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Journal articles on the topic 'Distributions à queues épaisses'

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Published: 4 June 2021

Last updated: 16 February 2022

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1

Taylor, Nicholas B., and Benjamin G. Heydecker. "Estimating probability distributions of dynamic queues." Transportation Planning and Technology 38, no. 1 (November 20, 2014): 3–27. http://dx.doi.org/10.1080/03081060.2014.976987.

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2

Wang, P. Patrick, and Vicky F. Locker. "Steady-State Distributions Of Parallel Queues." INFOR: Information Systems and Operational Research 39, no. 1 (February 2001): 89–106. http://dx.doi.org/10.1080/03155986.2001.11732428.

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3

Karpelevitch, F. I., and A. Ya Kreinin. "Joint distributions in Poissonian tandem queues." Queueing Systems 12, no. 3-4 (September 1992): 273–86. http://dx.doi.org/10.1007/bf01158803.

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4

Szczotka, Władysław. "Stationary representation of queues. I." Advances in Applied Probability 18, no. 3 (September 1986): 815–48. http://dx.doi.org/10.2307/1427189.

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The paper deals with the asymptotic behaviour of queues for which the generic sequence is not necessarily stationary but is asymptotically stationary in some sense. The latter property is defined by an appropriate type of convergence of probability distributions of the sequences to the distribution of a stationary sequence We consider six types of convergence of to The main result is as follows: if the sequence of the distributions converges in one of six ways then the sequence of distributions of the sequences converges in the same way, independently of initial conditions. Furthermore the limiting distribution is the same as the limiting distribution obtained by the weak convergence of the distributions Here wk and w∗k denote the waiting time of the kth unit in the queue generated by (v, u) and (v0, u0) respectively.

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5

Szczotka, Władysław. "Stationary representation of queues. I." Advances in Applied Probability 18, no. 03 (September 1986): 815–48. http://dx.doi.org/10.1017/s0001867800016086.

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The paper deals with the asymptotic behaviour of queues for which the generic sequence is not necessarily stationary but is asymptotically stationary in some sense. The latter property is defined by an appropriate type of convergence of probability distributions of the sequences to the distribution of a stationary sequence We consider six types of convergence of to The main result is as follows: if the sequence of the distributions converges in one of six ways then the sequence of distributions of the sequences converges in the same way, independently of initial conditions. Furthermore the limiting distribution is the same as the limiting distribution obtained by the weak convergence of the distributions Here wk and w∗ k denote the waiting time of the kth unit in the queue generated by ( v, u ) and ( v 0, u 0) respectively.

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6

Tarasov, V. N. "Analysis of queues with hyperexponential arrival distributions." Problems of Information Transmission 52, no. 1 (January 2016): 14–23. http://dx.doi.org/10.1134/s0032946016010038.

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7

Chaudhry, Mohan L., Indra, and Vijay Rajan. "Analytically Simple and Computationally Efficient Solution to Geo/G/1 and Geo/G/1/N Queues Involving Heavy-tailed Distributions for Service Times." Calcutta Statistical Association Bulletin 70, no. 1 (May 2018): 74–85. http://dx.doi.org/10.1177/0008068318770566.

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The previous studies pertaining to the queues Geo/G/1 and Geo/G/1/N involve light-tailed distributions for service time. However, due to applications of heavy-tailed distributions in computer science and financial engineering, these distributions are used for service time. This article provides a simple and computationally efficient solution to the queues Geo/G/1 and Geo/G/1/N involving heavy-tailed distributions for service times.

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8

Hunter, Jeffrey J. "Filtering of Markov renewal queues, IV: Flow processes in feedback queues." Advances in Applied Probability 17, no. 02 (June 1985): 386–407. http://dx.doi.org/10.1017/s0001867800015032.

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This paper is a continuation of the study of a class of queueing systems where the queue-length process embedded at basic transition points, which consist of ‘arrivals’, ‘departures’ and ‘feedbacks’, is a Markov renewal process (MRP). The filtering procedure of Çinlar (1969) was used in [12] to show that the queue length process embedded separately at ‘arrivals’, ‘departures’, ‘feedbacks’, ‘inputs’ (arrivals and feedbacks), ‘outputs’ (departures and feedbacks) and ‘external’ transitions (arrivals and departures) are also MRP. In this paper expressions for the elements of each Markov renewal kernel are derived, and thence expressions for the distribution of the times between transitions, under stationary conditions, are found for each of the above flow processes. In particular, it is shown that the inter-event distributions for the arrival process and the departure process are the same, with an equivalent result holding for inputs and outputs. Further, expressions for the stationary joint distributions of successive intervals between events in each flow process are derived and interconnections, using the concept of reversed Markov renewal processes, are explored. Conditions under which any of the flow processes are renewal processes or, more particularly, Poisson processes are also investigated. Special cases including, in particular, the M/M/1/N and M/M/1 model with instantaneous Bernoulli feedback, are examined.

