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Journal articles on the topic 'Markovian Arrival Process'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

HUNTER, JEFFREY J. "MARKOVIAN QUEUES WITH CORRELATED ARRIVAL PROCESSES." Asia-Pacific Journal of Operational Research 24, no. 04 (August 2007): 593–611. http://dx.doi.org/10.1142/s021759590700136x.

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In an attempt to examine the effect of dependencies in the arrival process on the steady state queue length process in single server queueing models with exponential service time distribution, four different models for the arrival process, each with marginally distributed exponential inter-arrivals to the queueing system, are considered. Two of these models are based upon the upper and lower bounding joint distribution functions given by the Fréchet bounds for bivariate distributions with specified marginals, the third is based on Downton's bivariate exponential distribution and fourthly the usual M/M/1 model. The aim of the paper is to compare conditions for stability and explore the queueing behavior of the different models.

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2

Cao, Jianyu, and Weixin Xie. "Joint arrival process of multiple independent batch Markovian arrival processes." Statistics & Probability Letters 133 (February 2018): 42–49. http://dx.doi.org/10.1016/j.spl.2017.09.012.

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3

Pérez-Ocón, Rafael, and Maria del Carmen Segovia. "Shock models under a Markovian arrival process." Mathematical and Computer Modelling 50, no. 5-6 (September 2009): 879–84. http://dx.doi.org/10.1016/j.mcm.2008.12.020.

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4

Neuts, Marcel F., LIU Dan, and NARAYANA Surya. "Local poissonification of the markovian arrival process." Communications in Statistics. Stochastic Models 8, no. 1 (January 1992): 87–129. http://dx.doi.org/10.1080/15326349208807216.

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5

Asmussen, Søren, and Ger Koole. "Marked point processes as limits of Markovian arrival streams." Journal of Applied Probability 30, no. 2 (June 1993): 365–72. http://dx.doi.org/10.2307/3214845.

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A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

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6

Asmussen, Søren, and Ger Koole. "Marked point processes as limits of Markovian arrival streams." Journal of Applied Probability 30, no. 02 (June 1993): 365–72. http://dx.doi.org/10.1017/s0021900200117371.

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A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

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7

Johnson, Mary A., and Surya Narayana. "Descriptors of arrival-process burstiness with application to the discrete Markovian arrival process." Queueing Systems 23, no. 1-4 (March 1996): 107–30. http://dx.doi.org/10.1007/bf01206553.

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8

Jin, Fang, Chengxun Wu, and Hui Ou. "Compound Binomial Model with Batch Markovian Arrival Process." Mathematical Problems in Engineering 2020 (November 28, 2020): 1–10. http://dx.doi.org/10.1155/2020/1932704.

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A compound binomial model with batch Markovian arrival process was studied, and the specific definitions are introduced. We discussed the problem of ruin probabilities. Specially, the recursion formulas of the conditional finite-time ruin probability are obtained and the numerical algorithm of the conditional finite-time nonruin probability is proposed. We also discuss research on the compound binomial model with batch Markovian arrival process and threshold dividend. Recursion formulas of the Gerber–Shiu function and the first discounted dividend value are provided, and the expressions of the total discounted dividend value are obtained and proved. At the last part, some numerical illustrations were presented.

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9

Ramírez-Cobo, Pepa, Rosa E. Lillo, and Michael P. Wiper. "Nonidentifiability of the Two-State Markovian Arrival Process." Journal of Applied Probability 47, no. 03 (September 2010): 630–49. http://dx.doi.org/10.1017/s0021900200006975.

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In this paper we consider the problem of identifiability for the two-state Markovian arrival process (MAP2). In particular, we show that the MAP2 is not identifiable, providing the conditions under which two different sets of parameters induce identical stationary laws for the observable process.

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10

Ramírez-Cobo, Pepa, Rosa E. Lillo, and Michael P. Wiper. "Nonidentifiability of the Two-State Markovian Arrival Process." Journal of Applied Probability 47, no. 3 (September 2010): 630–49. http://dx.doi.org/10.1239/jap/1285335400.

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In this paper we consider the problem of identifiability for the two-state Markovian arrival process (MAP2). In particular, we show that the MAP2 is not identifiable, providing the conditions under which two different sets of parameters induce identical stationary laws for the observable process.

