Analytical Study of Priority-Feedback Queue Network Model Comprising of Three Parallel Servers within Stochastic Conditions

We consider a queuing network with single-server nodes and heterogeneous customers. The number of customers, which can obtain service simultaneously, is restricted. Customers that cannot be admitted to the network upon arrival make repeated attempts to obtain service. The service time at the nodes is exponentially distributed. After service completion at a node, the serviced customer can transit to another node or leave the network forever. The main features of the model are the mutual dependence of processes of customer arrivals and retrials and the impatience and non-persistence of customers. Dynamics of the network are described by a multidimensional Markov chain with infinite state space, state inhomogeneous behavior and special structure of the infinitesimal generator. The explicit form of the generator is derived. An effective algorithm for computing the stationary distribution of this chain is recommended. The expressions for computation of the key performance measures of the network are given. Numerical results illustrating the importance of the account of the mentioned features of the model are presented. The model can be useful for capacity planning, performance evaluation and optimization of various wireless telecommunication networks, transportation and manufacturing systems.

What determines the average length of a queue, which stretches in front of a service station? The answer to this question clearly depends on the average rate at which jobs arrive at the queue and on the average rate of service. Somewhat less obvious is the fact that stochastic fluctuations in service and arrival times are also important, and that these are a major source of backlogs and delays. Strategies that could mitigate fluctuations-induced delays are, thus in high demand as queue structures appear in various natural and man-made systems. Here, we demonstrate that a simple service resetting mechanism can reverse the deleterious effects of large fluctuations in service times, thus turning a marked drawback into a favorable advantage. This happens when stochastic fluctuations are intrinsic to the server, and we show that service resetting can then dramatically cut down average queue lengths and waiting times. Remarkably, this strategy is also useful in extreme situations where the variance, and possibly even mean, of the service time diverge—as resetting can then prevent queues from “blowing up.” We illustrate these results on the M/G/1 queue in which service times are general and arrivals are assumed to be Markovian. However, the main results and conclusions coming from our analysis are not specific to this particular model system. Thus, the results presented herein can be carried over to other queueing systems: in telecommunications, via computing, and all the way to molecular queues that emerge in enzymatic and metabolic cycles of living organisms.

Paid curbside parking can be advantageously modeled as a network of interdependent queues. To this end, we introduce methods for analyzing a special class of networks of finite capacity queues where drivers arrive from an exogenous source, join the queue if there is an available parking space, or continue to search at an adjacent queue for an available space. Furthermore, we apply this model to estimate the proportion of drivers cruising in the neighborhood of Belltown, Seattle, WA, USA. Using occupancy approximated by parking transaction data, we estimate the percentage of drivers cruising for curbside parking by comparing the rate of drivers unable to find parking to bulk through-traffic measurement data. We find percentages of up to 50% for a Belltown's 1st Ave. depending on the time, day, and direction of travel. We then calculate a per vehicle travel-time cost to social welfare incurred by this proportion: upward of a 10% increase in travel time to all drivers along 1st Ave. Last, we introduce a simulation tool and test assumptions made when estimating interesting performance metrics like the probability of a block-face being full. Our results suggest that while assuming exponential service time distributions is not justified, mean rate solutions under a Markovian relaxation of the problem is comparable to service times representative of parking transaction data in simulation.

Simulation of IPA gradients in hybrid network systems

short-paper Share on Tail asymptotics of the stationary distribution for a two-node generalized Jackson network Author: Masakiyo Miyazawa Tokyo University of Science, Noda, Chiba, Japan Tokyo University of Science, Noda, Chiba, JapanView Profile Authors Info & Claims ACM SIGMETRICS Performance Evaluation ReviewVolume 42Issue 2September 2014pp 70–72https://doi.org/10.1145/2667522.2667545Published:04 September 2014Publication History 1citation26DownloadsMetricsTotal Citations1Total Downloads26Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Queueing Network Models (QNMs) with Finite Capacity provide powerful and realistic tools for the performance evaluation and prediction of discrete flow systems such as computer systems, communication networks and flexible manufacturing systems. Over recent years, there has been a great deal of progress towards the analysis and application of QNMs with finite capacity, and high quality research work has appeared in diverse scientific journals of learning and conference proceedings in the fields of Operations Research, Computer Science, Telecommunication Networks, Management and Industrial Engineering. However, there are still many important and interesting finite capacity queues and QNMs to be resolved, such as those involving multiple-job classes, bounds and theoretical properties, exact analysis, numerical solutions and approximate methods, as well as application studies to computer and distributed systems, high-speed networks and production systems.This volume of Annals of Operations Research on Queueing Networks with Finite Capacity presents 17 extended research papers, both from industry and academia, reflecting current original contributions in the theory and applications of finite capacity queues and networks worldwide. Earlier short versions of these papers were amongst those presented at QNets '95, The Third International Workshop on Queueing Networks with Finite Capacity, 6th-7th July, 1995, Ilkley, West Yorkshire, UK. The papers are classified into four parts covering exact analysis, approximate methods, error bounds and performance modelling studies.

