An Interpolation Approximation for Queueing Systems with Poisson Input

Abstract

For many queueing systems of practical interest, it is difficult to obtain exact analytical results. It is, however, often possible to obtain asymptotic results for light and heavy traffic. Heavy traffic limit theorems yield expressions for normalized quantities of interest. In light traffic, we can obtain, in addition to limits, more sensitive information by calculating what are effectively derivatives (of the quantity of interest) with respect to the arrival rate. We can then combine the light and heavy traffic results to yield a polynomial (in the arrival rate) as an approximation to the normalized quantity of interest. For instance, by utilizing the heavy traffic limit, the light traffic limit, and the first derivative, we can obtain a quadratic approximation. Then, by reversing the normalization process, we can obtain an approximation for the original quantity of interest. In this paper we present the details of the above approximation, focusing, via several examples, on applications of the method. We then compare the results of the approximation, for the examples considered, with exact and simulation results. In addition, we compare our results to some related approximations. For the examples considered, the approximation works extremely well.

Light traffic performance under the power of two load balancing strategy

“…This is in sharp contrast with other systems where the first light-traffic derivative depends on E σ 2 , e.g., M |G|1-like queues [8] and the discriminatory processor sharing model [3]. This is rather unexpected because the variance of σ is known to be a key factor (4)) are still valid if the servers all use the PS discipline; see [4] for details.…”

“…As already pointed out by Reiman and Simon [8,9], without additional information (e.g., heavy traffic information), we should not expect RApp(λ) to act as an accurate proxy for R(λ) in medium to heavy traffic. This lack of accuracy is certainly apparent in the simulation results reported below.…”

On the three-queue priority polling system with threshold service policy

“…By combing (19), we first rewrite h(y, z) = 0 and ϕ N (z) = 0 into a polynomial system with respect to y and z, which actually is of 2N degrees. Then we utilize the homotopy method to calculate all the roots inside {(y, z) : |y| ≤ 1, |z| ≤ 1} (see [10]).…”

“…[4] and [19] have developed an interpolation approximation of mean sojourn time of arbitrary traffic intensities by utilizing the light and heavy traffic asymptotics. Here we follow the same argument to interpolate the mean sojourn times.…”

Tail probability of transient and stationary waiting times in (max,+)-linear systems

“…This technique combined with the idea of computing a rational approximant from such MacLaurin expansions is applied to investigate various performance functions and system characteristics, see [13], [14], and [20]. Reiman and Simon [16], [17] consider Poisson driven queueing networks. Another approach that is proposed in Gong, Nananukul, and Yan [10] uses the Padé approximation method to approximate mean system times based on light traffic derivatives.…”