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11

Nishimura, Shoichi, and Hajime Sato. "EIGENVALUE EXPRESSION FOR A BATCH MARKOVIAN ARRIVAL PROCESS." Journal of the Operations Research Society of Japan 40, no. 1 (1997): 122–32. http://dx.doi.org/10.15807/jorsj.40.122.

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12

Okamura, H., T. Dohi, and K. S. Trivedi. "Markovian Arrival Process Parameter Estimation With Group Data." IEEE/ACM Transactions on Networking 17, no. 4 (August 2009): 1326–39. http://dx.doi.org/10.1109/tnet.2008.2008750.

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13

Ramírez-Cobo, Pepa, and Emilio Carrizosa. "A Note on the Dependence Structure of the Two-State Markovian Arrival Process." Journal of Applied Probability 49, no. 01 (March 2012): 295–302. http://dx.doi.org/10.1017/s0021900200009001.

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The Markovian arrival process generalizes the Poisson process by allowing for dependent and nonexponential interarrival times. We study the autocorrelation function of the two-state Markovian arrival process. Our findings show that the correlation structure of such a process has a very specific pattern, namely, it always converges geometrically to zero. Moreover, the signs of the autocorrelation coefficients are either constant or alternating.

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14

Ramírez-Cobo, Pepa, and Emilio Carrizosa. "A Note on the Dependence Structure of the Two-State Markovian Arrival Process." Journal of Applied Probability 49, no. 1 (March 2012): 295–302. http://dx.doi.org/10.1239/jap/1331216848.

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The Markovian arrival process generalizes the Poisson process by allowing for dependent and nonexponential interarrival times. We study the autocorrelation function of the two-state Markovian arrival process. Our findings show that the correlation structure of such a process has a very specific pattern, namely, it always converges geometrically to zero. Moreover, the signs of the autocorrelation coefficients are either constant or alternating.

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15

Hervé, Loïc, and James Ledoux. "Geometric ρ-Mixing Property of the Interarrival Times of a Stationary Markovian Arrival Process." Journal of Applied Probability 50, no. 02 (June 2013): 598–601. http://dx.doi.org/10.1017/s0021900200013590.

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In this note, the sequence of the interarrivals of a stationary Markovian arrival process is shown to be ρ-mixing with a geometric rate of convergence when the driving process is ρ-mixing. This provides an answer to an issue raised in the recent work of Ramirez-Cobo and Carrizosa (2012) on the geometric convergence of the autocorrelation function of the stationary Markovian arrival process.

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16

Hervé, Loïc, and James Ledoux. "Geometric ρ-Mixing Property of the Interarrival Times of a Stationary Markovian Arrival Process." Journal of Applied Probability 50, no. 2 (June 2013): 598–601. http://dx.doi.org/10.1239/jap/1371648964.

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In this note, the sequence of the interarrivals of a stationary Markovian arrival process is shown to be ρ-mixing with a geometric rate of convergence when the driving process is ρ-mixing. This provides an answer to an issue raised in the recent work of Ramirez-Cobo and Carrizosa (2012) on the geometric convergence of the autocorrelation function of the stationary Markovian arrival process.

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17

Choi, Bong Dae, Sung Ho Choi, Dan Keun Sung, Tae-Hee Lee, and Kyu-Seog Song. "Transient analysis of a queue with queue-length dependent MAP and its application to SS7 network." Journal of Applied Mathematics and Stochastic Analysis 12, no. 4 (January 1, 1999): 371–92. http://dx.doi.org/10.1155/s1048953399000325.

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We analyze the transient behavior of a Markovian arrival queue with congestion control based on a double of thresholds, where the arrival process is a queue-length dependent Markovian arrival process. We consider Markov chain embedded at arrival epochs and derive the one-step transition probabilities. From these results, we obtain the mean delay and the loss probability of the nth arrival packet. Before we study this complex model, first we give a transient analysis of an MAP/M/1 queueing system without congestion control at arrival epochs. We apply our result to a signaling system No. 7 network with a congestion control based on thresholds.

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18

Klemm, Alexander, Christoph Lindemann, and Marco Lohmann. "Modeling IP traffic using the batch Markovian arrival process." Performance Evaluation 54, no. 2 (October 2003): 149–73. http://dx.doi.org/10.1016/s0166-5316(03)00067-1.

