State-dependent Coupling of Quasireversible Nodes

The seminal paper of Jackson began a chain of research on queueing networks with product-form stationary distributions which continues strongly to this day. Hard on the heels of the early results on queueing networks followed a series of papers which discussed the relationship between product-form stationary distributions and the quasireversibility of network nodes. More recently, the definition of quasireversibility and the coupling mechanism between nodes have been extended so that they apply to some of the later product-form queueing networks incorporating negative customers, signals, and batch movements.In parallel with this research, it has been shown that some special queueing networks can have arrival and service parameters which depend upon the network state, rather than just the node state, and still retain a generalised product-form stationary distribution.In this paper we begin by offering an alternative proof of a product-form result of Chao and Miyazawa and then build on this proof by postulating a state-dependent coupling mechanism for a quasireversible network. Our main theorem is that the resultant network has a generalised product form stationary distribution. We conclude the paper with some examples.     

A new perspective on the normalization of invariant measures for loss networks and other product form systems

Loss networks have proved to be a vital tool in the study of telecommunication networks, computer performance, and resource allocation problems. For a large subset of these models, the invariant measure is of a product form. The existence of efficient procedures to normalize a product-form invariant measure is essential for analysis of the underlying system.Choudhury et al. [1—4] have recently presented a number of algorithms based upon Fourier analysis for the normalization of product-form invariant measures for specific systems. Bean and Stewart [5] discussed related work for the normalization of product-form invariant measures for closed queueing networks. In this paper, we present a simple unifying framework within which to study these algorithms. This framework is significant as it suggests a number of extensions to these algorithms and simplifies their derivation.     

On a class of approximations for closed queueing networks

This paper presents some analytical results concerning an approximation procedure for closed queueing networks. The procedure is well-known and has been found useful for product-form networks where large numbers of queues, jobs or job classes prohibit an exact analysis, as well as for networks which do not possess product-form. The procedure represents the mean sojourn time at a queue as a function of the throughput of the queue, and derives a set of fixed point equations for the throughputs of the various job classes. We begin by showing that under a mild regularity condition the fixed point equations have a unique solution. Then we show that derivatives of performance measures can be readily calculated, and that their simple form provides an interesting insight into capacity allocation in closed queueing networks.This work was supported in part by the Nuffield Foundation     

A general formulation for mean-value analysis in product-form batch-movement queueing networks

A number of recent papers have shown that many classes of queueing networks with batches of customers served and routed through the network have equilibrium distributions which factorise into product forms over the nodes of the network. In this paper we demonstrate how such networks are amenable to a mean-value analysis which generalises that used for single-movement networks.Since product-form stochastic Petri nets (SPNs) can be viewed as batch-movement queueing networks, our algorithm is also applicable to their analysis.     

Closed queueing networks with batch services

In this paper we study queueing networks which allow multiple changes at a given time. The model has a natural application to discrete-time queueing networks but describes also queueing networks in continuous time. It is shown that product-form results which are known to hold when there are single changes at a given instant remain valid when multiple changes are allowed.     

A generalization of Norton's theorem for queueing networks

A general framework for aggregation and decomposition of product form queueing networks with state dependent routing and servicing is presented. By analogy with electrical circuit theory, the stations are grouped into clusters of subnetworks such that the process decomposes into a global process and a local process. Moreover, the local process factorizes into the subnetworks. The global process and the local processes can be analyzed separately as if they were independent. The global process describes the behaviour of the queuing network in which each cluster is aggregated into a single station, whereas the local process describes the behaviour of the subnetworks as if they are not part of the queueing network. The decomposition and aggregation method formalized in this paper allows us to first analyze the global behaviour of the queueing network and subsequently analyze the local behaviour of the subnetworks of interest or to aggregate clusters into single stations without affecting the behaviour of the rest of the queueing network. Conditions are provided such that:

- the global equilibrium distribution for aggregated clusters has a product form;

- this form can be obtained by merely monitoring the global behaviour;

- the computation of a detailed distribution, including its normalizing constant, can be decomposed into the computation of a global and a local distribution;

- the marginal distribution for the number of jobs at the stations of a cluster can be obtained by merely solving local behaviour.

As a special application, Norton's theorem for queueing networks is extended to queueing networks with state dependent routing such as due to capacity constraints at stations or at clusters of stations and state dependent servicing such as due to service delays for clusters of stations.     

Maximum Entropy Condition in Queueing Theory

The main results in queueing theory are obtained when the queueing system is in a steady-state condition and if the requirements of a birth-and-death stochastic process are satisfied. The aim of this paper is to obtain a probabilistic model when the queueing system is in a maximum entropy condition. For applying the entropic approach, the only information required is represented by mean values (mean arrival rates, mean service rates, the mean number of customers in the system). For some one-server queueing systems, when the expected number of customers is given, the maximum entropy condition gives the same probability distribution of the possible states of the system as the birth-and-death process applied to an M/M/1 system in a steady-state condition. For other queueing systems, as M/G/1 for instance, the entropic approach gives a simple probability distribution of possible states, while no close expression for such a probability distribution is known in the general framework of a birth-and-death process.     

