A Note on Queueing Systems with Erlangian Service Time Distributions

Abstract

In [1] Conolly derived an important result (3.19) for the queueing system $GI/E_k/1$. This result has been used by him to obtain the joint distribution of the number of customers served during a busy period initiated by one customer and its length for the queueing system $GI/E_k/1$. In this note we shall derive their steady state properties by using the above result of Conolly. Earlier Wishart [6] and Conolly [2] had studied these features using different methods. The queueing system $GI/E_k/1$, studied in this note is the one in which (i) the time intervals between arrivals are independent and are identically distributed according to the law $dA(t) (0 < t < \infty)$, with mean $a$ and Laplace transform $\psi(\theta)$; (ii) the queue-discipline is "first come, first served"; (iii) there is only one counter and the service-times are independent and identically distributed having a $\chi^2$ distribution, with mean $b$ and $2k$ degrees of freedom i.e. $dB(t) = \lbrack e^{-kt/b}/(k - 1) !(kt/b)^{k - 1}k dt/b; \text{and}$ (iv) the service-times are independent of the inter-arrival times. Thus we may imagine that the service takes place in $k$ consecutive phases, where the time spent in each phase is distributed negative exponentially with mean $b/k$.