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Journal articles on the topic 'Interarrival time'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 February 2022

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1

Korolev, A., and P. R. Field. "Assessment of performance of the inter-arrival time algorithm to identify ice shattering artifacts in cloud particle probes measurements." Atmospheric Measurement Techniques Discussions 7, no. 10 (October 8, 2014): 10249–92. http://dx.doi.org/10.5194/amtd-7-10249-2014.

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Abstract. Shattering presents a serious obstacle to current airborne in-situ methods of characterizing the microphysical properties of ice clouds. Small shattered fragments result from the impact of natural ice crystals with the forward parts of aircraft-mounted measurement probes. The presence of these shattered fragments may result in a significant overestimation of the measured concentration of small ice crystals, contaminating the measurement of the ice particle size distribution (PSD). One method of identifying shattered particles is to use an interarrival time algorithm. This method is based on the assumption that shattered fragments form spatial clusters that have short interarrival times between particles, relative to natural particles, when they pass through the sample volume of the probe. The interarrival time algorithm is a successful technique for the classification of shattering artifacts and natural particles. This study assesses the limitations and efficiency of the interarrival time algorithm. The analysis has been performed using simultaneous measurements of 2-D optical array probes with the standard and antishattering "K-tips" collected during the Airborne Icing Instrumentation Experiment (AIIE). It is shown that the efficiency of the algorithm depends on ice particle size, concentration and habit. Additional numerical simulations indicate that the effectiveness of the interarrival time algorithm to eliminate shattering artifacts can be significantly restricted in some cases. Improvements to the interarrival time algorithm are discussed.

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2

Liu, Yi, Xiaoxia Luo, Jingxian Liu, Zongzhi Li, and Ryan Wen Liu. "Mixed Models of Single-Berth Interarrival Time Distributions." Journal of Waterway, Port, Coastal, and Ocean Engineering 144, no. 1 (January 2018): 04017034. http://dx.doi.org/10.1061/(asce)ww.1943-5460.0000421.

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3

Righter, Rhonda. "A note on losses in M/GI/1/n queues." Journal of Applied Probability 36, no. 04 (December 1999): 1240–43. http://dx.doi.org/10.1017/s0021900200018015.

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Let L n be the number of losses during a busy period of an M/GI/1/n queueing system. We develop a coupling between L n and L n+1 and use the resulting relationship to provide a simple proof that when the mean service time equals the mean interarrival time, EL n = 1 for all n. We also show that L n is increasing in the convex sense when the mean service time equals the mean interarrival time, and it is increasing in the increasing convex sense when the mean service time is less than the mean interarrival time.

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4

Righter, Rhonda. "A note on losses in M/GI/1/n queues." Journal of Applied Probability 36, no. 4 (December 1999): 1240–43. http://dx.doi.org/10.1239/jap/1032374770.

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Let Ln be the number of losses during a busy period of an M/GI/1/n queueing system. We develop a coupling between Ln and Ln+1 and use the resulting relationship to provide a simple proof that when the mean service time equals the mean interarrival time, ELn = 1 for all n. We also show that Ln is increasing in the convex sense when the mean service time equals the mean interarrival time, and it is increasing in the increasing convex sense when the mean service time is less than the mean interarrival time.

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5

ZHAO, RUIQING, and BAODING LIU. "RENEWAL PROCESS WITH FUZZY INTERARRIVAL TIMES AND REWARDS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11, no. 05 (October 2003): 573–86. http://dx.doi.org/10.1142/s0218488503002338.

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This paper considers a renewal process in which the interarrival times and rewards are characterized as fuzzy variables. A fuzzy elementary renewal theorem shows that the expected number of renewals per unit time is just the expected reciprocal of the interarrival time. Furthermore, the expected reward per unit time is provided by a fuzzy renewal reward theorem. Finally, a numerical example is presented for illustrating the theorems introduced in the paper.

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6

Xie, Qiang, Luping Xu, and Hua Zhang. "X-ray pulsar signal detection using photon interarrival time." Journal of Systems Engineering and Electronics 24, no. 6 (December 2013): 899–905. http://dx.doi.org/10.1109/jsee.2013.00104.

