Server waiting times in infinite supply polling systems with preparation times

We consider a system consisting of a single server serving a ¿xed number of stations. At each station there is an in¿nite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems, and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a suf¿cient condition for the existence of a limiting waiting-time distribution and we study the tail behavior of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system’s parameters to the performance of the server for both models

Queues and risk processes with dependencies

We study the generalization of the G/G/1 queue obtained by relaxing the assumption of independence between inter-arrival times and service requirements. The analysis is carried out for the class of multivariate matrix exponential distributions introduced in [12]. In this setting, we obtain the steady state waiting time distribution and we show that the classical relation between the steady state waiting time and the workload distributions re- mains valid when the independence assumption is relaxed. We also prove duality results with the ruin functions in an ordinary and a delayed ruin process. These extend several known dualities between queueing and risk models in the independent case. Finally we show that there exist stochastic order relations between the waiting times under various instances of correlation

A relative entropy rate method for path space sensitivity analysis of stationary complex stochastic dynamics

We propose a new sensitivity analysis methodology for complex stochastic dynamics based on the Relative Entropy Rate. The method becomes computationally feasible at the stationary regime of the process and involves the calculation of suitable observables in path space for the Relative Entropy Rate and the corresponding Fisher Information Matrix. The stationary regime is crucial for stochastic dynamics and here allows us to address the sensitivity analysis of complex systems, including examples of processes with complex landscapes that exhibit metastability, non-reversible systems from a statistical mechanics perspective, and high-dimensional, spatially distributed models. All these systems exhibit, typically non-gaussian stationary probability distributions, while in the case of high-dimensionality, histograms are impossible to construct directly. Our proposed methods bypass these challenges relying on the direct Monte Carlo simulation of rigorously derived observables for the Relative Entropy Rate and Fisher Information in path space rather than on the stationary probability distribution itself. We demonstrate the capabilities of the proposed methodology by focusing here on two classes of problems: (a) Langevin particle systems with either reversible (gradient) or non-reversible (non-gradient) forcing, highlighting the ability of the method to carry out sensitivity analysis in non-equilibrium systems; and, (b) spatially extended Kinetic Monte Carlo models, showing that the method can handle high-dimensional problems

Combined analysis of transient delay characteristics and delay autocorrelation function in the Geo(X)/G/1 queue

We perform a discrete-time analysis of customer delay in a buffer with batch arrivals. The delay of the kth customer that enters the FIFO buffer is characterized under the assumption that the numbers of arrivals per slot are independent and identically distributed. By using supplementary variables and generating functions, z-transforms of the transient delays are calculated. Numerical inversion of these transforms lead to results for the moments of the delay of the kth customer. For computational reasons k cannot be too large. Therefore, these numerical inversion results are complemented by explicit analytic expressions for the asymptotics for large k. We further show how the results allow us to characterize jitter-related variables, such as the autocorrelation of the delay in steady state