The paper deals with GI/GI/1 queueing system with impatient customers: the n-th customer joins the queue only if his patience time \(g_ n\) is greater than \(w_ n\) (workload of the server just before the arrival epoch \(T_ n\) of this customer). Let \(s_ n\) be the service time of the n-th customer, \(t_ n=T_ n-T_{n-1}\), \(\{g_ n\}\), \(\{t_ n\}\), \(\{s_ n\}\) be sequences consisting of i.i.d.r.v.’s, \(A(t)=P(t_ n\leq t)\), \(Et_ n=\lambda^{-1}\), \(a=\inf\{t:A(t)=1\}\), \(B(t)=P(s_ n\leq t)\), \(Es_ n=\mu^{-1}\), \(b=\sup\{t:B(t)=0\},\rho =\lambda\mu^{-1}\), \(G(t)=P(g_ n>t)\). Below we list some of the main results of the paper.
Result 1. If \(b\)-a\(<0\) then the state 0 is recurrent for the Markov chain \(\{w_ n\}\). If in addition \(1-\rho G(\infty)>0\) then \(\{w_ n\}\) is ergodic.
Result 2. Let \(\pi\) be a probability to reject an arbitrary taken customer: \(\pi =\lim_{n\to\infty }P(g_ n<w_ n)\). Then \(\pi=1-Em_ 1/E\ell_ 1\), where \(m_ 1 (\ell_ 1)\) is the number of customers successfully served (arrived to the system) during the busy period.
Result 3. Let \(V_{\infty} (W_{\infty})\) be the limit virtual (actual) waiting time. Then \(E \exp (-sV_{\infty})=P(V_{\infty}=0)+(1- \pi)a(s)E \exp (-sW_{\infty})\) where \(a(s)=\rho\), if \(s=0\) and \(a(s)=\lambda E[(1-\exp (-ss_ 1))/s],\) if \(s>0\). It follows that \(V_{\infty}=^{d}W_{\infty}\) for \(\{t_ n\}\) being exponentially distributed.