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19

Gupta, U. C., and Karabi Sikdar. "A finite capacity bulk service queue with single vacation and Markovian arrival process." Journal of Applied Mathematics and Stochastic Analysis 2004, no. 4 (January 1, 2004): 337–57. http://dx.doi.org/10.1155/s1048953304403025.

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Vacation time queues with Markovian arrival process (MAP) are mainly useful in modeling and performance analysis of telecommunication networks based on asynchronous transfer mode (ATM) environment. This paper analyzes a single-server finite capacity queue wherein service is performed in batches of maximum size “b” with a minimum threshold “a” and arrivals are governed by MAP. The server takes a single vacation when he finds less than “a” customers after service completion. The distributions of buffer contents at various epochs (service completion, vacation termination, departure, arbitrary and pre-arrival) have been obtained. Finally, some performance measures such as loss probability and average queue length are discussed. Numerical results are also presented in some cases.

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20

Matendo, Sadrac K. "Some performance measures for vacation models with a batch Markovian arrival process." Journal of Applied Mathematics and Stochastic Analysis 7, no. 2 (January 1, 1994): 111–24. http://dx.doi.org/10.1155/s1048953394000134.

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We consider a single server infinite capacity queueing system, where the arrival process is a batch Markovian arrival process (BMAP). Particular BMAPs are the batch Poisson arrival process, the Markovian arrival process (MAP), many batch arrival processes with correlated interarrival times and batch sizes, and superpositions of these processes. We note that the MAP includes phase-type (PH) renewal processes and non-renewal processes such as the Markov modulated Poisson process (MMPP).The server applies Kella's vacation scheme, i.e., a vacation policy where the decision of whether to take a new vacation or not, when the system is empty, depends on the number of vacations already taken in the current inactive phase. This exhaustive service discipline includes the single vacation T-policy, T(SV), and the multiple vacation T-policy, T(MV). The service times are i.i.d. random variables, independent of the interarrival times and the vacation durations. Some important performance measures such as the distribution functions and means of the virtual and the actual waiting times are given. Finally, a numerical example is presented.

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21

Dudin, Alexander N., and Srinivas R. Chakravarthy. "Multi-threshold control of the BMAP/SM/1/K queue with group services." Journal of Applied Mathematics and Stochastic Analysis 16, no. 4 (January 1, 2003): 327–47. http://dx.doi.org/10.1155/s1048953303000261.

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We consider a finite capacity queue in which arrivals occur according to a batch Markovian arrival process (BMAP). The customers are served in groups of varying sizes. The services are governed by a controlled semi-Markovian process according to a multithreshold strategy. We perform the steady-state analysis of this model by computing (a) the queue length distributions at departure and arbitrary epochs, (b) the Laplace-Stieltjes transform of the sojourn time distribution of an admitted customer, and (c) some selected system performance measures. An optimization problem of interest is presented and some numerical examples are illustrated.

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22

Kaspi, Haya, and David Perry. "On a duality between a non-Markoyian storage/production process and a Markovian dam process with state-dependent input and output." Journal of Applied Probability 26, no. 4 (December 1989): 835–44. http://dx.doi.org/10.2307/3214388.

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We consider a storage/production process where the production rate is state dependent, the demand arrival is a renewal process, and the sizes of the demands are i.i.d exponentially distributed random variables. The resulting content process is non-Markovian but regenerative. We construct a dual Markovian dam process with drift, jump rate and jump sizes that are state dependent and use it to compute the limiting one-dimensional distribution of the content process.

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23

Kaspi, Haya, and David Perry. "On a duality between a non-Markoyian storage/production process and a Markovian dam process with state-dependent input and output." Journal of Applied Probability 26, no. 04 (December 1989): 835–44. http://dx.doi.org/10.1017/s0021900200027704.

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We consider a storage/production process where the production rate is state dependent, the demand arrival is a renewal process, and the sizes of the demands are i.i.d exponentially distributed random variables. The resulting content process is non-Markovian but regenerative. We construct a dual Markovian dam process with drift, jump rate and jump sizes that are state dependent and use it to compute the limiting one-dimensional distribution of the content process.

