Busy period distribution in state-dependent queues
A state-dependent M/M/1 queueing system is considered. It is assumed that the arrival rate \(\lambda =\lambda (v(t))\) depends on the current work v(t) in the system and that the mean service time of a job arriving at time moment \(t^*\) is \(1/\mu (v(t^*))\). The arrival rate is fast while the mean service requests are supposed to be small. The density function \(p(x,t)=(\partial /\partial t)P_ r\{T<t| v(0)=x\}\) of residual length of a busy period is studied using singular perturbation techniques (here \(T=\inf \{t:v(t)=0\})\). Approximations to p(x,t) for two different cases are constructed: the first case has \(r(x)=\lambda (x)/\mu (x)<1\), \(x\in [0,\infty)\); the second case has \(r(x)<1\), \(x\in (0,\infty)\) and \(r(0)=1\).
scientific article; zbMATH DE number 4080589