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7

Jain, Sudha. "Estimating The Change Point Of Erlang Interarrival Time Distribution." INFOR: Information Systems and Operational Research 39, no. 2 (May 2001): 200–207. http://dx.doi.org/10.1080/03155986.2001.11732436.

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8

Field, P. R., A. J. Heymsfield, and A. Bansemer. "Shattering and Particle Interarrival Times Measured by Optical Array Probes in Ice Clouds." Journal of Atmospheric and Oceanic Technology 23, no. 10 (October 1, 2006): 1357–71. http://dx.doi.org/10.1175/jtech1922.1.

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Abstract Optical array probes are one of the most important tools for determining the microphysical structure of clouds. It has been known for some time that the shattering of ice crystals on the housing of these probes can lead to incorrect measurements of particle size distributions and subsequently derived microphysical properties if the resulting spurious particles are not rejected. In this paper it is shown that the interarrival times of particles measured by these probes can be bimodal—the “cloud” probes are more affected than the “precipitation” probes. The long interarrival time mode represents real cloud structure while the short interarrival time mode results from fragments of shattered ice particles. It is demonstrated for the flights considered here that if the fragmented particles are filtered using an interarrival time threshold of 2 × 10−4 s in three of the four cases and 1 × 10−5 s in the other, then the measured total concentration can be affected by up to a factor of 4 in situations where large particles are present as determined by the mass-weighted mean size exceeding 1 mm, or the exponential slope parameter falling below 30 cm−1. When the size distribution is narrow (mass weighted mean size <1 mm), ice water contents can be overestimated by 20%–30% for the cases presented here.

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9

Morin, Samantha, Myron Hlynka, and Shan Xu. "Comments on a Two Queue Network." International Journal of Statistics and Probability 6, no. 6 (September 20, 2017): 43. http://dx.doi.org/10.5539/ijsp.v6n6p43.

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A special customer must complete service from two servers, each with an $M/M/1$ queueing system. It is assumed that the two queueing systems have initiial numbers of customers $a$ and $b$ at the instant when the special customer arrives, and subsequent interarrival times and service times are independent. We find the expected total time (ETT) for the special customer to complete service. We show that even if the interarrival and service time parameters of two queues are identical, there exist examples (specific values of the parameters and initial lengths a and b) for which the special customer surprisingly has a lower expected total time to completion by joining the longer queue first rather than the shorter one.

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10

R, Elangovan. "Stochastic Model to Determine the Expected Time to Recruitment with Three Sources of Depletion of Manpower under Correlated Interarrival Times." Bonfring International Journal of Industrial Engineering and Management Science 4, no. 1 (February 28, 2014): 34–42. http://dx.doi.org/10.9756/bijiems.4802.

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11

Özaktaş, Hakan, Nureddin Kırkavak, and Ayşe Nilay Alpay. "A Paradox of the Average Waiting Time for the Case of a Single Bottleneck on the Commuters’ Route." Modelling and Simulation in Engineering 2021 (May 23, 2021): 1–9. http://dx.doi.org/10.1155/2021/2315987.

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Average waiting time is considered as one of the basic performance indicators for a bottleneck zone on a route for commuter traffic. It turns out that the average waiting time in a queue remains paradoxically unchanged regardless of how fast the queue dissolves for a single bottleneck problem. In this study, the paradox is verified theoretically for the deterministic case with constant arrival and departure rates. Consistent results with the deterministic case have also been obtained by simulation runs for which vehicle interarrival time is a random variable. Results are tabulated for interarrival times which have uniform, triangular, normal, and exponential distributions along with a statistical verification of the average waiting time paradox.

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12

Cidon, Israel, Roch Guréin, Asad Khamisy, and Moshe Sidi. "On Queues with Interarrival Times Proportional to Service Times." Probability in the Engineering and Informational Sciences 10, no. 1 (January 1996): 87–107. http://dx.doi.org/10.1017/s0269964800004198.

