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.     

A general approximation for the single product lot sizing model with queueing delays

The objective of this paper is to derive a general approximation for the single product lot sizing model with queueing delays, explicitly including a non-zero setup time. Most research focuses on bulk (batch) arrival and departure processes. In this paper we assume an individual arrival and departure process allowing the modelling of more realistic demand patterns. A general approximation of the expected lead time and the variance of the lead time is derived. The lead time probability distribution is approximated by means of a lognormal distribution. This allows the manufacturer to quote lead times satisfying a specified customer service level as a function of the lot size. The main result is a convex relationship of the expected lead time and the quoted lead time as a function of the lot size. The results are illustrated by means of numerical examples.     

On Markov-Additive Jump Processes

In 1995, Pacheco and Prabhu introduced the class of so-called Markov-additive processes of arrivals in order to provide a general class of arrival processes for queueing theory. In this paper, the above class is generalized considerably, including time-inhomogeneous arrival rates, general phase spaces and the arrival space being a general vector space (instead of the finite-dimensional Euclidean space). Furthermore, the class of Markov-additive jump processes introduced in the present paper is embedded into the existing theory of jump processes. The best known special case is the class of BMAP arrival processes.     

Approximations for Multi-Class Departure Processes

The exact analysis of a network of queues with multiple products is, in general, prohibited because of the non-renewal structure of the arrival and departure processes. Two-moment approximations (decomposition methods, Whitt [9] ) have been successfully used to study these systems. The performance of these methods, however, strongly depends on the quality of the approximations used to compute the squared coefficient variation (CV) of the different streams of products.In this paper, an approximation method for computing the squared coefficient of variation of the departure stream from a multi-class queueing system is presented. In particular, we generalize the results of Bitran and Tirupati [3] and Whitt [11] related to the interference effect.     

G／G／1排队系统的渐近泊松离去过程

本文给出了Ｇ／Ｇ／１排队系统的离去过程的有限维分布弱收敛到泊松过程的有限维分布的条件,特别给出了生灭排队系统及Ｇ／Ｍ／１排队系统的离去过程的有限维分布弱收敛到泊松过程的有限维分布的简单条件．     

M/G/1 queue with event-dependent arrival rates

Motivated by experiments on customers’ behavior in service systems, we consider a queueing model with event-dependent arrival rates. Customers’ arrival rates depend on the last event, which may either be a service departure or an arrival. We derive explicitly the performance measures and analyze the impact of the event-dependency. In particular, we show that this queueing model, in which a service completion generates a higher arrival rate than an arrival, performs better than a system in which customers are insensitive to the last event. Moreover, contrary to the M/G/1 queue, we show that the coefficient of variation of the service does not necessarily deteriorate the system performance. Next, we show that this queueing model may be the result of customers’ strategic behavior when only the last event is known. Finally, we investigate the historical admission control problem. We show that, under certain conditions, a deterministic policy with two thresholds may be optimal. This new policy is easy to implement and provides an improvement compared to the classical one-threshold policy.     

Traffic processes in a class of finite Markovian queues

This paper exposes the stochastic structure of traffic processes in a class of finite state queueing systems which are modeled in continuous time as Markov processes. The theory is presented for theM/E _k /φ/L class under a wide range of queue disciplines. Particular traffic processes of interest include the arrival, input, output, departure and overflow processes. Several examples are given which demonstrate that the theory unifies many earlier works, as well as providing some new results. Several extensions to the model are discussed.     

Departure Process of the MAP/SM/1 Queue

We study the departure process of a single server queue with Markovian arrival input and Markov renewal service time. We derive the joint transform of departure time and the number of departures and, based on this transform, we establish several expressions for burstiness (variance) and correlation (covariance sequence) of the departure process. These expressions reveal that burstiness and correlation of the arrival process have very little impact on the departure process when a queueing system is heavily loaded. In contrast, both burstiness and correlation of the service-time process greatly affect those of the departure process regardless of the load of the system. Finally, we show that, even when an arrival process is short-range dependent, the departure process could has long-range dependence if a service-time process is long-range dependent.     

