 
Summary: On Tandem Blocking Queues with a Common Retrial Queue
K. Avrachenkov
U. Yechiali
Abstract
We consider systems of tandem blocking queues having a common retrial queue.
The model represents dynamics of short TCP transfers in the Internet. Analytical re
sults are available only for a specific example with two queues in tandem. We propose
approximation procedures involving simple analytic expressions, based on Mean Value
Analysis (MVA) and on Fixed Point Approach (FPA). The mean sojourn time of a job
in the system and the mean number of visits to the orbit queue are estimated by the
MVA which needs as an input the fractions of blocked jobs in the primary queues. The
fractions of blocked jobs are estimated by FPA. Using a benchmark example of the
system with two primary queues, we conclude that the approximation works well in the
light traffic regime. We note that our approach becomes exact if the blocking proba
bilities are fixed. Finally, we consider two optimization problems regarding minimizing
mean total sojourn time of a job in the system: (i) finding the best order of queues,
and (ii) allocating a given capacity among the primary queues.
1 Introduction
Majority of TCP transfers in the Internet are small in volume, consisting of only few pack
ets [6]. The TCP congestion control mechanism does not have a chance to influence the