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We analyze a family of queueing systems where the interarrival time In+1 between customers n and n + 1 depends on the service time Bn of customer n. Specifically, we consider cases where the dependency between In+1 and Bn is a proportionality relation and Bn is an exponentially distributed random variable. Such dependencies arise in the context of packet-switched networks that use rate policing functions to regulate the amount of data that can arrive to a link within any given time interval. These controls result in significant dependencies between the amount of work brought in by customers/packets and the time between successive customers. The models developed in the paper and the associated solutions are, however, of independent interest and are potentially applicable to other environments.Several scenarios that consist of adding an independent random variable to the interarrival time, allowing the proportionality to be random and the combination of the two are considered. In all cases, we provide expressions for the Laplace-Stieltjes Transform of the waiting time of a customer in the system. Numerical results are provided and compared to those of an equivalent system without dependencies.

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13

Chaudhry, M. L., and U. C. Gupta. "Performance analysis of the discrete-time GI/Geom/1/N queue." Journal of Applied Probability 33, no. 1 (March 1996): 239–55. http://dx.doi.org/10.2307/3215281.

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This paper presents an analysis of the single-server discrete-time finite-buffer queue with general interarrival and geometric service time, GI/Geom/1/N. Using the supplementary variable technique, and considering the remaining interarrival time as a supplementary variable, two variations of this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both cases, steady-state distributions for outside observers as well as at random and prearrival epochs have been obtained. The waiting time analysis has also been carried out. Results for the Geom/G/1/N queue with LAS-DA have been obtained from the GI/Geom/1/N queue with EAS. We also give various performance measures. An algorithm for computing state probabilities is given in an appendix.

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14

Chaudhry, M. L., and U. C. Gupta. "Performance analysis of the discrete-time GI/Geom/1/N queue." Journal of Applied Probability 33, no. 01 (March 1996): 239–55. http://dx.doi.org/10.1017/s0021900200103894.

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This paper presents an analysis of the single-server discrete-time finite-buffer queue with general interarrival and geometric service time,GI/Geom/1/N. Using the supplementary variable technique, and considering the remaining interarrival time as a supplementary variable, two variations of this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both cases, steady-state distributions for outside observers as well as at random and prearrival epochs have been obtained. The waiting time analysis has also been carried out. Results for theGeom/G/1/Nqueue with LAS-DA have been obtained from theGI/Geom/1/Nqueue with EAS. We also give various performance measures. An algorithm for computing state probabilities is given in an appendix.

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15

Musson, R. M. W. "A Power-Law Function for Earthquake Interarrival Time and Magnitude." Bulletin of the Seismological Society of America 92, no. 5 (June 1, 2002): 1783–94. http://dx.doi.org/10.1785/0120000001.

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16

Li, Jinzhu, Qihe Tang, and Rong Wu. "Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model." Advances in Applied Probability 42, no. 04 (December 2010): 1126–46. http://dx.doi.org/10.1017/s0001867800004559.

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Consider a continuous-time renewal risk model with a constant force of interest. We assume that claim sizes and interarrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys a dependence structure described via the conditional tail probability of a claim size given the interarrival time before the claim. We focus on determining the impact of this dependence structure on the asymptotic tail probability of discounted aggregate claims. Assuming that the claim size distribution is subexponential, we derive an exact locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure. When the claim size distribution is extended regularly varying tailed, we show that this asymptotic formula is globally uniform.

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17

Li, Jinzhu, Qihe Tang, and Rong Wu. "Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model." Advances in Applied Probability 42, no. 4 (December 2010): 1126–46. http://dx.doi.org/10.1239/aap/1293113154.

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Consider a continuous-time renewal risk model with a constant force of interest. We assume that claim sizes and interarrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys a dependence structure described via the conditional tail probability of a claim size given the interarrival time before the claim. We focus on determining the impact of this dependence structure on the asymptotic tail probability of discounted aggregate claims. Assuming that the claim size distribution is subexponential, we derive an exact locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure. When the claim size distribution is extended regularly varying tailed, we show that this asymptotic formula is globally uniform.