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24

Ledoux, James. "Filtering and the EM-Algorithm for the Markovian Arrival Process." Communications in Statistics - Theory and Methods 36, no. 14 (October 22, 2007): 2577–93. http://dx.doi.org/10.1080/03610920701271038.

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25

Alfa, Attahiru Sule, and Marcel F. Neuts. "Modelling Vehicular Traffic Using the Discrete Time Markovian Arrival Process." Transportation Science 29, no. 2 (May 1995): 109–17. http://dx.doi.org/10.1287/trsc.29.2.109.

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26

Ren, Jiandong. "Analysis of Insurance Claim Settlement Process with Markovian Arrival Processes." Risks 4, no. 1 (March 11, 2016): 6. http://dx.doi.org/10.3390/risks4010006.

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27

CHAKRAVARTHY, SRINIVAS R. "A MULTI-SERVER QUEUEING MODEL WITH MARKOVIAN ARRIVALS AND MULTIPLE THRESHOLDS." Asia-Pacific Journal of Operational Research 24, no. 02 (April 2007): 223–43. http://dx.doi.org/10.1142/s0217595907001164.

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We consider a multi-server queueing model in which arrivals occur according to a Markovian arrival process (MAP). There is a single-server and additional (backup) servers are added or removed depending on sets of thresholds. The service times are assumed to be exponential and the servers are assumed to be homogeneous. A comparison of this model to the classical MAP/M/c queueing model through an optimization problem yields some interesting results that are useful in practical applications. For example, we notice that positively correlated arrival process appears to benefit with the threshold type queueing model. We also give the minimum delay costs and the associated maximum setup costs so that the threshold type queueing model is to be preferred over the classical MAP/M/c model.

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28

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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29

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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30

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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31

Fralix, Brian. "A TIME-DEPENDENT STUDY OF THE KNOCKOUT QUEUE." Probability in the Engineering and Informational Sciences 27, no. 3 (March 28, 2013): 309–17. http://dx.doi.org/10.1017/s0269964813000041.

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We examine the time-dependent behavior of a birth–death process, whose birth rates and death rates are decreasing and increasing, respectively, with respect to the current state. Such models can be used to describe Markovian queueing systems with exponential reneging, where potential arrivals balk with a certain probability that depends on the number of customers observed upon arrival. Our results are derived by interpreting the birth–death process as the queue-length process of what we refer to as the “knockout queue.”

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32

Koops, D. T., M. Saxena, O. J. Boxma, and M. Mandjes. "Infinite-server queues with Hawkes input." Journal of Applied Probability 55, no. 3 (September 2018): 920–43. http://dx.doi.org/10.1017/jpr.2018.58.

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Abstract In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and nonexponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy tailed, and we consider a heavy-traffic regime. We conclude by discussing how our results can be used computationally and by verifying the numerical results via simulations.

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33

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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34

Liu, Mei. "Analysis of retrial queue with heterogeneous servers and Markovian arrival process." Informatics 17, no. 1 (March 29, 2020): 29–38. http://dx.doi.org/10.37661/1816-0301-2020-17-1-29-38.

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Multi-server retrial queueing system with heterogeneous servers is analyzed. Requests arrive to the system according to the Markovian arrival process. Arriving primary requests and requests retrying from orbit occupy an available server with the highest service rate, if there is any available server. Otherwise, the requests move to the orbit having an infinite capacity. The total retrial rate infinitely increases when the number of requests in orbit increases. Service periods have exponential distribution. Behavior of the system is described by multi-dimensional continuous-time Markov chain which belongs to the class of asymptotically quasi-toeplitz Markov chains. This allows to derive simple and transparent ergodicity condition and compute the stationary probabilities distribution of chain states. Presented numerical results illustrate the dynamics of some system effectiveness indicators and the importance of considering of correlation in the requests arrival process.

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35

Neuts, Marcel F., and Attahiru Sule Alfa. "Pair formation in a Markovian arrival process with two event labels." Journal of Applied Probability 41, no. 04 (December 2004): 1124–37. http://dx.doi.org/10.1017/s002190020002088x.

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The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

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36

Kim, Jiseung, Alexander Dudin, Sergey Dudin, and Chesoong Kim. "Analysis of a semi-open queueing network with Markovian arrival process." Performance Evaluation 120 (April 2018): 1–19. http://dx.doi.org/10.1016/j.peva.2017.12.005.

