*Article* **Approximations in Performance Analysis of a Controllable Queueing System with Heterogeneous Servers**

#### **Dmitry Efrosinin 1,2, Natalia Stepanova 3, Janos Sztrik 4,\* and Andreas Plank 1**


Received: 21 September 2020; Accepted: 9 October 2020; Published: 16 October 2020

**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.

**Keywords:** heterogeneous servers; Markov decision process; policy-iteration algorithm; mean number of customers; decomposable semi-regenerative process