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18

Mi, Jie. "AVERAGE NUMBER OF EVENTS AND AVERAGE REWARD." Probability in the Engineering and Informational Sciences 14, no. 4 (October 2000): 485–510. http://dx.doi.org/10.1017/s0269964800144079.

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In a renewal process, the interarrival times are independent and identically distributed, and in a renewal reward process, the pairs of reward and interarrival time are i.i.d. Many useful results hold for these processes. This paper relaxes the assumption of identical distributions while keeping the assumption of independence. This paper explores the properties of the mean average number of occurrence of events and the mean average reward on any finite time interval. The paper also discusses the limiting properties of these two quantities and extends many results from the renewal process and renewal reward process to the more general counting process and reward sequence.

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19

Stamoulis, George D., and John N. Tsitsiklis. "On the settling time of the congested GI/G/1 queue." Advances in Applied Probability 22, no. 04 (December 1990): 929–56. http://dx.doi.org/10.1017/s000186780002320x.

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We analyze a stable GI/G/1 queue that starts operating at time t = 0 with N 0 ≠ 0 customers. First, we analyze the time required for this queue to empty for the first time. Under the assumption that both the interarrival and the service time distributions are of the exponential type, we prove that , where λ and μ are the arrival and the service rates. Furthermore, assuming in addition that the interarrival time distribution is of the non-lattice type, we show that the settling time of the queue is essentially equal to N 0/(μ –λ); that is, we prove that where is the total variation distance between the distribution of the number of customers in the system at time t and its steady-state distribution. Finally, we show that there is a similarity between the queue we analyze and a simple fluid model.

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20

Stamoulis, George D., and John N. Tsitsiklis. "On the settling time of the congested GI/G/1 queue." Advances in Applied Probability 22, no. 4 (December 1990): 929–56. http://dx.doi.org/10.2307/1427569.

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We analyze a stable GI/G/1 queue that starts operating at time t = 0 with N0 ≠ 0 customers. First, we analyze the time required for this queue to empty for the first time. Under the assumption that both the interarrival and the service time distributions are of the exponential type, we prove that , where λ and μ are the arrival and the service rates. Furthermore, assuming in addition that the interarrival time distribution is of the non-lattice type, we show that the settling time of the queue is essentially equal to N0/(μ –λ); that is, we prove that where is the total variation distance between the distribution of the number of customers in the system at time t and its steady-state distribution. Finally, we show that there is a similarity between the queue we analyze and a simple fluid model.

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21

Rahman, Md Mostafizur, and Attahiru Sule Alfa. "Computational Procedures for a Class of GI/D/kSystems in Discrete Time." Journal of Probability and Statistics 2009 (2009): 1–18. http://dx.doi.org/10.1155/2009/716364.

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A class of discrete time GI/D/ksystems is considered for which the interarrival times have finite support and customers are served in first-in first-out (FIFO) order. The system is formulated as a single server queue with new general independent interarrival times and constant service duration by assuming cyclic assignment of customers to the identical servers. Then the queue length is set up as a quasi-birth-death (QBD) type Markov chain. It is shown that this transformed GI/D/1 system has special structures which make the computation of the matrixRsimple and efficient, thereby reducing the number of multiplications in each iteration significantly. As a result we were able to keep the computation time very low. Moreover, use of the resulting structural properties makes the computation of the distribution of queue length of the transformed system efficient. The computation of the distribution of waiting time is also shown to be simple by exploiting the special structures.

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22

Gong, Wei-Bo, and Jian-Qiang Hu. "The MacLaurin series for the GI/G/1 queue." Journal of Applied Probability 29, no. 1 (March 1992): 176–84. http://dx.doi.org/10.2307/3214801.

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We derive the MacLaurin series for the moments of the system time and the delay with respect to the parameters in the service time or interarrival time distributions in the GI/G/1 queue. The coefficients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The light traffic derivatives can be obtained from these series. For the M/G/1 queue, we are able to recover the formulas for the moments of the system time and the delay, including the Pollaczek–Khinchin mean-value formula.

