Wladyslaw Szczotka

STATIONARY REPRESENTATION OF QUEUES. II

The paper deals with the asymptotic behaviour of queues for which the generic sequence (v, u)= {(vk, Uk), k i 1_} is not necessarily stationary but is asymptotically stationary in some sense. The latter property is defined by an appropriate type of convergence of probability distributions T (v,, u,), n 1, df of the sequences (va, u,) = {(v,,+k, U+k), k i 1) to the distribution T(vo, uo) of a stationary sequence (vo, u) df {(vo, u ), k > 1}. We consider six types of convergence of ?(v,, u,) to ?T(vo, u0). The main result is as follows: if the sequence of the distributions L(v,, un), nl 1, converges in one of six ways then the sequence of df

Exponential approximation of waiting time and queue size for queues in heavy traffic

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments

Asymptotic behaviour of random walks with correlated temporal structure

We introduce a continuous-time random walk process with correlated temporal structure. The dependence between consecutive waiting times is generated by weighted sums of independent random variables combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. We derive the corresponding diffusion limit and prove its subdiffusive character. Analysing the set of corresponding coupled Langevin equations, we verify the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking.

Correlated continuous-time random walks?scaling limits and Langevin picture

In this paper we analyze correlated continuous-time random walks introduced recently by Tejedor and Metzler (2010 J. Phys. A: Math. Theor. 43 082002). We obtain the Langevin equations associated with this process and the corresponding scaling limits of their solutions. We prove that the limit processes are self-similar and display anomalous dynamics. Moreover, we extend the model to include external forces. Our results are confirmed by Monte Carlo simulations.

Conditioned limit theorem for the pair of waiting time and queue line processes

AbstractThe pair (W(t), L(t)t⩾0, of the virtual waiting time and the queue line processes is considered in the GI/G/1 queueing system with the traffic intensity one. An asymptotic of

ASYMPTOTIC STATIONARITY OF QUEUEING PROCESSES

\left( {\frac{1}{{\sqrt t }}W(t), \frac{1}{{\sqrt t }}L(t)} \right)

Langevin Picture of Lévy Walks and Their Extensions

) conditioned on the event {T>t} is given ast→∞, whereT is the length of the first busy period. A similar result is also given in the situation whent runs over the arrival moments of customers.

ASYMPTOTIC STATIONARITY OF QUEUES IN SERIES AND THE HEAVY TRAFFIC APPROXIMATION

This study concerns the waiting time wk of the kth arrival to a single-server queueing system and the queue length lk just before the kth arrival. The first issue is whether the standard heavy-traffic limit distribution of these variables is the only possible limit. The second issue is the validity of the approximation 3, for large k, where [upsilon] is the average service time. The main results show that there are three types of heavy-traffic limiting distributions of the waiting times and queue lengths depending on whether the queueing systems are stable, marginally stable or unstable. Furthermore, these limit theorems justify the approximation 3 for the three heavy-traffic regimes and they characterize the asymptotic distribution of the difference 1. The results apply, in particular, to the GI[+45 degree rule]G[+45 degree rule]1 system and systems in which the service and interarrival times are stationary, regenerative, semi-stationary, asymptotically stationary and their sums satisfy certain functional limit laws. They also apply to queues that may not satisfy standard assumptions.