Interarrival Time Distributions Research Articles

Transient and Stationary Distributions for the GI / G / k Queue with Lebesgue-Dominated Inter-Arrival Time Distribution

In this paper, the multi-server queue with general service time distribution and Lebesgue-dominated iid inter-arival times is analyzed. This is done by introducing auxiliary variables for the remaining service times and then examining the embedded Markov chain at arrival instants. The concept of piecewise-deterministic Markov processes is applied to model the inter-arrival behaviour. It turns out that the transition probability kernel of the embedded Markov chain at arrival instants has the form of a lower Hessenberg matrix and hence admits an operator–geometric stationary distribution. Thus it is shown that matrix–analytical methods can be extended to provide a modeling tool even for the general multi-server queue.

A recursive method for the N policy G/M/1 queueing system with finite capacity

This paper studies a single removable server in a G/M/1 queueing system with finite capacity operating under the N policy. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distributions of the number of customers in the system. The method is illustrated analytically for exponential interarrival time distribution. Numerical results for various system performance measures are presented for four different interarrival time distributions such as exponential, 2-stage hyperexponential, 4-stage Erlang, and deterministic.

An analysis of system level power management algorithms and their effects on latency

The problem of power management for an embedded system is to reduce system level power dissipation by shutting off parts of the system when they are not being used and turning them back on when requests have to be serviced. Algorithms for this problem are online in nature; the algorithm must operate only with access to data that it has seen so far and without access to the complete data set or its characteristics. We present online algorithms to manage power for embedded systems and discuss their effects on system latency. We introduce competitive analysis as a formal framework for the evaluation of various power management algorithms. Competitive analysis does not depend on the distribution of interarrival times of requests. We present a nonadaptive online algorithm, analyze its behavior, and show that it is optimal. We also present a lower bound on the competitiveness of any adaptive algorithm. We show that no adaptive online algorithm can dissipate less than about 1.6 times the power dissipated by the optimal offline algorithm in the worst case. We also show that in order for any online algorithm to achieve this lower bound, it may have to maintain a complete history of the interarrival. times of the requests in the input sequence. Since this is not practical, we present a simple algorithm that uses only the last interarrival time to predict the arrival of the next request.

Token bucket characterization of long-range dependent traffic

The token bucket characterization provides a deterministic yet concise representation of a traffic source. In this paper, we study the impact of the long-range dependence (LRD) property of traffic generated by today's multimedia applications on the optimal dimensioning of token bucket parameters. To this aim, we empirically illustrate the difference between the token bucket characteristics of traffic exhibiting different degrees of time dependence but with identical macroscopic properties (i.e. inter-arrival time and packet size distributions). In addition, we use a statistical model to analytically determine optimal token bucket parameters under various optimization criteria. The statistical model is based on fractional Brownian motion and takes LRD into account. We apply this model to several aggregated MPEG video sources. We then assess the validity of these analytic results by comparing them to empirical results. We conclude that the analytic approach presented here is effective in optimally sizing token buckets for LRD traffic, and promises to be applicable under different traffic conditions and for various optimization criteria.

Distribution of number served during a busy period of GI/M/1/N queues-lattice path approach

Abstract This paper aims at transient analysis of finite queue GI/M/1/N by deriving the distribution of the number of customers served in a busy period. Lattice paths combinatorics is used, in which the process is split up at suitable renewable epochs and thus can be represented by a LP. The interarrival time distribution is approximated by two-phase Cox distribution, C2, that has Markovian property amenable to LP analysis. As distributions having rational Laplace–Stieltjes transforms and square coefficient of variation lying in [ 1 2 ,∞[ , forming a very wide class of distributions, can be approximated by C2, the use of C2 has yielded transient results applicable to almost any real life queueing system GI/M/1/N.

Lattice paths combinatorics applied to transient queue length distribution of C2/M/1 queues and busy period analysis of bulk queues C2b/M/1

Abstract This paper aims at deriving explicit transient queue length distribution for GI/M/1 system and busy period analysis of bulk queue GIb/M/1 through lattice paths (LPs) combinatorics. The general interarrival time distribution is approximated by two-phase Cox distribution, C2, that has Markovian property, enabling us to represent the processes by two-dimensional LPs. As distributions C2 cover a wide class of distributions that have rational Laplace–Stieltjes transforms (LSTs) with square coefficient of variation lying in [ 1 2 ,∞) , the results obtained are applicable to a large class of real life situations. Some numerical results for the C2b/M/1 model are also given.

