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Journal articles on the topic 'Distribution à queue épaisse'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Kim, Jeong-Sim. "QUEUE LENGTH DISTRIBUTION IN A QUEUE WITH RELATIVE PRIORITIES." Bulletin of the Korean Mathematical Society 46, no. 1 (January 31, 2009): 107–16. http://dx.doi.org/10.4134/bkms.2009.46.1.107.

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2

Rege, Kiran M., and Bhaskar Sengupta. "Queue-Length Distribution for the Discriminatory Processor-Sharing Queue." Operations Research 44, no. 4 (August 1996): 653–57. http://dx.doi.org/10.1287/opre.44.4.653.

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3

Choi, Bong Dae, Yeong Cheol Kim, Yang Woo Shin, and Charles E. M. Pearce. "The MX/G/1 queue with queue length dependent service times." Journal of Applied Mathematics and Stochastic Analysis 14, no. 4 (January 1, 2001): 399–419. http://dx.doi.org/10.1155/s104895330100034x.

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We deal with the MX/G/1 queue where service times depend on the queue length at the service initiation. By using Markov renewal theory, we derive the queue length distribution at departure epochs. We also obtain the transient queue length distribution at time t and its limiting distribution and the virtual waiting time distribution. The numerical results for transient mean queue length and queue length distributions are given.

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4

ZHANG, HONGBO, DINGHUA SHI, and ZHENTING HOU. "EXPLICIT SOLUTION FOR QUEUE LENGTH DISTRIBUTION OF M/T-SPH/1 QUEUE." Asia-Pacific Journal of Operational Research 31, no. 01 (February 2014): 1450001. http://dx.doi.org/10.1142/s0217595914500018.

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In this paper we study an M/T-SPH/1 queue system, where T-SPH denotes the continuous time phase type distribution defined on a birth and death process with countably many states. The queue model can be described by a quasi-birth-and-death (QBD) process with countable phases. For the QBD process, we give the computation scheme of the joint stationary distribution. Furthermore, the obtained results enable us to give the stationary queue length distribution for the M/T-SPH/1 queue.

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5

Zhang, Hongbo, Zhenting Hou, and Dinghua Shi. "Analysis of Stationary Queue Length Distribution for Geo/T-IPH/1 Queue." Journal of the Operations Research Society of China 1, no. 3 (September 2013): 415–24. http://dx.doi.org/10.1007/s40305-013-0026-7.

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6

Li, Jingwen, and Jihong Ou. "Characterizing the idle-period distribution of GI/G/1 queues." Journal of Applied Probability 32, no. 01 (March 1995): 247–55. http://dx.doi.org/10.1017/s0021900200102694.

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A variety of performance measures of aGI/G/1 queue are explicitly related to the idle-period distribution of the queue, suggesting that the system analysis can be accomplished by the analysis of the idle period. However, the ‘stand-alone' relationship for the idle-period distribution of theGI/G/1 queue (i.e. the counterpart of Lindley's equation) has not been found in the literature. In this paper we develop a non-linear integral equation for the idle period distribution of theGI/G/1 queue. We also show that this non-linear system defines a unique solution. This development makes possible the analysis of theGI/G/1 queue in a different perspective.

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7

Li, Jingwen, and Jihong Ou. "Characterizing the idle-period distribution of GI/G/1 queues." Journal of Applied Probability 32, no. 1 (March 1995): 247–55. http://dx.doi.org/10.2307/3214933.

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A variety of performance measures of a GI/G/1 queue are explicitly related to the idle-period distribution of the queue, suggesting that the system analysis can be accomplished by the analysis of the idle period. However, the ‘stand-alone' relationship for the idle-period distribution of the GI/G/1 queue (i.e. the counterpart of Lindley's equation) has not been found in the literature. In this paper we develop a non-linear integral equation for the idle period distribution of the GI/G/1 queue. We also show that this non-linear system defines a unique solution. This development makes possible the analysis of the GI/G/1 queue in a different perspective.

