Lévy-driven polling systems and continuous-state branching processes

In this paper we consider a ring of N = 1 queues served by a single server in a cyclic order. After having served a queue (according to a service discipline that may vary from queue to queue), there is a switch-over period and then the server serves the next queue and so forth. This model is known in the literature as a polling model. Each of the queues is fed by a non-decreasing Lévy process, which can be different during each of the consecutive periods within the server's cycle. The N-dimensional Lévy processes obtained in this fashion are described by their (joint) Laplace exponent, thus allowing for non-independent input streams. For such a system we derive the steady-state distribution of the joint workload at embedded epochs, i.e. polling and switching instants. Using the Kella-Whitt martingale, we also derive the steady-state distribution at an arbitrary epoch. Our analysis heavily relies on establishing a link between fluid (Lévy input) polling systems and multi-type Jirina processes (continuous-state discrete-time branching processes). This is done by properly defining the notion of the branching property for a discipline, which can be traced back to Fuhrmann and Resing. This definition is broad enough to contain the most important service disciplines, like exhaustive and gated

Queue lengths and workloads in polling systems

We consider a polling system: a queueing system of $N\ge 1$ queues with Poisson arrivals $Q_1,...,Q_N$ visited in a cyclic order (with or without switchover times) by a single server. For this system we derive the probability generating function $\mathscr Q(\cdot)$ of the joint queue length distribution at an arbitrary epoch in a stationary cycle, under no assumptions on service disciplines. We also derive the Laplace-Stieltjes transform $\mathscr W(\cdot)$ of the joint workload distribution at an arbitrary epoch. We express $\mathscr Q$ and $\mathscr W$ in the probability generating functions of the joint queue length distribution at visit beginnings, ${\mathscr V}_{b_i}(\cdot)$, and visit completions, ${\mathscr V}_{c_i}(\cdot)$, at $Q_i$, $i=1,...,N$. It is well known that ${\mathscr V}_{b_i}$ and ${\mathscr V}_{c_i}$ can be computed in a broad variety of cases. Furthermore, we establish a workload decomposition result

On the functional limits for partial sums under stable law

For the partial sums (Sn) of independent random variables we define a stochastic process sn(t)colon, equals(1/dn)¿k=[nt](Sk/k-µ) and prove that View the MathML source a.s. if and only if View the MathML source, for some sequence (dn) and distribution Gt. We also prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables attracted to an a-stable law with aset membership, variant(1,2]

Sample path properties of reflected Gaussian processes

We consider a stationary queueing process QX fed by a centered Gaussian process X with stationary increments and variance function satisfying classical regularity conditions. A criterion when, for a given function f, P(QX(t) > f(t) i.o.) equals 0 or 1 is provided. Furthermore, an Erdös–Révész type law of the iterated logarithm is proven for the last passage time ξ(t) = sup{s : 0 ≤ s ≤ t, QX(s) ≥ f(s)}. Both of these findings extend previously known results that were only available for the case when X is a fractional Brownian motion