Abstract
We consider a queueing system consisting of two non-identical exponential servers, where each server has its own dedicated queue and serves the customers in that queue FCFS. Customers arrive according to a Poisson process and join the queue promising the shortest expected delay, which is a natural and near-optimal policy for systems with non-identical servers. This system can be modeled as an inhomogeneous random walk in the quadrant. By stretching the boundaries of the compensation approach we prove that the equilibrium distribution of this random walk can be expressed as a series of product-forms that can be determined recursively. The resulting series expression is directly amenable for numerical calculations and it also provides insight in the asymptotic behavior of the equilibrium probabilities as one of the state coordinates tends to infinity.
Highlights
In this paper we analyze the performance of a system with two servers under the shortest expected delay (SED) routing policy
The initial solution is described in Lemma 6, the horizontal compensation step is described in Lemma 8, and the vertical compensation step is described in Lemma 7; see Fig. 5
We have studied a queueing system with two nonidentical servers with service rates 1 and s ∈ N, respectively, Poisson arrivals and the SED routing policy
Summary
In this paper we analyze the performance of a system with two servers under the shortest expected delay (SED) routing policy. Under the assumption of identical service rates, SED routing becomes join the shortest queue (JSQ) routing which is known to minimize the mean expected delay [29]. If on top of the number of waiting customers we can estimate the expected service times at the two queues, the natural choice is to route the customers according to SED, rather than JSQ. We show that the compensation approach can be further developed to overcome the obstacles caused by the inhomogeneous behavior of the random walk This leads to a solution for the stationary distribution in the form of a tree of product forms.
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