Poisson Arrival Process Research Articles

Computational schemes for two exponential servers where the first has a finite buffer

We consider a system consisting of two not necessarily identical exponential servers having a common Poisson arrival process. Upon arrival, customers inspect the flrst queue and join it if it is shorter than some threshold n. Otherwise, they join the second queue. This model was dealt with, among others, by Altman et al. (2004). We flrst derive an explicit expression for the Laplace-Stieltjes transform of the distribution underlying the arrival (renewal) process to the second queue. Second, we observe that given that the second server is busy, the two queue lengths are independent. Third, we develop two computational schemes for the stationary distribution of the two-dimensional Markov process underlying this model, one with a complexity of O(nlog‐ i1 ), the other with a complexity of O(lognlog 2 ‐ i1 ), where ‐ is the tolerance criterion.

Queueing analysis of data block synchronization mechanism in peer-to-peer based video streaming system

In a Peer-to-Peer (P2P) based video streaming system such as Coolstreaming, a single video stream is decomposed into multiple sub-streams. A client-peer node receives the sub-streams from multiple parent-peer nodes, combining them into the original video stream. Each client-peer node has a synchronization buffer and a cache buffer. Data blocks are stored in the synchronization buffer in a sub-stream basis, and then forwarded into the cache buffer according to their sequence numbers. In this buffering system, data-block synchronization plays a crucial role to guarantee video quality. In this paper, we consider the performance of data-block synchronization scheme with which data blocks are simultaneously forwarded just after all the data blocks composing a macro data block arrive at the synchronization buffer. We model the synchronization buffer as a multiple-buffer queueing system with homogeneous Poisson arrival processes, deriving the mean forwarding interval. We also consider the frame loss probability for multiple-path video streaming, investigating how the number of sub-streams decreases the frame loss probability. Numerical examples show that increasing the number of sub-streams makes the average forwarding interval large, while the frame loss probability at the bottleneck router is improved. It is also shown that increasing the synchronization buffer decreases the average forwarding interval.

Stochastic ship roll motion via path integral method

The response of ship roll oscillation under random ice impulsive loads modeled by Poisson arrival process is very important in studying the safety of ships navigation in cold regions. Under both external and parametric random excitations the evolution of the probability density function of roll motion is evaluated using the path integral (PI) approach. The PI method relies on the Chapman-Kolmogorov equation, which governs the response transition probability density functions at two close intervals of time. Once the response probability density function at an early close time is specified, its value at later close time can be evaluated. The PI method is first demonstrated via simple dynamical models and then applied for ship roll dynamics under random impulsive white noise excitation.

A polling model with smart customers

In this paper we consider a single-server, cyclic polling system with switch-over times. A distinguishing feature of the model is that the rates of the Poisson arrival processes at the various queues depend on the server location. For this model we study the joint queue length distribution at polling epochs and at server's departure epochs. We also study the marginal queue length distribution at arrival epochs, as well as at arbitrary epochs (which is not the same in general, since we cannot use the PASTA property). A generalised version of the distributional form of Little's law is applied to the joint queue length distribution at customer's departure epochs in order to find the waiting time distribution for each customer type. We also provide an alternative, more efficient way to determine the mean queue lengths and mean waiting times, using Mean Value Analysis. Furthermore, we show that under certain conditions a Pseudo-Conservation Law for the total amount of work in the system holds. Finally, typical features of the model under consideration are demonstrated in several numerical examples.

Stochastic ship roll motion via path integral method

The response of ship roll oscillation under random ice impulsive loads modeled by Poisson arrival process is very important in studying the safety of ships navigation in cold regions. Under both external and parametric random excitations the evolution of the probability density function of roll motion is evaluated using the path integral (PI) approach. The PI method relies on the Chapman-Kolmogorov equation, which governs the response transition probability density functions at two close intervals of time. Once the response probability density function at an early close time is specified, its value at later close time can be evaluated. The PI method is first demonstrated via simple dynamical models and then applied for ship roll dynamics under random impulsive white noise excitation.

ON THE PROBABILITY DISTRIBUTION OF JOIN QUEUE LENGTH IN A FORK-JOIN MODEL

In this article, we consider the two-node fork-join model with a Poisson arrival process and exponential service times of heterogeneous service rates. Using a mapping from the queue lengths in the parallel nodes to the join queue length, we first derive the probability distribution function of the join queue length in terms of joint probabilities in the parallel nodes and then study the exact tail asymptotics of the join queue length distribution. Although the asymptotics of the joint distribution of the queue lengths in the parallel nodes have three types of characterizations, our results show that the asymptotics of the join queue length distribution are characterized by two scenarios: (1) an exact geometric decay and (2) a geometric decay with the prefactor n−1/2.