Some notes on the reduction of network dimensionality in nested open queueing networks

In this paper, we study an open and nested tandem queueing network, where the population constraint within each subnetwork is controlled by a semaphore queue. The total number of customers that may be present in the subnetwork can not exceed a given value. Each node has a constant service time and the arrival process to the queueing network follows an arbitrary distribution.A major characteristic of this queueing network is that the low layer flow is halted by the state of the high layer. We develop a simple and equivalent queueing network that has the same performance characteristics as the original queueing network. Using this model, the waiting time on the queueing network can be easily derived. It is interesting to see how the simplification process can be applied to multi-layered queueing network.     

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.     

The departure process of discrete-time queueing systems with Markovian type inputs

This paper proposes a unified matrix-analytic approach to characterize the output processes of general discrete-time lossless/lossy queueing systems in which time is synchronized/slotted into fixed length intervals called slots. The arrival process can be continuous- or discrete-time Markovian processes. It can be either renewal or non-renewal. The service of a customer commences at the beginning of a slot, consumes a random number of slots, and completes at the end of a later slot. The service times are independent and follow a common and general distribution. Systems with and without server vacations are both treated in this paper. These queueing systems have potential applications in asynchronous transfer mode (ATM) networks, packet radio networks, etc. Since the output process of a node in a queueing network becomes an input process to some node at the next stage, the results of this paper can be used to facilitate end-to-end performance analysis which has attracted more and more attention in the literature. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the stationary distribution of queue lengths in a multi-class priority queueing system with customer transfers

This paper deals with a multi-class priority queueing system with customer transfers that occur only from lower priority queues to higher priority queues. Conditions for the queueing system to be stable/unstable are obtained. An auxiliary queueing system is introduced, for which an explicit product-form solution is found for the stationary distribution of queue lengths. Sample path relationships between the queue lengths in the original queueing system and the auxiliary queueing system are obtained, which lead to bounds on the stationary distribution of the queue lengths in the original queueing system. Using matrix-analytic methods, it is shown that the tail asymptotics of the stationary distribution is exact geometric, if the queue with the highest priority is overloaded.      

A queueing system with random server capacity and multiple control

The authors study queueing, input and output processes in a queueing system with bulk service and state dependent service delay. The input flow of customers, modulated by a semi-Markov process, is served by a single server that takes batches of a certain fixed size if available or waits until the queue accumulates enough customers for service. In the latter case, the batch taken for service is of random size dependent on the state of the system, while service duration depends both on the state of the system and on the batch size taken. The authors establish a necessary and sufficient condition for equilibrium of the system and obtain the following results: Explicit formulas for steady state distribution of the queueing process, intensity of the input and output processes, and mean values of idle and busy periods. They employ theory of semi-regenerative processes and illustrate the results by a number of examples. In one of them an optimization problem is discussed.     

Sojourn times in a processor sharing queue with service interruptions

We study the sojourn times of customers in an M/M/1 queue with the processor sharing service discipline and a server that is subject to breakdowns. The lengths of the breakdowns have a general distribution, whereas the on-periods are exponentially distributed. A branching process approach leads to a decomposition of the sojourn time, in which the components are independent of each other and can be investigated separately. We derive the Laplace–Stieltjes transform of the sojourn-time distribution in steady state, and show that the expected sojourn time is not proportional to the service requirement. In the heavy-traffic limit, the sojourn time conditioned on the service requirement and scaled by the traffic load is shown to be exponentially distributed. The results can be used for the performance analysis of elastic traffic in communication networks, in particular, the ABR service class in ATM networks, and best-effort services in IP networks.     

Generalized M/G/C/C state dependent queueing models and pedestrian traffic flows

The generality and usefulness ofM/G/C/C state dependent queueing models for modelling pedestrian traffic flows is explored in this paper. We demonstrate that the departure process and the reversed process of these generalizedM/G/C/C queues is a Poisson process and that the limiting distribution of the number of customers in the queue depends onG only through its mean. Consequently, the models developed in this paper are useful not only for the analysis of pedestrian traffic flows, but also for the design of the physical systems accommodating these flows. We demonstrate how theM/G/C/C state dependent model is incorporated into the modelling of large scale facilities where the blocking probabilities in the links of the network can be controlled. Finally, extensions of this work to queueing network applications where blocking cannot be controlled are also presented, and we examine an approximation technique based on the expansion method for incorporating theseM/G/C/C queues in series, merge, and splitting topologies of these networks.     

