Queueing Behavior Research Articles

Spectral decomposition approach for transient analysis of multi-server discrete-time queues

Discrete time queueing systems are extensively used in modeling the ATM environment at the cell level. This paper examines the transient behavior of a discrete time FCFS multiple server queue. For the arrival process we assume a general discrete time Markovian batch arrival process, where the batch size distribution of the arrivals in successive slots is governed by a finite state discrete time Markov chain. We show that once the spectral decomposition of the “probability generating matrix” of the arrival process is obtained, the complete solution in the transform domain may be given. Using the complex analysis technique and Cauchy's integral formula, we present an efficient numerical method for the numerical calculation of a few performance measures of interest, namely transient probability of queue being empty, and the mean of the queue length distribution. We also generalize our numerical method in the situations where the superposition of a number of independent arrival sources are fed to the queue. It is shown that the numerical complexity of obtaining the transient solution in this case can be substantially reduced by using the approach based on the Kronecker product. The finite buffer case is also analyzed.

Multiplexing digital television sources on ATM networks

This paper investigates the transmission of digital television signals on ATM networks. The key characteristic of an ATM network is its ability of statistically multiplexing variable bit rate sources in queueing systems called multiplexers. These queues are optimally located at the output port of the switching fabrics and emptied at constant bit rate by the destined transmission links. In that framework, the paper intends to analyze the multiplexing of digital television sources which are connected at a given ATM node and share the same transmission resources. Consequently, the study starts with analyzing the packet traffic generated when pooling television sources. The paper proceeds with modeling the queue behaviors and, eventually, with deriving the multiplexing performances. These latter parameters refer to three main topics namely, the cell losses and their configurations, the transmission jitter and the statistical multiplexing. Both theoretical approaches and simulations are employed to yield the final performances. As an important issue, substantial multiplexing gains is also put into evidence through theoretical and experimental arguments. Cell loss configurations are presented to suggest what policy to implement for cell loss burst correction. It is further shown how random transmission delay and jitter turn out to be minor problems when dealing with the transmission of television services.

A Markov service policy with application to the queueing study of a DQDB station

Abstract In this paper, a queueing system is formulated, analyzed and applied for the study of the queueing behavior of a DQDB station. At first, an appropriate process is defined to capture the station's perception of the state of the distributed global queue (DQDB network) and a queueing model is formulated. In the sequel, a general queueing system is considered; the packet service time requirement is assumed to depend on the state of an underlying Markov chain at the time when the packet is forwarded to the head of the queue. Finally, the process which captures the state of the global distributed queue is approximated and the resulting queueing model for the DQDB station is related to the general queueing system and analyzed. Results for the queue occupancy distribution of the DQDB station are derived and compared with simulations.

Stability of event synchronisation in distributed discrete event simulation

This paper is concerned with the behaviour of message queues in distributed discrete event simula tors . We view a logical process in a distributed simulation as comprising a message sequencer with associated message queues, followed by an event processor. We show that, with standard stochastic assumptions for message arrival and time-stamp processes, the message queues are unstable for conservative sequencing, and for conservative sequencing with maximum lookahead and hence for optimistic resequencing, and for any resequencing algorithm that does not employ interprocessor “flow control”. These results point towards certain fundamental limits on the performance of distributed simulation of open queueing networks.

On the transient behavior of the processor sharing queue

We analyze the transient behavior of the single server queue under the processor sharing discipline. Under fairly general assumptions, we give the rate of growth of the number of customers in the queue as well as the asymptotic behavior of the residual service times described in terms of a renormalized point process.

Busy-period and blocking behavior of finite queues with state-dependent Markov renewal arrivals

The busy-period length distributions and blocking probabilities are considered for finiteG/G/1/K queues with state-dependent Markov renewal arrivals. The Laplace-Stieltjes transforms of the distributions and blocking probabilities are given for the non-preemptive and last-come-first-served preemptive resume (or repeat) service disciplines. For Erlangian (or deterministic) service times in particular, it is proved that the busy-period length (the number of blocked customers) for the non-preemptive discipline is smaller (larger) than for the preemptive resume discipline.

On pathwise behavior of multiserver queues

In this paper, by exploiting recent results on the pathwise behavior of the workload process in single server, work conserving queues of theG/G/1/∞ type, we show that the workload of multiserver, work conserving queues ofG/G/m/∞ (m<∞) (andG/G/∞) queues satisfies an o(t) growth condition, provided that the time average of the work brought into the system is less thanm form < ∞ (and finite form=∞).

