On the dependence structure and bounds of correlated parallel queues and their applications to synchronized stochastic systems

Abstract

This paper studies the dependence structure and bounds of several basic prototypical parallel queueing systems with correlated arrival processes to different queues. The marked feature of our systems is that each queue viewed alone is a standard single-server queuing system extensively studied in the literature, but those queues are statistically dependent due to correlated arrival streams. The major difficulty in analysing those systems is that the presence of correlation makes the explicit computation of a joint performance measure either intractable or computationally intensive. In addition, it is not well understood how and in what sense arrival correlation will improve or deteriorate a system performance measure. The objective of this paper is to provide a better understanding of the dependence structure of correlated queueing systems and to derive computable bounds for the statistics of a joint performance measure. In this paper, we obtain conditions on arrival processes under which a performance measure in two systems can be compared, in the sense of orthant and supermodular orders, among different queues and over different arrival times. Such strong comparison results enable us to study both spatial dependence (dependence among different queues) and temporal dependence (dependence over different time instances) for a joint performance measure. Further, we derive a variety of upper and lower bounds for the statistics of a stationary joint performance measure. Finally, we apply our results to synchronized queueing systems, using the ideas combined from the theory of orthant and supermodular dependence orders and majorization with respect to weighted trees (Xu and Li (2000)). Our results reveal how a performance measure can be affected, favourably or adversely, by different types of dependencies.

Dependence orderings for some functionals of multivariate point processes

“…These results allow to obtain bounds for stochastic models with stationary (not necessary renewal) input stream. They extend results for example of Li and Xu [15,16]. As a byproduct we obtain regularity properties of sequences of stationary random variables which extend results for the iid case [18,28].…”

“…& Now, we establish comparison properties w.r.t. fU 1 n g; y; fU k n g: The first result generalizes Theorem 2.7 in [15].…”

“…The similar model was considered in [15]. Using the similar comment as in the previous example, dependence and variability in batch arrival process can be increased by dependence in batches or by dependence in nonbatch arrival process.…”

“…A number of examples of sequences comparable in a :¼ o sm ; o idcx orderings, including Markov-renewal sequences, stochastically monotone Markov chains can be found in [2][3][4]8,9,12,[15][16][17][20][21][22][23]29,32]. We write shortly H f ðm; nÞðBÞ for the above expression, and in particular, we write H f ðm; nÞðtÞ and H f ðm; nÞðIÞ for B j ¼ ð0; t j and B j ¼ I j ; j ¼ 1; y; r; respectively.…”

Multivariate risk model of phase type

“…(4.4) It follows from Proposition 5.5 in Li and Xu (2000) that the claim size vector in M is more dependent than the claim size vector in M in supermodular order. Thus, by Theorem 2.5, we have ψ sum (u) ≥ ψ sum (u) for any u ≥ 0.…”