Open Queueing Systems in Light Traffic

Abstract

Many quantities of interest in open queueing systems are expected values which can be viewed as functions of the arrival rate to the system. We are thus led to consider f(λ), where λ is the arrival rate, and f represents some quantity of interest. The aim of this paper is to investigate the behavior of f(λ), for λ near zero. This ‘light traffic’ information is obtained in the form of f(0) and its derivatives, f(n)(0), n ≥ 1. Focusing initially on Poisson arrival processes, we provide a method to calculate f(n)(0) for any ‘admissible’ function f. We describe a large class of queueing networks for which we show several standard quantities of interest to be admissible. The proof that the method yields the correct values for the derivatives involves an interchange of limits, whose justification requires a great deal of effort. The determination of f(n)(0) involves consideration of sample paths with n + 1 or fewer arrivals over all of time. These calculations are illustrated via several simple examples. These results can be extended to arrival processes which, although not Poisson, are ‘driven’ by a Poisson process. We carry out the details for phase type renewal processes and nonstationary Poisson processes.

Opinion propagation in bounded medium-sized populations

“…However due to the fact that the number of sample paths is uncountable and thus the probability of every individual path is zero, making this rigorous is non-trivial. For series expansions revolving around a Poisson process with a small rate, to which the examples in this work essentially belong, this was made rigorous by Reiman and Simon [32]. Important work extending this to a Palm calculus context was presented in [7].…”

Interpolation Approximations for M/G/infinity Arrival Processes

“…The right-hand derivatives at ρ = 0 of the various metrics of interest are evaluated using the Reiman-Simon technique [12]. For the system to be in the domain of applicability of the Reiman-Simon results, an assumption on finiteness of the exponential moment of σ is needed.…”

“…Such approximations become exact in the limit as the traffic intensity goes to zero and one respectively. In light traffic we take advantage of the fact that the M|G|∞ arrival process is obviously "Poisson driven", so that the Reiman-Simon theory [12] applies, under a bounded exponential moment assumption. The resulting light traffic limits of the queue with M|G|∞ arrivals differ from those of a classical GI|GI|1 queue.…”

The MacLaurin series for the GI/G/1 queue

“…These derivatives are called the light tra c derivatives and have been previously discussed in 9,8,12]. The light tra c derivatives can be used jointly with the heavy tra c limits for approximating the curve of the delay against the arrival rate.…”