The TCP window size process can be modeled as a piecewise-deterministic Markov process that increases linearly and experiences downward jumps at Poisson times. We present a transient analysis of this window size process. Our main result is the Laplace transform of the transient moments. Formulae for the integer and fractional moments are derived, as well as an explicit characterization of the speed of convergence to steady state. Central to our approach are the infinitesimal generator and Dynkin's martingale.

We consider a Markovian growth collapse process on the state space E = [0, ∞) which evolves as follows. Between random downward jumps the process increases with slope one. Both the jump intensity and the jump sizes depend on the current state of the process. We are interested in the behavior of the first hitting time τ y = inf{t ≥ 0|X t = y} as y becomes large and the growth of the maximum process M t = sup{X s |0 ≤ s ≤ t} as t → ∞. We consider the recursive sequence of equations Am n = m n−1 , m 0 ≡ 1, where A is the extended generator of the MGCP, and show that the solution sequence (which is essentially unique and can be given in integral form) is related to the moments of τ y . The Laplace transform of τ y can be expressed in closed form (in terms of an integral involving a certain kernel) in a similar way. We derive asymptotic results for the running maximum: (i) if m 1 (y) is of rapid variation, we have M t /m −1 (t)

We consider the length of a busy period in the M/D/∞ queue and show that it coincides with the sojourn time of the first customer in an M/D/1 processor-sharing queue. We further show that the busy period is intimately related with the stationary waiting time in the M/D/1 first-come-first-served queue. We present three characterizations for the distribution function of the busy period and an asymptotic expression for its tail distribution. The latter involves complexvalued branches of the Lambert W function.

We consider a risk model with threshold strategy, where the insurance company pays off a certain percentage of the income as dividend whenever the current surplus is larger than a given threshold. We investigate the ruin time, ruin probability and the total dividend, using methods and results from queueing theory.

We consider a make-to-stock production-inventory model with one machine that produces stock in a buffer. The machine is subject to breakdowns. During up periods, the machine fills the buffer at a level-dependent rate α(x) > 0. During down periods, the production rate is zero, and the demand rate is either β(x) > 0 or γ (x) > 0 when the inventory level is x; which of the two demand rates applies depends on a hysteretic control policy. Such a policy is used to avoid undesirable rapid switching. In the context of our paper hysteresis is introduced in the form of bang-bang control. Namely, there are two switchover levels such that when the buffer content reaches the higher level we change the drift downward and when the buffer content reaches the lower level we change the drift upward. We determine the conditions under which the steady-state distribution of the inventory level exists, and we derive that distribution. Other performance measures under consideration are the number of switches from β(·) to γ (·) per busy period, the busy period distribution, and the overshoot above a particular hysteretic level.