Interarrival Distributions Research Articles

Renewal convergence rates and correlation decay for homogeneous pinning models

A class of discrete renewal processes with exponentially decaying inter-arrival distributions coincides with the infinite volume limit of general homogeneous pinning models in their localized phase. Pinning models are statistical mechanics systems to which a lot of attention has been devoted both for their relevance for applications and because they are solvable models exhibiting a non-trivial phase transition. The spatial decay of correlations in these systems is directly mapped to the speed of convergence to equilibrium for the associated renewal processes. We show that close to criticality, under general assumptions, the correlation decay rate, or the renewal convergence rate, coincides with the inter-arrival decay rate. We also show that, in general, this is false away from criticality. Under a stronger assumption on the inter-arrival distribution we establish a local limit theorem, capturing thus the sharp asymptotic behavior of correlations.

Ruin Probabilities under Generalized Exponential Distribution

We consider a renewal risk model in which the claim inter-arrival distribution is generalized exponential (GE). We obtain the probability distribution for the ladder height distribution and use it to find the bounds for the ultimate ruin probabilities for individual claim amount distributions. The method suggested by Dufresne and Gerber (1989) is used for finding the bounds for ruin probabilities.

Ruin Probabilities under Generalized Exponential Distribution

We consider a renewal risk model in which the claim inter-arrival distribution is generalized exponential (GE). We obtain the probability distribution for the ladder height distribution and use it to find the bounds for the ultimate ruin probabilities for individual claim amount distributions. The method suggested by Dufresne and Gerber (1989) is used for finding the bounds for ruin probabilities.

A CASE STUDY OF INTER-ARRIVAL TIME DISTRIBUTIONS OF CONTAINER SHIPS

Given the recent prevalence of periodical container liners, the applicability of related models of widely applied queuing theory is left in doubt. Since they focus on marine liners, the distribution pattern is always considered to be a deterministic distribution for periodical schedule. In order to investigate the distribution patterns of container liners, observation units are segmented into single shipping liners, single berths, single container terminals and entire ports. The evolution of the ship arrival distribution pattern and the goodness-of-fit test for observation are based on the ship dynamic data of international harbors in Taiwan. This study aims to implement tests for public and dedicated container berths by comparing the variance in the patterns and characteristics of the ship arrival distributions. The ship inter-arrival distribution in the foregoing systems was found to follow the Erlang distribution; its distribution coefficient (k), changed by observation system, tends to decrease as the system's scale grows.

Probabilistic analysis of an algorithm to compute TCP packet round-trip time for intrusion detection

Estimating the length of a connection chain is challenging and critical in detecting stepping-stone intrusion. In this paper, we propose a novel method, called standard deviation-based clustering approach (SDBA), to estimate the length of an interactive connection chain by computing round-trip time (RTT). SDBA takes advantage of RTTs distribution and inter-arrival distribution of “send” packets. We prove that the probability of making a correct selection of RTT through SDBA is bounded by 1 − (1/ q 2), where q is a number related to standard deviation of RTTs distribution and send packets inter-arrival distribution. Experimental results showed that SDBA can compete against the best known algorithm in packet-matching rate and accuracy. This paper also presents the restrictions of SDBA.

Superposition of renewal processes and an application to multi-server queues

The aim of this paper is to compare the waiting times of customers in multiple-server queues, where the idle times are removed, with different numbers of servers. For this purpose we develop some results regarding the vector-valued marked point process whose points are arrival epochs of the superposition of renewal processes with different continuous inter-arrival distribution and the marks are the vectors of forward recurrence times of the various renewal processes at these arrival epochs.

Analysis and Computational Algorithm for Queues with State-Dependent Vacations I: G/M(n)/1/K

In this paper we study a queueing system with state-dependent services and state-dependent vacations, or simply G/M(n)/1/K. Since the service rate is state-dependent, this system includes G/M/c and G/M/c/K queues with various types of station vacations as special cases. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. The only input requirement is the Laplace-Stieltjes transform of the interarrival distribution as well as the state-dependent service rate and state-dependent vacation rate. In a subsequent companion paper, we study its dual system M(n)/G/1/K queue with state-dependent vacations.

The influence of demand variability on the performance of a make-to-stock queue

Variability, in general, has a deteriorating effect on the performance of stochastic inventory systems. In particular, previous results indicate that demand variability causes a performance degradation in terms of inventory related costs when production capacity is unlimited. In order to investigate the effects of demand variability in capacitated production settings, we analyze a make-to-stock queue with general demand arrival times operated according to a base-stock policy. We show that when demand inter-arrival distributions are ordered in a stochastic sense, increased arrival time variability indeed leads to an augmentation of optimal base-stock levels and to a corresponding increase in optimal inventory related costs. We quantify these effects through several numerical examples.

