Queueing and risk models with dependencies
The goal of the present paper is to explore a class of stochastic recursions \({W_{i + 1}} = \max ({W_i} + {B_i} - {A_i},0)\), \(i = 1,2,\dots\), in which there are no independence assumptions concerning \({B_i}\) and \({A_i}\). In Sections 2 and 3, the authors assume that \({A_i} = c({W_i} + {B_i}) + {E_i}\), where \(({E_i})\) is an independent sequence of exponentially distributed random variables (in both sections some relaxation of the exponentiality assumption is possible) and where \(0 < c < 1\). The key performance measure under consideration in Section 2 is the steady-state waiting time; its Laplace-Stieltjes transform (LST) is presented in Theorem 2.1. The key performance measure in Section 3 is the time \(\tau(x)\) until the system becomes empty, when the initial capital is \(x\). The LST (with respect to that initial capital \(x\)) of \(\tau(x)\) is derived in Theorem 3.1. Section 4 studies the Cramér-Lundberg insurance risk model. In this model, generally distributed claims arrive according to a Poisson process, and in between claims, the capital of the insurance company increases at a fixed premium rate. When the capital, right after the \(i\)th claim, equals \(y\), then the next interclaim time \({A_i} = \max (0,{E_i} - cy)\), where \(({E_i})\) is an independent sequence of \(\exp(\lambda)\) distributed random variables. In Theorem 4.1, the authors determine the LST of \(\tau(x)\), with respect to initial capital \(x\). In Section 5, they assume that \({B_i} = \max ({E_i} - c{W_i},0)\), where \(({E_i})\) is an independent sequence of \(\exp(\mu)\) distributed random variables. They derive the steady-state waiting time and workload LST (Theorem 5.1).
scientific article; zbMATH DE number 7613926
scientific article