Batch Renewal Arrival Process Subject to Geometric Catastrophes

Int. J. Appl. Comput. Math

Singh

2021

Self Cite

In this paper, we consider a stochastic model in which the population grows according to the batch Markovian arrival process and is subjected to renewal generated geometric catastrophes. Our analytical work starts from the vector generating function (VGF) of the population size at post-catastrophe epoch. We develop a methodology for extracting the population size distribution at post-catastrophe epoch from the VGF, which is based on the inversion of VGF using the roots method. The method is analytically quite simple and easy to implement. Further, we obtain the population size distribution at arbitrary, pre-catastrophe and pre-arrival epochs along with their factorial moments. To show the applicability and correctness of the proposed methodology, we match our results with the available ones in special cases and present several numerical examples for different inter-catastrophe time distributions. Moreover, we investigate the effect of key parameters on the system performance and display the results in the form of graphs along with a detailed description.

show abstract

“…After simplification ( 53 ) gives and using ( 56 ) in ( 54 ), we get The results given in ( 55 ) and ( 58 ) match exactly with the one given in [ 5 , Eq. 30].…”

Section: Special Casessupporting

confidence: 80%

Section: Special Casessupporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

“…In Barbhuiya et al. [ 5 ] and Kumar et al. [ 21 ], the authors investigate a geometric catastrophe population model in which population grows as per the batch renewal arrival process.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computational and Numerical Investigation of the Batch Markovian Arrival Process Subject to Renewal Generated Geometric Catastrophes

Int. J. Appl. Comput. Math

Singh

2021

Self Cite

show abstract

“…Meanwhile, the occurrence of a disaster simultaneously removes all the present customers in the system thus making the system idle. Disasters are also known by the terms catastrophic events (Barbhuiya et al [7]), mass exodus (Chen and Renshaw [12]) or queue flushing (Towsley and Tripathi [27]). Both, a negative customer and a disaster have no impact on the system when it is empty.…”

Section: Introductionmentioning

confidence: 99%

A Queueing System with Batch Renewal Input and Negative Arrivals

Infosys Science Foundation Series

Barbhuiya

2020

Self Cite

This paper studies an infinite buffer single server queueing model with exponentially distributed service times and negative arrivals. The ordinary (positive) customers arrive in batches of random size according to renewal arrival process, and join the queue/server for service. The negative arrivals are characterized by two independent Poisson arrival processes, a negative customer which removes the positive customer undergoing service, if any, and a disaster which makes the system empty by simultaneously removing all the positive customers present in the system. Using the supplementary variable technique and difference equation method we obtain explicit formulae for the steady-state distribution of the number of positive customers in the system at pre-arrival and arbitrary epochs. Moreover, we discuss the results of some special models with or without negative arrivals along with their stability conditions. The results obtained throughout the analysis are computationally tractable as illustrated by few numerical examples. Furthermore, we discuss the impact of the negative arrivals on the performance of the system by means of some graphical representations.

show abstract

A Renewal Generated Geometric Catastrophe Model with Discrete-Time Markovian Arrival Process

Methodol Comput Appl Probab

2020