It is well known that the M/G/1 busy-period density can be characterized by the Kendall functional equation for its Laplace transform. The Kendall functional equation can be solved iteratively to obtain transform values to use in numerical inversion algorithms. However, we show that the busy-period density can also be numerically inverted directly, without iterating a functional equation, explo...

We derive performance measures for burst arrival (e.g. messages of variable length packets) M / G / 1 queues with server vacations, controlled by the, so called, Randomly Timed Gated (RTG) protocol, operating as follows : Whenever the server returns from a ( general-type ) vacation and initiates a busy period, a Timer with random duration T is activated. If the server empties the queue before t...

Limit theorems are established and relatively simple closed-form approximations are developed for the busy-period distribution in single-server queues. For the M/G/1 queue, the complementary busy-period cdf is shown to be asymptotically equivalent as t → ∞ to a scaled version of the heavy-traffic limit (obtained as ρ → 1), where the scaling parameters are based on the asymptotics as t → ∞. We c...

We analyze the service times of customers in a stable M/M/1 queue in equilibrium depending on their position in a busy period. We give the law of the service of a customer at the beginning, at the end, or in the middle of the busy period. It enables as a by-product to prove that the process of instants of beginning of services is not Poisson. We then proceed to a more precise analysis. We consi...

The Israeli Queue is a polling-type system with N groups (queues) and un-limited size batch service, where the next group to be served is the one with the most senior customer. For such a system we derive (i) the mean size of each group, and (ii) the Laplace-Stieltjes Transform and mean of the length of a busy period starting with n ≥ 1 groups. Numerical calculations are presented and the param...