Discrete time queues with phase dependent arrivals

The queueing behavior of many communication systems is well modeled by a queueing system in which time is slotted, and the number of entities that arrive during a slot is dependent upon the state of a discrete time, discrete state Markov chain. Techniques for analyzing such systems have appeared in the literature from time to time, but distributions have been presented in only rare instances In the paper, the authors present the probability generating function (PGF) for joint and marginal buffer occupancy distributions of statistical time division multiplexing systems in this class. They discuss inversion of the PGF using discrete Fourier transforms, and also discuss a simple technique for obtaining moments of the queue length distribution. Numerical results, including queue length distributions for some special cases, are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An asymptotically exact decomposition of coupled Brownian systems

Brownian flow systems, i.e. multidimensional Brownian motion with regulating barriers, can model queueing and inventory systems in which the behavior of different queues is correlated because of shared input processes. The behavior of such systems is typically difficult to describe exactly. We show how Brownian models of such systems, conditioned on one queue length exceeding a large value, decompose asymptotically into smaller subsystems. This conditioning induces a change in drift of the system's net input process and its components. The results here are analogous to results for jump-Markov queues recently obtained by Shwartz and Weiss. The Brownian setting leads to a simple description of the component processes' asymptotic behaviour, as well as to explicit distributional results.

An asymptotically exact decomposition of coupled Brownian systems

Brownian flow systems, i.e. multidimensional Brownian motion with regulating barriers, can model queueing and inventory systems in which the behavior of different queues is correlated because of shared input processes. The behavior of such systems is typically difficult to describe exactly. We show how Brownian models of such systems, conditioned on one queue length exceeding a large value, decompose asymptotically into smaller subsystems. This conditioning induces a change in drift of the system's net input process and its components. The results here are analogous to results for jump-Markov queues recently obtained by Shwartz and Weiss. The Brownian setting leads to a simple description of the component processes' asymptotic behaviour, as well as to explicit distributional results.

A framework for an object-oriented approach to modeling basic queues in decision support systems (A framework for queues in a DDS)

We show that the analysis of queueing systems for decision support systems can be facilitated by describing the natural structure of queues as objects. Moreover, we show that the behavior of queues can be used to dynamically generate those objects.

Head of the line processor sharing for many symmetric queues with finite capacity

In this paper the steady-state behavior of many symmetric queues, under the head of the line processor-sharing discipline, is investigated. The arrival process to each of n queues is Poisson, with rateλA, and each queue hasr waiting spaces. A job arriving at a full queue is lost. The queues are served by a single exponential server, which has a mean rateμn, and splits its capacity equally amongst the jobs at the head of each nonempty queue. The normal traffic casep=λ/μ< 1 is considered, and it is assumed thatn≫1 andr= 0(1). A 2-term asymptotic approximation to the loss probabilityL is derived, and it is found thatL = 0(n−r), for fixedp. If6=(1−p)/p≪ 1, then the approximation is valid if nδ2≫ 1 and (r+ 1)2≪nδ, and in this caseL ∼r!/(nδ)r. Numerical values ofL are obtained forr = 1,2,3,4 and 5,n = 1000,500 and 200, and various values ofp< 1. Very small loss probabilities may be obtained with appropriate values of these parameters.

Queueing study of a 3-priority policy with distinct service strategies

A discrete-time, single server, 3-queue system is presented and analyzed. A distinct service strategy, namely the consistent-gated (c-G), 1-limited (L) and head-of-line (HoL), is applied to each of the queues (c-G/L/HoL policy). It is shown that this queueing system provides for an accurate analytical model for a DQDB station, as well as a means for an approximate evaluation of the correlation associated with key traffic processes in that network. In addition, the developed queueing system could be useful for the modeling of the queueing behavior of an ATM link shared by high-priority, low priority and control traffic. Through an asymptotic analysis under heavy low-priority traffic, the worst case performance for the high priority traffic is determined. Furthermore, it is illustrated that the asymptotic analysis provides for a potentially tight delay bounding technique. Finally, the delay performance of the developed queueing system is compared to that of a similar system in which one of the queues receives 1-limited service and the other two exhaustive (HoL/sup -//L/HoL/sup +/ policy). >

Transient analysis of a switched Poisson arrival queue under overload control

This paper studies the transient behavior of a Switched Poisson Arrival Queue (SPAQ) under overload control. The queue has infinite buffer capacity with an exponential server. A switched Poisson process is used to represent an aggregated voice/data packet arrival process in integrated services networks. By overload control, we mean to properly adapt the arrival process once the buffer contents exceed a designated level. The probability distribution of queue length as a function of time is obtained. The temporal effect of the overload control is measured in two forms. While in overload, we measure the amount of time for the queue to fall into an underload period. While in underload, we measure the amount of time for the queue to rise to an overload period. a proper design of the control will not only reduce the fall time but also increase the rise time, to protect high priority and time constrained services. Our study provides important information to overload control design on packet switching networks. We also explore the transient queueing behavior as affected by time stochastic properties of the underlying two-state Markov chain for the arrival process.

