarding of native packets is not delayed if no coding opportunities are available. Finally, we use computer simulations to verify the accuracy of our analytical model.

Auxiliary variables for Bayesian inference in multi-class queueing networks

Queueing networks describe complex stochastic systems of both theoretical and practical interest. They provide the means to assess alterations, diagnose poor performance and evaluate robustness across sets of interconnected resources. In the present paper, we focus on the underlying continuous-time Markov chains induced by these networks, and we present a flexible method for drawing parameter inference in multi-class Markovian cases with switching and different service disciplines. The approach is directed towards the inferential problem with missing data, where transition paths of individual tasks among the queues are often unknown. The paper introduces a slice sampling technique with mappings to the measurable space of task transitions between the service stations. This can address time and tractability issues in computational procedures, handle prior system knowledge and overcome common restrictions on service rates across existing inferential frameworks. Finally, the proposed algorithm is validated on synthetic data and applied to a real data set, obtained from a service delivery tasking tool implemented in two university hospitals.

A Numerical Approach to Stability of Multiclass Queueing Networks

The Multi-class Queueing Network (McQN) arises as a natural multi-class extension of the traditional (single-class) Jackson network. In a single-class network subcriticality (i.e. subunitary nominal workload at every station) entails stability, but this is no longer sufficient when jobs/customers of different classes (i.e. with different service requirements and/or routing scheme) visit the same server; therefore, analytical conditions for stability of McQNs are lacking, in general. In this note we design a numerical (simulation-based) method for determining the stability region of a McQN, in terms of arrival rate(s). Our method exploits certain (stochastic) monotonicity properties enjoyed by the associated Markovian queue-configuration process. Stochastic monotonicity is a quite common feature of queueing models and can be easily established in the single-class framework (Jackson networks); recently, also for a wide class of McQNs, including first-come-first-serve (FCFS) networks, monotonicity properties have been established. Here, we provide a minimal set of conditions under which the method performs correctly. Eventually, we illustrate the use of our numerical method by presenting a set of numerical experiments, covering both single and multi-class networks.

OptiSpot: minimizing application deployment cost using spot cloud resources.

The spot instance model is a virtual machine pricing scheme in which some resources of cloud providers are offered to the highest bidder. This leads to the formation of a spot price, whose fluctuations can determine customers to be overbid by other users and lose the virtual machine they rented. In this paper we propose OptiSpot, a heuristic to automate application deployment decisions on cloud providers that offer the spot pricing model. In particular, with our approach it is possible to determine: (i) which and how many resources to rent in order to run a cloud application, (ii) how to map the application components to the rented resources, and (iii) what spot price bids to use to minimize the total cost while maintaining an acceptable level of performance. To drive the decision making, our algorithm combines a multi-class queueing network model of the application with a Markov model that describes the stochastic evolution of the spot price and its influence on virtual machine reliability. We show, using a model developed for a real enterprise application and historical traces of the Amazon EC2 spot instance prices, that our heuristic finds low cost solutions that indeed guarantee the required levels of performance. The performance of our heuristic method is compared to that of nonlinear programming and shown to markedly accelerate the finding of low-cost optimal solutions.

Approximate Reduction of Heterogenous Nonlinear Models With Differential Hulls

We present a model reduction technique for a class of nonlinear ordinary differential equation (ODE) models of heterogeneous systems, where heterogeneity is expressed in terms of classes of state variables having the same dynamics structurally, but which are characterized by distinct parameters. To this end, we first build a system of differential inequalities that provides lower and upper bounds for each original state variable, but such that it is homogeneous in its parameters. Then, we use two methods for exact aggregation of ODEs to exploit this homogeneity, yielding a smaller model of size independent of the number of heterogeneous classes. We apply this technique to two case studies: a multiclass queuing network and a model of epidemics spread.

Ergodic control of multi-class $M/M/N+M$ queues in the Halfin–Whitt regime

We study a dynamic scheduling problem for a multi-class queueing network with a large pool of statistically identical servers. The arrival processes are Poisson, and service times and patience times are assumed to be exponentially distributed and class dependent. The optimization criterion is the expected long time average (ergodic) of a general (nonlinear) running cost function of the queue lengths. We consider this control problem in the Halfin–Whitt (QED) regime, that is, the number of servers $n$ and the total offered load $\mathbf{r}$ scale like $n\approx\mathbf{r}+\hat{\rho}\sqrt{\mathbf{r}}$ for some constant $\hat{\rho}$. This problem was proposed in [Ann. Appl. Probab. 14 (2004) 1084–1134, Section 5.2]. The optimal solution of this control problem can be approximated by that of the corresponding ergodic diffusion control problem in the limit. We introduce a broad class of ergodic control problems for controlled diffusions, which includes a large class of queueing models in the diffusion approximation, and establish a complete characterization of optimality via the study of the associated HJB equation. We also prove the asymptotic convergence of the values for the multi-class queueing control problem to the value of the associated ergodic diffusion control problem. The proof relies on an approximation method by spatial truncation for the ergodic control of diffusion processes, where the Markov policies follow a fixed priority policy outside a fixed compact set.