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8

Shin, Yang Woo, and Chareles E. M. Pearce. "The BMAP/G/1 vacation queue with queue-length dependent vacation schedule." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 40, no. 2 (October 1998): 207–21. http://dx.doi.org/10.1017/s0334270000012479.

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AbstractWe treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution

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9

Tan, Xiaoming, and Charles Knessl. "Sojourn time distribution in some processor-shared queues." European Journal of Applied Mathematics 4, no. 4 (December 1993): 437–48. http://dx.doi.org/10.1017/s0956792500001224.

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We develop a technique for obtaining asymptotic properties of the sojourn time distribution in processor-sharing queues. We treat the standard M/M/1-PS queue and its finite capacity version, the M/M/1/K-PS queue. Using perturbation methods, we construct asymptotic expansions for the distribution of a tagged customer's sojourn time, conditioned on that customer's total required service. The asymptotic limit assumes that (i) the traffic intensity is close to one for the infinite capacity model, and (ii) that the system's capacity is large for the finite capacity queue.

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10

Kim, Jeongsim. "AN APPROXIMATION FOR THE QUEUE LENGTH DISTRIBUTION IN A MULTI-SERVER RETRIAL QUEUE." Journal of the Chungcheong Mathematical Society 29, no. 1 (February 15, 2016): 95–102. http://dx.doi.org/10.14403/jcms.2016.29.1.95.

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11

Kim, Jeongsim, Bara Kim, and Jerim Kim. "Moments of the queue size distribution in the MAP/G/1 retrial queue." Computers & Operations Research 37, no. 7 (July 2010): 1212–19. http://dx.doi.org/10.1016/j.cor.2009.04.007.

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12

Kim, Bara, and Jeongsim Kim. "Queue size distribution in a discrete-time D-BMAP/G/1 retrial queue." Computers & Operations Research 37, no. 7 (July 2010): 1220–27. http://dx.doi.org/10.1016/j.cor.2009.04.016.

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13

Kim, Sunggon. "Approximate queue length distribution of a discriminatory processor sharing queue with impatient customers." Journal of the Korean Statistical Society 43, no. 1 (March 2014): 105–18. http://dx.doi.org/10.1016/j.jkss.2013.04.001.

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14

Ishizaki, Fumio. "Loss probability in a finite queue with service interruptions and queue length distribution in the corresponding infinite queue." Performance Evaluation 63, no. 7 (July 2006): 682–99. http://dx.doi.org/10.1016/j.peva.2005.06.006.

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15

Li, Jun, and Yiqiang Q. Zhao. "ON THE PROBABILITY DISTRIBUTION OF JOIN QUEUE LENGTH IN A FORK-JOIN MODEL." Probability in the Engineering and Informational Sciences 24, no. 4 (August 19, 2010): 473–83. http://dx.doi.org/10.1017/s0269964810000112.

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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.

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16

M. Ferreira, Manuel Alberto. "The M|D|∞ queue busy period distribution." Journal of Mathematics and Technology 5, no. 1 (March 30, 2014): 26–32. http://dx.doi.org/10.7813/jmt.2014/5-1/5.

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17

Kim, Jeongsim, and Bara Kim. "Waiting time distribution in an retrial queue." Performance Evaluation 70, no. 4 (April 2013): 286–99. http://dx.doi.org/10.1016/j.peva.2012.12.003.

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18

Wang, P. Patrick. "Queue Length Distribution of an Unreliable Machine." OPSEARCH 37, no. 2 (June 2000): 99–123. http://dx.doi.org/10.1007/bf03398602.

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19

Boon, M. A. A., and E. M. M. Winands. "HEAVY-TRAFFIC ANALYSIS OF K-LIMITED POLLING SYSTEMS." Probability in the Engineering and Informational Sciences 28, no. 4 (June 27, 2014): 451–71. http://dx.doi.org/10.1017/s0269964814000096.