Some case studies of skewed (and other ab-normal) data distributions arising in low-level environmental research.

Three general classes of skewed data distributions have been encountered in research on background radiation, chemical and radiochemical blanks, and low levels of 85Kr and 14C in the atmosphere and the cryosphere. The first class of skewed data can be considered to be theoretically, or fundamentally skewed. It is typified by the exponential distribution of inter-arrival times for nuclear counting events for a Poisson process. As part of a study of the nature of low-level (anti-coincidence) Geiger-Muller counter background radiation, tests were performed on the Poisson distribution of counts, the uniform distribution of arrival times, and the exponential distribution of inter-arrival times. The real laboratory system, of course, failed the (inter-arrival time) test--for very interesting reasons, linked to the physics of the measurement process. The second, computationally skewed, class relates to skewness induced by non-linear transformations. It is illustrated by non-linear concentration estimates from inverse calibration, and bivariate blank corrections for low-level 14C-12C aerosol data that led to highly asymmetric uncertainty intervals for the biomass carbon contribution to urban "soot". The third, environmentally, skewed, data class relates to a universal problem for the detection of excursions above blank or baseline levels: namely, the widespread occurrence of ab-normal distributions of environmental and laboratory blanks. This is illustrated by the search for fundamental factors that lurk behind skewed frequency distributions of sulfur laboratory blanks and 85Kr environmental baselines, and the application of robust statistical procedures for reliable detection decisions in the face of skewed isotopic carbon procedural blanks with few degrees of freedom.

Approximations and bounds for the variance of steady-state waiting times in a GI/G/1 queue

The focus of this paper is on developing bounds for the variance of steady-state waiting times in a GI/G/1 queue in terms of the first three moments of the interarrival and service time distributions. A review of the existing bounds and approximations is provided, and new bounds are derived. Numerical experience reveals that the bounds complement existing approximations in providing refinements that increase the accuracy of those approximations, especially at high traffic intensities.

Evaluating focused factory benefits with queuing theory

In this paper, we study the benefits of a focused factory using lead-time as a performance measure. Specifically, we model a production process, using multi-class (multiple product types), general interarrival and processing time distributions with multiple machines ( GI/G/c) queuing models for deriving each product’s mean lead-time. We also perform simulations for estimating the standard deviation of lead times. There are two product types: a standard and a customized product. The customized product has a more variable demand pattern than the standard product, and also requires additional processing time (setup and run time) in its production process. We assume that management is willing to sacrifice the lead-time performance of the customized product in favor of improved performance for the standard product. The paper shows that focusing the factory is more attractive for plants operating at higher utilization, and manufacturing products that have higher processing time and demand variability differentials between product types.

Option Pricing with a General Marked Point Process

This paper examines the impact of a random number of price changes on the options valuation. The model introduces the structure of the general marked point processes (MPP). This kind of model allows us to take account of more general distributions of interarrival times than usual jump-diffusion models. In particular, stock price variations can be correlated with the times of transactions. Thus, the investors can decide to trade according to the history of market values and the sizes of price variations can also depend on the past times of transactions. By using the special decomposition of predictable processes, with respect to a marked point process, the determination of all risk-neutral probabilities is detailed. Derivative prices are calculated in this context with different basic examples.

Internet-Type Queues with Power-Tailed Interarrival Times and Computational Methods for Their Analysis

Internet traffic flows have often been characterized as having power-tailed (long-tailed, fat-tailed, heavy-tailed) packet interarrival times or service requirements. In this work, we focus on the development of complete and computationally efficient steady-state solutions of queues with power-tailed interarrival times when the service times are assumed exponential. The classical method for obtaining the steady-state probabilities and delay-time distributions for the G/M/1 (G/M/c) queue requires solving a root-finding problem involving the Laplace-Stieltjes transform of the interarrival-time distribution function. Then the exponential service assumption is combined with the derived geometric arrival-point probabilities to get both the limiting general-time state and delay distributions. However, in situations where there is a power tail, the interarrival transform is typically quite complicated and never analytically tractable. In addition, it is possible that there is only a degenerate steady-state system-size probability distribution. Thus, an alternative approach to obtaining a steady-state solution is typically needed when power-tailed interarrivals are present. We exploit the exponentiality of the steady-state delay distributions for the G/M/1 and G/M/c queues, using level-crossings and a transform approximation method, to develop an alternative root-finding problem when there are power-tailed interarrival times. Extensive computational results are given.

