Sojourn time in a Processor Sharing queue with batch arrivals
Fabrice M. Guillemin, Veronica Quintuna Rodriguez, Alain D. Simonian · Stochastic Models · 2018
For the Processor Sharing queue with batch job arrivals, the sojourn time W of a single job is investigated. This queuing model is motivated by the evaluation of Cloud Computing or Virtualized Network systems where the treatment of micro-services within requests determines the global system performance. We first show that the distribution of sojourn time W can be obtained from an infinite linear differential system; the structure of this system, however, makes the explicit derivation of this distribution generally difficult. When further assuming that the batch size has a geometric distribution with some given parameter , this differential system can be analyzed via a single generating function which is shown to verify a second-order partial differential equation involving a boundary term at point . Solving this partial differential equation for with required analyticity properties determines the one-sided Laplace transform for given u. Writing in terms of a multivariate hypergeometric function enables one to extend its analyticity domain to a cut-plane , for negative constants and . By means of a Laplace inversion of for a suitable value of u, the complementary distribution function is then given an explicit integral representation. This enables us to show that the tail of this distribution has an exponential decay with rate , together with a sub-exponential factor. A convergence in distribution is also asserted for W in heavy load condition, the limit distribution exhibiting a sub-exponential behavior itself. Using our exact results for the sojourn time of a single job, we finally discuss an approximation for the distribution of the sojourn time of an entire batch when assuming that the batch size is not too large.