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Large-scale service systems with packing constraints and heterogeneous servers

机译：具有打包约束和异构服务器的大规模服务系统

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摘要

A service system with multiple types of arriving customers and multiple types of servers is considered. Several customers (possibly of different types) can be placed for concurrent service into same server, subject to "packing" constraints, which depend on the server type. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered such that the customer arrival rates grow to infinity. We consider two variants of the model. For the infinite-server model, we prove asymptotic optimality of the Greedy Random (GRAND) algorithm in the sense of minimizing the weighted (by type) number of occupied servers in steady-state. (This version of GRAND generalizes that introduced in for the homogeneous systems, with all servers of same type.) We then introduce a natural extension of GRAND algorithm for finite-server systems with blocking. Assuming subcritical system load, we prove existence, uniqueness, and local stability of the large-scale system equilibrium point such that no blocking occurs. This result strongly suggests a conjecture that the steady-state blocking probability under the algorithm vanishes in the large-scale limit.
机译：考虑具有多种类型的到达客户和多种类型的服务器的服务系统。可以将多个客户（可能是不同类型）放置在同一服务器中进行并发服务，这取决于服务器的“打包”约束。即使同一台服务器同时提供服务，不同客户的服务时间也是独立的。大规模渐近状态被认为使得顾客到达率增长到无穷大。我们考虑模型的两个变体。对于无限服务器模型，我们从最小化稳态下占用服务器的加权（按类型）数量的意义上证明了贪婪随机（GRAND）算法的渐近最优性。 （此版本的GRAND概括了针对同类系统引入的所有类型相同的服务器。）然后，我们为具有阻塞的有限服务器系统引入了GRAND算法的自然扩展。假设亚临界系统负载，我们证明了大型系统平衡点的存在，唯一性和局部稳定性，因此不会发生阻塞。该结果有力地提出了一个猜想，即该算法下的稳态阻塞概率在大规模限制下消失了。

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