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23

Gong, Wei-Bo, and Jian-Qiang Hu. "The MacLaurin series for the GI/G/1 queue." Journal of Applied Probability 29, no. 01 (March 1992): 176–84. http://dx.doi.org/10.1017/s0021900200106722.

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We derive the MacLaurin series for the moments of the system time and the delay with respect to the parameters in the service time or interarrival time distributions in the GI/G/1 queue. The coefficients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The light traffic derivatives can be obtained from these series. For the M/G/1 queue, we are able to recover the formulas for the moments of the system time and the delay, including the Pollaczek–Khinchin mean-value formula.

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24

Harel, Arie. "Convexity results for single-server queues and for multiserver queues with constant service times." Journal of Applied Probability 27, no. 2 (June 1990): 465–68. http://dx.doi.org/10.2307/3214668.

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We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates.These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

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25

Harel, Arie. "Convexity results for single-server queues and for multiserver queues with constant service times." Journal of Applied Probability 27, no. 02 (June 1990): 465–68. http://dx.doi.org/10.1017/s0021900200038948.

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We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates. These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

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26

Helbing, Dirk, Martin Treiber, and Arne Kesting. "Understanding interarrival and interdeparture time statistics from interactions in queuing systems." Physica A: Statistical Mechanics and its Applications 363, no. 1 (April 2006): 62–72. http://dx.doi.org/10.1016/j.physa.2006.01.048.

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27

Siew Khew, Koh, Chin Ching Herny, Tan Yi Fei, Pooi Ah Hin, Goh Yong Kheng, Lee Min Cherng, and Ng Tan Ching. "Repairable Queue with Non-exponential Interarrival Time and Variable Breakdown Rates." International Journal of Engineering & Technology 7, no. 2.15 (April 6, 2018): 76. http://dx.doi.org/10.14419/ijet.v7i2.15.11218.

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This paper considers a single server queue in which the service time is exponentially distributed and the service station may breakdown according to a Poisson process with the rates γ and γ' in busy period and idle period respectively. Repair will be performed immediately following a breakdown. The repair time is assumed to have an exponential distribution. Let g(t) and G(t) be the probability density function and the cumulative distribution function of the interarrival time respectively. When t tends to infinity, the rate of g(t)/[1 – G(t)] will tend to a constant. A set of equations will be derived for the probabilities of the queue length and the states of the arrival, repair and service processes when the queue is in a stationary state. By solving these equations, numerical results for the stationary queue length distribution can be obtained.

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28

Gao, Qingwu, Na Jin, and Juan Zheng. "Uniform Estimate of the Finite-Time Ruin Probability for All Times in a Generalized Compound Renewal Risk Model." Advances in Decision Sciences 2012 (October 25, 2012): 1–17. http://dx.doi.org/10.1155/2012/936525.

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We discuss the uniformly asymptotic estimate of the finite-time ruin probability for all times in a generalized compound renewal risk model, where the interarrival times of successive accidents and all the claim sizes caused by an accident are two sequences of random variables following a wide dependence structure. This wide dependence structure allows random variables to be either negatively dependent or positively dependent.

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Assaf, David, Yuliy Baryshnikov, and Wolfgang Stadje. "Optimal strategies in a risk selection investment model." Advances in Applied Probability 32, no. 02 (June 2000): 518–39. http://dx.doi.org/10.1017/s0001867800010065.

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We study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each of them the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital. Between the investments he spends money at some constant rate. The objective is to avoid bankruptcy as long as possible. For the case e>d we characterize a strategy maximizing the probability that ruin never occurs. It is proved that the optimal value function is a concave function of the initial capital and uniquely determined as the solution of a fixed point equation for some intricate operator. In general, two-point distributions suffice; furthermore, we show that the cautious strategy of always taking the deterministic amount e is optimal if the interarrival times are hyperexponential, and, in the case of bounded interarrival times, is optimal ‘from some point on’, i.e. whenever the current capital exceeds a certain threshold. In the case e = 0 we consider a class of natural objective functions for which the optimal strategies are non-stationary and can be explicitly determined.

