On The Waiting Time for A M/M/1 Queue with Impatience

Abstract

This paper focuses on the problem of modeling the correspondence pattern for ordinary people. Suppose that letters arrive at a rate $λ$ and are answered at a rate $μ$. Furthermore, we assume that, for a constant $T$, a letter is disregarded when its waiting time exceeds $T$, and the remains are answered in {\it last in first out} order. Let $W_n$ be the waiting time of the $n$-th {\it answered} letter. It is proved that $W_n$ converges weekly to $W_T$, a non-negative random variable which possesses a density with {\it power-law} tail when $λ=μ$ and with exponential tail otherwise. Note that this may provide a reasonable explanation to the phenomenons reported by Oliveira and Barabási in \cite{OB}.

10 pages