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37

Economou, A., and A. Gómez-Corral. "The Batch Markovian Arrival Process Subject to Renewal Generated Geometric Catastrophes." Stochastic Models 23, no. 2 (May 8, 2007): 211–33. http://dx.doi.org/10.1080/15326340701300761.

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38

Neuts, Marcel F. "Descriptors for point processes based on runs: the Markovian arrival process." Journal of Applied Mathematics and Stochastic Analysis 9, no. 4 (January 1, 1996): 469–88. http://dx.doi.org/10.1155/s104895339600041x.

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This paper is part of a broader investigation of properties of a point process that can be identified by imagining that the process is involved in a competition for the generation of runs of events. The general purpose of that methodology is to quantify the prevalence of gaps and bursts in realizations of the process. The Markovian arrival process (MAP) is highly versatile in qualitative behavior and its analysis is numerically tractable by matrix-analytic methods. It can therefore serve well as a benchmark process in that investigation. In this paper, we consider the MAP and a regular grid competing for runs of lengths at least r1 and r2, respectively. A run of length r in one of the processes is defined as a string of r successive events occurring without an intervening event in the other process.This article is dedicated to the memory of Roland L. Dobrushin.

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39

Neuts, Marcel F., and Attahiru Sule Alfa. "Pair formation in a Markovian arrival process with two event labels." Journal of Applied Probability 41, no. 4 (December 2004): 1124–37. http://dx.doi.org/10.1239/jap/1101840557.

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The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

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40

Ramírez-Cobo, Pepa, Xavier Marzo, Alba V. Olivares-Nadal, José Álvarez Francoso, Emilio Carrizosa, and M. Fernanda Pita. "The Markovian arrival process: A statistical model for daily precipitation amounts." Journal of Hydrology 510 (March 2014): 459–71. http://dx.doi.org/10.1016/j.jhydrol.2013.12.033.

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41

Naumov, Valeriy A., Yuliya V. Gaidamaka, and Konstantin E. Samouylov. "On Two Interacting Markovian Queueing Systems." Mathematics 7, no. 9 (September 1, 2019): 799. http://dx.doi.org/10.3390/math7090799.

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In this paper, we study a Markovian queuing system consisting of two subsystems of an arbitrary structure. Each subsystem generates a multi-class Markovian arrival process of customers arriving to the other subsystem. We derive the necessary and sufficient conditions for the stationary distribution to be of product form and consider some particular cases of the subsystem interaction for which these conditions can be easily verified.

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42

Sharifnassab, Arsalan, John N. Tsitsiklis, and S. Jamaloddin Golestani. "Fluctuation Bounds for the Max-Weight Policy with Applications to State Space Collapse." Stochastic Systems 10, no. 3 (September 2020): 223–50. http://dx.doi.org/10.1287/stsy.2019.0038.

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We consider a multihop switched network operating under a max-weight scheduling policy and show that the distance between the queue length process and a fluid solution remains bounded by a constant multiple of the deviation of the cumulative arrival process from its average. We then exploit this result to prove matching upper and lower bounds for the time scale over which additive state space collapse (SSC) takes place. This implies, as two special cases, an additive SSC result in diffusion scaling under non-Markovian arrivals and, for the case of independent and identically distributed arrivals, an additive SSC result over an exponential time scale.

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43

Kumar, Nitin, and U. C. Gupta. "A Renewal Generated Geometric Catastrophe Model with Discrete-Time Markovian Arrival Process." Methodology and Computing in Applied Probability 22, no. 3 (January 21, 2020): 1293–324. http://dx.doi.org/10.1007/s11009-019-09768-8.

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Montoro-Cazorla, Delia, Rafael Pérez-Ocón, and Maria del Carmen Segovia. "Replacement policy in a system under shocks following a Markovian arrival process." Reliability Engineering & System Safety 94, no. 2 (February 2009): 497–502. http://dx.doi.org/10.1016/j.ress.2008.06.007.

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Nielsen, Bo Friis, L. A. Fredrik Nilsson, Uffe H⊘gsbro Thygesen, and Jan E. Beyer. "Higher Order Moments and Conditional Asymptotics of the Batch Markovian Arrival Process." Stochastic Models 23, no. 1 (February 7, 2007): 1–26. http://dx.doi.org/10.1080/15326340601141844.

