Open queueing networks in discrete time-some limit theorems

A finite number of nodes, each with a single server and infinite buffers, is considered in discrete time. The service may be FIFO and the service times are constant. The external arrivals and the routing decision variables form a general stationary sequence. Stability of the system is proved under these assumptions. Extension to multiple servers at a node and general stationary distributions holds. If the external input is i.i.d. and the routing is Markovian then stochastic ordering, continuity of stationary distributions, rates of convergence, a functional CLT and a functional LIL and various other limit theorems for the queue length process are also proved. Generalizations to multiple servers at nodes, customers with priority, multiple customer classes, general service length and Markov modulated external arrival cases are discussed.     

Asymptotic analysis for closed multichain queueing networks with bottlenecks

We consider a two-chain exponential queueing network with a large number of customers that consists of one infinite-server (IS) station and two processor-sharing (PS) or FCFS single-server stations. The asymptotic behavior of the partition function is studied for such a network when one or both PS (FCFS) nodes are heavily loaded. The results are derived using methods of multidimensional complex analysis (the theory of homologies and residues) and the saddle-point method.     

Asymptotic analysis for closed multiclass queueing networks in critical usage

We consider a class of closed multiclass queueing networks containing First-Come-First-Serve (FCFS) and Infinite Server (IS) stations. These networks have a productform solution for their equilibrium probabilities. We study these networks in an asymptotic regime for which the number of customers and the service rates at the FCFS stations go to infinity with the same order. We assume that the regime is in critical usage, whereby the utilizations of the FCFS servers slowly approach one. The asymptotic distribution of the normalized queue lengths is shown to be in many cases a truncated multivariate normal distribution. Traffic conditions for which the normalized queue lengths arealmost asymptotically independent are determined. Asymptotic expansions of utilizations and expected queue lengths are presented. We show through an example how to obtain asymptotic expansions of performance measures when the networks are in mixed usage and how to apply the results to networks with finite data.Supported partially by NSF grant NCR93-04601.     

Fleet-sizing and service availability for a vehicle rental system via closed queueing networks

In this paper, we address the problem of determining the optimal fleet size for a vehicle rental company and derive analytical results for its relationship to vehicle availability at each rental station in the company’s network of locations. This work is motivated by the recent surge in interest for bicycle and electric car sharing systems, one example being the French program Vélib (2010). We first formulate a closed queueing network model of the system, obtained by viewing the system from the vehicle’s perspective. Using this framework, we are able to derive the asymptotic behavior of vehicle availability at an arbitrary rental station with respect to fleet size. These results allow us to analyze imbalances in the system and propose some basic principles for the design of system balancing methods. We then develop a profit-maximizing optimization problem for determining optimal fleet size. The large-scale nature of real-world systems results in computational difficulties in obtaining this exact solution, and so we provide an approximate formulation that is easier to solve and which becomes exact as the fleet size becomes large. To illustrate our findings and validate our solution methods, we provide numerical results on some sample networks.     

Towards better multi-class parametric-decomposition approximations for open queueing networks

Methods are developed for approximately characterizing the departure process of each customer class from a multi-class single-server queue with unlimited waiting space and the first-in-first-out service discipline. The model is (GT _i /GI _i )/1 with a non-Poisson renewal arrival process and a non-exponential service-time distribution for each class. The methods provide a basis for improving parametric-decomposition approximations for analyzing non-Markov open queueing networks with multiple classes. For example, parametric-decomposition approximations are used in the Queueing Network Analyzer (QNA). The specific approximations here extend ones developed by Bitran and Tirupati [5]. For example, the effect of class-dependent service times is considered here. With all procedures proposed here, the approximate variability parameter of the departure process of each class is a linear function of the variability parameters of the arrival processes of all the classes served at that queue, thus ensuring that the final arrival variability parameters in a general open network can be calculated by solving a system of linear equations.     

Deadlock properties of queueing networks with finite capacities and multiple routing chains

Blocking in queueing network models with finite capacities can lead to deadlock situations. In this paper, deadlock properties are investigated in queueing networks with multiple routing chains. The necessary and sufficient conditions for deadlockfree queueing networks with blocking are provided. An optimization algorithm is presented for finding deadlock-free capacity assignments with the least total capacity. The optimization algorithm maps the queueing network into a directed graph and obtains the deadlock freedom conditions from a specified subset of cycles in the directed graph. In certain network topologies, the number of deadlock freedom conditions can be large, thus, making any optimization computationally expensive. For a special class of topologies, so-calledtandem networks, it is shown that a minimal capacity assignment can be directly obtained without running an optimization algorithm. Here, the solution to the minimal capacity assignment takes advantage of the regular topology of tandem networks.This work was supported by the National Science Foundation under Grant No. CCR-90-11981.     