Approximations of lead-time distributions in an assemble-to-order system under a base-stock policy

In this paper, we consider an assemble-to-order production system in which a product with multiple components is produced under a base-stock policy. The demand follows a Poisson arrival process, and the production time at each facility is exponentially distributed.Our main results are closed-form formulae for approximating the lead-time distributions in equilibrium. The approximation formulae for the lead-time distributions can be obtained with a linear combination of bounds, in which certain constants are obtained from auxiliary properties and simulation experiments. To determine the quality of the approximations, we obtained sample lead-time distributions from many simulations. Kolmogorov–Smirnov tests show that our formulae are very good fits of the sample distributions. The approximations are good over the entire range of traffic intensities and are even more accurate in light or heavy traffic.

Admission control and pricing in a queue with batch arrivals

We investigate a problem of admission control in a queue with batch arrivals. We consider a single server with exponential service times and a compound Poisson arrival process. Each arriving batch computes its expected benefit and decides whether or not to enter the system. The controller’s problem is to set state dependent prices for arriving batches. Once prices have been set we formulate the admission control problem, derive properties of the value function, and obtain the optimal admission policy.

The Erlang multirate loss model with Batched Poisson arrival processes under the bandwidth reservation policy

The Erlang Multirate Loss Model (EMLM) has been widely used as a springboard in the study of multirate loss systems. In this paper, we study an extension of the EMLM in order to ensure QoS guarantee per service-class in the heterogeneous environment of telecom networks. The proposed model is named Batched Poisson EMLM under Bandwidth Reservation (BR) policy (BP-EMLM/BR), since its input process is a Batched Poisson arrival process; that is, calls of each service-class arrive to a fixed-capacity link as batches. The distribution of the batch size can be general. The batch blocking discipline is the partial batch blocking, i.e. depending on the available link bandwidth, a part of an arriving batch can be accepted while the rest of it is discarded. The Call Admission Control is based on the BR policy, whereby we can achieve our goal to guarantee specific QoS for each service-class at call-level. In order to apply the BR policy in the BP-EMLM/BR, we examine two methods also used in the EMLM: (a) the Roberts’ method and (b) the Stasiak–Glabowski (S&G) method. The proposed model does not have a product form solution and therefore we propose approximate but recursive formulas for the calculation of various performance measures, such as time and call congestion probabilities as well as link utilization. A comparison between the analytical and simulation results shows that both methods provide satisfactory results when equalization between time congestion probabilities of all service-classes is desired. The S&G method, however, performs much better than the Roberts’ method in the case where more than two service-classes are accommodated in the link.

Spectrum Sensing for Cognitive Radios with Transmission Statistics: Considering Linear Frequency Sweeping

The spectrum sensing performance of Cognitive Radios (CRs) considering noisy signal measurements and the time domain transmission statistics of the Primary User (PU) is considered in this paper. When the spectrum is linearly swept in the frequency domain continuously to detect the presence of the PU the time-domain statistics of the PU plays an important role in the detection performance. This is true especially when the PU's bandwidth is much smaller than the CR's scanning frequency range. We model the transmission statistics that is the temporal characteristics of the PU as a Poisson arrival process with a random occupancy time. The spectrum sensing performance at the CR node is then theoretically analyzed based on noisy envelope detection together with the time domain spectral occupancy statistics. The miss detection and false alarm probabilities are derived from the considered spectral occupancy model and the noise model, and we present simulation results to verify our theoretical analysis. We also study the minimum required sensing time for the wideband CR to reliably detect the narrowband PU with a given confidence level considering its temporal characteristics.

Uncertainty and Predictability of Urban Link Travel Time

Travel times that vehicles experience in urban road networks are intrinsically uncertain because of the stochastic character of delays at signalized intersections. The ability to capture delay characteristics at signalized intersections is critical for estimating and predicting travel times on urban links. Much research has been done to predict travel time on urban links on the basis of traffic state information (e.g., volumes or speeds). However, the results have not been promising. One important reason is that delays experienced by vehicles on urban links are uncertain because of both traffic conditions and traffic control at intersections. This paper addresses the causes of travel time uncertainty. A probabilistic delay distribution model with stochastic arrivals and departures is proposed to investigate delay uncertainty in both undersaturated and oversaturated conditions. The delay distributions with the Poisson arrival process and with the binomial arrival process are compared. Results show that different arrival patterns have little influence on the delay distribution in undersaturated conditions, although they have significant influence on the delay distribution in oversaturated conditions. The delay distributions under different degrees of saturation are overlapping, which indicates that it is difficult to determine the traffic state for a given single-valued delay and vice versa. The “width” of the delay distribution based on percentiles is used to quantify the delay uncertainty at signalized intersections. The dynamics of delay uncertainty and of delay uncertainty under different degrees of saturation are investigated.