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Assaf, David, Yuliy Baryshnikov, and Wolfgang Stadje. "Optimal strategies in a risk selection investment model." Advances in Applied Probability 32, no. 2 (June 2000): 518–39. http://dx.doi.org/10.1239/aap/1013540177.

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We study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each of them the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital. Between the investments he spends money at some constant rate. The objective is to avoid bankruptcy as long as possible. For the case e>d we characterize a strategy maximizing the probability that ruin never occurs. It is proved that the optimal value function is a concave function of the initial capital and uniquely determined as the solution of a fixed point equation for some intricate operator. In general, two-point distributions suffice; furthermore, we show that the cautious strategy of always taking the deterministic amount e is optimal if the interarrival times are hyperexponential, and, in the case of bounded interarrival times, is optimal ‘from some point on’, i.e. whenever the current capital exceeds a certain threshold. In the case e = 0 we consider a class of natural objective functions for which the optimal strategies are non-stationary and can be explicitly determined.

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31

Kim, Woo Sung, and Kyungsu Park. "Waiting time distribution in single-channel deterministic flow lines with discrete interarrival time distributions." European J. of Industrial Engineering 16, no. 2 (2022): 1. http://dx.doi.org/10.1504/ejie.2022.10039905.

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32

Hoang, Nguyen Huy, and Bao Quoc Ta. "Ruin Probabilities of Continuous-Time Risk Model with Dependent Claim Sizes and Interarrival Times." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 28, Supp01 (August 28, 2020): 69–80. http://dx.doi.org/10.1142/s0218488520400061.

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In this paper we investigate an insurance continuous-time risk model when the claim sizes and inter-arrival times are m-dependent random variables. We provide an upper exponential bound for the ruin probability.

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33

Kim, Bara, and Jeongsim Kim. "The waiting time distribution for a correlated queue with exponential interarrival and service times." Operations Research Letters 46, no. 2 (March 2018): 268–71. http://dx.doi.org/10.1016/j.orl.2018.02.001.

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34

Huisman, Tijs, and Richard J. Boucherie. "The sojourn time distribution in an infinite server resequencing queue with dependent interarrival and service times." Journal of Applied Probability 39, no. 03 (September 2002): 590–603. http://dx.doi.org/10.1017/s0021900200021823.

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We consider an infinite server resequencing queue, where arrivals are generated by jumps of a semi-Markov process and service times depend on the jumps of this process. The stationary distribution of the sojourn time, conditioned on the state of the semi-Markov process, is obtained both for the case of hyperexponential service times and for the case of a Markovian arrival process. For the general model, an accurate approximation is derived based on a discretisation of interarrival and service times.

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35

Huisman, Tijs, and Richard J. Boucherie. "The sojourn time distribution in an infinite server resequencing queue with dependent interarrival and service times." Journal of Applied Probability 39, no. 3 (September 2002): 590–603. http://dx.doi.org/10.1239/jap/1034082130.

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We consider an infinite server resequencing queue, where arrivals are generated by jumps of a semi-Markov process and service times depend on the jumps of this process. The stationary distribution of the sojourn time, conditioned on the state of the semi-Markov process, is obtained both for the case of hyperexponential service times and for the case of a Markovian arrival process. For the general model, an accurate approximation is derived based on a discretisation of interarrival and service times.

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36

Tin, Pyke. "A queueing system with Markov-dependent arrivals." Journal of Applied Probability 22, no. 3 (September 1985): 668–77. http://dx.doi.org/10.2307/3213869.

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This paper considers a single-server queueing system with Markov-dependent interarrival times, with special reference to the serial correlation coefficient of the arrival process. The queue size and waiting-time processes are investigated. Both transient and limiting results are given.

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37

Tin, Pyke. "A queueing system with Markov-dependent arrivals." Journal of Applied Probability 22, no. 03 (September 1985): 668–77. http://dx.doi.org/10.1017/s0021900200029417.