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9

Hunter, Jeffrey J. "Filtering of Markov renewal queues, IV: Flow processes in feedback queues." Advances in Applied Probability 17, no. 2 (June 1985): 386–407. http://dx.doi.org/10.2307/1427147.

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This paper is a continuation of the study of a class of queueing systems where the queue-length process embedded at basic transition points, which consist of ‘arrivals’, ‘departures’ and ‘feedbacks’, is a Markov renewal process (MRP). The filtering procedure of Çinlar (1969) was used in [12] to show that the queue length process embedded separately at ‘arrivals’, ‘departures’, ‘feedbacks’, ‘inputs’ (arrivals and feedbacks), ‘outputs’ (departures and feedbacks) and ‘external’ transitions (arrivals and departures) are also MRP. In this paper expressions for the elements of each Markov renewal kernel are derived, and thence expressions for the distribution of the times between transitions, under stationary conditions, are found for each of the above flow processes. In particular, it is shown that the inter-event distributions for the arrival process and the departure process are the same, with an equivalent result holding for inputs and outputs. Further, expressions for the stationary joint distributions of successive intervals between events in each flow process are derived and interconnections, using the concept of reversed Markov renewal processes, are explored. Conditions under which any of the flow processes are renewal processes or, more particularly, Poisson processes are also investigated. Special cases including, in particular, the M/M/1/N and M/M/1 model with instantaneous Bernoulli feedback, are examined.

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10

Boxma, O. J., and V. Dumas. "Fluid queues with long-tailed activity period distributions." Computer Communications 21, no. 17 (November 1998): 1509–29. http://dx.doi.org/10.1016/s0140-3664(98)00219-9.

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11

Takács, Lajos. "Limit distributions for queues and random rooted trees." Journal of Applied Mathematics and Stochastic Analysis 6, no. 3 (January 1, 1993): 189–216. http://dx.doi.org/10.1155/s1048953393000176.

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In this paper several limit theorems are proved for the fluctuations of the queue size during the initial busy period of a queuing process with one server. These theorems are used to find the solutions of various problems connected with the heights and widths of random rooted trees.

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12

Ciucu, Florin, Felix Poloczek, and Amr Rizk. "Queue and Loss Distributions in Finite-Buffer Queues." Proceedings of the ACM on Measurement and Analysis of Computing Systems 3, no. 2 (June 19, 2019): 1–29. http://dx.doi.org/10.1145/3341617.3326146.

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13

Ciucu, Florin, Felix Poloczek, and Amr Rizk. "Queue and Loss Distributions in Finite-Buffer Queues." ACM SIGMETRICS Performance Evaluation Review 47, no. 1 (December 17, 2019): 65–66. http://dx.doi.org/10.1145/3376930.3376972.

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14

Fakinos, D. "Equilibrium queue size distributions for semi-reversible queues." Stochastic Processes and their Applications 36, no. 2 (December 1990): 331–37. http://dx.doi.org/10.1016/0304-4149(90)90099-e.

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15

Kijima, Masaaki, and Naoki Makimoto. "Computation of quasi-stationary distributions in Markovian queues." Computers & Industrial Engineering 27, no. 1-4 (September 1994): 429–32. http://dx.doi.org/10.1016/0360-8352(94)90326-3.

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16

Miyazawa, Masakiyo, and Ronald W. Wolff. "Symmetric queues with batch departures and their networks." Advances in Applied Probability 28, no. 01 (March 1996): 308–26. http://dx.doi.org/10.1017/s0001867800027385.

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Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length distribution are obtained for this network.

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17

Miyazawa, Masakiyo, and Ronald W. Wolff. "Symmetric queues with batch departures and their networks." Advances in Applied Probability 28, no. 1 (March 1996): 308–26. http://dx.doi.org/10.2307/1427923.

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Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length distribution are obtained for this network.

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18

Harrison, Peter G., and Edwige Pitel. "Response time distributions in tandem G-networks." Journal of Applied Probability 32, no. 1 (March 1995): 224–46. http://dx.doi.org/10.2307/3214932.