Entropy maximisation and queueing network models

Over recent years it has become increasingly evident that classical queueing theory cannot easily handle complex queueing systems and networks with many interacting elements. As a consequence, alternative ideas and tools, analogous to those applied in the field of Statistical Mechanics, have been proposed in the literature. In this context, the principles of Maximum Entropy (ME) and Minimum Relative Entropy (MRE), a generalisation, provide consistent methods of inference for characterising the form of an unknown but true probability distribution, based on information expressed in terms of known to exist true expected values or when, in addition, there exists a prior estimate of the unknown distribution. This paper traces the progress achieved so far towards the creation of ME and MRE product-form approximations and related algorithms for the performance analysis of general Queueing Network Models (QNMs) and indicates potential research extensions in the area.Earlier research on entropy maximisation and QNMs was sponsored by the UK Science and Engineering Research Council (SERC) with grant GR/D/12422, while recent and current work is supported by SERC grants GR/F/29271 and GR/H/18609. Research on Complex I/O subsystems was funded by Metron Technology Ltd., UK, under grant JJCA415.     

Modeling a class of state-dependent routing in flexible manufacturing systems

We develop a closed queueing network model for flexible manufacturing systems (FMSs), where parts routing follows a probabilistic shortest-queue (PSQ) scheme, i.e. parts are routed to the shortest queue (or the most empty station) with the highest probability. We allow limited local buffer at each work station. We prove that with the PSQ routing, the Markovian queue-length process satisfies time reversibility and has a product-form equilibrium distribution. An algorithm is developed to compute the solutions to the model. The model can be used as a performance evaluation tool to study FMSs.     

A matrix-analytic solution for the DBMAP/PH/1 priority queue

Priority queueing models have been commonly used in telecommunication systems. The development of analytically tractable models to determine their performance is vitally important. The discrete time batch Markovian arrival process (DBMAP) has been widely used to model the source behavior of data traffic, while phase-type (PH) distribution has been extensively applied to model the service time. This paper focuses on the computation of the DBMAP/PH/1 queueing system with priorities, in which the arrival process is considered to be a DBMAP with two priority levels and the service time obeys a discrete PH distribution. Such a queueing model has potential in performance evaluation of computer networks such as video transmission over wireless networks and priority scheduling in ATM or TDMA networks. Based on matrix-analytic methods, we develop computation algorithms for obtaining the stationary distribution of the system numbers and further deriving the key performance indices of the DBMAP/PH/1 priority queue. AMS subject classifications: 60K25 · 90B22 · 68M20 The work was supported in part by grants from RGC under the contracts HKUST6104/04E, HKUST6275/04E and HKUST6165/05E, a grant from NSFC/RGC under the contract N_HKUST605/02, a grant from NSF China under the contract 60429202.     

An M/G/1 Retrial Queueing System with Two-Phase Service and Preemptive Resume

An M/G/1 retrial queueing system with additional phase of service and possible preemptive resume service discipline is considered. For an arbitrarily distributed retrial time distribution, the necessary and sufficient condition for the system stability is obtained, assuming that only the customer at the head of the orbit has priority access to the server. The steady-state distributions of the server state and the number of customers in the orbit are obtained along with other performance measures. The effects of various parameters on the system performance are analysed numerically. A general decomposition law for this retrial queueing system is established.     

Computation of the steady state distribution for multi-server retrial queues with phase type service process

We consider a multi-server retrial queueing system with the Batch Markovian Arrival Process and phase type service time distribution. Such a general queueing system suits for modeling and decision making in many real life objects including modern wireless communication networks. Behavior of such a system is described by the level dependent multi-dimensional Markov chain. Blocks of the generator of this chain, which is the block structured matrix of infinite size, can have large size if the number of servers is large and distribution of service time is not exponential. Due to this fact, the existing in literature algorithms allow to compute key performance measures of such a system only for a small number of servers. Here we describe the algorithm that allows to compute the stationary distribution of the system for larger number of servers and numerically illustrate its advantage. Importance of taking into account correlation in the arrival process is numerically demonstrated.     

Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse

Certain diffusion processes known as semimartingale reflecting Brownian motions (SRBMs) have been shown to approximate many single class and some multiclass open queueing networks under conditions of heavy traffic. While it is known that not all multiclass networks with feedback can be approximated in heavy traffic by SRBMs, one of the outstanding challenges in contemporary research on queueing networks is to identify broad categories of networks that can be so approximated and to prove a heavy traffic limit theorem justifying the approximation. In this paper, general sufficient conditions are given under which a heavy traffic limit theorem holds for open multiclass queueing networks with head-of-the-line (HL) service disciplines, which, in particular, require that service within each class is on a first-in-first-out (FIFO) basis. The two main conditions that need to be verified are that (a) the reflection matrix for the SRBM is well defined and completely- S, and (b) a form of state space collapse holds. A result of Dai and Harrison shows that condition (a) holds for FIFO networks of Kelly type and their proof is extended here to cover networks with the HLPPS (head-of-the-line proportional processor sharing) service discipline. In a companion work, Bramson shows that a multiplicative form of state space collapse holds for these two families of networks. These results, when combined with the main theorem of this paper, yield new heavy traffic limit theorems for FIFO networks of Kelly type and networks with the HLPPS service discipline. This revised version was published online in June 2006 with corrections to the Cover Date.