Analysis of the asymmetric shortest queue problem

Abstract

Abstract. In this paper we study a system consisting of two parallel servers with different service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. On arrival a job joins the shortest queue and in case both queues have equal lengths, he joins the first queue with probability q and the second one with probability 1-q, where q is an arbitrary number between 0 and 1. In a previous paper we showed for the symmetric problem, that is for equal service rates and q =1,1, that the equilibrium distribution of the lengths of the two queues can be exactly represented by an infinite sum of product form solutions by using an elementary compensation procedure. The main purpose of the present paper is to prove a similar product form result for the asymmetric problem by using a generalization of the compensation procedure. Furthermore, it is shown that the product form representation leads to a numerically efficient algorithm. Essentially, the method exploits the convergence properties of the series of product forms. Because of the fast convergence an efficient method is obtained with upper and lower bounds for the exact solution. For states further away from the origin the convergence is faster. This aspect is also exploited in the paper.

Shortest expected delay routing for Erlang servers

“…The primary conclusion of the present paper may be that the compensation method, which was originally developed for the shortest queue problem with exponentially distributed service times and Poisson arrivals (d. [1]), can indeed be extended to the case of Erlang distributed service times with expected delay routing. The compensation steps themselves become more complicated and also the finding of starting values 0'0 requires more skill.…”

“…Remarkably, a appears to be the feasible 0'0 for the case r = 1 (see [1]). In this section it will be investigated whether also for the general case the solution of the instantaneous jockeying problem produces the feasible ao's.…”

“…In this way the current solution always needs compensation to turn it into a solution on the other boundary, however, after compensation it stops being a solution on the first boundary, which requires again compensation, etc. In [1] it has been proved that the constructed solution converges absolutely.…”

“…This may have been tried in the course of the years, but the idea was only successful when it was combined with an idea of how to construct subsequent terms of the linear combination (ef. [1]). The essence of the latter idea is that there is an uncountable number of product forms satisfying the equilibrium equations for states in the inner area; of these product forms a selection should be made such that a particular linear combination of this selection not only satisfies the equilibrium equations for inner states, but also for all boundary states.…”

“…The essence of the latter idea is that there is an uncountable number of product forms satisfying the equilibrium equations for states in the inner area; of these product forms a selection should be made such that a particular linear combination of this selection not only satisfies the equilibrium equations for inner states, but also for all boundary states. In [1] it has been worked out in detail how this selection can be made via a compensation argument: alternately, product forms are added which make the current solution a solution for the equilibrium equations on the horizontal and on the vertical boundary respectively. In this way the current solution always needs compensation to turn it into a solution on the other boundary, however, after compensation it stops being a solution on the first boundary, which requires again compensation, etc.…”