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The Laplace transform of the probability distribution of the end-to-end delay in tandem networks is obtained where the first and/or second queue are G-queues, i.e. they have negative arrivals. For the most general case the method is based on the solution of a boundary value problem on a closed contour in the complex plane, which itself reduces to the solution of a Fredholm integral equation of the second kind. We also consider the dependence or independence of the sojourn times at each queue in the two special cases where only one of the queues is a G-queue, the other having no negative arrivals.

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19

Harrison, Peter G., and Edwige Pitel. "Response time distributions in tandem G-networks." Journal of Applied Probability 32, no. 01 (March 1995): 224–46. http://dx.doi.org/10.1017/s0021900200102682.

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The Laplace transform of the probability distribution of the end-to-end delay in tandem networks is obtained where the first and/or second queue are G-queues, i.e. they have negative arrivals. For the most general case the method is based on the solution of a boundary value problem on a closed contour in the complex plane, which itself reduces to the solution of a Fredholm integral equation of the second kind. We also consider the dependence or independence of the sojourn times at each queue in the two special cases where only one of the queues is a G-queue, the other having no negative arrivals.

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20

Gelenbe, Erol, Peter Glynn, and Karl Sigman. "Queues with negative arrivals." Journal of Applied Probability 28, no. 1 (March 1991): 245–50. http://dx.doi.org/10.2307/3214756.

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We study single-server queueing models where in addition to regular arriving customers, there are negative arrivals. A negative arrival has the effect of removing a customer from the queue. The way in which this removal is specified gives rise to several different models. Unlike the standard FIFOGI/GI/1 model, the stability conditions for these new models may depend upon more than just the arrival and service rates; the entire distributions of interarrival and service times may be involved.

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21

Gelenbe, Erol, Peter Glynn, and Karl Sigman. "Queues with negative arrivals." Journal of Applied Probability 28, no. 01 (March 1991): 245–50. http://dx.doi.org/10.1017/s0021900200039589.

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We study single-server queueing models where in addition to regular arriving customers, there are negative arrivals. A negative arrival has the effect of removing a customer from the queue. The way in which this removal is specified gives rise to several different models. Unlike the standard FIFO GI/GI/1 model, the stability conditions for these new models may depend upon more than just the arrival and service rates; the entire distributions of interarrival and service times may be involved.

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22

Ayesta, Urtzi. "A Unifying Conservation Law for Single-Server Queues." Journal of Applied Probability 44, no. 04 (December 2007): 1078–87. http://dx.doi.org/10.1017/s0021900200003752.

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We develop a conservation law for a multi-class GI/GI/1 queue operating under a general work-conserving scheduling discipline. For single-class single-server queues, conservation laws have been obtained for both nonanticipating and anticipating disciplines with general service time distributions. For multi-class single-server queues, conservation laws have been obtained for (i) nonanticipating disciplines with exponential service time distributions and (ii) nonpreemptive nonanticipating disciplines with general service time distributions. The unifying conservation law we develop generalizes already existing conservation laws. In addition, it covers popular nonanticipating multi-class time-sharing disciplines such as discriminatory processor sharing (DPS) and generalized processor sharing (GPS) with general service time distributions. As an application, we show that the unifying conservation law can be used to compare the expected unconditional response time under two scheduling disciplines.

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23

Ayesta, Urtzi. "A Unifying Conservation Law for Single-Server Queues." Journal of Applied Probability 44, no. 4 (December 2007): 1078–87. http://dx.doi.org/10.1239/jap/1197908826.

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We develop a conservation law for a multi-class GI/GI/1 queue operating under a general work-conserving scheduling discipline. For single-class single-server queues, conservation laws have been obtained for both nonanticipating and anticipating disciplines with general service time distributions. For multi-class single-server queues, conservation laws have been obtained for (i) nonanticipating disciplines with exponential service time distributions and (ii) nonpreemptive nonanticipating disciplines with general service time distributions. The unifying conservation law we develop generalizes already existing conservation laws. In addition, it covers popular nonanticipating multi-class time-sharing disciplines such as discriminatory processor sharing (DPS) and generalized processor sharing (GPS) with general service time distributions. As an application, we show that the unifying conservation law can be used to compare the expected unconditional response time under two scheduling disciplines.

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24

Li, Junping, and Anyue Chen. "Decay property of stopped Markovian bulk-arriving queues." Advances in Applied Probability 40, no. 01 (March 2008): 95–121. http://dx.doi.org/10.1017/s0001867800002391.

