Sojourn Times in the M/PH/1 Processor Sharing Queue

Abstract

We give in this paper an algorithm to compute the sojourn time distribution in the processor sharing, single server queue with Poisson arrivals and phase type distributed service times. In a first step, we establish the differential system governing the conditional sojourn times probability distributions in this queue, given the number of customers in the different phases of the PH distribution at the arrival instant of a customer. This differential system is then solved by using a uniformization procedure and an exponential of matrix. The proposed algorithm precisely consists of computing this exponential with a controlled accuracy. This algorithm is then used in practical cases to investigate the impact of the variability of service times on sojourn times and the validity of the so-called reduced service rate (RSR) approximation, when service times in the different phases are highly dissymmetrical. For two-stage PH distributions, we give conjectures on the limiting behavior in terms of an M/M/1 PS queue and provide numerical illustrative examples.

The uniform estimation of the M/G/1 processor sharing response time distribution

“…For easy comparisons, however, we do not hesitate to select the same example of a two-stage hyper-exponential ðH 2 Þ service time distribution cf. Subsection 4.2 in Reference [14]. The methodical difference is based on two consequences of our derivations.…”

“…This case was extensively treated in Reference [14] and based on a strict evaluation of the underlying differentialdifference equation. The reported example of a two-stage hyper-exponential distribution forms an interesting application of Internet traffic when customers of quite different mean service times, e.g.…”

“…But these again depend on the generally difficult solution of the M/G/1 FCFS waiting time distribution seen in References [5,10,11]. Apart from crude bounds, there are sufficiently verified results in the original domain for M/M/1 and M/PH/1 only [12][13][14]. In any sense, the uniform but sufficient accurate approximations of the PS response time in the original domain remain an important topic.…”

Waiting times for M/M systems under state-dependent processor sharing

“…For the M/GI/1 − PS system with k permanent requests, corresponding results are given in [28,32]. Recently, for the M/PH/1 − PS system in [26], a numerical algorithm is presented-based on a uniformization procedure-for computing sojourn time characteristics. For the M/M/2 − PS system in [27] the second moment of V (τ ) is given.…”

Exact Sojourn Time Distribution in an Online IPTV Recording System

“…We are interested in computing the exact sojourn time a customer spends in the system in order to completely download the entire file. The method we will apply actually consists of solving a system of differential equations and is inspired by the work of Sericola et al in [7] or Masuyama and Takine in [8]. However, our system is more complex since it integrates impatience and different service behaviors according to the actual number of customers in the system.…”