Head of the line processor sharing for many symmetric queues with finite capacity

Abstract

In this paper the steady-state behavior of many symmetric queues, under the head of the line processor-sharing discipline, is investigated. The arrival process to each of n queues is Poisson, with rateλA, and each queue hasr waiting spaces. A job arriving at a full queue is lost. The queues are served by a single exponential server, which has a mean rateμn, and splits its capacity equally amongst the jobs at the head of each nonempty queue. The normal traffic casep=λ/μ< 1 is considered, and it is assumed thatn≫1 andr= 0(1). A 2-term asymptotic approximation to the loss probabilityL is derived, and it is found thatL = 0(n ^−r), for fixedp. If6=(1−p)/p≪ 1, then the approximation is valid if nδ²≫ 1 and (r+ 1)²≪nδ, and in this caseL ∼r!/(nδ)^r. Numerical values ofL are obtained forr = 1,2,3,4 and 5,n = 1000,500 and 200, and various values ofp< 1. Very small loss probabilities may be obtained with appropriate values of these parameters.