Infinite-server queueing model with MAPkGk Markov arrival streams, random volume of customers in random environment subject to catastrophe

TL;DR

This paper analyzes an infinite-server queue in a semi-Markov environment with multiple Markov arrival streams, random customer resources, and catastrophes, deriving distributions and moments for system states.

Contribution

It introduces a novel queueing model incorporating multiple Markov arrival streams, random resources, and catastrophes, with explicit distributional results.

Findings

Derived transient and stationary distributions of customer numbers and resources.

Established moments of accumulated resources in the system.

Applied Danzig collective marks and renewal theory methods for analysis.

Abstract

In this paper the infinite server queue model in semi-Markov random environment with k Markov arrival streams, random resources of customers, and catastrophes is considered. After catastrophes occur, all customers in the model are flashed out and the system jumps into recovery station. After the recovery time the model works from the empty state. The transient and stationary joint distributions of numbers of different types of customers in the model at moment t, numbers of different types of served in some interval customers, volume of accumulated resources in the model at moment t, and total volume of served resources in an interval for the model without catastrophes are found. The transient and stationary joint distributions of numbers of different types of customers in the model at moment t, and volume of accumulated resources in the model at moment t and their moments for the model with catastrophes are obtained. All results are obtained using Danzig collective marks method and renewal theory methods.