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Matendo, Sadrac K. "Some performance measures for vacation models with a batch Markovian arrival process." Journal of Applied Mathematics and Stochastic Analysis 6, no. 4 (January 1, 1993): 412. http://dx.doi.org/10.1155/s1048953393000358.

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Schellhaas, Helmut. "Single server queues with a batch Markovian arrival process and server vacations." OR Spektrum 15, no. 4 (December 1994): 189–96. http://dx.doi.org/10.1007/bf01719449.

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Chydzinski, Andrzej, and Blazej Adamczyk. "Transient and Stationary Losses in a Finite-Buffer Queue with Batch Arrivals." Mathematical Problems in Engineering 2012 (2012): 1–17. http://dx.doi.org/10.1155/2012/326830.

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We present an analysis of the number of losses, caused by the buffer overflows, in a finite-buffer queue with batch arrivals and autocorrelated interarrival times. Using the batch Markovian arrival process, the formulas for the average number of losses in a finite time interval and the stationary loss ratio are shown. In addition, several numerical examples are presented, including illustrations of the dependence of the number of losses on the average batch size, buffer size, system load, autocorrelation structure, and time.

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Banik, A. D., M. L. Chaudhry, and James J. Kim. "A Note on the Waiting-Time Distribution in an Infinite-Buffer GI[X]/C-MSP/1 Queueing System." Journal of Probability and Statistics 2018 (September 2, 2018): 1–10. http://dx.doi.org/10.1155/2018/7462439.

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This paper deals with a batch arrival infinite-buffer single server queue. The interbatch arrival times are generally distributed and arrivals are occurring in batches of random size. The service process is correlated and its structure is presented through a continuous-time Markovian service process (C-MSP). We obtain the probability density function (p.d.f.) of actual waiting time for the first and an arbitrary customer of an arrival batch. The proposed analysis is based on the roots of the characteristic equations involved in the Laplace-Stieltjes transform (LST) of waiting times in the system for the first, an arbitrary, and the last customer of an arrival batch. The corresponding mean sojourn times in the system may be obtained using these probability density functions or the above LSTs. Numerical results for some variants of the interbatch arrival distribution (Pareto and phase-type) have been presented to show the influence of model parameters on the waiting-time distribution. Finally, a simple computational procedure (through solving a set of simultaneous linear equations) is proposed to obtain the “R” matrix of the corresponding GI/M/1-type Markov chain embedded at a prearrival epoch of a batch.

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Krishnamoorthy, Achyutha, Anu Nuthan Joshua, and Dmitry Kozyrev. "Analysis of a Batch Arrival, Batch Service Queuing-Inventory System with Processing of Inventory While on Vacation." Mathematics 9, no. 4 (February 20, 2021): 419. http://dx.doi.org/10.3390/math9040419.

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A single-server queuing-inventory system in which arrivals are governed by a batch Markovian arrival process and successive arrival batch sizes form a finite first-order Markov chain is considered in this paper. Service is provided in batches according to a batch Markovian service process, with consecutive service batch sizes forming a finite first-order Markov chain. A service starts for the next batch on completion of the current service, provided that inventory is available at that epoch; otherwise, there will be a delay in starting the next service. When the service of a batch is completed, the inventory decreases by 1 unit, irrespective of batch size. A control policy in which the server goes on vacation when a service process is frozen until a quorum can initiate the next batch service is proposed to ensure idle-time utilization. During the vacation, the server produces inventory (items) for future services until it hits a specified level L or until the number of customers in the system reaches a maximum service batch size N, with whichever occurring first. In the former case, a server stays idle once the processed inventory level reaches L until the number of customers reaches (or even exceeds because of batch arrival) a maximum service batch size N. The time required for processing one unit of inventory follows a phase-type distribution. In this paper, the steady-state probability vector of this infinite system is computed. The distributions of inventory processing time in a vacation cycle, idle time in a vacation cycle, and vacation cycle length are found. The effect of correlation in successive inter-arrival times and service times on performance measures for such a queuing system is illustrated with a numerical example. An optimization problem is considered. The proposed system is then compared with a queuing-inventory system without the Markov-dependent assumption on successive arrivals as well as service batch sizes using numerical examples.

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