Transforming Renewal Processes for Simulation of Nonstationary Arrival Processes

Simulation models of real-life systems often assume stationary (homogeneous) Poisson arrivals. Therefore, when nonstationary arrival processes are required, it is natural to assume Poisson arrivals with a time-varying arrival rate. For many systems, however, this provides an inaccurate representation of the arrival process that is either more or less variable than Poisson. In this paper we extend techniques that transform a stationary Poisson arrival process into a nonstationary Poisson arrival process (NSPP) by transforming a stationary renewal process into a nonstationary, non-Poisson (NSNP) arrival process. We show that the desired arrival rate is achieved and that when the renewal base process is either more or less variable than Poisson, then the NSNP process is also more or less variable, respectively, than an NSPP. We also propose techniques for specifying the renewal base process when presented properties of, or data from, an arrival process and illustrate them by modeling real arrival data.

Sequential Testing to Guarantee the Necessary Sample Size in Clinical Trials

This article addresses the sequential sampling issues related to attainment of a given number of subjects in a Poisson arrival process. The problem of guaranteeing the necessary sample size is formulated as that of open-ended hypothesis testing. To implement the procedure a one-sided stopping boundary must be specified. The formulas for the probabilities of general stopping time distribution are derived. For the linear boundary they are in the class of Lagrangian Poisson distributions. A locally optimal test is obtained.

A Continuous-Time Approach to Robbins' Problem of Minimizing the Expected Rank

Let X1, X2, …, Xn be independent random variables uniformly distributed on [0,1]. We observe these sequentially and have to stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation? What is the value of the expected rank (as a function of n) and what is the limit of this value when n goes to ∞? This full-information expected selected-rank problem is known as Robbins' problem of minimizing the expected rank, and its general solution is unknown. In this paper we provide an alternative approach to Robbins' problem. Our model is similar to that of Gnedin (2007). For this, we consider a continuous-time version of the problem in which the observations follow a Poisson arrival process on ℝ+ × [0,1] of homogeneous rate 1. Translating the previous optimal selection problem in this setting, we prove that, under reasonable assumptions, the corresponding value function w(t) is bounded and Lipschitz continuous. Our main result is that the limiting value of the Poisson embedded problem exists and is equal to that of Robbins' problem. We prove that w(t) is differentiable and also derive a differential equation for this function. Although we have not succeeded in using this equation to improve on bounds on the optimal limiting value, we argue that it has this potential.

A Continuous-Time Approach to Robbins' Problem of Minimizing the Expected Rank

Let X 1, X 2, …, X n be independent random variables uniformly distributed on [0,1]. We observe these sequentially and have to stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation? What is the value of the expected rank (as a function of n) and what is the limit of this value when n goes to ∞? This full-information expected selected-rank problem is known as Robbins' problem of minimizing the expected rank, and its general solution is unknown. In this paper we provide an alternative approach to Robbins' problem. Our model is similar to that of Gnedin (2007). For this, we consider a continuous-time version of the problem in which the observations follow a Poisson arrival process on ℝ+ × [0,1] of homogeneous rate 1. Translating the previous optimal selection problem in this setting, we prove that, under reasonable assumptions, the corresponding value function w (t) is bounded and Lipschitz continuous. Our main result is that the limiting value of the Poisson embedded problem exists and is equal to that of Robbins' problem. We prove that w (t) is differentiable and also derive a differential equation for this function. Although we have not succeeded in using this equation to improve on bounds on the optimal limiting value, we argue that it has this potential.

Optimal Buffer Allocation in Tandem Queues with Communication Blocking

In this letter, we consider an m-node tandem queue (queues in series) with a Poisson arrival process and either deterministic or non-overlapping service times. With the assumption that each node has a finite buffer except for the first node, we show the non-increasing convex property of stationary waiting time with respect to the finite buffer capacities. We apply it to an optimization problem which determines the smallest buffer capacities subject to probabilistic constraints on stationary waiting times.