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We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λCis obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λC]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λC]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λC-transient and some deep properties are revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

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25

Li, Junping, and Anyue Chen. "Decay property of stopped Markovian bulk-arriving queues." Advances in Applied Probability 40, no. 1 (March 2008): 95–121. http://dx.doi.org/10.1239/aap/1208358888.

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We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λC is obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λC]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λC]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λC-transient and some deep properties are revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

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26

Walraevens, Joris, Dieter Claeys, and Tuan Phung-Duc. "Asymptotics of queue length distributions in priority retrial queues." Performance Evaluation 127-128 (November 2018): 235–52. http://dx.doi.org/10.1016/j.peva.2018.10.004.

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27

Kijima, Masaaki. "Quasi-Stationary Distributions of Single-Server Phase-Type Queues." Mathematics of Operations Research 18, no. 2 (May 1993): 423–37. http://dx.doi.org/10.1287/moor.18.2.423.

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28

Myers, Daniel S., and Mary K. Vernon. "Estimating queue length distributions for queues with random arrivals." ACM SIGMETRICS Performance Evaluation Review 40, no. 3 (December 4, 2012): 77–79. http://dx.doi.org/10.1145/2425248.2425268.

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29

Sigman, Karl, and Ward Whitt. "Heavy-traffic limits for nearly deterministic queues: stationary distributions." Queueing Systems 69, no. 2 (July 30, 2011): 145–73. http://dx.doi.org/10.1007/s11134-011-9253-y.

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30

Henderson, W., B. S. Northcote, and P. G. Taylor. "Geometric equilibrium distributions for queues with interactive batch departures." Annals of Operations Research 48, no. 5 (October 1994): 493–511. http://dx.doi.org/10.1007/bf02033316.

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31

Chao, Xiuli, Qi-Ming He, and Sheldon Ross. "Tollbooth tandem queues with infinite homogeneous servers." Journal of Applied Probability 52, no. 04 (December 2015): 941–61. http://dx.doi.org/10.1017/s0021900200113002.

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In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

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32

Chao, Xiuli, Qi-Ming He, and Sheldon Ross. "Tollbooth tandem queues with infinite homogeneous servers." Journal of Applied Probability 52, no. 4 (December 2015): 941–61. http://dx.doi.org/10.1239/jap/1450802745.

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In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

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33

Chang, Cheng-Shang. "On the input-output map of a G/G/1 queue." Journal of Applied Probability 31, no. 4 (December 1994): 1128–33. http://dx.doi.org/10.2307/3215337.

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In this note, we consider G/G/1 queues with stationary and ergodic inputs. We show that if the service times are independent and identically distributed with unbounded supports, then for a given mean of interarrival times, the number of sequences (distributions) of interarrival times that induce identical distributions on interdeparture times is at most 1. As a direct consequence, among all the G/M/1 queues with stationary and ergodic inputs, the M/M/1 queue is the only queue whose departure process is identically distributed as the input process.

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34

Chang, Cheng-Shang. "On the input-output map of a G/G/1 queue." Journal of Applied Probability 31, no. 04 (December 1994): 1128–33. http://dx.doi.org/10.1017/s0021900200099654.

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In this note, we consider G/G/1 queues with stationary and ergodic inputs. We show that if the service times are independent and identically distributed with unbounded supports, then for a given mean of interarrival times, the number of sequences (distributions) of interarrival times that induce identical distributions on interdeparture times is at most 1. As a direct consequence, among all the G/M/1 queues with stationary and ergodic inputs, the M/M/1 queue is the only queue whose departure process is identically distributed as the input process.

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35

König, Dieter, and Volker Schmidt. "Stationary queue-length characteristics in queues with delayed feedback." Journal of Applied Probability 22, no. 2 (June 1985): 394–407. http://dx.doi.org/10.2307/3213782.

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A class of two-node queueing networks with general stationary ergodic governing sequence is considered. This means that, in particular, a non-Poissonian arrival process and dependent service times, as well as a non-Bernoulli feedback mechanism are admitted. A mixing condition ensures that the limiting distributions of the number of customers in the nodes observed in continuous time as well as at certain embedded epochs can be expressed by the Palm distributions of appropriately chosen marked point processes. This gives the possibility of connecting the classical concept of embedding with a general point-process approach. Furthermore, it leads to simple proofs of relationships between the limiting distributions. An example is given to illustrate how these relationships can be used to derive explicit formulas for various stationary queueing characteristics.

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36

König, Dieter, and Volker Schmidt. "Stationary queue-length characteristics in queues with delayed feedback." Journal of Applied Probability 22, no. 02 (June 1985): 394–407. http://dx.doi.org/10.1017/s0021900200037852.