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In this paper, we study a two-queue polling model with zero switchover times and k-limited service (serve at most ki customers during one visit period to queue i, i=1, 2) in each queue. The arrival processes at the two queues are Poisson, and the service times are exponentially distributed. By increasing the arrival intensities until one of the queues becomes critically loaded, we derive exact heavy-traffic limits for the joint queue-length distribution using a singular-perturbation technique. It turns out that the number of customers in the stable queue has the same distribution as the number of customers in a vacation system with Erlang-k2 distributed vacations. The queue-length distribution of the critically loaded queue, after applying an appropriate scaling, is exponentially distributed. Finally, we show that the two queue-length processes are independent in heavy traffic.

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20

König, D., and M. Miyazawa. "Relationships and decomposition in the delayed bernoulli feedback queueing system." Journal of Applied Probability 25, no. 1 (March 1988): 169–83. http://dx.doi.org/10.2307/3214243.

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For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

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21

König, D., and M. Miyazawa. "Relationships and decomposition in the delayed bernoulli feedback queueing system." Journal of Applied Probability 25, no. 01 (March 1988): 169–83. http://dx.doi.org/10.1017/s0021900200040730.

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For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

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22

Wijayanti, Devi, Sugito Sugito, and Hasbi Yasin. "ANALISIS MODEL ANTREAN NON-POISSON DAN UKURAN KINERJA SISTEM BERBASIS GUI WEB INTERAKTIF MENGGUNAKAN R-SHINY (Studi Kasus: Bus di Terminal Penggaron Kota Semarang)." Jurnal Gaussian 9, no. 4 (December 7, 2020): 444–53. http://dx.doi.org/10.14710/j.gauss.v9i4.29010.

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Since September 1, 2018, The Semarang City Government has diverted intercity bus stop within the province from Terboyo Terminal to Penggaron Terminal, resulting in an imbalance of movement and capacity of the Penggaron Terminal which causes queue of bus. Non-Poisson queue is a queue model in which the arrival and service distribution do not have a Poisson distribution or do not have an Exponential distribution. The study was conducted on buses entering the Penggaron Bus Station with the destination of Jepara, Kedungjati, Juwangi, Yogyakarta, Kudus/Pati/Lasem, Pekalongan/Tegal, and Purwokerto/Purworejo. The main goal of this project is to identify the queue model of Non-Poisson and calculate the measure of system performance using the GUI R. One of the programs in R that can create an interactive web-based GUI (Graphical User Interface) is R-Shiny. R-Shiny is a toolkit of R programs that can be used to create online programs. The distribution test obtained using the EasyFit program. The bus queue model of Jepara is (DAGUM/GEV/4):(GD/∞/∞), the queue model of Kedungjati is (GPD/ DAGUM/1):(GD/∞/∞), the queue model of Juwangi is (GEV/ GEV/1):(GD/∞/∞), the queue model of Yogyakarta is (DAGUM/ DAGUM/1) : (GD/∞/∞), the queue model of Kudus/Pati/Lasem is (DAGUM/GEV/1):(GD/∞/∞), the queue model of Pekalongan/Tegal is (GEV/DAGUM/1):(GD/∞/∞), and the queue model of Purwokerto/Purworejo is (GPD/DAGUM/1) : (GD/∞/∞).

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23

Daikoku, Kentaro, Hiroyuki Masuyama, Tetsuya Takine, and Yutaka Takahashi. "ALGORITHMIC COMPUTATION OF THE TRANSIENT QUEUE LENGTH DISTRIBUTION IN THE BMAP/D/c QUEUE." Journal of the Operations Research Society of Japan 50, no. 1 (2007): 55–72. http://dx.doi.org/10.15807/jorsj.50.55.

