Sojourn Time Estimation in an M/G/∞ Queue with Partial Information

Abstract

We propose an estimator for the cumulative distribution function G of the sojourn time in a steady-state M/G/∞ queueing system, when the available data consists of the arrival and departure epochs alone, without knowing which arrival corresponds to which departure. The estimator generalizes an estimator proposed in Brown (1970), and is based on a functional relationship between G and the distribution function of the time between a departure and the rth latest arrival preceding it. The estimator is shown to outperform Brown's estimator, especially when the system is heavily loaded.

Skampling for the Flow Duration Distribution

“…The effect of the magnitude of λ q should also be noted here, since it is a parameter we can control. For λ q close to 0, busy periods are very likely to be trivial, consisting of a single flow, and with B = D. The estimator D(t) will reflect this, though this is not immediately evident from the expression (5). The downside is that there will be very few busy period samples available.…”

“…Theoretical properties of the estimator D(t) are considered in [25], and an extension to the times between departures and and the rth preceding arrivals in [5].…”

“…III). The generalization of the Brown estimator derived in [5] could be considered to improve the estimation in the tail as mentioned in Sec. III.…”

“…Indeed, there has been a growing interest in recent years (e.g. [5], [6]) in statistical inference problems where partial measurements of queues can be used to infer various queueing parameters. Of the available approaches relevant to the estimation of service distributions, we select three where the required data is amenable to collection by simple counters acting on the raw packet arrival information, based respectively on busy periods (BP), queue length (QL), and "arrivals-to-departures" (AD) information.…”

The $M/G/\infty$ estimation problem revisited

“…The idea of pairing departure points with arrivals to the left was also exploited in the subsequent work by [4]. The authors use distances from departure points to the rth nearest arrival to the left, and show consistency of the proposed estimators.…”