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A class of two-node queueing networks with general stationary ergodic governing sequence is considered. This means that, in particular, a non-Poissonian arrival process and dependent service times, as well as a non-Bernoulli feedback mechanism are admitted. A mixing condition ensures that the limiting distributions of the number of customers in the nodes observed in continuous time as well as at certain embedded epochs can be expressed by the Palm distributions of appropriately chosen marked point processes. This gives the possibility of connecting the classical concept of embedding with a general point-process approach. Furthermore, it leads to simple proofs of relationships between the limiting distributions. An example is given to illustrate how these relationships can be used to derive explicit formulas for various stationary queueing characteristics.

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37

Whitt, Ward. "LIMITS FOR CUMULATIVE INPUT PROCESSES TO QUEUES." Probability in the Engineering and Informational Sciences 14, no. 2 (April 2000): 123–50. http://dx.doi.org/10.1017/s0269964800142019.

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We establish functional central limit theorems (FCLTs) for a cumulative input process to a fluid queue from the superposition of independent on–off sources, where the on periods and off periods may have heavy-tailed probability distributions. Variants of these FCLTs hold for cumulative busy-time and idle-time processes associated with standard queueing models. The heavy-tailed on-period and off-period distributions can cause the limit process to have discontinuous sample paths (e.g., to be a non-Brownian stable process or more general Lévy process) even though the converging processes have continuous sample paths. Consequently, we exploit the Skorohod M1 topology on the function space D of right-continuous functions with left limits. The limits here combined with the previously established continuity of the reflection map in the M1 topology imply both heavy-traffic and non-heavy-traffic FCLTs for buffer-content processes in stochastic fluid networks.

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38

Boxma, O. J., D. Perry, and F. A. van der Duyn Schouten. "FLUID QUEUES AND MOUNTAIN PROCESSES." Probability in the Engineering and Informational Sciences 13, no. 4 (October 1999): 407–27. http://dx.doi.org/10.1017/s0269964899134028.

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This paper is devoted to the analysis of a fluid queue with a buffer content that varies linearly during periods that are governed by a three-state semi-Markov process. Two cases are being distinguished: (i) two upward slopes and one downward slope, and (ii) one upward slope and two downward slopes. In both cases, at least one of the period distributions is allowed to be completely general. We obtain exact results for the buffer content distribution, the busy period distribution, and the distribution of the maximal buffer content during a busy period. The results are obtained by establishing relations between the fluid queues and ordinary queues with instantaneous input and by using level crossing theory.

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39

Lee, Ho Woo, Se Won Lee, and Jongwoo Jeon. "Using factorization in analyzing D-BMAP/G/1 queues." Journal of Applied Mathematics and Stochastic Analysis 2005, no. 2 (January 1, 2005): 119–32. http://dx.doi.org/10.1155/jamsa.2005.119.

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We demonstrate how one can use the factorization property to derive the queue-length distributions of the discrete-time BMAP/G/1 queues with complex operational behavior during the idle period. The procedure demonstrated in this paper can be applied to the analysis of many other discrete-time BMAP/G/1 queues with more behavioral complexities.

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40

Shanthikumar, J. George, and Ushio Sumita. "On the busy-period distributions of M/G/1/K queues by state-dependent arrivals and FCFS/LCFS-P service disciplines." Journal of Applied Probability 22, no. 4 (December 1985): 912–19. http://dx.doi.org/10.2307/3213958.

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The busy-period distributions of M/G/1/K queues with state-dependent arrival rates are considered. Two recursion formulas for the Laplace–Stieltjes transforms of the busy periods under the FCFS and preempt resume LCFS service disciplines are obtained. It is shown that the busy-period distributions for the two service disciplines are, in general, different, in contrast to the fact that they coincide for ordinary M/G/1 queues. For deterministic service times and arrival rates non-increasing in the number of customers in the system, stochastic ordering between these two busy periods is also established.

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41

Shanthikumar, J. George, and Ushio Sumita. "On the busy-period distributions of M/G/1/K queues by state-dependent arrivals and FCFS/LCFS-P service disciplines." Journal of Applied Probability 22, no. 04 (December 1985): 912–19. http://dx.doi.org/10.1017/s0021900200108149.

