The stationary local sojourn time in single server tandem queues with renewal input (Q1566524)

The paper deals with the stationary local sojourn times in a series of \(m+1\) FCFS single server queues (tandem queues), where the arrival process is a renewal process and the successive service times at the single server systems are governed by different (general) distributions. However, the successive service times may be mutually dependent. Firstly, as a starting point, earlier results of the author are presented, including general recurrence relations, equivalent packet tandem queues, overall distributions and unitary sojourn times. Using these results, analytical expressions (Cauchy contour integrals) for the local sojourn time of an arbitrary customer and an approximation are derived. The formulae and the approximation are specialized in case an input is a superposition of \(N\) Poisson traffic streams and if the service times of the \(i\)th stream (\(i=1,\dots{},N\)) are constant \(T_i\) at each of the \(m+1\) stages (packet switching model, \(T_i=\) packet length). In particular, formulae/approximations for the mean and variance of the local sojourn time are given. The numerical results presented as an example of three packet streams show that the formulae and proposed approximations work very well. Finally, a general approximation in case of a Poisson input is given and its goodness is illustrated by an example.