On the reversibility of queueing networks

This paper relates the reversibility of certain discrete state Markovian queueing networks — the class of quasi-reversible networks — to the reversibility of the underlying switching process. Quasi-reversible networks are characterized by a product form equilibrium state distribution.When the state can be represented by customer totals at each node, the reversibility of the state process is equivalent to the reversibility of the switching process. More complicated quasi-reversible networks require additional conditions, to ensure the reversibility of the network state process.     

Calculating the normalization constants in queueing networks

Many queueing network models have a product-form solution for their steady-state probability distributions. However, the calculation of the normalization constants involved in the solutions is nontrivial. Recently, Gordon proposed a method to derive the closed form formula for normalization constants for certain closed networks. In this paper, we describe a simpler method using Z-transform. In the cases of networks of multiple server queues or of single server queues with equal traffic intensities, the computation involved in proposed approach is much simpler than that in Gordon's paper. For multichain closed networks, we propose to use FFT (Fast-Fourier-Transform) to calculate the normalization constants.     

Approximate analysis of arbitrary configurations of open queueing networks with blocking

An algorithm for analyzing approximately open exponential queueing networks with blocking is presented. The algorithm decomposes a queueing network with blocking into individual queues with revised capacity, and revised arrival and service processes. These individual queues are then analyzed in isolation. Numerical experience with this algorithm is reported for three-node and four-node queueing networks. The approximate results obtained were compared against exact numerical data, and they seem to have an acceptable error level.Supported in part by a grant from CAIP Center, Rutgers University.Supported in part by the National Science Foundation under Grant DCR-85-02540.     

Diffusion approximations for open queueing networks with service interruptions

This paper establishes functional central limit theorems describing the heavy-traffic behavior of open single-class queueing networks with service interruptions. In particular, each station has a single server which is alternatively up and down. There are two treatments of the up and down times. The first treatment corresponds to fixed up and down times and leads to a reflected Brownian motion, just as when there are no service interruptions, but with different parameters. To represent long rare interruptions, the second treatment has growing up and down times with the up and down times being of ordern andn ^1/2, respectively, when the traffic intensities are of order 1-n^–1/2. In this case we establish convergence in the SkorohodM ₁ topology to a multidimensional reflection of multidimensional Brownian motion plus a multidimensional jump process.     

Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks

Fluid models have recently become an important tool for the study of open multiclass queueing networks. We are interested in a family of such models, which we refer to as head-of-the-line proportional processor sharing (HLPPS) fluid models. Here, the fraction of time spent serving a class present at a station is proportional to the quantity of the class there, with all of the service going into the first customer of each class. To study such models, we employ an entropy function associated with the state of the system. The corresponding estimates show that if the traffic intensity function is at most 1, then such fluid models converge exponentially fast to equilibria. When the traffic intensity function is strictly less than 1, the limit is always the empty state and occurs after a finite time. A consequence is that generalized HLPPS networks with traffic intensity strictly less than 1 are positive Harris recurrent. Related results for FIFO fluid models of Kelly type were obtained in Bramson [4].Partially supported by NSF Grants DMS-93-00612 and DMS-93-04580. The paper was written while the author was in residence at the Institute for Advanced Study.     

Convergence to equilibria for fluid models of FIFO queueing networks

The qualitative behavior of open multiclass queueing networks is currently a topic of considerable activity. An important goal is to formulate general criteria for when such networks possess equilibria, and to characterize these equilibria when possible. Fluid models have recently become an important tool for such purposes. We are interested here in a family of such models, FIFO fluid models of Kelly type. That is, the discipline is first-in, first-out, and the service rate depends only on the station. To study such models, we introduce an entropy function associated with the state of the system. The corresponding estimates show that if the traffic intensity function is at most 1, then such fluid models converge exponentially fast to equilibria with fixed concentrations of customer types throughout each queue. When the traffic intensity function is strictly less than 1, the limit is always the empty state and occurs after a finite time. A consequence is that generalized Kelly networks with traffic intensity strictly less than 1 are positive Harris recurrent, and hence possess unique equilibria.1991Mathematics Subject Classification, 60K25, 68M20, 90B10. Partially supported by NSF Grant DMS-93-00612.     

Simple bounds for closed queueing networks

Consider a closed Jackson type network in which each queue has a single exponential server. Assume that N customers are moving among k queues. We establish simple closed form bounds on the network throughput (both lower and upper), which are sharper than those that are currently available. Numerical evaluation indicates that the improvements are significant.     

Deadlock free buffer allocation in closed queueing networks

Blocking queueing networks are of much interest in performance analysis due to their realistic modeling capability. One important feature of such networks is that they may have deadlocks which can occur if the node capacities are not sufficiently large. A necessary and sufficient condition for the node capacities is presented such that the network is deadlock free. An algorithm is given for buffer allocation in blocking queueing networks such that no deadlocks will occur assuming that the network has the special structure called cacti-graph. Additional algorithm which takes linear time in the number of nodes, is presented to find cycles in cacti networks.Akyildiz's work was supported in part by School of Information and Computer Science, ICS, of Georgia Tech and by the Air Force Office of the Scientific Research (AFOSR) under Grant AFOSR-88-0028.     

