Travel times in queueing networks and network sojourns

We first describe expected values of sojourn times for semi-stationary (or synchronous) networks. This includes sojourn times for units and sojourn times for the entire network. A typical sojourn time of a unit is the time it spends in a sector (set of nodes) while it travels through the network, and a typical network sojourn time is the busy period of a sector. Our results apply to a wide class of networks including Jackson networks with general service times, general Markov or regenerative networks, and networks with batch processing and concurrent movement of units. The results also shed more light on when Little's law for general systems, holds for expectations as well as for limiting averages. Next, we describe the expectation of a unit's travel time on a general route in a basic Markov network process (such as a Jackson process). Examples of travel times are the time it takes for a unit to travel from one sector to another, and the time between two successive entrances to a node by a unit. Finally, we characterize the distributions of the sojourn times at nodes on certain overtake-free routes and the travel times on such routes for Markov network processes.This research was supported in part by the Air Force Office of Scientific Research under contract 89-0407 and NSF grant DDM-9007532.     

Analysis of effective service time with age dependent interruptions and its application to optimal rollback policy for database management

A stochastic model is developed describing a service system subject to inhomogeneous Poisson interruptions with age dependent interruption periods. By studying the probabilistic flow of the underlying multivariate Markov process, the Laplace transform of the effective service time is explicitly obtained. For general renewal interruptions, only the expected effective service time is derived. As an application, an optimal checkpoint policy is examined for database management. It is shown that an optimal policy maximizing the ergodic availability of the database is to implement a checkpoint as soon as the cumulative uptime of the database reaches a prespecified constantk ^*. A computational procedure is then developed for findingk ^* and numerical results are exhibited.This work was supported in part by the National Science Foundation under Grant No. ECS-8600992 and by the IBM Program of Support for Education in the Management of Information Systems.     

Stability and instability of routings through a queueing network

We consider a network of N nodes with given distances in which customers arrive at one of the nodes and request transportation to one of the other nodes. There is a finite number of servers (or transporters) with possibly different capacities and speed available. We show that there exists a critical arrival rate κ_c, such that if the arrival rate is larger than κ_c then the number of customers in the system increases to ∞ with linear speed no matter which routing strategy for the transporters is chosen (even if we allow the strategy to depend on the future arrival process). If, on the other hand, κ < _c then there exists even a fixed periodic routing strategy which keeps the system stable (in a sense to be made precise). We prefer to start with a deterministic set-up which allows very general arrival processes and look at the stochastic case only in the final section where we get more refined results by assuming that the arrival process has integrable stationary ergodic increments (which still allows batch arrivals and long-range dependence). Potential applications include the control of elevators in highrise buildings. This revised version was published online in June 2006 with corrections to the Cover Date.     

Markovian network processes: Congestion-dependent routing and processing

A Markovian network process describes the movement of discrete units among a set of nodes that process the units. There is considerable knowledge of such networks, often called queueing networks, in which the nodes operate independently and the routes of the units are independent. The focus of this study, in contrast, is on networks with dependent nodes and routings. Examples of dependencies are parallel processing across several nodes, blocking of transitions because of capacity constraints on nodes, alternate routing of units to avoid congestion, and accelerating or decelerating the processing rate at a node depending on downstream congestion. We introduce a general network process representing the numbers of units at the nodes and derive its equilibrium distribution. This distribution takes the form of a product of functions of vectors in which the arguments of the functions satisfy an interchangeability property. This new type of distribution may apply to other multi-variate processes as well. A basic idea in our approach is a linking of certain micro-level balance properties of the network routing to the processing rates at the nodes. The link is via routing-balance partitions of nodes that are inherent in any network. A byproduct of this approach is a general characterization of blocking of transitions without the restriction that the process is reversible, which had been a standard assumption. We also give necessary and sufficient conditions under which a unit moving in the network sees a time average for the unmoved units (called the MUSTA property). Finally, we discuss when certain flows between nodes in an open network are Poisson processes.This research was sponsored in part by Air Force Office of Scientific Research contract 84-0367.     

Markov network processes with product form stationary distributions

This study concerns the equilibrium behavior of a general class of Markov network processes that includes a variety of queueing networks and networks with interacting components or populations. The focus is on determining when these processes have product form stationary distributions. The approach is to relate the marginal distributions of the process to the stationary distributions of “node transition functions” that represent the nodes in isolation operating under certain fictitious environments. The main result gives necessary and sufficient conditions on the node transition functions for the network process to have a product form stationary distribution. This result yields a procedure for checking for a product form distribution and obtaining such a distribution when it exits. An important subclass of networks are those in which the node transition rates have Poisson arrival components. In this setting, we show that the network process has a product form distribution and is “biased locally balanced” if and only if the network is “quasi-reversible” and certain traffic equations are satisfied. Another subclass of networks are those with reversible routing. We weaken the known sufficient condition for such networks to be product form. We also discuss modeling issues related to queueing networks including time reversals and reversals of the roles of arrivals and departures. The study ends by describing how the results extend to networks with multi-class transitions. This revised version was published online in June 2006 with corrections to the Cover Date.     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