Heterogeneous bursty traffic dispersion over multiple server clusters

This letter presents performance analysis of multiple subnets, each representing a cluster of computing server nodes, that are introduced with non-uniformly distributed bursty packet arrivals. In particular, we study the case of a multi-state Markov-modulated arrival process, heterogeneously dispersed among designated queues. Cluster processing is modeled by employing a batch service discipline. The probability generating functions of the interarrival times distributions are utilized to derive closed-form expressions for each of the queue size distributions.

A cause of self-similarity in TCP traffic

We analyse the statistical properties of aggregated traffic flows generated by TCP, in order to clarify a possible cause of self-similarity in Internet traffic. Using ns-2 simulation, we first show that aggregated traffic flows by TCP can be characterized by phase transition phenomena between non-congested and congested phases in statistical physics. Interestingly, although the traffic exhibits self-similarity with the Hurst parameter H ≈ 1.0 at the critical point between them, it is close to Poissonian away from the critical point. This result is consistent with results from real WAN traffic measurement. The main contribution of our work is to show that TCP itself contains a mechanism for generating self-similarity, without assuming self-similarity or long-range dependency (LRD) in the application layer (e.g. packet inter-arrival, connection arrival, and file size distribution). Moreover, we find that the value of the Hurst parameter at the critical point is determined by the simple feedback control called stop-and-wait flow control. Namely, it appears even without any packet retransmission events and is independent of the explicit rate increment algorithm such as slow-start and congestion avoidance. Additionally, we demonstrate that the value of the Hurst parameter at the critical point depends on the constant bit rate algorithm and topology of network. Finally, we indicate that the time series of the round trip time follow the same statistics at the critical state. Copyright © 2005 John Wiley & Sons, Ltd.

Proper Framework for the Application of Power Management Algorithms

A power aware system can reduce its energy dissipation by dynamically powering off during idle periods and powering on again upon a new service request arrival. We minimize the dissipated energy, by selecting the optimal waiting interval before powering off, under consideration of the expected time of the next arrival. This approach has been already proposed in the past, using the idle times distribution, rather than the interarrival periods captured at the moment of service completion. Algorithms proposed in the literature utilize the history of idle periods or assume a vanishing service time. There has been no clear proposition on how service time affects the time instance of our power off decision; rather, whenever service time has been significant, a "blurred" image of the system's characteristic and a corresponding approximated optimal policy occurred. We clearly show analytically and experimentally that the idle times distribution should not be used as a primary design input, since it is the product of two separate inputs; the interarrival times and the service times. We give insight to our problem, using a mechanical equivalent established at the moment of service completion of all pending requests and show through analytical examples how service time affects our power-off decision. We explain the paradox of being advantageous to wait for intervals more than the shutdown threshold (which is a system characteristic) and show how the introduction of idle period lengths instead of interarrival periods "blurs" the input distribution, leading to non-optimal decisions. Our contribution is to define and solve the proper problem, solely relying on the interarrival distribution. Further, we examine the problem under the framework of competitive analysis. We show how the interarrival distribution that maximizes the competitive ratio, being an exponential distribution, intervenes with power management; it renders the optimization procedure worthless through its "memoryless property". Exponential interarrivals, irrespective of the service time pattern, are the marginal case where we cannot obtain energy gains. In all other cases the framework we promote ensures considerable advantages compared to other approaches in the literature. Moreover, it leads to a self contained module, implementable in software or hardware, which is based on an iterative formula and thus reduces power management calculations significantly. Here we exploit all operational features of the problem in proposing an implementation which spreads computations over the whole of the waiting period. We extensively compare our results numerically both against claimed expectations and against previous proposals. The outcome fully supports our framework as the one most appropriate for the application of power management.

On the joint distributions of surplus immediately before ruin and the deficit at ruin for Erlang(2) risk processes

In this note, we present some results for the joint distributions of the surplus immediately before and after ruin under a risk process in which the claim inter-arrival distribution is an Erlang(2) distribution.