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24

Kim, Jerim, Bara Kim, and Sung-Seok Ko. "Tail Asymptotics for the Queue Size Distribution in an M/G/1 Retrial Queue." Journal of Applied Probability 44, no. 04 (December 2007): 1111–18. http://dx.doi.org/10.1017/s0021900200003788.

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We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.

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25

KIM, JEONGSIM. "TAIL ASYMPTOTICS FOR THE QUEUE SIZE DISTRIBUTION IN AN MX/G/1 RETRIAL QUEUE." Journal of applied mathematics & informatics 33, no. 3_4 (May 30, 2015): 343–50. http://dx.doi.org/10.14317/jami.2015.343.

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26

Tang, Yinghui, and Xiaowo Tang. "THE QUEUE-LENGTH DISTRIBUTION FOR M x /G/1 QUEUE WITH SINGLE SERVER VACATION." Acta Mathematica Scientia 20, no. 3 (July 2000): 397–408. http://dx.doi.org/10.1016/s0252-9602(17)30647-1.

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27

Kouvatsos, Demetres D. "A maximum entropy queue length distribution for the G/G/1 finite capacity queue." ACM SIGMETRICS Performance Evaluation Review 14, no. 1 (May 1986): 224–36. http://dx.doi.org/10.1145/317531.317555.

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28

Kim, Jerim, Bara Kim, and Sung-Seok Ko. "Tail Asymptotics for the Queue Size Distribution in an M/G/1 Retrial Queue." Journal of Applied Probability 44, no. 4 (December 2007): 1111–18. http://dx.doi.org/10.1239/jap/1197908829.

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We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.

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29

Kim, Bara, Jeongsim Kim, and Jerim Kim. "Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue." Queueing Systems 66, no. 1 (June 4, 2010): 79–94. http://dx.doi.org/10.1007/s11134-010-9179-9.

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30

Kim, Jerim, Jeongsim Kim, and Bara Kim. "Tail asymptotics of the queue size distribution in the M/M/m retrial queue." Journal of Computational and Applied Mathematics 236, no. 14 (August 2012): 3445–60. http://dx.doi.org/10.1016/j.cam.2012.03.027.

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31

Gu, Jianxiong, Yingyuan Wei, Yinghui Tang, and Miaomiao Yu. "Queue size distribution of Geo/G/1 queue under the Min(N,D)-policy." Journal of Systems Science and Complexity 29, no. 3 (March 18, 2016): 752–71. http://dx.doi.org/10.1007/s11424-016-4180-y.

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32

Chydzinski, Andrzej. "On the Queue Length Distribution in BMAP Systems." Journal of Communications Software and Systems 2, no. 4 (April 4, 2017): 275. http://dx.doi.org/10.24138/jcomss.v2i4.272.

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Batch Markovian Arrival Process – BMAP – is a teletraffic model which combines high ability to imitate complexstatistical behaviour of network traces with relative simplicity in analysis and simulation. It is also a generalization of a wide class of Markovian processes, a class which in particular include the Poisson process, the compound Poisson process, the Markovmodulated Poisson process, the phase-type renewal process and others. In this paper we study the main queueing performance characteristic of a finite-buffer queue fed by the BMAP, namely the queue length distribution. In particular, we show a formula for the Laplace transform of the queue length distribution. The main benefit of this formula is that it may be used to obtain both transient and stationary characteristics. To demonstrate this, several numerical results are presented.

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33

Koh, Siew Khew, Ah Hin Pooi, and Yi Fei Tan. "New Approach for Finding Basic Performance Measures of Single Server Queue." Journal of Probability and Statistics 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/851738.

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Consider the single server queue in which the system capacity is infinite and the customers are served on a first come, first served basis. Suppose the probability density functionf(t)and the cumulative distribution functionF(t)of the interarrival time are such that the ratef(t)/1-F(t)tends to a constant ast→∞, and the rate computed from the distribution of the service time tends to another constant. When the queue is in a stationary state, we derive a set of equations for the probabilities of the queue length and the states of the arrival and service processes. Solving the equations, we obtain approximate results for the stationary probabilities which can be used to obtain the stationary queue length distribution and waiting time distribution of a customer who arrives when the queue is in the stationary state.