A multiclass network with non-linear, non-convex, non-monotonic stability conditions

We consider a stochastic queueing network with fixed routes and class priorities. The vector of class sizes forms a homogeneous Markov process of countable state space Z⁶ ₊. The network is said “stable” (resp.“unstable”) if this Markov process is ergodic (resp. transient). The parameters are the traffic intensities of the different classes. An unusual condition of stability is obtained thanks to a new argument based on the characterization of the “essential states”. The exact stability conditions are then detected thanks to an associated fluid network: we identify a zone of the parameter space in which diverging, fluid paths appear. In order to show that this is a zone of instability (and that the network is stable outside this zone), we resort to the criteria of ergodicity and transience proved by Malyshev and Menshikov for reflected random walks in Z⁶ ₊ (Malyshev and Menshikov, 1981). Their approach allows us to neglect some “pathological” fluid paths that perturb the dynamics of the fluid model. The stability conditions thus determined have especially unusual characteristics: they have a quadratic part, the stability domain is not convex, and increasing all the service rates may provoke instability (Theorem 1.1 and section 7). This revised version was published online in June 2006 with corrections to the Cover Date.     

The stability of a non-homogeneous queueing system with regenerative input

A queueing system with nonidentical service channels, a regenerative input, and a renewal service process at each channel is considered. The ergodic conditions are found when the basic process describing the system (and including the queue-size and the waiting time vector) is regenerative with finite expectation of the regeneration cycle length. The asymptotic properties of the embedded renewal process of the regeneration points are used. The zero-delayed and delayed cases are considered separately. Some queueing network applications are discussed. Supported by the Nordic Council of Ministers. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

A queueing model of a production system with two machines,one operator and priorities

This paper studies a priority queueing model of a production system in which one operator serves two types of units with overlapping service times. The two types of units arrive in independent Poisson processes. There are two machines in the system. Units of type 1 receive two consecutive types of services at machine #1: the handwork performed by the operator and the automatic machining without the operator. Units of type 2 receive only the handwork performed by the operator at machine #2. The operator attends the two machines according to a strict-priority discipline which always gives units of type 2 higher priority than units of type 1. At each machine the handwork times have a general distribution, and at machine #1 the machining times have an exponential distribution. The Laplace-Stieltjes transform of the queue-size distributions and the waiting time distributions for a stationary process are obtained.     

B.A. Sevastyanov’s famous theorem

In 1956, at the Third All-Union Mathematical Congress, Boris Aleksandrovich Sevastyanov gave a talk on the ergodic theorem proved by him for Markov processes and on its application to queueing systems. In 1957, this result was published in the journal Teoriya Veroyatnostei i Ee Primeneniya (Theory of Probability and Its Applications). An important corollary to the ergodic theorem is a generalization of Erlang’s well-known formula to a queueing system with a Poisson input flow and an arbitrary distribution of the service time. This result of Sevastyanov has served as a starting point for numerous studies on the problem, which was later called the insensitivity (invariance) problem for queueing systems with losses. There are hundreds of references to this result of Sevastyanov.     

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.     

An adaptive Euler-Maruyama scheme for SDEs: convergence and stability

J. C. Mattingly ^{The understanding of adaptive algorithms for stochastic differentialequations (SDEs) is an open area, where many issues relatedto both convergence and stability (long-time behaviour) of algorithmsare unresolved. This paper considers a very simple adaptivealgorithm, based on controlling only the drift component ofa time step. Both convergence and stability are studied. Theprimary issue in the convergence analysis is that the adaptivemethod does not necessarily drive the time steps to zero withthe user-input tolerance. This possibility must be quantifiedand shown to have low probability. The primary issue in thestability analysis is ergodicity. It is assumed that the noiseis nondegenerate, so that the diffusion process is elliptic,and the drift is assumed to satisfy a coercivity condition.The SDE is then geometrically ergodic (averages converge tostatistical equilibrium exponentially quickly). If the driftis not linearly bounded, then explicit fixed time step approximations,such as the Euler–Maruyama scheme, may fail to be ergodic.In this work, it is shown that the simple adaptive time-steppingstrategy cures this problem. In addition to proving ergodicity,an exponential moment bound is also proved, generalizing a resultknown to hold for the SDE itself.}     

Cox Point Processes Driven by Ornstein–Uhlenbeck Type Processes

The paper is devoted to the development of Cox point processes driven by nonnegative processes of Ornstein–Uhlenbeck (OU) type. Starting with multivariate temporal processes we develop formula for the cross pair correlation function. Further filtering problem is studied by means of two different approaches, either with discretization in time or through the point process densities with respect to the Poisson process. The first approach is described mainly analytically while in the second case we obtain numerical solution by means of MCMC. The Metropolis–Hastings birth–death chain for filtering can be also used when estimating the parameters of the model. In the second part we try to develop spatial and spatio-temporal Cox point processes driven by a stationary OU process. The generating functional of the point process is derived which enables evaluation of basic characteristics. Finally a simulation algorithm is given and applied.      