Transient analysis of multi-server queues with Markov-modulated Poisson arrivals and overload control

This paper studies the transient behavior of a Markov-molulated Poisson arrival queue under overload control. The queue has finite or infinite buffer capacity with multiple exponential servers. A Markov-modulated Poisson process is used to represent an aggregated voice or video packet arrival process in integrated service networks. By overload control, we mean to properly adapt the arrival process once the buffer contents exceed a designated level. The probability distribution of queue length as a function of time is obtained. The temporal effect of the overload control is measured in two forms. While in overload, we measure the amount of time for the queue to fall into underload. While in underload, we measure the amount of time for the queue to rise to overload. A proper design of the control will not only reduce the fall time but also increase the rise time. We also explore the transient queueing behavior as affected by time stochastic properties of the underlying Markov chain for the arrival process.

On pathwise behavior of queues

In this paper we study the sample-path behavior of the workload process in a single server queue without imposing the stationarity hypothesis. In particular we study the stability or worst case growth properties and show that the growth rate is of the form o( t α, α > 1, depending on the rate of convergence of the pathwise average of the work brought into the system. We then study the existence of empirical means and in particular show that a pathwise version of the Pollaczek-Khinchine formula plays an important role. Finally we give conditions under which our results hold knowing the rate or intensity of the arrival process. Our results highlight the limitations of pure sample-path techniques.

An Approach to Analyzing the Behavior of Some Queueing Networks

Several researchers have reported difficulties in analyzing the behavior of single queues and networks of queues. This is so even in the case of closed product-form networks, for which an exact solution and efficient solution algorithms are known. The difficulty arises because the exact solution could not, by itself, be used for such analysis as proving properties of the network, relating performance measures to one another, and characterizing some interesting behavior. This paper proposes an approach to surmounting such difficulties. The idea is to analyze an approximate solution based on Schweitzer's approximation, and interpret the results as approximate relationships among the exact performance measures. This approach is applied to three problems concerning the interaction among job classes, the mean arrival and variance of queue length, and thrashing. The reliability of the approach is tested by applying it to an optimal routing problem, for which the exact solution is known. The results are illustrated with problems drawn from computer systems.

Transient behaviour of finite capacity tandem queues with blocking

ABSTRACT The transient output process from tandem qucueing systems with no wailing positions is analysed. Our system is subject to blocking; that is, a unit cannot move to the second queue until it completes service at the first queue and the second queue is not busy. Expressions arc derived for the expected output time of the nth unit when the system is initially empty and these expressions are used to derive the throughput of these systems in equilibrium. For the systems studied, placing a faster server ahead of a slower server maximizes the throughput.

A mathematical model for the behavior of pedestrians

The movement of pedestrians is supposed to show certain regularities which can be best described by an “algorithm” for individual behavior and is easily simulated on computers. This behavior is assumed to be determined by an intended velocity, by several attractive and repulsive effects, and by fluctuations. The movement of pedestrians is dependent on decisions, which have the purpose of optimizing their behavior and can be explicitly modeled. Some interesting applications of the model to real situations are given, especially to formation of groups, behavior in queues, avoidance of collisions, and selection processes among behavioral alternatives.

Queueing behaviour of two interconnected buffers of a packet network with application to the evaluation of packet routeing policies

AbstractIn this paper, the behaviour of two interconnected nodes of a packet communication network is analysed. The packet arrival process to the first node is described by an i.i.d. discrete time process. The packet arrival process to the second node consists of two mutually independent streams; one stream carries the packet output process from node 1; the other stream carries a packet traffic described by a dependent discrete time process. For this system, the following quantities are derived: (a) the first two moments of the buffer occupancy and the mean packet delay in the first node; (b) a description of the packet output process of the first node; (c) the first two moments of the buffer occupancy and the mean packet delay in the second node.In the sequel, the behaviour of the buffers is investigated under some simple routeing policies in node 1. The objective at this point is to evaluate the queueing behaviour of both buffers under some routeing policies and it is not to seek for the optimal packet routeing policy in the first node. It is shown how the routeing policy in node 1 can be incorporated in the description of the packet output process of node 1 and, then, the behaviour of buffer 2 be accurately evaluated.The numerical results show that the queueing behaviour of buffer 2 cannot be accurately evaluated by incorporating a Bernoulli approximation on the output process of node 1. Furthermore, it is shown that different routeing policies in the first node result in different intensities of the queueing problems in the second node, although the packet output rate from the first node is constant for all routeing policies considered. It turns out that a routeing policy which best randomizes the packet output process results in the least significant queueing problems in the second node. The latter implies that routeing decisions should be based not only on the desired packet traffic rates (which achieve optimality in the load distribution among the nodes), but also on the packet randomization affect of the routeing policy considered.