Some related paradoxes of queuing theory: new cases and a unifying explanation

In this paper, we study a number of closely related paradoxes of queuing theory, each of which is based on the intuitive notion that the level of congestion in a queuing system should be directly related to the stochastic variability of the arrival process and the service times. In contrast to such an expectation, it has previously been shown that, in all Hk/G/1 queues, PW (the steady-state probability that a customer has to wait for service) decreases when the service-time becomes more variable. An analagous result has also been proved for ploss (the steady-state probability that a customer is lost) in all Hk/G/1 loss systems. Such theoretical results can be seen, in this paper, to be part of a much broader scheme of paradoxical behaviour which covers a wide range of queuing systems. The main aim of this paper is to provide a unifying explanation for these kinds of behaviour. Using an analysis based on a simple, approximate model, we show that, for an arbitrary set of nGI/Gk/1 loss systems (k=1,..., n), if the interarrival-time distribution is fixed and ‘does not differ too greatly’ from the exponential distribution, and if the n systems are ordered in terms of their ploss values, then the order that results whenever cA<1 is the exact reverse of the order that results whenever cA>1, where cA is the coefficient of variation of the interarrival time. An important part of the analysis is the insensitivity of the ploss value in an M/G/1 loss system to the choice of service-time distribution, for a given traffic intensity. The analysis is easily generalised to other queuing systems for which similar insensitivity results hold. Numerical results are presented for paradoxical behaviour of the following quantities in the steady state: ploss in the GI/G/1 loss system; PW and Wq (the expected queuing time of customers) in the GI/G/1 queue; and pK (the probability that all K machines are in the failed state) in the GI/G/r machine interference model. Two of these examples of paradoxical behaviour have not previously been reported in the literature. Additional cases are also discussed.

Higher order approximations for the single server queue with splitting, merging and feedback

The performance evaluation of many modern communication and manufacturing systems has been made possible by modeling them as queueing networks. Due to the limitations of exact algorithms to analyze general queueing networks, researchers have been focusing on developing approximations. Most of the approximations are lower order approximations in the sense that only a few parameters of the interarrival and service time distributions are used. This could lead to large errors in many conditions. In this paper, higher order approximations are developed for the single server queue with splitting, merging and feedback phenomena, which can then be integrated along with parametric decomposition to develop higher order approximations for queueing networks. Not only are the higher order approximations more accurate, but they can also be used to estimate the higher moments of the performance measures.

GX/GY /1 double ended queue: diffusion approximation

A double ended queueing system in which either customers which arrive in a group of random size or idle servers which may also serve in batches is studied. The arriving customers form a queue for service as well as idle servers (taxis) queue up for customers (passengers). Four possible ways to discretize continuous probability distribution of number of customers in the system have been discussed. The expressions for probability distribution of number of customers in the system, mean number of customers and mean number of servers in the system have been obtained in terms of means and variances of interarrival time distributions and group size distributions of customers and taxis.

A Correction Method for Spray Intensity Measurements Obtained via Phase Doppler Interferometry

A phase Doppler interferometer (PDI) was used to measure the intensity (i.e., the expected number of droplets measured per unit time) of a swirling methanol spray flame. Gaps observed in the data, which have been previously reported (McDonell and Samuelsen 1995), correspond to periods where the PDI was inactive. The intensity of the spray as a function of position within the flame was determined from the distribution of interarrival times (time between subsequent droplets entering the probe volume of the PDI), and a statistical simulation was developed to correct for the regularly occurring intervals in which the PDI does not accept data. The duration of this dead time was determined to be ca 5.2 ms for all points in the flame for which data were collected. The intensity of the spray varied by over two orders of magnitude throughout the flame. A statistical model is presented to correct for the observed behavior of the PDI, and the good agreement between the simulations and the experimental data indicate ...

