Upper and lower bounds for the waiting time in the symmetric shortest queue system

Abstract

van der, J. (1992). Upper and lower bounds for the waiting time in the symmetric shortest queue system. (Memorandum COSOR; Vol. 9209). Eindhoven: Technische Universiteit Eindhoven. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. ABSTRACTIn this paper we compare the exponential symmetric shortest queue system with two related systems: the shortest queue system with threshold jockeying and the shortest queue system with threshold blocking. The latter two systems are easier to analyse and are shown to give tight lower and upper bounds respectively for the mean waiting time in the shortest queue system. The approach also gives bounds for the distribution of the total number of jobs in the system.

The symmetric longest queue system

“…The proof follows by induction. For n =0 inequality (1) trivially holds. Assuming (II) to hold for n we prove it for 11 + 1.…”

Stochastic bounds for randomized load balancing

“…In order to circumvent this problem we next borrow ideas from the JSQ analysis (see Adan et al [1]), whereby the original Markov chain is transformed by suitably redirecting transitions such that the new generator matrix has some regular structure. Concretely, we introduce a threshold parameter T such that, in the transformed Markov chains (one for getting lower bounds and another for getting upper bounds), the following condition must hold…”

“…Note that SQ(d) reduces to a simple uniform random selection when d = 1, and to JSQ when d = N , where N is the total number of servers. A fundamental qualitative result is that SQ(2) yields an exponential improvement over SQ (1) in terms of delay, yet with a conceivably small overhead cost. This result is known as the 'power-of-two' result [5].…”

“…Lower and upper bounds on delay were obtained for the particular case when d = N , i.e., JSQ. The main idea is to transform the original Markov chain with the inherent irregular structure into Markov chains with some regular structure (see Adan et al [1], Lui et al [4], or Zhao and Grassmann [7]). To get a lower bound, for instance, the transformation consists in redirecting some transitions between the states of the original Markov chain in such a way that the new system is less loaded than the original one.…”

Bounds for performance characteristics: a systematic approach via cost structures

“…Based on these precedences it appears to be easy to produce suitable modi cations. In fact, the approach originates from earlier work in 19,21,20,23,24,3,26,4], and attempts to unify the techniques used in these references.…”