Functional limit theorems for a new class of non-stationary shot noise processes

We study a class of non-stationary shot noise processes which have a general arrival process of noises with non-stationary arrival rate and a general shot shape function. Given the arrival times, the shot noises are conditionally independent and each shot noise has a general (multivariate) cumulative distribution function (c.d.f.) depending on its arrival time. We prove a functional weak law of large numbers and a functional central limit theorem for this new class of non-stationary shot noise processes in an asymptotic regime with a high intensity of shot noises, under some mild regularity conditions on the shot shape function and the conditional (multivariate) c.d.f. We discuss the applications to a simple multiplicative model (which includes a class of non-stationary compound processes and applies to insurance risk theory and physics) and the queueing and work-input processes in an associated non-stationary infinite-server queueing system. To prove the weak convergence, we show new maximal inequalities and a new criterion of existence of a stochastic process in the space $\mathbb{D}$ given its consistent finite dimensional distributions, which involve a finite set function with the superadditive property.     

Networks of infinite-server queues with nonstationary Poisson input

In this paper we focus on networks of infinite-server queues with nonhomogeneous Poisson arrival processes. We start by introducing a more general Poisson-arrival-location model (PALM) in which arrivals move independently through a general state space according to a location stochastic process after arriving according to a nonhomogeneous Poisson process. The usual open network of infinite-server queues, which is also known as a linear population process or a linear stochastic compartmental model, arises in the special case of a finite state space. The mathematical foundation is a Poisson-random-measure representation, which can be obtained by stochastic integration. It implies a time-dependent product-form result: For appropriate initial conditions, the queue lengths (numbers of customers in disjoint subsets of the state space) at any time are independent Poisson random variables. Even though there is no dependence among the queue lengths at each time, there is important dependence among the queue lengths at different times. We show that the joint distribution is multivariate Poisson, and calculate the covariances. A unified framework for constructing stochastic processes of interest is provided by stochastically integrating various functionals of the location process with respect to the Poisson arrival process. We use this approach to study the flows in the queueing network; e.g., we show that the aggregate arrival and departure processes at a given queue (to and from other queues as well as outside the network) are generalized Poisson processes (without necessarily having a rate or unit jumps) if and only if no customer can visit that queue more than once. We also characterize the aggregate arrival and departure processes when customers can visit the queues more frequently. In addition to obtaining structural results, we use the stochastic integrals to obtain explicit expressions for time-dependent means and covariances. We do this in two ways. First, we decompose the entire network into a superposition of independent networks with fixed deterministic routes. Second, we make Markov assumptions, initially for the evolution of the routes and finally for the entire location process. For Markov routing among the queues, the aggregate arrival rates are obtained as the solution to a system of input equations, which have a unique solution under appropriate qualifications, but not in general. Linear ordinary differential equations characterize the time-dependent means and covariances in the totally Markovian case.     

Branching/queueing networks: Their introduction and near-decomposability asymptotics

A new class of models, which combines closed queueing networks with branching processes, is introduced. The motivation comes from MIMD computers and other service systems in which the arrival of new work is always triggered by the completion of former work, and the amount of arriving work is variable. In the variant of branching/queueing networks studied here, a customer branches into a random and state-independent number of offspring upon completing its service. The process regenerates whenever the population becomes extinct. Implications for less rudimentary variants are discussed. The ergodicity of the network and several other aspects are related to the expected total number of progeny of an associated multitype Galton-Watson process. We give a formula for that expected number of progeny. The objects of main interest are the stationary state distribution and the throughputs. Closed-form solutions are available for the multi-server single-node model, and for homogeneous networks of infinite-servers. Generally, branching/queueing networks do not seem to have a product-form state distribution. We propose a conditional product-form approximation, and show that it is approached as a limit by branching/queueing networks with a slowly varying population size. The proof demonstrates an application of the nearly complete decomposability paradigm to an infinite state space. This revised version was published online in June 2006 with corrections to the Cover Date.     

具有可变到达率的多重休假Geo~(λ_1,λ_2)/G/1排队分析

本文考虑顾客到达与服务员休假相关的多重休假离散时间排队系统,用更新过程及u-变换分析了系统的队长性质.分别得到系统在三种时点(n~-,n~+,n)处的队长分布的递推解,进而揭示了在不同到达率条件下系统队长分布不再具有随机分解特性,得到了系统在四种时点(n~-,n~+,n,离去时点D_n)处稳态队长分布的重要关系(不同于连续时间排队系统).     