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??For the infinite Jackson network, assume that the net input rates are greater than the service rates for some nodes. Via solving the new throughput equation, the stochastic comparable processes are obtained by coupling method, and furthermore the limits for the queueing length in all nodes are also obtained. Despite the whole network is non-ergodic, it is possible to get the maximal ergodic subnetwork.     

Tandem behavior of a telecommunication system with finite buffers and repeated calls

The tandem behavior of a telecommunication system with finite buffers and repeated calls is modeled by the performance of a finite capacityG/M/1 queueing system with general interarrival time distribution, exponentially distributed service time, the first-come-first-served queueing discipline and retrials. In this system a fraction of the units which on arrival at a node of the system find it busy, may retry to be processed, by merging with the incoming arrival units in that node, after a fixed delay time. The performance of this system in steady state is modeled by a queueing network and is approximated by a recursive algorithm based on the isolation method. The approximation outcomes are compared against those from a simulation study. Our numerical results indicate that in steady state the non-renewal superposition arrival process, the non-renewal overflow process, and the non-renewal departure process of the above system can be approximated with compatible renewal processes.     

Spectral representations of sum- and max-stable processes

To each max-stable process with α-Fréchet margins, α ∈ (0,2), a symmetric α-stable process can be associated in a natural way. Using this correspondence, we deduce known and new results on spectral representations of max-stable processes from their α-stable counterparts. We investigate the connection between the ergodic properties of a stationary max-stable process and the recurrence properties of the non-singular flow generating its spectral representation. In particular, we show that a stationary max-stable process is ergodic iff the flow generating its spectral representation has vanishing positive recurrent component. We prove that a stationary max-stable process is ergodic (mixing) iff the associated SαS process is ergodic (mixing). We construct non-singular flows generating the max-stable processes of Brown and Resnick.     

Uniform concentration inequality for ergodic diffusion processes observed at discrete times

In this paper a concentration inequality is proved for the deviation in the ergodic theorem for diffusion processes in the case of discrete time observations. The proof is based on geometric ergodicity of diffusion processes. We consider as an application the nonparametric pointwise estimation problem of the drift coefficient when the process is observed at discrete times.     

A large closed queueing network with autonomous service and bottleneck

This paper studies the queue-length process in a closed Jackson-type queueing network with the large number N of homogeneous customers by methods of the theory of martingales and by the up- and down-crossing method. The network considered here consists of a central node (hub), being an infinite-server queueing system with exponentially distributed service times, and k single-server satellite stations (nodes) with generally distributed service times with rates depending on the value N. The service mechanism of these k satellite stations is autonomous, i.e., every satellite server j serves the customers only at random instants that form a strictly stationary and ergodic sequence of random variables. Assuming that the first k-1 satellite stations operate in light usage regime the paper considers the cases where the kth satellite station is a bottleneck node. The approach of the paper is based both on development of the method from the paper by Kogan and Liptser [16], where a Markovian version of this model has been studied, and on development of the up- and down-crossing method. This revised version was published online in June 2006 with corrections to the Cover Date.     

Shift ergodicity for stationary Markov processes

In this paper shift ergodicity and related topics are studied for certain stationary processes. We first present a simple proof of the conclusion that every stationary Markov process is a generalized convex combination of stationary ergodic Markov processes. A direct consequence is that a stationary distribution of a Markov process is extremal if and only if the corresponding stationary Markov process is time ergodic and every stationary distribution is a generalized convex combination of such extremal ones. We then consider space ergodicity for spin flip particle systems. We prove space shift ergodicity and mixing for certain extremal invariant measures for a class of spin systems, in which most of the typical models, such as the Voter Models and the Contact Models, are included. As a consequence of these results we see that for such systems, under each of those extremal invariant measures, the space and time means of an observable coincide, an important phenomenon in statistical physics. Our results provide partial answers to certain interesting problems in spin systems.     

Large Closed Queueing Networks in Semi-Markov Environment and Their Application

The paper studies closed queueing networks containing a server station and k client stations. The server station is an infinite server queueing system, and client stations are single-server queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by a strictly stationary and ergodic sequence of random variables. The total number of units in the network is N. The expected times between departures in client stations are (N μ _j )⁻¹. After a service completion in the server station, a unit is transmitted to the jth client station with probability p _j (j=1,2,…,k), and being processed in the jth client station, the unit returns to the server station. The network is assumed to be in a semi-Markov environment. A semi-Markov environment is defined by a finite or countable infinite Markov chain and by sequences of independent and identically distributed random variables. Then the routing probabilities p _j (j=1,2,…,k) and transmission rates (which are expressed via parameters of the network) depend on a Markov state of the environment. The paper studies the queue-length processes in client stations of this network and is aimed to the analysis of performance measures associated with this network. The questions risen in this paper have immediate relation to quality control of complex telecommunication networks, and the obtained results are expected to lead to the solutions to many practical problems of this area of research.