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.

中文翻译：

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具有异构服务器的可控排队系统的性能分析中的近似值

本文研究了一种可控的多服务器异构排队系统，其中服务器以不同的服务速率运行而没有抢占，即服务时间不中断。最佳控制策略以在系统中的平均客户数达到其最小值的方式在服务器之间分配客户。马尔可夫决策模型和策略迭代算法用于计算最优分配策略和相应的平均性能特征。当忽略慢速服务器的弱影响时，最佳策略的阈值类型定义为阈值级别的序列，该阈值级别指定了任何慢速服务器使用的队列长度。为了避免对具有大量服务器的系统进行耗时的计算，我们在这里集中于对最佳阈值的启发式评估，并将此解决方案与实际值进行比较。我们还开发了基于等效异构排队系统的近似值的简单下限和上限方法，该方法优先考虑了在最佳策略下运行的系统中的平均客户数。最后，仿真技术用于提供启发式解决方案对到达间隔和服务时间分布形式变化的敏感性分析。