Moment-Generating Algorithm for Response Time in Processor Sharing Queueing Systems

Abstract

Abstract. Response times are arguably the most representative and important metric for measuring the performance of modern computer systems. Further, service level agreements (SLAs), ranging from data centres to smartphone users, demand quick and, equally important, predictable response times. Hence, it is necessary to calculate moments, at least, and ideally response time distributions, which is not straightforward. A new moment-generating algorithm for calculating response times analytically is obtained, based on M/M/1 processor sharing (PS) queueing models. This algorithm is compared against existing work on response times in M/M/1-PS queues and extended to M/M/1 discriminatory PS queues. Two real-world case studies are evaluated.

Higher response time moments for M/M/1 discriminatory processor sharing queues

“…When K = 1, substituting equal α i weights into the aforementioned PDE from [7] and simplifying duly yields equation (3). For K = 2, a mechanized algorithm to solve the Kims' equations for the second moments was obtained for multiple job types [17] that allowed arbitrary weighted α i , mean service requirements 1/µ i and arrival rates λ i . Explicit expressions for the first two moments of class 1 jobs, which assume equal weights α 1 = α 2 = 0.5, were found to be:…”

“…To obtain the general k th moment in M/M/1-DPS queues, we extend the work of [17] by forming a novel automated algorithm, implemented in Wolfram's Mathematica. The numerical algorithm is based on the direct approach of solving the moment-equations obtained by differentiating the Kim's joint transform PDE [7] repeatedly -k times for the k th moment; the details are given in figure 1.…”

“…The corresponding multi-class response time moments are summarized in tables 3 and 4. The applicability of modeling multiple arrivals and the flexibility of setting arbitrary weights allows the automated algorithm in figure 1 to improve existing work; previously, either all α i weights were equal, as with Chis and Harrison's work [17], or different weights were used to approximate response time for up to two moments only, as with Kim and Kim's work [7]. Table 3: Class 1 response time moments from TCP dataset 2.…”

“…With such work focusing primarily on reducing mean response times, it is important to consider higher moments to understand the effects of variance and skewness on response time distributions. Similarly, existing work for approximating response times in M/M/1-DPS queues [3,7,12,17] is limited by the number of moments obtained or the number of job classes considered. Hence, this paper provides an automated algorithm to iteratively calculate higher response time moments (i.e.…”

“…• Extend the two-moment equations introduced by Kim and Kim [7] (utilized in an algorithm by Chis and Harrison [17]) via an automated algorithm.…”