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34

Wu, De-An, and Hideaki Takagi. "Processor-sharing and random-service queues with semi-Markovian arrivals." Journal of Applied Probability 42, no. 02 (June 2005): 478–90. http://dx.doi.org/10.1017/s0021900200000474.

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We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-lcustomer who, upon his arrival, meetskcustomers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-lcustomer who, upon his arrival, meetsk+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

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35

M. Ferreira, Manuel Alberto. "M|G|∞ queue busy period with PME distribution." Journal of Mathematics and Technology 4, no. 2 (November 30, 2013): 5–8. http://dx.doi.org/10.7813/jmt.2013/4-2/1.

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36

M. Ferreira, Manuel Alberto. "M|G|∞ queue busy period with PME distribution." Journal of Mathematics and Technology 5, no. 1 (March 30, 2014): 5–8. http://dx.doi.org/10.7813/jmt.2014/5-1/1.

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37

Kerbl, Bernhard, Jörg Müller, Michael Kenzel, Dieter Schmalstieg, and Markus Steinberger. "A scalable queue for work distribution on GPUs." ACM SIGPLAN Notices 53, no. 1 (March 23, 2018): 401–2. http://dx.doi.org/10.1145/3200691.3178526.

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38

Ferreira, M. A. M. "The M|D|_\infty queue busy cycle distribution." International Mathematical Forum 9 (2014): 81–87. http://dx.doi.org/10.12988/imf.2014.311249.

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39

Yue, Meng, Zhijun Wu, and Jingjie Wang. "Detecting LDoS attack bursts based on queue distribution." IET Information Security 13, no. 3 (May 1, 2019): 285–92. http://dx.doi.org/10.1049/iet-ifs.2018.5097.

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40

Goldenshluger, Alexander. "Nonparametric estimation of the service time distribution in the M/G/∞ queue." Advances in Applied Probability 48, no. 4 (December 2016): 1117–38. http://dx.doi.org/10.1017/apr.2016.67.

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AbstractThe subject of this paper is the problem of estimating the service time distribution of the M/G/∞ queue from incomplete data on the queue. The goal is to estimate G from observations of the queue-length process at the points of the regular grid on a fixed time interval. We propose an estimator and analyze its accuracy over a family of target service time distributions. An upper bound on the maximal risk is derived. The problem of estimating the arrival rate is considered as well.

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41

Wei, Yingyuan, Yinghui Tang, and Miaomiao Yu. "Recursive Solution of Queue Length Distribution for Geo/G/1 Queue with Delayed Min(N, D)-Policy." Journal of Systems Science and Information 8, no. 4 (August 26, 2020): 367–86. http://dx.doi.org/10.21078/jssi-2020-367-20.

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AbstractIn this paper we consider a discrete-time Geo/G/1 queue with delayed Min(N, D)-policy. Using renewal process theory, total probability decomposition technique and z-transform, we study the transient and equilibrium properties of the queue length from an arbitrary initial state, and obtain both the recursive expressions of the transient state queue length distribution and the steady state queue length distribution at arbitrary time epoch n+. Furthermore, we derive the important relations between equilibrium queue length distributions at different time epochs n–, n and n+. Finally, we give some numerical examples about capacity decision in queueing systems to demonstrate the application of the analytical results reported in this paper.

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42

Gall, Pierre Le. "The stationary G/G/s queue." Journal of Applied Mathematics and Stochastic Analysis 11, no. 1 (January 1, 1998): 59–71. http://dx.doi.org/10.1155/s1048953398000057.

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43

Kempa, Wojciech M. "A Direct Approach to Transient Queue-Size Distribution in a Finite-Buffer Queue with AQM." Applied Mathematics & Information Sciences 7, no. 3 (May 1, 2013): 909–15. http://dx.doi.org/10.12785/amis/070308.

