On cycle maxima, first passage problems and extreme value theory for queues

The distribution of the maximum ⫫(C) = sup_0≤t≤C V(t) of the virtual waiting time during a cycle C is studied for a variety of queueing models. For the M/M/1 queue, the idea is a generalization of the ladder height representation of the steady-state limit V(∞), and the results are explicit in terms of the failure rate r(u) and the density. For queues with a general Markovian arrival process and phase-type service times, the basic idea is to represent the distribution of ⫫(C) by means of a multivariate version r(u) of the failure rate which again is related to generalized ladder heights. The fundamental step in the evaluation of r(u) is the determination of a set of first passage probabilities, which can be done either by solving a set of linear equations, or by deriving a matrix Ricatti differential equation having an explicit matrix-exponential solution; both approaches require the steady-state characteristics. For extreme value theory in the form of studying the asymptotic behaviour of ⫫(t) as t → ∞, essentially only the tail characteristics of ⫫(C) are needed, and are derived by a change of measure via exponential families. The paper also contains new material on steady state solutions of queues with a Markovian arrival process, some basic formulas in extreme value theory for semi-regenerative processes and a simple proof of Takács’ formula for the distribution of ⫫(C) in the M/G/1 case.