A survey of product form queueing networks with blocking and their equivalences

Queueing network models have been extensively used to represent and analyze resource sharing systems, such as production, communication and information systems. Queueing networks with blocking are used to represent systems with finite capacity resources and with resource constraints. Different blocking mechanisms have been defined and analyzed in the literature to represent distinct behaviors of real systems with limited resources. Exact product form solutions of queueing networks with blocking have been derived, under special constraints, for different blocking mechanisms. In this paper we present a survey of product form solutions of queueing networks with blocking and equivalence properties among different blocking network models. By using such equivalences we can extend product form solutions to queueing network models with different blocking mechanisms. The equivalence properties include relationships between open and closed product form queueing networks with different blocking mechanisms.This work has been partially supported by CNR Project Research Funds Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo and by MURST Project Research Funds Performability hw/sw di sistemi distribuiti e paralleli.     

Asymptotic marginal independence in large networks of loss systems

Network models in which each node is a loss system frequently arise in telephony. Models with several hundred nodes are common. Suppose a customer requires a server from each of several nodes. It would be convenient if the probability that the required servers are all free were approximately a product, where each term is the probability a required node has a free server. We present some theorems to support this approximation. Most of the theorems are restricted to nodes with one server. Some of the difficulties in analyzing nodes with multiple servers are described.     

Maximizing throughput in queueing networks with limited flexibility

We study a queueing network where customers go through several stages of processing, with the class of a customer used to indicate the stage of processing. The customers are serviced by a set of flexible servers, i.e., a server is capable of serving more than one class of customers and the sets of classes that the servers are capable of serving may overlap. We would like to choose an assignment of servers that achieves the maximal capacity of the given queueing network, where the maximal capacity is λ^∗ if the network can be stabilized for all arrival rates λ < λ^∗ and cannot possibly be stabilized for all λ > λ^∗. We examine the situation where there is a restriction on the number of servers that are able to serve a class, and reduce the maximal capacity objective to a maximum throughput allocation problem of independent interest: the total discrete capacity constrained problem (TDCCP). We prove that solving TDCCP is in general NP-complete, but we also give exact or approximation algorithms for several important special cases and discuss the implications for building limited flexibility into a system.     

Bottleneck analysis in multiclass closed queueing networks and its application

Asymptotic behavior of queues is studied for large closed multi-class queueing networks consisting of one infinite server station with K classes and M processor sharing (PS) stations. A simple numerical procedure is derived that allows us to identify all bottleneck PS stations. The bottleneck station is defined asymptotically as the station where the number of customers grows proportionally to the total number of customers in the network, as the latter increases simultaneously with service rates at PS stations. For the case when K=M=2, the set of network parameters is identified that corresponds to each of the three possible types of behavior in heavy traffic: both PS stations are bottlenecks, only one PS station is a bottleneck, and a group of two PS stations is a bottleneck while neither PS station forms a bottleneck by itself. In the last case both PS stations are equally loaded by each customer class and their individual queue lengths, normalized by the large parameter, converge to uniformly distributed random variables. These results are directly generalized for arbitrary K=M. Generalizations for K≠M are also indicated. The case of two bottlenecks is illustrated by its application to the problem of dimensioning bandwidth for different data sources in packet-switched communication networks. An engineering rule is provided for determining the link rates such that a service objective on a per-class throughput is satisfied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Symmetry property of the throughput in closed tandem queueing networks with finite buffers

In this paper we consider closed tandem queueing networks with finite buffers and blocking before service. With this type of blocking, a server is allowed to start processing a job only if there is an empty space in the next buffer. It was recently conjectured that the throughput of such networks is symmetrical with respect to the population of the network. That is, the throughput of the network with population N is the same as that with population C − N, where C is the total number of buffer spaces in the network. The main purpose of this paper is to prove this result in the case where the service time distributions are of phase type (PH-distribution). The proof is based on the comparison of the sample paths of the network with populations N and C − N. Finally, we also show that this symmetry property is related to a reversibility property of this class of networks.     

Asymptotic analysis of a large closed queueing network with discriminatory processor sharing

In this paper the steady-state behavior of a closed queueing network with multiple classes and large populations is investigated. One of the two nodes of the network simply introduces random delays and the discipline in the other node is discriminatory processor sharing. The network is not product-form, so not even the steady-state behavior is known. We assume that the usage is moderately heavy, and obtain two-term asymptotic approximations to the mean number of jobs, and the mean sojourn time, of each class of jobs in the processor node. We also obtain the leading term in the asymptotic approximation to the joint distribution of the number of jobs in the processor node, which is a zero-mean multivariate Gaussian distribution around a line through the origin.