Justifying Diffusion Approximations for Stochastic Processing Networks Under a Moment Condition

Stochastic processing networks (SPN) are, in general, difficult objects to study analytically. The diffusion approximation refers to using the stationary distribution of the diffusion limit as an approximation of the diffusion-scaled process (say, the workload) in the original SPN. To validate such an approximation amounts to justifying the interchange of two limits, t→∞ and k→∞, with t being the time index and k, the scaling parameter. Here, we show this interchange of limits is justified for a broad class of SPN under a p*-th moment condition on the primitive data, interarrival and service times; and we provide an explicit characterization of the required order (p*), which depends naturally on the desired order of convergence of the workload process. To illustrate the generality of this moment condition, we first use it to establish the justification for resource sharing networks, where, to be processed each job needs to concurrently occupy multiple resources (servers), whereas each resource is shared among different job classes following a so-called proportional fair allocation scheme. We then show the same approach applies to most multi-class queueing networks that are known to have diffusion limits.

A Lyapunov view on positive Harris recurrence of multiclass queueing networks

In this paper we prove the positive Harris recurrence of the underlying Markov process of a multiclass queueing network. Using the fact that under certain conditions the stability of a fluid network is equivalent to the existence of a continuous Lyapunov function, we propose a new method to conclude that the underlying Markov process is positive Harris recurrent if the associated fluid network model is stable by using explicitly the Lyapunov function of the fluid network.

Maximum Stable Throughput of Network-Coded Multiple Broadcast Sessions for WirelessTandem Random Access Networks

This paper presents an analytical study of the stable throughput for multiple broadcast sessions in a multi-hop wireless tandem network with random access. Intermediate nodes leverage on the broadcast nature of wireless medium access to perform inter-session network coding among different flows. This problem is challenging due to the interaction among nodes, and has been addressed so far only in the saturated mode where all nodes always have packet to send, which results in infinite packet delay. In this paper, we provide a novel model based on multi-class queueing networks to investigate the problem in unsaturated mode. We devise a theoretical framework for computing maximum stable throughput of network coding for a slotted ALOHA-based random access system. Using our formulation, we compare the performance of network coding and traditional routing. Our results show that network coding leads to high throughput gain over traditional routing. We also define a new metric, network unbalance ratio (NUR), that indicates the unbalance status of the utilization factors at different nodes. We show that although the throughput gain of the network coding compared to the traditional routing decreases when the number of nodes tends to infinity, NUR of the former outperforms the latter. We carry out simulations to confirm our theoretical analysis.

Validity of Heavy-Traffic Steady-State Approximations in Multiclass Queueing Networks: The Case of Queue-Ratio Disciplines

A class of stochastic processes known as semi-martingale reflecting Brownian motions (SRBMs) is often used to approximate the dynamics of heavily loaded queueing networks. In two influential papers, Bramson [Bramson M (1998) State space collapse with applications to heavy-traffic limits for multiclass queueing networks. Queueing Systems 30:89–148] and Williams [Williams RJ (1998b) Diffusion approximations for open multiclass queueuing networks: Sufficient conditions involving state space collapse. Queueing Systems 30:27–88] laid out a general and structured approach for proving the validity of such heavy-traffic approximations, in which an SRBM is obtained as a diffusion limit from a sequence of suitably normalized workload processes. However, for multiclass networks it is still not known in general whether the steady-state distribution of the SRBM provides a valid approximation for the steady-state distribution of the original network. In this paper we study the case of queue-ratio disciplines and provide a set of sufficient conditions under which the above question can be answered in the affirmative. In addition to standard assumptions made in the literature towards the stability of the pre- and post-limit processes and the existence of diffusion limits, we add a requirement that solutions to the fluid model are attracted to the invariant manifold at a linear rate. For the special case of static-priority networks such linear attraction is known to hold under certain conditions on the network primitives. The analysis elucidates interesting connections between stability of the pre- and post-limit processes, their respective fluid models and state-space collapse, and identifies the respective roles played by all of the above in establishing validity of heavy-traffic steady-state approximations.

Fluid limits to analyze long-term flow rates of a stochastic network with ingress discarding

We study a simple rate control scheme for a multiclass queuing network for which customers are partitioned into distinct flows that are queued separately at each station. The control scheme discards customers that arrive to the network ingress whenever any one of the flow's queues throughout the network holds more than a specified threshold number of customers. We prove that if the state of a corresponding fluid model tends to a set where the flow rates are equal to target rates, then there exist sufficiently high thresholds that make the long-term average flow rates of the stochastic network arbitrarily close to these target rates. The same techniques could be used to study other control schemes. To illustrate the application of our results, we analyze a network resembling a 2-input, 2-output communications network switch.

Stability of multi-class queueing networks with infinite virtual queues

We generalize the standard multi-class queueing network model by allowing both standard queues and infinite virtual queues which have an infinite supply of work. We pose the general problem of finding policies which allow some of the nodes of the network to work with full utilization, and yet keep all the standard queues in the system stable. Toward this end we show that re-entrant lines, systems of two re-entrant lines through two service stations, and rings of service stations can be stabilized with priority policies under certain parameter restrictions. The analysis throughout the paper depends on model and policy and illustrates the difficulty in solving the general problem.