Random Walk with a Heavy-Tailed Jump Distribution

The classical random walk of which the one-step displacement variable u has a first finite negative moment is considered. The R.W. possesses an unique stationary distribution; x is a random variable with this distribution. It is assumed that the right-hand and/or the left-hand tail of the distribution of u are heavy-tailed. For the type of heavy-tailed distribution considered in this study a contraction factor Δ(a) exists with Δ(a) ↓ 0 for a ↑ 1, and a↑1 is equivalent with E{u} ↑ 0. It is shown that Δ(a)x converges in distribution for a↑1. It is the analysis of the tail of this limiting distribution of Δ(a)x which is the main purpose of the present study in particular when u is a mix of stochastic variables ui, ie1,…,N, each ui having its own heavy tail characteristics for its right- and left-hand tails. For an important case it is shown that for the tail of the distribution of Δ(a)x an asymptotic expression in the variables Δ(a) and t for Δ(a) ↓ 0 and t→∞ can be derived. For this asymptotic relation the dominating term is completely determined by the heavier tail of the 2N tails of the ui; the other terms of the asymptotic relation show the influence of less heavy tails and, depending on t, the terms may have a contribution which is not always negligible. The study starts with the derivation of a functional equation for the L.S.-transform of the distribution of x and that of the excess distribution of the stationary idle time distribution. For several important cases this functional equation could be solved and thus has led to the above mentioned asymptotic result. The derivation of it required quite some preparation, because it needed an effective description of the heavy-tailed jump sequence u. It was obtained by prescribing the heavy-tailed distributions of [u]+emax (0,u) and [u]−emin (0,u). It is shown that the random walk can serve as a model for the actual waiting process of a GI/G/1 queueing model; in that case the distribution of u is that of the difference of the service time and the interarrival time. The analysis of the present study then describes the heavy-traffic theory for the case with heavy-tailed service- and/or interarrival time distribution.

Matrix product-form solutions of stationary probabilities in tandem queues

Fujimoto et al., have proved that the tail of the joint queue length distribution in a two-stage tandem queueing system has the geometric decay property. We continue to investigate the properties of stationary distributions in this tandem queueing system. Under the same conditions proposed by them, it is further shown that the stationary probability of the saturated states in the PH/PH/c1 A' /PH/c2 queue has a linear combination of product-forms. The method of linear combination of product-forms is presented in a QBD process with a countable number of phases in each level. We show that each component of these products can be expressed in terms of roots of the associated characteristic polynomials which involve only the Laplace-Stieltjes transforms of the interarrival and service time distributions.

Derivation of the N-step interdeparture time distribution in GI/ G/1 queueing systems

The departure process of a queueing system has been studied since the 1960s. Due to its inherent complexity, closed form solutions for the distribution of the departure process are nearly intractable. In this paper, we derive a closed form expression for the distribution of interdeparture time in a GI/ G/1 queueing model. Without loss of generality, we consider an embedded Markov chain in a general K M / G/1 queueing system, in which the interarrival time distribution is Coxian and service time distribution is general. Closed form solutions of the equilibrium distribution are derived for this model and the Laplace–Stieltjes transform (LST) of the distribution of interdeparture times is presented. An algorithmic computing procedure is given and numerical examples are provided to illustrate the results. With the analysis presented, we provide a novel analytic tool for studying the departure process in a general queueing model.

Point processes with finite-dimensional conditional probabilities

We study the structure of point processes N with the property that the P (θ t N∈· | F t ) vary in a finite-dimensional space where θ t is the shift and F t the σ -field generated by the counting process up to time t . This class of point processes is strictly larger than Neuts’ class of Markovian arrival processes. On the one hand, it allows for more general features like interarrival distributions which are matrix-exponential rather than phase type, on the other the probabilistic interpretation is a priori less clear. Nevertheless, the properties are very similar. In particular, finite-dimensional distributions of interarrival times, moments, Laplace transforms, Palm distributions, etc., are shown to be given by two fundamental matrices C , D just as for the Markovian arrival process. We also give a probabilistic interpretation in terms of a piecewise deterministic Markov process on a compact convex subset of R p , whose jump times are identical to the epochs of N .

Effects of Multiple Node Crossing on ATM Traffic Performance

AbstractIt is commonly recognized that traffic statistics change when a stream of ATM cells crosses several network nodes. In particular, some cells may experience a shorter delay than the previous ones, and thus the peak rate of the traffic stream may increase. In order to control this “cell clumping” effect, on one side spacing techniques have been proposed to restore the original peak rate of the sources at the outlets of each network node, while on the other side bounds to keep into account the peak rate increase have been suggested. Although some contributes have evaluated the effects of multiple node crossing on the distribution of the cell interarrival time, little attention has been paid to the effect of these modifications in terms of cell loss probability. Scope of this paper is to show, by adopting a realistic simulation model, that multiple node crossing produces a negligible effect on cell loss probability, when the admitted traffic is suitably regulated by a connection admission control rule. Moreover, a notable result is that, in the case of large buffers at the network nodes and traffic loads well over the limits imposed by the CAC rule, the per node cell loss probability eventually reduces as network nodes are crossed.