Abstract
The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of customers in the system reaches its minimal value. The Markov decision model and the policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, is of threshold type defined as a sequence of threshold levels which specifies the queue lengths for the usage of any slower server. To avoid time-consuming calculations for systems with a large number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this solution with the real values. We develop also the simple lower and upper bound methods based on approximation by an equivalent heterogeneous queueing system with a preemption to measure the mean number of customers in the system operating under the optimal policy. Finally, the simulation technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of inter-arrival and service time distributions.
Highlights
The study of multi-server queueing systems in most cases assumes the servers to be homogeneous when the individual service rates are the same for all the servers in the system
As it was shown in [18,19], taking into account the incompleteness of the theoretical proof noticed by Vericourt and Zhou in [20], the optimal allocation policy, which minimizes the mean number of customers in heterogeneous queueing system without preemption, belongs to a set of structural policies
As established in previous section, the estimation of the optimal threshold policy is possible by means of a simple heuristic solution
Summary
The study of multi-server queueing systems in most cases assumes the servers to be homogeneous when the individual service rates are the same for all the servers in the system. The case of infinitely many servers was proposed by Shenker and Weinrib [17], where an asymptotic analysis of large heterogeneous queueing systems is performed As it was shown in [18,19], taking into account the incompleteness of the theoretical proof noticed by Vericourt and Zhou in [20], the optimal allocation policy, which minimizes the mean number of customers in heterogeneous queueing system without preemption, belongs to a set of structural policies.
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