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44

Takine, Tetsuya. "A new recursion for the queue length distribution in the stationary BMAP/G/1 queue." Communications in Statistics. Stochastic Models 16, no. 2 (January 2000): 335–41. http://dx.doi.org/10.1080/15326340008807590.

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45

Luo, Chuanyi, Yinghui Tang, and Cailiang Li. "Transient Queue Size Distribution Solution of Geom / G / 1 Queue with Feedback-A Recursive Method." Journal of Systems Science and Complexity 22, no. 2 (May 2, 2009): 303–12. http://dx.doi.org/10.1007/s11424-009-9165-7.

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46

Kim, Sunggon, Jongwoo Kim, and Eui Yong Lee. "Stationary distribution of queue length in G / M / 1 queue with two-stage service policy." Mathematical Methods of Operations Research 64, no. 3 (September 6, 2006): 467–80. http://dx.doi.org/10.1007/s00186-006-0096-y.

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47

Wu, De-An, and Hideaki Takagi. "Processor-sharing and random-service queues with semi-Markovian arrivals." Journal of Applied Probability 42, no. 2 (June 2005): 478–90. http://dx.doi.org/10.1239/jap/1118777183.

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We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-l customer who, upon his arrival, meets k customers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-l customer who, upon his arrival, meets k+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

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48

Ferrandiz, Josep M. "The BMAP/GI/1 queue with server set-up times and server vacations." Advances in Applied Probability 25, no. 01 (March 1993): 235–54. http://dx.doi.org/10.1017/s0001867800025258.

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Using Palm-martingale calculus, we derive the workload characteristic function and queue length moment generating function for the BMAP/GI/1 queue with server vacations. In the queueing system under study, the server may start a vacation at the completion of a service or at the arrival of a customer finding an empty system. In the latter case we will talk of a server set-up time. The distribution of a set-up time or of a vacation period after a departure leaving a non-empty system behind is conditionally independent of the queue length and workload. Furthermore, the distribution of the server set-up times may be different from the distribution of vacations at service completion times. The results are particularized to the M/GI/1 queue and to the BMAP/GI/1 queue (without vacations).

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Ferrandiz, Josep M. "The BMAP/GI/1 queue with server set-up times and server vacations." Advances in Applied Probability 25, no. 1 (March 1993): 235–54. http://dx.doi.org/10.2307/1427504.

Full text

Abstract:

Using Palm-martingale calculus, we derive the workload characteristic function and queue length moment generating function for the BMAP/GI/1 queue with server vacations. In the queueing system under study, the server may start a vacation at the completion of a service or at the arrival of a customer finding an empty system. In the latter case we will talk of a server set-up time. The distribution of a set-up time or of a vacation period after a departure leaving a non-empty system behind is conditionally independent of the queue length and workload. Furthermore, the distribution of the server set-up times may be different from the distribution of vacations at service completion times. The results are particularized to the M/GI/1 queue and to the BMAP/GI/1 queue (without vacations).

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50

Debicki, Krzysztof, Michel Mandjes, and Miranda van Uitert. "A TANDEM QUEUE WITH LÉVY INPUT: A NEW REPRESENTATION OF THE DOWNSTREAM QUEUE LENGTH." Probability in the Engineering and Informational Sciences 21, no. 1 (December 15, 2006): 83–107. http://dx.doi.org/10.1017/s0269964807070064.

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Abstract:

In this article we present a new representation for the steady-state distribution of the workload of the second queue in a two-node tandem network. It involves the difference of two suprema over two adjacent intervals. In the case of spectrally positive Lévy input, this enables us to derive the Laplace transform and Pollaczek–Khintchine representation of the workload of the second queue. Additionally, we obtain the exact distribution of the workload in the case of Brownian and Poisson input, as well as some insightful formulas representing the exact asymptotics for α-stable Lévy inputs.

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