Queueing Systems 50, 109–130, 2005 c  2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands. Sojourn Times in the M/ PH /1 Processor Sharing Queue BRUNO SERICOLA Bruno.Sericola@irisa.fr INRIA-IRISA, Campus de Beaulieu, 35042 Rennes, Cedex, France FABRICE GUILLEMIN Fabrice.Guillemin@rd.francetelecom.com JACQUELINE BOYER Jacqueline.Boyer@rd.francetelecom.com France Telecom, Division R&D, 22300 Lannion, France Received 3 June 2004; Revised 28 January 2005 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. Keywords: phase type distribution, processor sharing discipline, sojourn time, asymptotic estimates AMS subject classification: 60K25, 68M20 1. Introduction Over the past few years, the study of processor sharing disciplines has gained renewed interest in relation with the problematic of bandwidth sharing of elastic flows in packet telecommunication networks [20]. As a matter of fact, the processor sharing discipline, which has been studied for decades in the queueing literature [14], ideally represents, at the expense of several simplifying assumptions (no latency in rate adaptation, same round trip times, etc.), how bandwidth is shared among flows controlled by TCP (Transmission Control Protocol). Almost 90% of the total volume of data transmitted through the Internet are nowadays controlled by this transport protocol. A flow may actually correspond to a single TCP connection (micro-flow) or to a group of TCP connections (macro-flow) having some characteristics in common. In the latter case, the different TCP connections of a flow may belong to the same session or have some common addressing information (for instance the same prefix in the destination address information element in packet headers). Moreover, several studies have shown that when observing a transmission link, flows may reasonably be assumed to arrive as