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This paper considers a single-server queueing system with Markov-dependent interarrival times, with special reference to the serial correlation coefficient of the arrival process. The queue size and waiting-time processes are investigated. Both transient and limiting results are given.

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38

Dvurečenskij, A., and G. A. Ososkov. "On a modified counter with prolonging dead time." Journal of Applied Probability 22, no. 3 (September 1985): 678–87. http://dx.doi.org/10.2307/3213870.

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Emitted particles arrive at the counter with prolonging dead time so that the interarrival times and the lengths of impulses in any dead time are independent but not necessarily identically distributed random variables, and whenever the counter is idle then the following evolution starts from the beginning. For this class of counters we derive the probability laws of the numbers of particles arriving at the counters during their dead times, and the Laplace transform of the cycle, respectively.

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39

Dvurečenskij, A., and G. A. Ososkov. "On a modified counter with prolonging dead time." Journal of Applied Probability 22, no. 03 (September 1985): 678–87. http://dx.doi.org/10.1017/s0021900200029429.

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Emitted particles arrive at the counter with prolonging dead time so that the interarrival times and the lengths of impulses in any dead time are independent but not necessarily identically distributed random variables, and whenever the counter is idle then the following evolution starts from the beginning. For this class of counters we derive the probability laws of the numbers of particles arriving at the counters during their dead times, and the Laplace transform of the cycle, respectively.

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40

Wolff, Ronald W., and Chia-Li Wang. "Idle period approximations and bounds for the GI/G/1 queue." Advances in Applied Probability 35, no. 03 (September 2003): 773–92. http://dx.doi.org/10.1017/s0001867800012532.

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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.

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41

Wolff, Ronald W., and Chia-Li Wang. "Idle period approximations and bounds for the GI/G/1 queue." Advances in Applied Probability 35, no. 3 (September 2003): 773–92. http://dx.doi.org/10.1239/aap/1059486828.

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Abstract:

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.

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42

Di Crescenzo, Antonio, and Alessandra Meoli. "On a jump-telegraph process driven by an alternating fractional Poisson process." Journal of Applied Probability 55, no. 1 (March 2018): 94–111. http://dx.doi.org/10.1017/jpr.2018.8.

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AbstractThe basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

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43

Shubinskii, I. B., and N. P. Vasil'ev. "Active fault protection in computer systems where job processing time is comparable with interarrival time." Cybernetics and Systems Analysis 27, no. 4 (1992): 508–12. http://dx.doi.org/10.1007/bf01130359.

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44

Bertsimas, Dimitris J., and Garrett Van Ryzin. "Stochastic and dynamic vehicle routing with general demand and interarrival time distributions." Advances in Applied Probability 25, no. 04 (December 1993): 947–78. http://dx.doi.org/10.1017/s0001867800025842.

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We analyze a class of stochastic and dynamic vehicle routing problems in which demands arrive randomly over time and the objective is minimizing waiting time. In our previous work ([6], [7]), we analyzed this problem for the case of uniformly distributed demand locations and Poisson arrivals. In this paper, using quite different techniques, we are able to extend our results to the more realistic case where demand locations have an arbitrary continuous distribution and arrivals follow only a general renewal process. Further, we improve significantly the best known lower bounds for this class of problems and construct policies that are provably within a small constant factor relative to the optimal solution. We show that the leading behavior of the optimal system time has a particularly simple form that offers important structural insight into the behavior of the system. Moreover, by distinguishing two classes of policies our analysis shows an interesting dependence of the system performance on the demand distribution.

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45

Bertsimas, Dimitris J., and Garrett Van Ryzin. "Stochastic and dynamic vehicle routing with general demand and interarrival time distributions." Advances in Applied Probability 25, no. 4 (December 1993): 947–78. http://dx.doi.org/10.2307/1427801.

