An infinite-server queueing model MMAPkGk in semi-Markov random environment with marked MAP arrival and subject to catastrophes

TL;DR

This paper develops a comprehensive mathematical model for an infinite-server queue with complex arrival and catastrophe processes, deriving generating functions and equations for analyzing system behavior over time.

Contribution

It introduces a novel queueing model with semi-Markov environment, marked MAP arrivals, and catastrophes, providing new analytical tools for performance analysis.

Findings

Derived joint PGFs for transient and stationary distributions.

Obtained Laplace transforms of total accumulated resources.

Established differential and renewal equations for queue size distributions.

Abstract

In the present paper the infinite-server MMAPkGk queueing model with random resource vector of customers, marked MAP arrival and semi-Markov (SM) arrival of catastrophes is considered. The joint generating functions (PGF) of transient and stationary distributions of number of busy servers and numbers of different types served customers, as well as Laplace transformations (LT) of joint distributions of total accumulated resources in the model at moment and total accumulated resources of served customers during time interval are found. The basic differential and renewal equations for transient and stationary PGF of queue sizes of customers are found.