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The busy-period distributions of M/G/1/K queues with state-dependent arrival rates are considered. Two recursion formulas for the Laplace–Stieltjes transforms of the busy periods under the FCFS and preempt resume LCFS service disciplines are obtained. It is shown that the busy-period distributions for the two service disciplines are, in general, different, in contrast to the fact that they coincide for ordinary M/G/1 queues. For deterministic service times and arrival rates non-increasing in the number of customers in the system, stochastic ordering between these two busy periods is also established.

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42

Glynn, Peter W., and Ward Whitt. "A new view of the heavy-traffic limit theorem for infinite-server queues." Advances in Applied Probability 23, no. 01 (March 1991): 188–209. http://dx.doi.org/10.1017/s0001867800023399.

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This paper presents a new approach for obtaining heavy-traffic limits for infinite-server queues and open networks of infinite-server queues. The key observation is that infinite-server queues having deterministic service times can easily be analyzed in terms of the arrival counting process. A variant of the same idea applies when the service times take values in a finite set, so this is the key assumption. In addition to new proofs of established results, the paper contains several new results, including limits for the work-in-system process, limits for steady-state distributions, limits for open networks with general customer routes, and rates of convergence. The relatively tractable Gaussian limits are promising approximations for many-server queues and open networks of such queues, possibly with finite waiting rooms.

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43

Glynn, Peter W., and Ward Whitt. "A new view of the heavy-traffic limit theorem for infinite-server queues." Advances in Applied Probability 23, no. 1 (March 1991): 188–209. http://dx.doi.org/10.2307/1427517.

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Abstract:

This paper presents a new approach for obtaining heavy-traffic limits for infinite-server queues and open networks of infinite-server queues. The key observation is that infinite-server queues having deterministic service times can easily be analyzed in terms of the arrival counting process. A variant of the same idea applies when the service times take values in a finite set, so this is the key assumption. In addition to new proofs of established results, the paper contains several new results, including limits for the work-in-system process, limits for steady-state distributions, limits for open networks with general customer routes, and rates of convergence. The relatively tractable Gaussian limits are promising approximations for many-server queues and open networks of such queues, possibly with finite waiting rooms.

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44

Izady, N., and D. Worthington. "Approximate analysis of non-stationary loss queues and networks of loss queues with general service time distributions." European Journal of Operational Research 213, no. 3 (September 2011): 498–508. http://dx.doi.org/10.1016/j.ejor.2011.03.029.

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45

Phatarfod, R. M. "The geometricity of the limiting distributions in queues and dams." Journal of Applied Probability 30, no. 2 (June 1993): 438–45. http://dx.doi.org/10.2307/3214852.

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There are a number of cases in the theories of queues and dams where the limiting distribution of the pertinent processes is geometric with a modified initial term — herein called zero-modified geometric (ZMG). The paper gives a unified treatment of the various cases considered hitherto and some others by using a duality relation between random walks with impenetrable and with absorbing barriers, and deriving the probabilities of absorption by using Waldian identities. Thus the method enables us to distinguish between those cases where the limiting distribution would be ZMG and those where it would not.

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46

Phatarfod, R. M. "The geometricity of the limiting distributions in queues and dams." Journal of Applied Probability 30, no. 02 (June 1993): 438–45. http://dx.doi.org/10.1017/s0021900200117449.

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Abstract:

There are a number of cases in the theories of queues and dams where the limiting distribution of the pertinent processes is geometric with a modified initial term — herein called zero-modified geometric (ZMG). The paper gives a unified treatment of the various cases considered hitherto and some others by using a duality relation between random walks with impenetrable and with absorbing barriers, and deriving the probabilities of absorption by using Waldian identities. Thus the method enables us to distinguish between those cases where the limiting distribution would be ZMG and those where it would not.

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47

Braband, Jens. "Waiting time distributions for M/M/N processor sharing queues." Communications in Statistics. Stochastic Models 10, no. 3 (January 1994): 533–48. http://dx.doi.org/10.1080/15326349408807309.

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48

Daw, Andrew, and Jamol Pender. "On the distributions of infinite server queues with batch arrivals." Queueing Systems 91, no. 3-4 (February 15, 2019): 367–401. http://dx.doi.org/10.1007/s11134-019-09603-4.

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49

Schassberger, R., and H. Daduna. "Sojourn times in queuing networks with multiserver modes." Journal of Applied Probability 24, no. 2 (June 1987): 511–21. http://dx.doi.org/10.2307/3214274.

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50

Schassberger, R., and H. Daduna. "Sojourn times in queuing networks with multiserver modes." Journal of Applied Probability 24, no. 02 (June 1987): 511–21. http://dx.doi.org/10.1017/s0021900200031144.

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