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.     

Poincaré Inequality on the Path Space of Poisson Point Processes

Quasi-invariance is proved for the distributions of Poisson point processes under a random shift map on the path space. This leads to a natural Dirichlet form of jump type on the path space. Differently from the O–U Dirichlet form on the Wiener space satisfying the log-Sobolev inequality, this Dirichlet form merely satisfies the Poincaré inequality but not the log-Sobolev one.     

The Periodic BMAP/PH/c Queue

In queueing theory, most models are based on time-homogeneous arrival processes and service time distributions. However, in communication networks arrival rates and/or the service capacity usually vary periodically in time. In order to reflect this property accurately, one needs to examine periodic rather than homogeneous queues. In the present paper, the periodic BMAP/PH/c queue is analyzed. This queue has a periodic BMAP arrival process, which is defined in this paper, and phase-type service time distributions. As a Markovian queue, it can be analysed like an (inhomogeneous) Markov jump process. The transient distribution is derived by solving the Kolmogorov forward equations. Furthermore, a stability condition in terms of arrival and service rates is proven and for the case of stability, the asymptotic distribution is given explicitly. This turns out to be a periodic family of probability distributions. It is sketched how to analyze the periodic BMAP/M _t /c queue with periodically varying service rates by the same method.     

Queue-length balance equations in multiclass multiserver queues and their generalizations

A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: The distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for multidimensional queue-length processes for a quite general network of multiclass multiserver queues. We formally derive those balance equations under a general framework. They are called distributional relationships and are obtained for any external arrival process and state-dependent routing as long as certain stationarity conditions are satisfied and external arrivals and service completions do not simultaneously occur. We demonstrate the use of these balance equations, in combination with PASTA, by (1) providing very simple derivations of some known results for polling systems and (2) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a nonstationary framework.     

On the equivalence of three types of blocking in non-Markovian tandem queues

Definitions of blocking types for tandem queueing systems are given where each stage consists of multiple servers. Some of them are shown to be identical with respect to service initiation epochs at each stage. Such identical relationships are proved using sample path correspondence, and are therefore valid irrespective of the arrival process and the service time distributions.     

An Application of Little's Result to the G/G/1 (LCFS/P) Queue

This paper considers a quite general single-server queueing system, under a last-come-first-served queue discipline, with pre-emption and arbitrary restarting policy. Expressions are given for the queue-size limiting distribution when the system is considered at arrival (or departure) epochs and in continuous time, by using very simple arguments.     

Convexity of single stage queueing systems with bulk arrival

In this paper we study the convexity of the waiting time, workload and the number of jobs in single stage queueing systems with respect to the bulk size of the arrival process. In particular we show that the number of jobs in a single server queueing systemG ^[x ]/GI/1 and in a multiple server queueing systemG ^[x]/M/c with bulk sizesx=(x ₁ ,x ₂ ,x ₃ ,...) is componentwise convex inx. This is in the sense of the sample path convexity introduced in Shaked and Shanthikumar [11]. These results have applications in the stochastic comparison of bulk arrival queueing systems.Research supported in part by NSF grant DDM-9113008.     

A Markov Renewal Based Model for Wireless Networks

We propose a queueing network model which can be used for the integration of the mobility and teletraffic aspects that are characteristic of wireless networks. In the general case, the model is an open network of infinite server queues where customers arrive according to a non-homogeneous Poisson process. The movement of a customer in the network is described by a Markov renewal process. Moreover, customers have attributes, such as a teletraffic state, that are driven by continuous time Markov chains and, therefore, change as they move through the network. We investigate the transient and limit number of customers in disjoint sets of nodes and attributes. These turn out to be independent Poisson random variables. We also calculate the covariances of the number of customers in two sets of nodes and attributes at different time epochs. Moreover, we conclude that the arrival process per attribute to a node is the sum of independent Poisson cluster processes and derive its univariate probability generating function. In addition, the arrival process to an outside node of the network is a non-homogeneous Poisson process. We illustrate the applications of the queueing network model and the results derived in a particular wireless network.