Improvement and Evaluation of Scenario-Type Hypothesis Object Tracking

Event Abstract Back to Event Improvement and Evaluation of Scenario-Type Hypothesis Object Tracking Masakazu Murata1*, Yoshiaki Taniguchi2, Go Hasegawa2 and Hirotaka Nakano2 1 Osaka University, Graduate School of Information Science and Technology, Japan 2 Osaka University, Cyberscience Center, Japan Object tracking is one of the major applications of sensor networks. We have proposed an object tracking method called Scenario-type Hypothesis Tracking (SHT) [1]. In our method, a monitoring region is divided into multiple micro cells, and sensors are placed at borders among micro cells (hereinafter referred to as gates), as shown in Fig. 1(a). A sensor is assumed to be able to detect objects and their movement directions. For example, the sensor is realized by putting pair of binary sensors. Sensor information are gathered to a tracking server through wireless or wired networks. The tracking server maintains virtual micro cells and virtual objects which indicate a variety of hypotheses of object trajectory estimation. A virtual object is generated, moved, and removed in virtual micro cells appropriately according to sensor information. By using SHT, it is possible to handle a variety of scenarios including behavior of objects and structure of micro cells, for example, slowdown of objects’ velocities when objects cross each other in narrow passageway, and extension of travel distance of objects in a micro cell due to obstacles. However, the number of virtual objects exponentially increases against parameters, structure of micro cells, and arrival rate of objects [1]. In addition, we should take into account loss of sensor information due to packet loss, failure of sensing, and functional limitation of sensors. In this article, we propose two improvement methods for SHT. In the first improved method, when predicted sensor information is not obtained at the tracking server, it generates virtual objects for all possibilities of the object’s trajectories to handle loss of sensor information. In the second improved method, new virtual objects are generated from selected virtual objects which have high likelihood, to decrease the number of generated virtual objects while keeping object tracking accuracy. Hereinafter, due to space limitation, we only show one of preliminary simulation results. In the simulation, a monitoring region is composed of 2x2 square micro cells where each micro cell has four gates for each edge. Objects arrive to randomly selected gate following Poisson arrival process, pass through the monitoring region by randomly selecting gates, and depart from the monitoring region. Among intra-micro cells, sensor information is dropped with some probability. Fig. 1(b) shows the tracking accuracy against information loss probability, where the tracking accuracy is the ratio of the number of successful trajectory estimation to the number of objects. Fig. 1(c) shows the virtual objects generation ratio, which means that the ratio of the number of generated virtual objects to the number of arrival objects. In these figures, improved SHT-1 means SHT with the first improved method, and improved SHT-2 means SHT with both improved methods. As shown in Fig. 1(b), it is shown that our improved methods achieve higher tracking accuracy in comparison with SHT regardless of information loss probability. In addition, Fig. 1(c) shows that the number of generated virtual objects in improved SHT-2 is reduced in comparison with improved SHT-1.This work was partly supported by the KAKENHI (2170075, 1804950) of MEXT in Japan. fig 1

LONG RANGE DEPENDENCE, UNBALANCED HAAR WAVELET TRANSFORMATION AND CHANGES IN LOCAL MEAN LEVEL

Long range dependent (LRD) stationary time series have historically served to model real time series with apparent changes in local mean level. A natural tool to study changes in local mean level is the unbalanced Haar wavelet transformation (UHT). In this work, UHT is used to study changes in local mean level in LRD models and several real and simulated time series exhibiting LRD. In particular, simulations for LRD models suggest that changes in local mean level occur at times essentially governed by a homogeneous Poisson arrival process, and only the local mean levels themselves inherit the LRD property of the original time series. These properties are compared with the analogous ones in several real and simulated time series. The results are mixed though generally in favor of LRD models. The approach based on UHT is also compared to several alternatives such as defining changes in local mean level through kernel smoothing. The interest throughout is mainly in very long time series such as those collected in the studies of data traffic over Internet.

FEC recovery performance for video streaming services over wired-wireless networks

This paper considers the packet recovery performance of forward error correction (FEC) for video streaming services over wired-wireless networks. Focusing on a wireless base station, we model it as a single-server queueing system with a Markovian service process in which the state of the server alternates between Good and Bad states. The system has two independent input processes: one is a general renewal input process and the other is a Poisson arrival process. We analyze the packet- and block-level loss probabilities to investigate the recovery performance of FEC at block level. The analysis is validated with simulation experiments driven by real traffic traces. Numerical examples show that the block-loss probability is greatly affected by the system capacity and the mean Bad-state period, and that the recovery performance of FEC deteriorates according to the fluctuation in the packet transmission rate at a wireless base station.

Fluid analysis of an input control problem

A two-station network with controllable inputs and sequencing control, proposed by Wein (Oper. Res. 38:1065---1078, 1990), is analyzed. A control is sought to minimize holding cost subject to a throughput constraint. In a Lagrangian formulation, input vanishes in the fluid limit. Several alternative fluid models, including workload formulations, are analyzed to develop a heuristic policy for the stochastic network. Both the fluid heuristic and Wein's diffusion solution are compared with the optimal policy by solving the dynamic program. Examples with up to six customer classes, using Poisson arrival and service processes, are presented. The fluid heuristic does well at sequencing control but the diffusion gives additional, and better, information on input control. The fluid analysis, in particular whether the fluid priorities are greedy, aids in determining whether the fluid heuristic contains useful information.