The retrial bulk arrival queue with unreliable server and negative customers is considered. On arrival of the group of customers, one of the customers gets the service immediately if the server is idle and other customers join the retrial orbit. There is a provision to opt additional service after completion of the essential service of the customers. The server may fail due to arrival of negative customers during any stage of the service. After completion of the service, the customer may again join the queue as a feedback customer to get another regular/optional service or depart from the system. The non-persistent (impatient) phenomenon also occurs because of the delayed in repair/repair time of the failed server. By using the supplementary variable approach, various measures of queueing and reliability characteristics are analysed. To facilitate the comparative study of the performance metrics of the system, the maximum entropy principle is used. The numerical results for various performance indices and optimal cost are obtained.

This paper studies the optimal policy for joint control of admission, routing, service, and jockeying in a queueing system consisting of two exponential servers in parallel. Jobs arrive according to a Poisson process. Upon each arrival, an admission/routing decision is made, and the accepted job is routed to one of the two servers with each being associated with a queue. After each service completion, the servers have an option of serving a job from its own queue, serving a jockeying job from another queue, or staying idle. The system performance is inclusive of the revenues from accepted jobs, the costs of holding jobs in queues, the service costs and the job jockeying costs. To maximize the total expected discounted return, we formulate a Markov decision process (MDP) model for this system. The value iteration method is employed to characterize the optimal policy as a hedging point policy. Numerical studies verify the structure of the hedging point policy which is convenient for implementing control actions in practice.

A recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks

On the batch arrival batch service queue with finite buffer under server’s vacation: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si94.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>Y</mml:mi></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>N</mml:mi></mml:math> queue

Queueing networks constitute a large family of stochastic models, involving jobs that enter a network, compete for service, and eventually leave the network upon completion of service. Since the early 1990s, substantial attention has been devoted to the question of when such networks are stable. This monograph presents a summary of such work. Emphasis is placed on the use of fluid models in showing stability, and on examples of queueing networks that are unstable even when the arrival rate is less than the service rate. The material of this volume is based on a series of nine lectures given at the Saint-Flour Probability Summer School 2006, and is also being published in the Springer Lecture Notes series.

This paper establishes structural properties for the throughput of a large class of queueing networks with i.i.d. new-better-than-used service times. The main result obtained in this paper is applied to a wide range of networks, including tandems, cycles and fork-join networks with general blocking and starvation (as well as certain networks with splitting and merging of traffic streams), to deduce the concavity of their throughput as a function of system parameters, such as buffer and initial job configurations, and blocking and starvation parameters. These results have important implications for the optimal design and control of such queueing networks by providing exact solutions, reducing the search space over which optimization need be performed, or establishing the convergence of optimization algorithms. In order to obtain results for such disparate networks in a unified manner, we introduce the framework of constrained discrete event systems (CDES), which enables us to characterize any permutable and non-interruptive queueing network through its constraint set. The main result of this paper establishes comparison properties of the event occurrence processes of CDES as a function of the constraint sets, which are then translated into the above-mentioned concavity of the throughput as a function of system parameters in the context of queueing networks.

An arbitrary configuration of an open queueing network with exponential service times and finite buffers is analyzed. We offer an iterative procedure for approximating the marginal occupancy probabilities for each queue of the system. The method decomposes the queueing network into individual queues and analyzes each in isolation using information from only its nearest neighbors. Based upon the SIMP approximation previously used for tandem queues, it replaces each server's service time with a clearance time, which includes blocking, and each server's arrival rate by an equivalent acceptance rate. The procedure is easy to implement and requires modest memory and computer time. Extensive numerical experiments, performed for various topologies, yield accurate results compared with those obtained by exact or simulation methods.

Product form was first introduced by Jackson [Operations Research, 1957; Management Science, 1963] who considered only negative exponential service distributions with queue length dependent service rates. Posner and Bernholtz [Operations Research, 1968] showed that one has product form in the case where there are many classes of customers, service times are exponential with class queue length dependent service rates, and lag times (the time for a customer to travel between two servers is a random variable with continuously differentiable distribution function). Baskett, Chandy, Muntz and Palacios [JACM, 1975] extended these results to networks with multiple classes of customers and service time distributions which have rational Laplace transforms vanishing at ∞. Chandy, Howard and Towsley [JACM, 1977] extended this further to the case where the service time distribution functions are continuously differentiable. They also introduced the concept of station balance and established some important relationships between station balance, local balance and product form. In particular, for the closed queueing network they showed that if each queue in the network satisfies local balance in isolation with Poisson arrivals then one has product formsolution for the network. This paper extends previous results to arbitrary service with finite mean (the nondifferentiable case included). This is done by introducing supplementary variables for the remaining service requirements at the various queues in the system. By using generalized function (in the sense of Schwartz) a new technique is presented for handling queueing networks in which service distributions may be general.

We study a multi-class queueing network which consists of a finite capacity node (node 0) linked to M parallel finite capacity nodes (nodes 1 to M). M classes of customers are assumed. All customers first join node 0. A class i customer after completion of its service at node 0 always joins the ith node. All service times and inter-arrival times are assumed to be exponentially distributed. The service priority at node 0 is head-of-line with preemption. When node i (i=1, 2, ..., M) is full, node 0 cannot process class i customers. In addition to the service priority at node 0, push-out is employed. That is, a customer that arrives at node 0 when the node is full, takes the space of a customer which has the lowest priority among the customers already in the node. If all customers in the node have a higher or equal priority, then the arriving customer is lost. This queueing network is analyzed approximately by decomposing it into individual nodes, and then analyzing each node separately. Node 0 is analyzed using a class by class decomposition. The approximation algorithm has been validated using simulation, and the approximate results have a good error. >