Study of a semi-open queueing network with hysteresis control of service regimes

Ciro D'Apice; Alexander Dudin; Sergei Dudin; Rosanna Manzo; Ciro D'Apice; Alexander Dudin; Sergei Dudin; Rosanna Manzo

doi:10.3934/math.2025144

AIMS Mathematics

2025, Volume 10, Issue 2: 3095-3123. doi: 10.3934/math.2025144

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Study of a semi-open queueing network with hysteresis control of service regimes

1.
Dipartimento di Scienze Aziendali - Management & Innovation Systems, University of Salerno, Via Giovanni Paolo II, 132, Fisciano 84084, Salerno, Italy
2.
Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus
3.
Department of Political and Communication Sciences, University of Salerno, Via Giovanni Paolo II, 132, Fisciano 84084, Salerno, Italy

Received: 16 January 2025 Revised: 05 February 2025 Accepted: 11 February 2025 Published: 19 February 2025
MSC : 60K25, 60K30, 60M20

Semi-open queueing networks are suitable for modeling complex manufacturing, health care, and logistics systems. Such networks are different from more well-known open queueing networks because the number of users, that can be serviced in the network simultaneously is restricted by a finite constant. The network loses customers who arrive when its capacity reaches its limit. This paper examined an analytical model characterized by features like the possibility to capture potential correlations in the arrival process by assuming the marked Markov arrival process and modify service rates in the network's nodes depending on the number of users currently processed in the network. A hysteresis strategy for dynamic service rate selection was assumed. Fixing the thresholds of this strategy, the behavior of the network was determined by a continuous-time multidimensional Markov chain with a finite state that is a quasi-birth-and-death process. An explicit formula for the generator of this process was obtained. Expressions for the computation of network performance measures were derived. Numerical results highlight the dependence of some measures on thresholds defining the control policy, and their use to optimize the system is illustrated.
- queueing network,
- hysteresis control,
- multidimensional Markov chains,
- modeling
Citation: Ciro D'Apice, Alexander Dudin, Sergei Dudin, Rosanna Manzo. Study of a semi-open queueing network with hysteresis control of service regimes[J]. AIMS Mathematics, 2025, 10(2): 3095-3123. doi: 10.3934/math.2025144

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Abstract

Semi-open queueing networks are suitable for modeling complex manufacturing, health care, and logistics systems. Such networks are different from more well-known open queueing networks because the number of users, that can be serviced in the network simultaneously is restricted by a finite constant. The network loses customers who arrive when its capacity reaches its limit. This paper examined an analytical model characterized by features like the possibility to capture potential correlations in the arrival process by assuming the marked Markov arrival process and modify service rates in the network's nodes depending on the number of users currently processed in the network. A hysteresis strategy for dynamic service rate selection was assumed. Fixing the thresholds of this strategy, the behavior of the network was determined by a continuous-time multidimensional Markov chain with a finite state that is a quasi-birth-and-death process. An explicit formula for the generator of this process was obtained. Expressions for the computation of network performance measures were derived. Numerical results highlight the dependence of some measures on thresholds defining the control policy, and their use to optimize the system is illustrated.

References

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