Steady-state analysis of an ordered hunting scheduled multiserver queueing system with Markovian arrival process, infinite buffer, heterogeneous servers, and phase-type distribution of service

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Abstract
This paper analyzes a infinite-capacity queueing system operating with multiple distinct servers. Phase-type distributions with different irreducible representations characterize the service times. The servers are indexed in a fixed order. The entry of customers is modeled according to a Markovian arrival process. Upon arrival, a customer is placed in the infinite-capacity buffer if all servers are busy. Conversely, if multiple servers are idle, the arriving customer occupies the free server with the lowest index. Jockeying between servers during service is prohibited. This system is a natural generalization of the classical system with Markovian arrival process, phase-type service distribution and N servers (MAP/PH/N) to the case where the servers can be distinct. Such a system frequently arises as a model for various real-world systems with heterogeneous equipment. A multidimensional Markov chain with an infinitesimal generator having a special block structure characterizes the state dynamics of the system. We present an explicit expression for the generator and derive the tractable ergodicity condition. The system states exhibit a matrix-geometric stationary distribution. Additionally, analytical expressions to evaluate the core system performance metrics are established. Closed-form formulations for the Laplace–Stieltjes transforms and the initial moments of the distribution regarding both sojourn and waiting times of a generic customer are derived. Finally, numerical examples are presented. These examples illustrate the dependence of the system's key performance measures on the arrival rate and the service rate of the slowest server. They also confirm the importance of accounting for correlation in the arrival process and volatility of service times and demonstrate how the results can be used for optimization purposes.

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