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We analyze a class of stochastic and dynamic vehicle routing problems in which demands arrive randomly over time and the objective is minimizing waiting time. In our previous work ([6], [7]), we analyzed this problem for the case of uniformly distributed demand locations and Poisson arrivals. In this paper, using quite different techniques, we are able to extend our results to the more realistic case where demand locations have an arbitrary continuous distribution and arrivals follow only a general renewal process. Further, we improve significantly the best known lower bounds for this class of problems and construct policies that are provably within a small constant factor relative to the optimal solution. We show that the leading behavior of the optimal system time has a particularly simple form that offers important structural insight into the behavior of the system. Moreover, by distinguishing two classes of policies our analysis shows an interesting dependence of the system performance on the demand distribution.

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46

Kempa, Wojciech M. "ON TIME-TO-BUFFER OVERFLOW DISTRIBUTION IN A SINGLE-MACHINE DISCRETE-TIME SYSTEM WITH FINITE CAPACITY." Mathematical Modelling and Analysis 25, no. 2 (March 18, 2020): 289–302. http://dx.doi.org/10.3846/mma.2020.10433.

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A model of a single-machine production system with finite magazine capacity is investigated. The input flow of jobs is organized according to geometric distribution of interarrival times, while processing times are assumed to be generally distributed. The closed-form formula for the generating function of the time to the first buffer overflow distribution conditioned by the initial buffer state is found. The analytical approach based on the idea of embedded Markov chain, the formula of total probability and linear algebra is applied. The corresponding result for next buffer overflows is also given. Numerical examples are attached as well.

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47

HUILLET, THIERRY. "ON THE WAITING TIME PARADOX AND RELATED TOPICS." Fractals 10, no. 02 (June 2002): 173–88. http://dx.doi.org/10.1142/s0218348x02001142.

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Consider a pure recurrent positive renewal process generated by some interarrival waiting time. The waiting time paradox reveals that, asymptotically, the time interval covering one's arrival in the file is statistically longer than the typical waiting time. Special properties are known to hold, were the waiting time to be infinitely divisible, two particular subclasses of interest being the exponential power mixtures' and the Lévy's ones. These models are revisited in some detail. Questions related to these problems are investigated and special examples of interest are underlined.

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48

Peköz, Erol A., Sheldon M. Ross, and Sridhar Seshadri. "How Nearly do Arriving Customers See Time-Average Behavior?" Journal of Applied Probability 45, no. 04 (December 2008): 963–71. http://dx.doi.org/10.1017/s0021900200004915.

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Customers arriving at a queue do not usually see its time-average behavior unless arrivals occur according to a Poisson process. In this article we study how nearly customers see time-average behavior. We give total variation error bounds for comparing the distance between the time- and customer-average distributions of a queueing system in terms of properties of the interarrival distribution. Some refinements are given for special cases and numerical computations are used to demonstrate the performance of the inequalities.

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49

Peköz, Erol A., Sheldon M. Ross, and Sridhar Seshadri. "How Nearly do Arriving Customers See Time-Average Behavior?" Journal of Applied Probability 45, no. 4 (December 2008): 963–71. http://dx.doi.org/10.1239/jap/1231340227.

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Customers arriving at a queue do not usually see its time-average behavior unless arrivals occur according to a Poisson process. In this article we study how nearly customers see time-average behavior. We give total variation error bounds for comparing the distance between the time- and customer-average distributions of a queueing system in terms of properties of the interarrival distribution. Some refinements are given for special cases and numerical computations are used to demonstrate the performance of the inequalities.

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50

Cao, Xi-Ren. "State aggregation and discrete-state Markov chains embedded in a class of point processes." Journal of Applied Probability 32, no. 1 (March 1995): 39–51. http://dx.doi.org/10.2307/3214919.

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One result that is of both theoretical and practical importance regarding point processes is the method of thinning. The basic idea of this method is that under some conditions, there exists an embedded Poisson process in any point process such that all its arrival points form a sub-sequence of the Poisson process. We extend this result by showing that on the embedded Poisson process of a uni- or multi-variable marked point process in which interarrival time distributions may depend on the marks, one can define a Markov chain with a discrete state that characterizes the stage of the interarrival times. This implies that one can construct embedded Markov chains with countable state spaces for the state processes of many practical systems that can be modeled by such point processes.

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