Idle period approximations and bounds for the GI/G/1 queue

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

Idle period approximations and bounds for the GI/G/1 queue

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

Cluster processes: a natural language for network traffic

We introduce a new approach to the modeling of network traffic, consisting of a semi-experimental methodology combining models with data and a class of point processes (cluster models) to represent the process of packet arrivals in a physically meaningful way. Wavelets are used to examine second-order statistics, and particular attention is paid to the modeling of long-range dependence and to the question of scale invariance at small scales. We analyze in depth the properties of several large traces of packet data and determine unambiguously the influence of network variables such as arrival patterns, durations, and volumes of transport control protocol (TCP) flows and internal flow structure. We show that session-level modeling is not relevant at the packet level. Our findings naturally suggest the use of cluster models. We define a class where TCP flows are directly modeled, and each model parameter has a direct meaning in network terms, allowing the model to be used to predict traffic properties as networks and traffic evolve. The class has the key advantage of being mathematically tractable, in particular, its spectrum is known and can be readily calculated, its wavelet spectrum deduced, interarrival distributions can be obtained, and it can be simulated in a straightforward way. The model reproduces the main second-order features, and results are compared against a simple black box point process alternative. Discrepancies with the model are discussed and explained, and enhancements are outlined. The elephant and mice view of traffic flows is revisited in the light of our findings.

On the ergodic distribution of oscillating queueing systems

This paper examines a new class of queueing systems and proves a theorem on the existence of the ergodic distribution of the number of customers in such a system. An ergodic distribution is computed explicitly for the special case of a G/M−M/1 system, where the interarrival distribution does not change and both service distributions are exponential. A numerical example is also given.

On the time to ruin for Erlang(2) risk processes

In this paper, we consider a Sparre Andersen risk process for which the claim inter-arrival distribution is Erlang(2). Our purpose is to find expressions for moments of the time to ruin, given that ruin occurs. To do this, we define an auxiliary function φ along the lines of Gerber and Shiu [N. Am. Actu. J. 2 (1998) 48] and Gerber and Landry [Ins.: Math. Econ. 22 (1998) 263]. Our method of solution differs from that of Willmot and Lin [Ins.: Math. Econ. 25 (1999) 570; Ins.: Math. Econ. 27 (2000) 19] who consider this problem for the classical risk model, in that we first solve for the auxiliary function φ.

Evaluation of Reservation Mechanisms for Optical Burst Switching

Summary In this paper, we give an overview and classification of optical burst switching schemes and present burst reservation concepts. The performance of various basic reservation mechanisms proposed in literature is compared. Furthermore, a new analysis is introduced that allows to calculate the loss probabilities of a two-class system based on the reservation mechanism just-enough-time (JET) for arbitrary offsets. Finally, a variety of new results is presented including the dependence of burst loss probabilities on offset, burst length distribution, and interarrival distribution.

Algorithmic Discussions Of Distributions Of Numbers Of Busy Channels For GI/Geom/m/m Queues

The high speed multi-access communication channels such as BISDN and ATM operate on a discrete-time basis rather than on a continuous-time basis. The usual Erlang loss formula developed for continuous-time is not valid for discrete-time queues, though models such as these can be derived from the corresponding discrete-time models. In this paper, we consider the discrete-time GI/Geom/m/m queue. Two variants of this model viz. late-arrival system with delayed access and early arrival system have been discussed and analyzed. Distributions of numbers of busy channels at various epochs have been obtained using the supplementary variable technique. The performance measures such as probability of loss, and average number of busy channels have been presented for interarrival distributions: geometric, deterministic, arbitrary, etc. It is hoped that the results obtained in this paper may provide useful information to designers of telecommunication systems, practitioners, and others.

Stochastic optimal control under Poisson-distributed observations

Optimal control problems for linear, stochastic continuous-time systems are considered, in which the time domain is decomposed into a finite set of N disjoint random intervals of the form [t/sub i/, t/sub i+1/), in which a complete state observation is taken at each instant t/sub i/, 0/spl les/i/spl les/N-1. Two optimal control problems termed, respectively, the (piecewise) time-invariant control and time-variant control are considered in this framework. Concerning the observation point process, we first consider the general situation in which the increment intervals are i.i.d.r.v.s with unspecified probabilistic distributions. The (piecewise) time-invariant solution is thoroughly developed in this general case, and computations are illustrated using Erlang as the observations interarrival distribution. Next, the problem is specialized so increments are exponentially distributed, and the particular optimal control structure that results from this assumption is presented. Finally, and still under the Poisson assumption and for the time-variant case, we show that the control problem is closely related to linear quadratic Gaussian regulation with an exponentially discounted cost. The optimal control is made again of a sequence of piecewise open-loop controls corresponding, in this case, to linear feedback of the state predictor based on the most recent information on each interval. The feedback gains are time-varying matrices obtained from a sequence of algebraic Riccati equations, which are also computed off-line.